Four-Vectors (4-Vectors) & Scalar Invariants of Special Relativity: A Study of Elegant Physics, The Ultimate Four-Vector Reference

This remains a work in progress.
Please, send comments/corrections to John
SRQM 4-Vector and Lorentz Scalar Diagram

A vector is a mathematical object which has both magnitude and direction. It is a powerful tool for describing physical phenomena. A common 3-vector is the velocity vector (v_x, v_y, v_z), which tells you in what direction and how fast something is moving. One might use the (x, y, z) coordinates to write down the velocity vector of some object in the laboratory. That would be an example of a rectilinear coordinate system. Another person might use a coordinate system that is rotated wrt. the first observer, with components (y', x', z). The same vector might also be described by the (r, θ, φ) spherical coordinate system. Within a given coordinate system, each component is typically orthogonal to each other component. While these different coordinate systems will usually have different numbers in the vector 3-tuple, they nevertheless describe the same vector and the same physics. Hence, the vector can be considered the "primary" element, which is then described by any number of different coordinate systems, which simply represent one point-of-view of the given vector.

The extension of 3-vectors to that of 4-vectors is a simple idea. Let's imagine some <event> in spacetime. The location of the <event> in the Newtonian world would be it's 3-position (x,y,z), and the time (t) at which it occurs. In the Newtonian world these are totally separate ideas. SR unites them into a single object. The location of the <event> in the SR world would be it's 4-position (ct,x,y,z). All that we have done is to insert the time (t) into the vector as another component. The factor of (c) is put with it to make the dimensional units work out right. ( [m/s]*[s] = [m]). So, each component now has overall units of [m] for this 4-vector. This rather simple idea, combined with the postulates of SR, lead to some amazing results and elegant simplifications of physical concepts...

There are two postulates which lead to all of SR-Special Relativity:
(1) The laws of physics are the same for all inertial reference frames. This means the form of the physical laws should not change for different inertial observers. This can be also restated as "All inertial observers measure the same interval magnitude between two <event>'s". I say it this way because all of experimental physics ultimately boils down to taking a measurement. Yet another way to say it is that the result of any Poincaré Transformation (includes Lorentz Transformations) leaves the Invariant Interval unchanged.
(2) The speed of light (c) in vacuum is the same for all inertial reference frames. This is the result of millions of independent measurements, all confirming the same observation. This differentiates SR from classical Galilean invariance, which also obeys the first postulate.

4-vectors are tensorial entities which display Poincaré Invariance, meaning they leave invariant the differential squared interval (ds)² = (cdt)²-dx²-dy²-dz². A consequence of this invariant measurement is that any physical equation which is written in Poincaré Invariant form is automatically valid for any inertial reference frame, regardless of how coordinate systems are arranged. Transformations which leave these vectors unchanged include fixed translations through space and/or time, rotations through space, and boosts (coordinate systems moving with constant velocity) through spacetime. Since 4-vectors are tensors, and Poincaré Invariant, they can be used to describe and explain the physical properties that are observed in nature. Although the vector components may change from one reference frame to another, the 4-vector itself is an invariant, meaning that it gives valid physical information for all inertial observers. Likewise, the scalar products of Lorentz Invariant 4-vectors are themselves invariant quantities, known as Lorentz Scalars. Lorentz Invariance is a subset of the more general Poincaré Invariance.

The reason that I really like 4-vectors and their notation is that they beautifully and elegantly display the relations between lots of different physical properties. They also devolve very nicely into the limiting/approximate Newtonian cases of {|v|<<c} by letting {γ →1 and dγ/dt →0}. SR tells us that several different physical properties are actually dual aspects of the same thing, with the only real difference being one's point of view, or reference frame. Examples include: (Time , Space), (Energy , Momentum), (Power , Force), (Frequency , WaveNumber), (ChargeDensity , CurrentDensity), (EM-ScalarPotential , EM-VectorPotential), (Time Differential, Spatial Gradient), etc. Also, things are even more related than that. The 4-Momentum is just a constant times 4-Velocity. The 4-WaveVector is just a constant times 4-Momentum. In addition, the very important conservation/continuity equations seem to just fall out of the notation. The universe apparently has some simple laws which can be easy to write down by using a little math and a super notation.

Abbreviations

QM = Quantum Mechanics SR = Special Relativity
SM = Statistical Mechanics GR = General Relativity

Units of Measure - (SI variant, mksC)

Useful SR Quantities

Velocity
v_group or v or u: v = cβ = c²/v_p = group velocity = <event> velocity= [0..c], {u is historically used in SR notation}
v_phase or v_p: v_p = c/β = c²/v = phase velocity = celerity = [c..Infinity]

Minkowski Flat (Pseudo-Euclidian) Spacetime Metric:
η_μν = η^μν = SR g^μν = Diag[+1,-1,-1,-1]

Dimensionless SR Factors:
β = (v/c) = (v_group/c) = (c/v_phase)= [0..1]: Relativistic Beta factor, the fraction of the speed of light c
β = (u/c): Vector form of Beta factor, u is the velocity
γ[u] = dt/dτ: Lorentz Gamma Scaling Factor (Relativistic Gamma factor)
γ = (1 / √[1-(v/c)²] ) = (1 / √[1-(u·u/c²)] ): Lorentz Gamma Scaling Factor (~1 for v<<c), (>>1 for v~c)
γ = (1 / √[1-β²] ) = (1 / √[1-β·β] ): Lorentz Gamma Scaling Factor (~1 for β<<1), (>>1 for β~1)
γ = (1 / √[1-β²] ) = 1/√[(1+|β|)/(1-|β|)] : Useful for Doppler Shift Eqns
φ = Ln[γ(1+ β)] ~ Atanh[β]: BoostParameter/Rapidity (which remains additive in SR, unlike v)
e^φ = γ(1+β) = √[(1+β)/(1-β)]
β = Tanh[φ], γ = Cosh[φ], γβ = Sinh[φ], φ = Rapidity (which remains strictly additive in SR, unlike v)
D = 1 / [γ(1 - β Cos[θ] )] = 1 / [γ(1 - β·n )]: Relativistic Doppler Factor (sometimes called a relativistic beaming factor)
D₊ = γ(1 + β Cos[θ] ): Forward jet Doppler shift
D_- = γ(1 - β Cos[θ] ): Counter-jet Doppler shift

Temporal Factors:
τ = t / γ : Proper Time = Rest Time (time as measured in a frame at rest)
dτ = dt / γ : Differential of Proper Time
d/dτ = γ d/dt = U·∂ : Differential wrt Proper Time

Useful SR Formulas:
V·V = Vo·Vo : Invariant interval is often easier to calculate in rest frame coordinates
√[1+x] ~ (1+x/2) for x ~ 0 : Math relation often used to simplify Relativistic eqns. to Newtonian eqns.
1/√[1+x] ~ (1-x/2) for x ~ 0 : Math relation sometimes used to simplify Relativistic eqns. to Newtonian eqns.
δ^uv = Delta function = (1 if u = v, 0 if u ≠ v)
γ = (1 / √[1-(u·u/c²)]) = c/√[c²-v²] = c/√[c²-u·u]
γ² = c²/(c²-v²) = c²/(c²-u·u) = 1/(1- β²)
c²/γ² = (c²-v²)
v γ = c √[γ²-1]
β γ = √[γ²-1]
(1-β²)γ² = 1
(1-β²) = 1/γ²
β²γ² = γ²-1
β²γ² +1 = γ²
β²γ = (γ-1/γ)
c² dγ = γ³ v dv
d(γ v) = c² dγ / v = γ³ dv
dγ = γ³ v dv / c² = γ³ β dβ
dβ = dv / c
dγ/dv = γ³ v / c²
d(γ^-1)/dv = - γ v / c²
γ' = dγ/dt = (γ³ v dv/dt)/c² = (γ³ u·a)/c² = (u·a_r)/c²
γ'' = dγ'/dt = d²γ/dt² = (γ³/c²)*[(3γ²/c²)(u·a)² + (u'·a) + (u·a')]
u² = u²
u·u' = uu' = ua
(|u x a|)² + (u·a)² = u²a²
sin² + cos² = 1
(∇·∇)[1/r] = Δ[1/r] = -4πδ³(r)
(∇·∇)[1/|r-r'|] = Δ[1/|r-r'|] = -4πδ³(r-r') Green's function for Poisson's Eqn

SR Notation Used

**NOTE**
All results below use the time-positive SR Minkowski Metric η_μν = Diag[+1,-1,-1,-1].
If you wish to do GR, with other metrics g_μν, then some results below may need GR modification, such as the GR √[-g] for whichever metric you are using...
You have been warned.

There are several different SR notations available that are, mathematically speaking, equivalent.
However, some are easier to employ than others. I have used that one which seems the most practical and least error-prone.
If you mix notations, you will get errors! Always check notation conventions in SR & 4-Vector references, they are all relative ;-)

Minkowski SR Metric (time 0-component positive), for which {in Cartesian coordinates} η_μν = η^μν = SR g_μν = SR g^μν = Diag[+1,-1] = Diag[+1,-1,-1,-1]
Signature[η^μν] = -2

Generic 4-Vector Definition:
A = A^u = (a⁰,a) = (a⁰,a¹,a²,a³)    time (a⁰) in the 0th coord. ( some alternate notations use time as a⁴ )

Specific coordinate system representations:
A = A^u = (a⁰,a) = (a⁰,a¹,a²,a³) => (a^t,a^x,a^y,a^z) {for rectangular/Cartesian coords}
A = A^u = (a⁰,a) = (a⁰,a¹,a²,a³) => (a^t,a^r,a^θ,a^z) {for cylindrical coords}
A = A^u = (a⁰,a) = (a⁰,a¹,a²,a³) => (a^t,a^r,a^θ,a^φ) {for spherical coords}

Note that the superscripted variables are not exponents, they are upper tensor indices

Intervals:
TimeLike/Temporal (+ interval) = 0 coordinate ( some alternate notations use time as - interval and space as the + interval)
LightLike/Null         (0 interval)
SpaceLike/Spatial    (- interval) = 1,2,3 coordinates

Temporal Components: Future(+), Now(0), Past(-)

4-Vector Name: always references the "Spatial" 3-vector component (basically trying to extend the Newtonian 3-vector to SR 4-vector)
4-Vector Magnitude: usually references the "temporal" scalar component (because many vectors in the rest frame only have a temporal component)
4-Vector Tensor Indices: I use the convention of [Greek symbols dim{0..3} = time+space], [Latin symbols dim{1..3} = only space]
4-Vector Symbols: A = A^μ = (a⁰,a) = (a⁰,aⁱ) = (a⁰,a¹+a²+a³) = (a⁰,a¹,a²,a³), where the raised index indicates dimension, not exponent
4-Vector Definition: A = A^μ , always references the upper tensor index unless otherwise noted
4-Vector c-Factor: almost always applied to "Temporal" scalar component, as necessary to give consistent dimensional units for all vector components (a⁰,a¹,a²,a³) <==> (ct,x,y,z) = (ct,x)
*Note* c-Factor can be on the top, as ( ct , x , y , z ) = [m], or on bottom, as ( E/c , p_x , p_y , p_z ) = [kg m s^-1]. It all depends on the particular 4-Vector and its components.
*Note* P = (E/c,p) = (mc,p); the 4-Momentum is a good case showing top or bottom, with E = mc²
4-Vector Computer HTML Representation:
SR 4-vector = {BOLD UPPERCASE} = A
time scalar component = {regular lowercase} = a⁰
space 3-vector component = {bold lowercase} = a = aⁱ = (a¹,a²,a³)

Contraction & Dilation Relativistic Component: v --> v_o in a rest-frame, typically v = γ v_o (dilation) or v = (1/γ) v_o (contraction)
eg.
t = γ t_o (time dilation) - pertains to temporal separation between two <event>'s
L = (1/γ) L_o (length contraction) - pertains to the spatial separation between two parallel world lines

Generally, time-like quantities get dilated, space-like quantities get contracted by motion
Also, I typically denote "at-rest" invariant quantities with a "naught", or "_o", i.e.:
L_o (invariant rest length = proper length), relativistic length L = (1/γ) L_o
V_o (invariant rest volume), relativistic volume V = (1/γ) V_o
m_o (invariant rest mass), relativistic mass m = γ m_o
E_o (invariant rest energy), relativistic energy E = γ E_o
ω_o (invariant rest ang-frequency), relativistic ang-frequency ω = γ ω_o
ρ_o (invariant rest charge-density), relativistic charge-density ρ = γ ρ_o
n_o (invariant rest number-density), relativistic number-density n = γ n_ot_o (invariant rest time = proper time), relativistic time t = γ t_o = γ τ
etc.
This avoids the confusion of some texts which use just "m" as invariant mass, or just "ρ" as invariant charge-density.
It also helps to avoid confusion such as:
If the mass m of an object increases with velocity, wouldn't it have be a black hole in some reference frames (near c), since the mass increases with velocity.
Answer - no. The rest mass m_o does not change. The relativistic mass is simply an "apparent" mass, how the object is velocity-related to an observer, not how much "stuff" is in it...
The apparent increase is fully due to the gamma factor( γ ), which is simply an indication of the amount of relative motion.

Imaginary unit: ( i ) used only for QM phenomena, not for SR frame transformations or metric. To follow up on a quote from MTW " ict was put to the sword ".
This allows all the purely SR stuff to use only real numbers. Imaginary/complex stuff apparently only enters the scene via QM.
( some alternate notations use the imaginary unit ( i ) in the components/frame transformations/metric )

So, in summary, this notation allows:
easy recovery of Newtonian cases by allowing (γ→1, dγ→0) when (|v|<<c)
easy separation of SR vs Newtonian concepts, with the Newtonian 3-vector (a) extending naturally into the SR 4-vector (A)
easy separation of SR vs QM concepts, no ict's -- ( i ) only enters into QM concepts, such as Photon Polarization, Quantum Probability Current, etc.
easy separation of relativistic quantities vs. invariant quantities, E = γ E_o
reduction in number of minus signs (-), eg. U·U = c², P·P = (m_oc)²: the square magnitudes of velocity, momentum, wavevector, and other velocity-based vectors are positive

Minkowski SR Spacetime Metric

The main assumption of SR, or GR for that matter, is that the structure of spacetime is described by a metric g_μν. A metric tells how the spacetime is put together, or how distances are measured within the spacetime. These distances are known as intervals. In GR, the metric may take a number of different values, depending on various circumstances which determine its curvature. We are interested in the flat/pseudo-Euclidean spacetime of SR, also known as the Minkowski Metric, for which g_μν => η_μν = η^μν = Diag[+1,-1,-1,-1].

SR 4-Vector Basics

η_μν = SR g_μν = SR g^μν = DiagonalMatrix[1,-1,-1,-1]: Minkowski Spacetime Metric-the "flat" spacetime of SR {in Cartesion coordinates}

A = A^μ = (a⁰,aⁱ) = (a⁰,a¹,a²,a³) => (a^t,a^x,a^y,a^z) : Typical SR 4-vector (using all upper indices)
       A_μ = (a₀,a_i) = (a₀,a₁,a₂,a₃) => (a_t,a_x,a_y,a_z) : Typical SR 4-covector (using all lower indices)

We can always get the alternate form by applying the Minkowski Metric Tensor: A^μ = η^μνA_ν and A_μ = η_μν A^ν
Basically, this has the effect of putting a minus sign on the space component(s)

A = A^μ = (a⁰,aⁱ) = (a⁰,a¹,a²,a³) = (a⁰, a¹, a², a³) = (a⁰,a) :Typical SR 4-vector (all upper indices)
       A_μ = (a₀,a_i) = (a₀,a₁,a₂,a₃) = (a⁰,-a¹,-a²,-a³) = (a⁰,-a): Typical SR 4-covector (converted using Metric Tensor)

It is occasionally convenient to choose a particular basis to simplify component calculations
Typical bases are rectangular, cylindrical, spherical

A = A^μ = (a⁰,aⁱ) = (a⁰,a¹,a²,a³) = (a⁰,a) = (a⁰,a¹,a²,a³)      = (a⁰,a) => (a^t,a^x,a^y,a^z) :Typical SR 4-vector (choosing the rectangular basis)
       A_μ = (a₀,a_i) = (a₀,a₁,a₂,a₃) = (a₀,-a) = (a⁰,-a¹,-a²,-a³) = (a⁰,-a) => (a_t,a_x,a_y,a_z) = (a^t,-a^x,-a^y,-a^z) : Typical SR 4-covector (choosing the rectangular basis)

B = B^μ = (b⁰,b¹,b²,b³) = (b⁰,b) = (b^t,b^x,b^y,b^z) : Another typical SR 4-vector

A·B = η_μν A^μ B^ν = A_ν B^ν = A^μ B_μ = +a⁰b⁰-a¹b¹-a²b²-a³b³ = (+a⁰b⁰-a·b): The Scalar Product or Invariant Product relation, used to make SR invariants

c(A + B) = (cA + cB) scalar multiplication
A·A = A² = (+a₀² - a₁² - a₂² - a₃²) = (+a₀² - a·a) magnitude squared, which can be { - , 0 , + }
A = |A| = √|A²| >= 0 absolute magnitude or length, which can be { 0 , + }
A·B = B·A commutative, with the exception of the (∂) operator, since it only acts to the right
A·(B + C) = A·B + A·C distributive
d(A·B) = d(A)·B + A·d(B) differentiation
B = d(A)/dθ, where θ is a Lorentz Scalar Invariant

A_proj = (A·B)/(B·B) B = Projection of A along B

A_|| = (A·B)/(B·B) B = Component of A parallel to B
A_⊥ = A - A_||
A_⊥ =A - (A·B)/(B·B) B = Component of A perpendicular to B

Let A = (a⁰,a) be a general 4-Vector and T = U/c = γ(c, u)/c = γ(1, β) = (γ, γβ) be the unit-temporal 4-Vector
Then...
(A·T) = (a⁰,a)·γ(1, β) = γ(a⁰-a·β) = 1(a⁰_o-a·0) = a⁰_o
which is a Lorentz Invariant way to get (a⁰_o), the rest temporal component of A

Let A = (a⁰,a) be a general 4-Vector and S = γ_βn(β·n, n) be the unit-spatial 4-Vector
Then...
-(A·S) = -(a⁰,a)·γ_βn(β·n, n) = -γ_βn(a⁰β·n, a·n) = -1(a⁰a·0 - a·n) = a·n
which is a Lorentz Invariant way to get (a·n), the rest spatial component of A along the n direction

If A is a time-like vector, then you can also do the following:
(A·T)² - (A·A) = (a⁰_o)² - (a⁰a⁰-a·a) = (a⁰_o)² - (a⁰)² + (a·a) = (a·a)
Sqrt[(A·T)² - (A·A)] = Sqrt[(a·a)] = |a|
which is the Lorentz Invariant way to get the magnitude |a|, the spatial component magnitude of a time-like A

Also:
(A·A) = (a⁰)² - (a·a) = invariant = (a⁰_o)² ,where (a⁰_o) is the Lorentz Scalar Invariant "temporal rest value" for those vectors that can be at rest, and just the invariant for others
(a⁰)² = (a⁰_o)² + (a·a)

Consider the f ollowing identity:
γ = 1/Sqrt[1-β²]    :    γ² = 1 + γ²β²
Multiply by the square of a Lorentz Scalar Invariant (a⁰_o):
γ²(a⁰_o)² = (a⁰_o)² + γ²(a⁰_o)²β²

Compare Terms:
(a⁰)² = (a⁰_o)² + (a·a)
γ²(a⁰_o)² = (a⁰_o)² + γ²(a⁰_o)²β²

Notice the following correspondences:
a⁰ = γa⁰_o, The temporal component is the Gamma Factor times the rest value
a = γa⁰_oβ = a⁰β, The spatial component is temporal component times t he Beta Factor

Try it with the 4-Momentum P = (E/c,p)
(E/c) = γ(E_o/c) or E = γE_o
p = γ(E_o/c)β = γ(E_o/c)(v/c) = γ(E_ov/c²) = γ(m_ov) = γm_ov
p = (E/c)β = (mc)β = (mv) = γm_ov

If β=1 then |p| = E/c or E = |p|c, which is correct for photons
It also shows that as β → 1: γ → Infinity, (a⁰_o) → 0, a⁰ = (γa⁰_o) → some finite value

Special Relativity is interesting in that it is one area of physics where {Infinity*Zero = Finite Value} for certain variables.
eg. E = γE_o
For photons, the rest energy E_o = 0, the gamma factor γ = Infinity, but the overall energy of photon E = finite value for a given obs erver.
One can do this with any SR rest value variable. Always pair γ={1..Infinity} with a_o={large..0} to get a finite value a = γa_o={something finite}

These correspondences can also be generated by letting A = LorentzScalar (a⁰_o) * TemporalUnit 4-Vector T
A = (a⁰_o)T = (a⁰_o)γ(1,β) = γ(a⁰_o)(1,β) = (γa⁰_o)(1,β) = (a⁰)(1,β) = (a⁰,a⁰β) = (a⁰,a)
If β→0 , then A --> (a⁰,a⁰β) → (a⁰_o,0), which has (A·A) = (a⁰_o)² as expected
If β→1 , then A --> (a⁰,a⁰n) → a⁰(1,n), which has (A·A) = (a⁰)² (1² - n·n) = 0 as expected

==========
if A·A = const
then
dA¹dA²dA³ / A₀
dA⁰dA²dA³ / A₁
dA⁰dA¹dA³ / A₂
dA⁰dA¹dA² / A₃
are all scalar invariants
from Jacobian derivation
============

if A_μ dX^μ = invariant for any dX^μ, then A_μ is a 4-vector

η_μν Λ^μ_α Λ^ν_β = η_αβ This is basically the reason why the Scalar Product relation gives invariants
η_μν (A'^μ)(B'^ν) = η_μν (Λ^μ_αA^α)(Λ^ν_βB^β) = η_μν (Λ^μ_αΛ^ν_β)(A^α)(B^β) = η_αβ (A^α)(B^β) = η_αβ (A^αB^β)
Thus, A'·B' = A·B, where the primed 4-vectors are just Lorentz Transformed versions of the unprimed ones

Λ^α_μ Λ^μ_β = d^α_β

A'^μ = Λ^μ_ν A^ν: Lorentz Transform (Transformation tensor which gives relations between alternate boosted inertial reference frames)

Λ^μ'_ν = (∂X^μ'/∂X^ν)

Λ^μ_ν = (for x-boost)

Complex SR 4-Vectors

A few 4-vectors are known to have complex components. The Polarization 4-vector and ProbabilityCurrent 4-vector are a couple of these.
It will be assumed that all physical 4-vectors may potentially be complex, although, as far as I know, these only come into play via QM...

i = √[-1] :Imaginary Unit
e₀: Unit vector in the temporal direction (typically not used since the temporal unit is always considered a scalar)
e₁, e₂, e₃ :Unit Vectors in the spatial x, y, z directions (used instead of i, j, k so that there is no confusion with the imaginary unit i)

Note that for the following 4-vectors, the superscript is the tensor index, not exponentiation.

A = (a⁰_c + a¹_c e₁+ a²_c e₂+ a³_c e₃): Complex 4-vector has complex components, 1 along time and 3 along space
Scalar[A] = a⁰_c: Just the time component
Vector[A] = a¹_c e₁ + a²_c e₂ + a³_c e₃: Just the spatial components
A = Scalar[A] + Vector[A]

A = ( (a⁰_r + a⁰_i ) + (a¹_r + a¹_i ) e₁ + (a²_r + a²_i ) e₂ + (a³_r + a³_i ) e₃ ): Complex 4-vector has real + imaginary components, 1 each along time and 3 each along space
Re[A] = ( (a⁰_r ) + (a¹_r ) e₁ + (a²_r ) e₂ + (a³_r ) e₃ ): Only the real components
Im[A] = ( (a⁰_i ) + (a¹_i ) e₁ + (a²_i ) e₂ + (a³_i ) e₃ ): Only the imaginary components
A = Re[A] + i Im[A]

A = (a⁰_r + i a⁰_i,a_r + i a_i) : Standard 4-vector
A^* = (a⁰_r - i a⁰_i,a_r - i a_i): Complex conjugate 4-vector, just changes the sign of the imaginary component

A = (a⁰_r + i a⁰_i,a_r + i a_i) : A^* = (a⁰_r - i a⁰_i,a_r - i a_i)
B = (b⁰_r + i b⁰_i,b_r+ i b_i) : B^* = (b⁰_r - i b⁰_i,b_r- i b_i)

A·B = [( a⁰_r b⁰_r - a_r·b_r ) - ( a⁰_i b⁰_i - a_i·b_i )] + i [( a⁰_r b⁰_i - a_r·b_i ) + ( a⁰_i b⁰_r - a_i·b_r )] : General scalar product
A·A = [( a⁰_r² - a_r·a_r ) - ( a⁰_i² - a_i·a_i )] + 2i [( a⁰_r a⁰_i - a_r·a_i )] = |A|² : Scalar product of 4-vector with itself gives the magnitude squared
A·A^* = [( a⁰_r² + a⁰_i² ) - ( a_r·a_r + a_i·a_i )] = Re[A·A^*]: Scalar product of 4-vector with its complex conjugate is Real, thus Im[A·A^*] = 0

∂·B = [( ∂/c∂t_r b⁰_r + ∇_r·b_r ) - ( ∂/c∂t_i b⁰_i + ∇_i·b_i )] + i [( ∂/c∂t_r b⁰_i + ∇_r·b_i ) + ( ∂/c∂t_i b⁰_r + ∇_i·b_r )]
= [( ∂/c∂t_r b⁰_r + ∇_r·b_r ) - ( ∂/c∂t_i b⁰_i + ∇_i·b_i )]
= Re[∂·B]
The 4-Divergence of a Complex 4-Vector is Real, assuming that:
The real gradient acts only on real spaces & the imaginary gradient acts only on imaginary spaces, thus Im[∂·B] = 0
I believe this is due to the physical functions being complex analytic functions.

Fundamental/Universal Mathematical Constants

Fundamental/Universal Physical Constants (Lorentz Scalars)

c = Speed of Light Const = 1/√[ε_oμ_o] ~ 2.99729x10⁸ [m/s]
h = Planck's Constant - relates particle to wave - Action constant
ћ = (h/2π) = Planck's Reduced Const , aka. Dirac's Const - same idea as transforming between cycles and radians for angles
In essence, the reduced Planck constant is a conversion factor between phase (in radians) and action (in joule-seconds)
k_B = Boltzmann's Const ~ 1.3806504(24)×10⁻²³ [J/ ºK] relates temperature to energy
m_o = Rest Mass Const (varies with particle type)
q = Electric Charge Const (varies with particle type)

Note:
I do not set various fundamental physical constants to dimensionless unity, (i.e. c = h = G = k_B = 1).
While doing so may make the mathematics/geometry a bit easier, it ultimately obscures the physics.
While pure 4-Vectors may be Math, SR 4-Vectors is Physics. I prefer to keep the dimensional units.
Also, it is much easier to set them to unity in a final formula than to figure out where they go later if you need them.

Fundamental/Universal Physical SR 4-Vectors (Lorentz Vectors)

Fundamental/Universal Relations

Flux 4-Vectors, 4-Vector "Charges", and the Continuity/Conservation Equation

SR Path to QM

d/dτ[R] = (i ћ / m_o) ∂ <Event> motion ~ spacetime structure --- depends on i ћ / m_o

So, the following assumptions within SR-Special Relativity lead to QM-Quantum Mechanics:

R·R = (Δ s)² = (ct)²-r·r = (ct)²-|r|² : dR·dR = (ds)² = (c dt)²-dr·dr = (c dt)²-|dr|² : Invariant Interval
U·U = c²
P·P = (m_oc)²
K·K = (m_oc / ћ)² = (ω_o/c)²
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂²/c²∂t²-∇·∇ = -(m_oc / ћ)² : Klein-Gordon Relativistic Wave Eqn.
Each relation may seem simple, but there is a lot of complexity generated by each level.
*see QM from SR (Quantum Mechanics derived from Special Relativity)*

This can be further explored:
∂·∂ + (m_oc / ћ)² = 0

Derived Physical Constants (Scalar Products of Lorentz Vectors = Lorentz Scalars)

Invariants & Conservation Laws

There is an important distinction between an invariant quantity and a conserved quantity.
An invariant quantity has the same value wrt. all inertial systems, but may possibly change upon physical interaction for a system of multiple particles
(e.g. a fission/fusion reaction "redistributes" the rest masses).
A conserved quantity maintains the same value both before and after an interaction, although the component values may appear different in different frames.

In 4-vector notation:
An invariant quantity is a Lorentz Scalar, the dot product of two 4-Vectors, A·B = invariant = same value for all inertial observers.
A conserved quantity is a component of a 4-Vector that has 4-Divergence = 0, ∂·V = 0.

Relativistic Invariant Quantities (Lorentz Scalars~A·B), although perhaps there might be some that are simply just scalars...
also known as Relativistic Covariance = Relativistic Invariance = Lorentz Invariance
Lorentz Scalars = World Scalars = Invariant Scalars = Lorentz Invariants

Conserved Quantities (components of V, such that the 4-Divergence ∂·V = 0 )

Derived Equations

thus, for photonic frequency Doppler shifts,
if {n toward and β toward obs}, then ν_obs = ν_emit √[(1+|β|)/(1-|β|)] -->Doppler BlueShift
if {n toward and β 90° to obs}, then ν_obs = ν_emit / γ -->the transverse Doppler effect
if {n toward and β away from obs}, then ν_obs = ν_emit √[(1-|β|)/(1+|β|)] -->Doppler RedShift

Relativistic Stat-Mech (SM)/Thermodynamic stuff

U = γ(c, u), P = (E/c,p), d(P) = (dE/c,dp)
U·d(P) = γ(c dE/c-u·dp) = γ(dE-u·dp) = γ(T dS - P dV + μ dN) = (T_o dS_o - P_o dV_o + μ_o dN_o) = 0 ??

V*P (particle superstructure = Vol*Press)
μ*N (particle structure = ChemPoten*ParticleNum)
T*S (particle substructure = Temp*Entropy)

Time t = γ t_o
Length L = L_o/γ

Heat Q = q/γ
dq = T_odS_o
InertialMassDen(of radiation field) q = P/vV = γ q

Total Particle Number N = N_o is an invariant, because the NumberDensity n varies as n = γ n_o, but this is balanced by Volume V = V_o/γ
NumberDenstiy n = γ n_o where NumberFlux 4-Vector N = (cn,n_f) = n_o γ(c, u) = n_oU,n_o = N_o/(Δ_x_o*Δ_y_o*Δ_z_o)
N = n * V = (γ n_o)*(V_o/γ) = n_o* V_o = N_o
N·N = (n_oc)²

Total Entropy S = S_o is an invariant, because the EntropyDensity s varies as s = γ s_o, but this is balanced by Volume V = V_o/γ
EntropyDensity s = γ s_o where EntropyFlux 4-Vector S = (cs,s_f) = s_o γ(c, u) = s_oU,s_o = S_o/(Δ_x_o*Δ_y_o*Δ_z_o)
S = s * V = (γ s_o)*(V_o/γ) = s_o* V_o = S_o
S·S = (s_oc)²

Relativistic Analytical-Mech

Action S = S(ct,x,y,z)
dS/dτ = 0
dS/dτ = U·∂(S) = γ(∂S/∂t + u·∇(S)) = 0
see Menzel pg.172

Newtonian Approximations

√[1+x] ~ (1+x/2) for x<<1 This mathematical formula is used to derive the Newtonian limit of the various relativistic entities
γ = (1 / √[1-(v/c)²] )
γ --> 1 for v<<c

All of the formulas below can also be generated from the 4-Velocity Relation and multiplying by the appropriate Lorentz scalar:
U·U = γ[u]²(c²-u·u) = c²
γ²(1-β·β) = 1
γ² = 1 + γ²β²
γ = ±√[1 + γ²β²]
We choose the positive root since γ is always positive
γ = √[1 + γ²β²]
γ ~ [1 + γ²β²/2] for (γ²β² << 1)

4-Momentum
P = (E/c, p)
P·P = (E_o/c)² = (m_oc)²
E² = E_o² + p·p c²
E = E_o√[ 1 + p·p c² / E_o²]
E ~ E_o( 1 + p·p c² / 2 E_o² + ...) for | p·p c² | << | E_o² | discarding higher order terms...
E ~ ( E_o + p·p c² / 2 E_o ) for | p·p c² | << | E_o² |
E ~ ( E_o + p·p / 2 m_o ) for | p·p c² | << | E_o² | where E_o = m_oc²
E ~ ( E_o + |p|² / 2 m_o ) for | p c | << | E_o |
Total Energy = Rest Energy + Newtonian Momentum term
alternately:
γ ~ [1 + γ²β²/2]
γE_o ~ E_o[1 + γ²β²/2]
E ~ [E_o + γ²E_oβ²/2]
E ~ [E_o + γ²m_oc²β²/2]
E ~ [E_o + γ²m_o²c²β²/2m_o]
E ~ [E_o + γ²m_o²v²/2m_o]
E ~ [E_o + p²/2m_o]

4-WaveVector
K = (ω/c, k)
K·K = (ω_o/c)² = (E_o/ћc)² = (m_oc / ћ)²
ω² = ω_o² + k·k c²
ω = ω_o √[ 1 + k·k c² / ω_o² ]
ω ~ ω_o( 1 + k·k c² / 2 ω_o² + ...) for | k·k c² | << | ω_o² | and choosing the positive root and discarding higher order terms...
ω ~ ( ω_o + k·k c² / 2 ω_o ) for | k·k c² | << | ω_o² |
ω ~ ( ω_o + ћ k·k / 2 m_o ) for | k·k c² | << | ω_o² | where ω_o = m_oc² / ћ
ω ~ ( ω_o + ћ |k|² / 2 m_o ) for | k c | << | ω_o |
Total Angular Frequency = Rest Angular Frequency + Newtonian Wave Number term

4-Gradient (Wave equation)
∂ = ∂/∂x_μ = (∂/c∂t, -∇) = (∂_t/c, -∇)
∂·∂ = (∂_to/c)² = (- i m_oc / ћ )²: Klein-Gordon Relativistic Wave eqn.
∂_t² = ∂_to² + ∇·∇ c²
∂_t = ±∂_to√[ 1 + ∇·∇ c² / ∂_to²]
∂_t ~ ∂_to( 1 + ∇·∇ c² / 2 ∂_to² + ...) for | ∇·∇ c² | << | ∂_to² | and choosing the positive root and discarding higher order terms...
∂_t ~ ( ∂_to + ∇·∇ c² / 2 ∂_to ) for | ∇·∇ c² | << | ∂_to² |
∂_t ~ ( ∂_to - ћ ∇·∇ / i 2 m_o ) for | ∇·∇ c² | << | ∂_to² | where ∂_to = - i m_oc² / ћ
∂_t ~ ( ∂_to - ћ |∇|² / i 2 m_o ) for | ∇ c | << | ∂_to |
or, in more standard form
i ћ ∂_t ~ ( i ћ ∂_to - ћ² |∇|² / 2 m_o ) for | ∇ c | << | ∂_to | where i ћ ∂_to = E_o, the rest energy of the potential V
i ћ ∂_t ~ ( V(x,t) - ћ² |∇|² / 2 m_o ) for | ∇ c | << | ∂_to |
Time dependent Schroedinger equation is just the Newtonian approximation of the Klein-Gordon Relativistic Wave eqn.

4-ProbabilityCurrentDensity (change in form of Probability Density)
J = (cρ, j) = (iћ/2m_o)(ψ*∂[ψ]-∂[ψ*]ψ)
taking the temporal component, the relativistic probability density
ρ = (iћ/2m_oc²)(ψ* ∂_t[ψ]-∂_t[ψ*] ψ)
assume wave solution in following general form:
ψ = A f [k] e^(-iωt) and ψ* = A* f [k]* e^(+iωt)
then
∂_t[ψ] = (-iω)A f [k] e^(-iωt) = (-iω)ψ and ∂_t[ψ*] = (+iω)A* f [k]* e^(+iωt) = (+iω)ψ*
then
ρ = (iћ/2m_oc²)(ψ* ∂_t[ψ]-∂_t[ψ*] ψ)
ρ = (iћ/2m_oc²)((-iω)ψ*ψ-(+iω)ψ*ψ)
ρ = (iћ/2m_oc²)((-2iω)ψ*ψ)
ρ = (ћω/m_oc²)(ψ*ψ)
now use the Newtonian form of ω from above
ρ ~ [ћ( ω_o + ћ |k|² / 2 m_o )/m_oc²](ψ*ψ)
ρ ~ [(ћω_o/m_oc²) + (ћћ |k|² / 2 m_om_oc²)](ψ*ψ)
ρ ~ [(ћω_o/m_oc²) + (ћω_oћω_o |k|²c² / 2 ω_oω_om_oc²m_oc² )](ψ*ψ)
ρ ~ [(1) + ( |k|²c² / 2 ω_o²)](ψ*ψ), but | k c | << | ω_o |
ρ ~ [(1) + (~0)](ψ*ψ) because 2nd term is very small in non-relativistic limit
ρ ~ (ψ*ψ)
The standard probability density (ψ*ψ) is the Newtonian approximation of the temporal component of the 4-ProbabilityCurrent
Alternately, use ω = γω_o
ρ = (ћω/m_oc²)(ψ*ψ)
ρ = (ћγω_o/m_oc²)(ψ*ψ)
ρ = (γ)(ψ*ψ)
ρ ~ (ψ*ψ) where γ->1 in the Newtonian limit

Relativistic Quantum Mechanics

∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂²/c²∂t²-∇·∇ = -(m_oc / ћ)²: Klein-Gordon Relativistic Wave eqn.
D_EM = (∂/c∂t + iq/ћ V_EM/c, -∇ + iq/ћ a_EM) = ∂ + (iq/ћ)A_EM
D_EM·D_EM = -(m_oc / ћ)²: Klein-Gordon Relativistic Wave eqn. in electromagnetic potentials
(∂ + (iq/ћ)A_EM)·(∂ + (iq/ћ)A_EM) = -(m_oc / ћ)²: Klein-Gordon Relativistic Wave eqn. w/ electromagnetic potentials
(∂·∂) + (iq/ћ)(∂·A_EM + A_EM·∂) + (iq/ћ)²(A_EM·A_EM) = -(m_oc / ћ)²: Klein-Gordon Relativistic Wave eqn. w/ electromagnetic potentials
if (∂·A_EM + A_EM·∂) = 0
then (∂·∂) + (iq/ћ)²(A_EM·A_EM) = -(m_oc / ћ)²
(∂/c∂t,-∇)·(∂/c∂t,-∇) + (iq/ћ)²((V_EM/c, a_EM)·(V_EM/c, a_EM)) = -(m_oc / ћ)²
(∂²/c²∂t²-∇·∇) + (iq/ћ)²( (V_EM/c)²-(a_EM·a_EM) ) = -(m_oc / ћ)²
(∂²/c²∂t²+(iq/cћ)²(V_EM²)-(∇·∇+(iq/ћ)²(a_EM·a_EM) ) = -(m_oc / ћ)²

The Klein-Gordon equation is more general than the Schrödinger equation, but simplifies to the Schrödinger equation in the (φ/c)<<1 limit.
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂²/c²∂t²-∇·∇ = -(m_oc / ћ)²: Klein-Gordon Relativistic Wave eqn.
∂²/c²∂t² = ∇·∇-(m_oc / ћ)²
∂²/c²∂t² = (im_oc / ћ)²+∇·∇
(i ћ)²∂²/c²∂t² = (i ћ)²(im_oc / ћ)²+(i ћ)²∇·∇
(i ћ)²∂²/c²∂t² = (m_oc)²+(i ћ)²∇·∇
(i ћ)²∂²/∂t² = (m_oc²)²*[1 + (i ћ/m_oc)²∇·∇]
(i ћ)∂/∂t = ± (m_oc²)*Sqrt[1 + (i ћ/m_oc)²∇·∇]
(i ћ)∂/∂t ~ ± (m_oc²)*[1 + (1/2)*(i ћ/m_oc)²∇·∇ + ...] for ( ћ)²*∇·∇<<(m_oc)² ,generally a very good approx. for non-relativistic systems
(i ћ)∂/∂t ~ ± [(m_oc²) + (i² ћ²/2m_o)∇·∇ + ...]
choosing the positive root and discarding higher order terms...
(i ћ)∂/∂t ~ (m_oc²) - ( ћ²/2m_o)|∇|²
(i ћ)∂/∂t ~ - ( ћ²/2m_o)|∇|² becomes the time dependent Schrödinger eqn. for a free particle

Also, extensions into EM fields (or other types of relativistic potentials) can be made using D = ∂ + iq/ћ A_EM where A_EM is the EM vector potential and q is the EM charge,
and allowing D·D = -(m_oc/ћ)² to be the more correct EM quantum wave equation.

D·D = -(m_oc/ћ)²
(∂ + iq/ћ A_EM)·(∂ + iq/ћ A_EM) + (m_oc/ћ)² = 0

let A'_EM = iq/ћ A_EM
let M = m_oc/ћthen (∂ + A'_EM)·(∂ + A'_EM) + (M)² = 0

∂·∂ + ∂·A'_EM + 2 A'_EM·∂ + A'_EM·A'_EM + (M)² = 0
now the trick is that factor of 2, it comes about by keeping track of tensor notation...
a weakness of strick 4-vector notation

let the 4-Vector potential be a conservative field, then ∂·A_EM = 0
(∂·∂) + 2(A'_EM·∂) + (A'_EM·A'_EM) + (M)² = 0

expanding to temporal/spatial components...
( ∂_t²/c²-∇·∇ ) + 2(φ'/c ∂_t/c - a'·∇ ) + ( φ'²/c²- a'·a' ) + (M)² = 0

gathering like components
( ∂_t²/c² + 2φ'/c ∂_t/c + φ'²/c² ) - (∇·∇ + 2 a'·∇ + a'·a' ) + (M)² = 0
( ∂_t² + 2φ'∂_t + φ'² ) - c²(∇·∇ + 2 a'·∇ + a'·a' ) + c²(M)² = 0
( ∂_t + φ' )² - c²(∇ + a' )² + c²(M)² = 0

multiply everything by (i ћ)²
(i ћ)²( ∂_t + φ' )² - c²(i ћ)²(∇ + a' )² + c²(i ћ)²(M)² = 0

put into suggestive form
(i ћ)²( ∂_t + φ' )² = - c²(i ћ)²(M)² + c²(i ћ)²(∇ + a' )²
(i ћ)²( ∂_t + φ' )² = i²c²(i ћ)²(M)² + c²(i ћ)²(∇ + a' )²
(i ћ)²( ∂_t + φ' )² = i²c²(i ћ)²(M)² [1 + c²(i ћ)²(∇ + a' )²/ i²c²(i ћ)²(M)² ]
(i ћ)²( ∂_t + φ' )² = i²c²(i ћ)²(M)² [1 + (∇ + a' )²/ i²(M)² ]

take Sqrt of both sides
(i ћ)( ∂_t + φ' ) = ic(i ћ)(M) Sqrt[1 + (∇ + a' )²/ i²(M)² ]

use Newtonian approx Sqrt[1+x] ~ ±[1+x/2] for x<<1
(i ћ)( ∂_t + φ' ) ~ ic(i ћ)(M) ±[1 + (∇ + a' )²/2 i²(M)² ]
(i ћ)( ∂_t + φ' ) ~ ±[ic(i ћ)(M) + ic(i ћ)(M)(∇ + a' )²/2 i²(M)² ]
(i ћ)( ∂_t + φ' ) ~ ±[c(i² ћ)(M) + c( ћ)(∇ + a' )²/2(M) ]

remember M = m_oc/ћ
(i ћ)( ∂_t + φ' ) ~ ±[c(i² ћ)(m_oc/ћ) + c( ћ)(∇ + a' )²/2(m_oc/ћ) ]
(i ћ)( ∂_t + φ' ) ~ ±[c(i²)(m_oc) + (ћ)²(∇ + a' )²/2(m_o) ]
(i ћ)( ∂_t + φ' ) ~ ±[-(m_oc²) + (ћ)²(∇ + a' )²/(2m_o) ]

remember A'_EM = iq/ћ A_EM
(i ћ)( ∂_t + iq/ћφ ) ~ ±[-(m_oc²) + (ћ)²(∇ + iq/ћa )²/2m_o ]
(i ћ)( ∂_t ) + (i ћ)(iq/ћ)(φ) ~ ±[-(m_oc²) + (ћ)²(∇ + iq/ћa )²/2m_o ]
(i ћ)( ∂_t ) + (i²)(qφ ) ~ ±[-(m_oc²) + (ћ)²(∇ + iq/ћa )²/2m_o ]
(i ћ)( ∂_t ) -(qφ ) ~ ±[-(m_oc²) + (ћ)²(∇ + iq/ћa )²/2m_o ]
(i ћ)( ∂_t ) ~ (qφ )±[-(m_oc²) + (ћ)²(∇ + iq/ћa )²/2m_o ]

take the negative root
(i ћ)( ∂_t ) ~ (qφ ) + [(m_oc²) - (ћ)²(∇ + iq/ћa )²/2m_o ]

Here is the general Newtonian result
(i ћ)( ∂_t ) ~ (qφ ) + (m_oc²) - (ћ)²(∇ + iq/ћa )²/2m_o
or
(i ћ)( ∂_t ) ~ (qφ ) + (m_oc²) + [( ћ / i )∇ + qa ]²/2m_o

call (qφ ) + (m_oc²) = V[x]
(i ћ)( ∂_t ) ~ V[x] - (ћ)²(∇ + iq/ћa )²/2m_o
typically the vector potential a is zero in most non-relativistic settings
(i ћ)( ∂_t ) ~ V[x] - (ћ)²(∇)²/2m_o

And there you have it, the Schrodinger Equation with a potential
The assumptions for non-relativistic equation were:
Conservative field A_EM, then ∂·A_EM = 0
(∇ + a' )²/ i²(M)² = (∇ + a' )²/ i²(m_oc/ћ)² = (ћ)²(∇ + a' )²/ i²(m_oc)² is near zero
i.e. (ћ)²(∇ + a' )² << (m_oc)², a good approximation for low-energy systems
Arbitrarily chose vector potential a=0
Or keep it around for a near-Pauli equation (we would just have to track spins, not included in this derivation)

Interesting Relations

(K = m_o/ћU = ω_o/c²U) gives (c²/v_phase n = u) Both the wave vector and particle velocity point in the same direction; along the worldline. The product of the phase velocity and the particle velocity always equals c². ( v_phase * u = c² ). In the case of photons, the phase velocity = particle velocity = c. In the case of matter particles, the phase velocity v_phase = c²/u > c and particle velocity u<c. What does this mean? Suppose that you have a collection of particles traveling at identical velocities that all flash at the same time. The v_phase is the speed at which the flash moves in other reference frames, and can be considered the speed of propagation of simultaneity. For particles which are at rest, the v_phase is infinite, which makes sense since they all appear to flash simultaneously. v_phase (the phase velocity) is sometimes known as the celerity.

(∂·∂)A_EM = μ_o J+∂(∂·A_EM) Inhomogeneous Maxwell Equation
(∂·∂)A_EM = μ_o J Homogeneous Maxwell/Lorentz Equation (if ∂·A_EM = 0 Lorenz Gauge)
∂·J = ∂ρ/∂t +∇·j = 0 Conservation of EMcurrent
Psi = a E e^-iK·R Photon Wave Equation (Solution to Maxwell Equation)
E·K = 0 The Polarization of a photon is orthogonal to the WaveVector of that photon

V·U_obs = γ_v(c,v)·γ[u_obs] u_obs = γ_vγ[u_obs](c²-v·u_obs)
V·U_obs[u_obs = 0]/c² = γ_v (RestFrame Invariant expression for relative gamma factor)
P·U_obs = E/c γ[u_obs]c-p·γ[u_obs] u_obs = γ[u_obs](E-p·u_obs)
P·U_obs[u_obs = 0] = E (RestFrame Invariant expression for energy)
K·U_obs = w/c γ[u_obs]c-k·γ[u_obs] u_obs = γ[u_obs](w-k·u_obs)
K·U_obs[u_obs = 0] = w (RestFrame Invariant expression for angular frequency)
R·U_obs = ct γ[u_obs]c-r·γ[u_obs] u_obs = γ[u_obs](c²t-p·u_obs)
R·U_obs[u_obs = 0]/c² = t (RestFrame Invariant expression for time)
J·U_obs = cp γ[u_obs]c-j·γ[u_obs] u_obs = γ[u_obs](pc²-j·u_obs)
J·U_obs[u_obs = 0]/c² = ρ (RestFrame Invariant expression for ElecChargeDensity)

F^uv = ∂^uA^v-∂^vA^u Electromagnetic Field Tensor (F⁰ⁱ = -Eⁱ,F^ij = e^ijkB^k)
L = -1/4 F_uvF^uv - J_u A^u : Lagrangian Density for EM field

L = -m_oc²/γ -V: Relativistic Lagrangian function of a Particle in a Conservative Potential
V_EM = q U·A_EM/γ: Potential of EM field
L_EM = -m_oc²/γ - q U·A_EM/γ = - (P·P/m_o + qU·A_EM)/γ = - (m_oU·U + qU·A_EM)/γ

d/dτ = U·∂ = γ d/dt
U·∂/γ = ∂/∂t + u·∇ = d/dt : Convective Derivative

Larmor formula can be written in Lorentz invariant form
P = -( q²/ 6πε_oc³)(A·A) = -(μ_oq²)/(6πc)(A·A) Guassian units?
= ( q²/ 6πε_oc³) γ⁶/ (u'² - (u x u')²/c²)
= ( q²/ 6πε_oc³) γ⁶/ (β'² - (β x β')²)
= (2q²/ 3c(1-β'²)³) γ⁶/ ( β'² - ( β x β')²) SI Units?

alternate Larmor formula:
P = (-2/3)(q²/ m_o²c³)(F·F) SI units?
P = -( q²/ 6πε_om_o²c³)(F·F) Guassian units?

Relativistic Power radiated by moving charge by Abraham-Lorentz-Dirac force
P = (μ_oq²a²γ⁶)/(6πc)

====
Liénard-Wiechert potentials - potential due to a moving charge
A^μ(x) = (q/c4πε_o) U^μ/ ( R_νU^ν ) where R_ν is a null vector (R_νR^ν = 0)
A_EM = (q/c4πε_o) U / (R·U) where (R·R = 0, the definition of a light signal)
= (q/c4πε_o) U / ( cγ ( |r|-r·u/c ) )
= (q/c²4π ε_o)(c,u)/( |r|-r·u/c )
and therefore
φ_EM = (q / 4 π ε_o ) 1/[ r - r·u/c]_ret
a_EM = (μ_o q / 4 π) [u]/[ r - r·u/c]_ret
where terms in square brackets [] indicate retarded quantities
(R·U) = (ct,r)·γ(c,u) = γ(c²t - r·u) = cγ(ct - r·u/c)
t_ret = t - |x-x'|/c: (retarded time)
r_u = r - r u/c = the virtual present radius vector; i.e., the radius vector directed from the position the charge would occupy at time t' if it had continued with uniform velocity from its retarded position to the field point.
=====

F = - grad V(x): Particle moving in conservative force field
mc² + V(x) = E = const: Relativistic energy conservation in conservative force fields
T = mc²-m_oc² = (γ[u]-1) m_oc² = (γ-1) m_oc² Relativistic Kinetic Energy:
F·dX/dt = dT/dt: Also holds in Relativistic Mechanics
F·U = (m_oA+(dm_o/dτ)U)·U = c²(dm_o/dτ) = γc²(dm_o/dt)

Relativistic Perfect Fluids, where dissipative effects (viscosity, heat conduction, etc.) are neglected.
Particle 4-Flow N is a conservative quantity whose balance eqn. is ∂·N = 0
N = (cn, n_f) = n_o γ(c, u) = n(c, u) = n_oU
∂·N = ∂n/∂t +∇·(nu) = 0, where n=n_o γ
∂·N = ∂n_o γ/∂t +∇·(n_o γu) = 0
In non-relativistic limit this becomes ∂n_o/∂t +∇·( n_ou) = 0

T^αβ = ((ne+p)/c²)U^α U^β - p η^αβ
∂_βT^αβ = 0, Consevation of Energy-Momentum Tensor

Everyday Special Relativistic Effects

Things to look-up

An lamellar vector field is a synonym for an irrotational vector field.[1] The adjective "lamellar" derives from the noun "lamella", which means a thin layer. In Latin, lamella is the diminutive of lamina (but do not confuse with laminar flow). The lamellae to which "lamellar flow" refers are the surfaces of constant potential.

An irrotational vector field which is also solenoidal is called a Laplacian vector field.

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field.

In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero:

∇·v = 0

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as:

v = ∇ x A

automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

∇·v = ∇·(∇ x A) = 0

The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ x A. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)

In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:

∇ x v = 0
∇·v = 0

Since the curl of v is zero, it follows that v can be expressed as the gradient of a scalar potential (see irrotational field) φ:

v = ∇ φ (1)

Therefore, the potential of a Laplacian field satisfies Laplace's equation.

In fluid dynamics, a potential flow is a velocity field which is described as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of a gradient always being equal to zero (since the curl of a gradient is equivalent to take the cross product of two parallel vectors, which is zero).

In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.

Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow.

For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable.

A velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region and is irrotational. In such a case,

∇ x u = 0

where u denotes the flow velocity of the fluid. As a result, u can be represented as the gradient of a scalar function Φ:

u = ∇ Φ

A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant.

Unlike a stream function, a velocity potential can exist in three-dimensional flow.

see Cosmological Physics
Relativistic Euler Equations:
dv/dt = - 1/[γ²(ρ + p/c²)](∇ p + p'v/c²): Conservation of Momentum
d/dt[γ²(ρ + p/c²)] = p'/c² - γ²(ρ + p/c²)∇·v: Conservation of Energy
where p' = ∂ p/∂ t
∂·J = 0 where J = n_oU (J is the Number Flux here)
Relativistic Enthalpy w = (ρ + p/c²)

d/dt[γw/n] = p'/γnc²
Thus, in steady flow, γ * (enthalpy/particle) = const.

In non-relativistic limit these reduce to
dv/dt = - 1/[ρ](∇ p): Conservation of Momentum
d/dt[(ρ)] = - (ρ)∇·v: Conservation of Mass
p = Pressure
ΔE = - p ΔV
E = ρ c² V
ΔV / V = - Δρ_o/ρ_o

Relativistic Bernoulli's eqn.
γ w / ρ_o = const

Examples of Covariant equations

Lagrangian/Hamiltonian Formalisms:

Quantum Commutation & SR Uncertainty Relations

[X^u,P^v] = - i ћ η^uv
this gives
[ x , p_x ] = [ y , p_y ] = [ z , p_z ] = (i ћ)
[ ct , E/c ] = [ t , E ] = (- i ћ) :assuming that one can treat the time as an operator...

both of these yield the familiar uncertainty relations:
Generalized Uncertainty relation: (Δ A) * (Δ B) > = (1/2) |< i[A,B] >| see Sudbury pg. 59 for a great derivation

(Δ x * Δ p_x > = ћ / 2) and (Δ t * Δ E > = ћ / 2)
or more generally
(Δ R^u * Δ P^v > = ћ δ^uv / 2)
or
(Δ R^u * Δ K^v > = δ^uv / 2)
(Δ x * Δ k_x > = 1/2) and (Δ t * Δ w > = 1/2)

[ R^u , R^v ] = R^uR^v - R^vR^u = 0 : All position coordinates commute
[ P^u , P^v ] = P^uP^v - P^vP^u = 0 : All momentum coordinates commute

While I'm at it, a small comment about the quantum uncertainty relation. A great many books state that the quantum uncertainty relations mean that a "particle" cannot simultaneously have precise properties of position and momentum. I disagree with that interpretation. The uncertainty relations, the mathematical structure of the argument, say nothing about "simultaneous" measurements. They do say something about "sequential" measurements. A measurement of one variable places the system in a state such that if the next measurement is that of a non-commuting variable of the first, then the uncertainty must be of a minimum>0 amount. Also, note that the uncertainty relations are not necessarily about the size of h. Nor are they about the factor of ( i ) in the commutation relation. It would appear that they are about the metric g^μν itself, which has a non-zero result for sequential, non-commuting measurements.

Also, a comment on the EPR results. Based on SR, one cannot say that the measurement of one particle immediately "collapses" the physical state of the other. Since the two entangled particles can be setup such that they are space-like separated at the <event>'s of their respective measurement, there exist coordinate frames in which the measurement of the 1st particle occurs before that of the 2nd, exactly at the same time as the 2nd, and after that of the 2nd. Thus, how is the first particle to "know" that it must collapse the wavefunction of the 2nd, or that it must itself be collapsed by the 2nd? The answer is of course that the one measurement does not affect the space-like separated other measurement.
--------
need to derive:
(Δ phi_x * Δ L_x > = ћ / 2)
where phi_x is angle about x, and L_x is angular momentum about x

CPT & SR Phase Connection

The Phase is a Lorentz Scalar Invariant - all observers must agree on its value.

K∙X= (ω/c,k)∙(ct,x) = (ωt -k∙x) = φ: Phase of Wave

We take the point of view of an observer operating on a particle at 4-Position X which has an initial 4-WaveVector K.
The 4-Position X of the particle, the operation's <event>, will not change: we are applying the various operations only to the particle's 4-Momentum K.

Note that for matter particles:
K= (ω_o/c)T, where T is the Unit-Temporal 4-Vector T= γ(1,β), which defines the particle's worldline at each point. The gamma factor ( γ ) will be unaffected in the following operations since it uses the square of β:
γ=1/Sqrt(1-β∙β).

For photonic particles, K= (ω/c)N
, where N is the "Unit"-Null 4-Vector N = (1,n) and n is a unit-spatial 3-vector. All operations listed below
work similarly on the Null 4-Vector.

Do a Time Reversal Operation: T
The particle's temporal direction is reversed & complex-conjugated:
T_T= -T* = γ(-1,β)*

Do a Parity Operation (Space Reflection): P
Only the spatial directions are reversed:
T_P= γ(1,-β)

Do a Charge Conjugation Operation: C
Feynman showed this is the equivalent of world-line reversal & complex-conjugation
This is the Feynman–Stueckelberg Interpretation:
T_C= γ(-1,-β)*

Charge Conjugation actually changes all internal quantum #'s - charge, lepton #, etc.

Pairwise combinations:
T_TP=T_PT=T_C= γ(-1,-β)*

T_TC=T_CT=T_P= γ(1,-β)

T_PC=T_CP=T_T= γ(-1,β)*,

which shows that a CP <event> is mathematically the same as a T <event>

T_CPT = T = γ(1,β)
T_CC = T = γ(1,β)
T_PP = T = γ(1,β)
T_TT = T = γ(1,β)

It is only the combination of all three ops: (C,P,T) , or repeats of the same op, that leave the Unit-Temporal 4-Vector, and thus the Phase, Invariant

Light Cone

| time-like interval(+)

/ light-like interval(0)
worldline

      |
      |       c
\   future /
  \   |   /
   \ | /         -- space-like interval(-)
      \|/now
      /|\
    / |  \         elsewhere
/   |   \
/   past   \
      |       -c

(0,0) Zero-Null Vector

(+a,0) Future Pointing Pure TimeLike
(-a,0) Past Pointing Pure TimeLike
(0,b) Pure SpaceLike

(a,b) |a|>|b| TimeLike
(a,b) |a| =|b| Photonic-LightLike
(a,b) |a|<|b| SpaceLike

Any TimeLike 4-Vector (a,b) may be boosted into a Pure TimeLike (ka,0) state
Any SpaceLike 4-Vector (a,b) may be boosted into a Pure SpaceLike (0,kb) state

<Event> Tracking

R = R^μ = (ct, r)
U = dR/dτ = γ(c, u)
A = dU/dτ = γ(c dγ/dt, dγ/dt u+γ a) = dU/dτ = γ dU/dt =d²R/dτ²

R·R= (Δs)² = (ct)²-r·r = (ct)²-|r|² = (cτ)²
R·R = 0 for photonic signal w source at origin

U·U = c², in general, all <event>'s move into the future at the speed of light

R·R = (cτ)², the distance squared a particle travels along its own worldline
d[R·R]/dτ = d[(cτ)²]/dτ
d[R]/dτ·R +R·d[R]/dτ = 2c²τ
U·R+R·U = 2c²τ
R·U = c²τ
R·U = (ct,r)·γ(c,u ) = γ(c²t - r·u) = cγ(ct - r·u/c) = c²t_o = c²τ

d[R·U]/dτ = d[c²τ]/dτ
d[R]/dτ·U+R·d[U]/dτ = c²
U·U + R·A = c²
but U·U = c² so R·A = 0 { Remember, this is a derivation for a single particle with no interactions, so A = 0 }

d[U·U]/dτ = d[c²]/dτ = 0
U·d[U]/dτ +d[U]/dτ·U = 0
U·A +A·U = 0
U·A = 0, a nice general result that a particle's acceleration is perpendicular to its worldline.

A·A = -a² = -γ⁴[a² + (γ/c)²(u·a)²]

Poincare' Invariance/Group Theory:

So far, Poincare Invariance appears to be an absolute conservation law of all quantum field theories, as well as being a basis for Special Relativity. A number of quantum field theories are based on the complex (charged) scalar (Klein-Gordon) quantum field - which is mathematically the simplest QFT that still contains a continuous global [U(1)] internal symmetry. A real (Hermetian) scalar QFT is mathematically still simpler, but the absence of "charge" renders it uninteresting for most purposes.

Poincare group (aka inhomogeneous Lorentz group) and its representations
The set of Lorentz transforms and spacetime translations (Λ,A) such that:

X'^μ = Λ^μ_ν X^ν + A^μ

with conditions:
Det[Λ] = +1 (excludes discrete transforms of space inversion => proper)
Λ⁰₀ >= +1 (excluded discrete transforms of time inversion => orthochronous, preserve direction of time)

Λ^μ_ν (a Lorentz Transform - maps spacetime onto itself and therefore preserves the inner product)
Λ^μ_ν Λ_μλ = g_νλ (the Minkowski Metric)
A^μ = (Space-time Translation)

Unitary Operators representing these transforms:
U(A,1) = Exp[ i P·A ]
U(0,Λ) = Exp[ i M^μν Λ_μν ]

Poincare group has 10 generators (spacetime 4-generators)
P^μ (4 generators of space-time translation = Conservation of 4-Momentum)
M^μν (6 generators of Lorentz group = 3 orbital angular momenta + 3 Lorentz boosts)

[ P^μ, P^ν ] = 0 (Energy/Momentum commutes with itself)

[ M^μν, P^σ ] = - i ( P^μ g^νσ - P^ν g^μσ )
or
[ M_μν, P_σ ] = i ( g_νσ P_μ - g_μσ P_ν ) {one of these has a sign error I think}

[ M^μν, M^ρσ ] = -i ( M^νσ g^μρ - M^μσ g^νρ + M^ρν g^μσ - M^ρμ g^νσ )

Then, define the spatial 3-generators:
"Spatial Rotation" generators J_i = -(1/2) ε_ijk M^jk (for i=1,2,3), are Hermetian, (M^jk)^† = M^jk
"Lorentz Boost" generators K_i = M_i0 (for i=1,2,3), are anti-Hermetian, (M_i0)^† = - M_i0

[ J_i , P_k ] = i ε_iklP^l
[ J_i , P₀ ] = 0 (Spin commutes Energy)
[ K_i , P_k ] = i P₀g_ik
[ K_i , P₀ ] = - i P_i
[ J_m , J_n ] = i ε_mnkJ^k
[ J_m , K_n ] = i ε_mnkK^k
[ K_m , K_n ] = - i ε_mnkJ^k

Covariance of physical laws under Poincare trans. imply that all quantities defined in Minkowski space-time must belong to a representation of the Poincare group. By def., the states that describe elementary particles belong to irreducible representations of the Poincare group. These representations can be classified by the eigenvalues of the Casimir operators, which are the functions of the generators that commute with all the generators. This property implies that the eigenvalues of the Casimir operators remain invariant under group transforms.

Poincare Algebra ISO(1,3)
There are two Casimir operators of the Poincare group. They lead, respectively, to mass and spin. Thus, mass and spin are inevitable properties of particles in a universe where SR is valid.

(1) P² = η_μν P^μ P^ν = P_μ P^μ with corresponding eigenvalues P² = m²
which measure the invariant mass of field configurations.
In the real world we observe only time-like or light-like four-momenta, i.e. particles with positive or zero mass. Furthermore, the temporal components are always positive.
With dimensional units this would be P² = m²c²

(2) W² = η_μν W^μ W^ν = W_μ W^μ with corresponding eigenvalues W² = ( w₀² - w·w ) = - (w·w) = - (P₀²j²) = - m² s(s+1),
which measure the invariant spin of the particle, where there are (2s+1) spin states
(or 2 polarization/helicity states for massless fields)
with W_μ as the Pauli-Lubanski (mixed) Spin-Momentum four vector
With dimensional units this would be W² = - m²c²ћ² s(s+1)

Note: Massless representation give P² = m² = 0 and W² = - m² s(s+1) = 0
For instance, for a photonic P^μ = E(1,0,0,1), one has W^μ = M₁₂ P^μ
so that M₁₂ takes the possible eigenvalues ± s

W_σ = (1/2) ε_σμνρ M^μν P^ρ
or
W_σ = - (1/2) ε_μνρσ M^μν P^ρ
such that

[ W_σ , P^μ ] = 0
[ M_μν , W_σ ] = -i ( W_μ g_νσ - W_ν g_μσ )
[ W_λ , W_σ ] = i ε_λσαβ W^α P^β

Further,
W = (w⁰,w) = (p·j , P₀j - p x k)
w⁰ = p·j
w = P₀j - p x k
where
j = (M₃₂,M₁₃,M₂₁) are the 3 components of angular momentum, where [J₁,J₂] = i J₃ and cyclic permutations
k = (M₀₁,M₀₂,M₀₃) are boosts in 3 Cartesian directions

Wigner's classification: (non-negative energy irreducible unitary representations of the Poincare group)
The irreducible unitary representations of the Poincare' group are classified according to the eigenvalues of P² and W²
They fall into several classes:

1a) P² = m² > 0 and P₀ > 0: Massive particle
1b) P² = m² > 0 and P₀ < 0: Massive anti-particle??
2a) P² = 0 and P₀ > 0: Photonic
2b) P² = 0 and P₀ < 0: Photonic??
3) P² = 0 and P₀ = 0: P in the 4-Zero, the vacuum
4) P² = m² < 0: Tachyonic

A complete set of commuting observables is composed of P², the 3 components of p, W², and one of the 4 components of W^μ

The eigenvalues of P² (mass) and W² (spin) distinguish (possibly together with other quantum numbers) different particles. This is the general result for finite-mass quantum fields that are invariant under the Poincare transformation.

In the case of the scalar field, it is straightforward to identify the particle content of its Hilbert space.
A 1-particle state |k> = a^t(k)|0> is characterized by the eigenvalues
p⁰|k> = ћω(k)|k>, p|k> = ћk|k>, W²|k> = 0
thus showing that the quanta of such a quantum field may be identified with particles of definite energy-momentum and mass m, carrying a vanishing spin (in the massive case) or helicity (in the massless case). Relativistic QFT's are thus the natural framework in which to describe all the relativistic quantum properties, including the processes of their annihilation and creation in interactions, or relativistic point-particles. It is the Poincare invariance properties, the relativistic covariance of such systems, that also justifies, on account of Noether's theorem, this physical interpretation.
One has to learn how to extend the above description to more general field theories whose quanta are particles of nonvanishing spin or helicity. One then has to consider collections of fields whose components also mix under Lorentz transforms.

One may list the representations which are invariant under parity and correspond to the lowest spin/helicity content possible.

(0,0)	φ	scalar field
(1/2,0) (+) (0,1/2)	ψ	Dirac spinor
(1/2,1/2)	A_μ	vector field
(1,0) (+) (0,1)	F^uv = ∂^uA^v-∂^vA^u	EM field tensor

Consider an arbitrary spacetime vector x^μ
Construct the 2 x 2 Hermitian matrix X = X^†

X = x^μσ_μ =	( x⁰ + x³	x¹ - i x² )
X = x^μσ_μ =							( x¹ + i x²	x⁰ - x³ )

then Det[X] = x² = x·x = η_μν x^μ x^ν

see Proceedings of the Third International Workshop on Contemporary Problems in Physics, By Jan Govaerts, M. Norbert Hounkonnou, Alfred Z. Msezane
see Conceptual Foundations of Modern Particle Physics, Robert Eugene Marshak
see Fundamentals of Neutrino Physics and Astrophysics, Carlo Giunti
see Kinematical Theory of Spinning Particles, Martin Rivas

All of the relativistic wave equations can be derived from a common source, the relativistic mass-energy relation, inc. spin, in an EM field

4-Momentum inc. Spin
Ps = Σ·P = Σ^μ_ν P^ν = η_αβ Σ^μα P^β = Ps^μ
Σ^μ_ν is a Pauli Spin Matrix Tensor = Diag[σ⁰,σ]
Σ^μν is a Pauli Spin Matrix Tensor = Diag[σ⁰,-σ]

Ps = Diag[σ⁰,-σ]·P = Diag[σ⁰,-σ]·(E/c,p) = (σ⁰E/c,σ·p)

Ps = (ps⁰,ps) = (σ⁰E/c,σ·p)
where σ⁰ is an identity matrix of appropriate spin dimension and σ is the Pauli Spin Matrix Vector

4-Momentum inc. Spin in External Field
where:
H = E_t = Hamiltonian = TotalEnergyOfSystem
p_T = Total3-MomentumOfSystem

P_T= P + qA
P = P_T - qA

Ps = (ps⁰,ps) = (σ⁰ E/c,σ·p) = [σ⁰(E_t/c-qφ/c) , σ·(p_t-qa)]
Ps·Ps = (ps⁰)² - (ps)² = [σ⁰(E/c)]² - [σ·(p)]² = [σ⁰(E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)² = (E_o/c)²

The 4-Total Momentum (inc. External Field Minimal-Coupling and Spin)
P_s = Σ·P = Σ·(P_T-qA) = [σ⁰(E_T/c-qφ/c),σ·(p_T-qa)]
where Σ = Σ^μν are the Pauli Spin Matrices, and taking the Einstein summation gives the σ⁰ and σ

P_s·P_s =( Σ·P )² = [ Σ·(P_T-qA)]² = [σ⁰(E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)²
( Σ·P )² = (m_oc)²
( Σ·∂ )² = -(m_oc/ћ)²
( Σ·∂ )² + (m_oc/ћ)² = 0
( Σ·(D-(i/h)qA) )² + (m_oc/ћ)² = 0

Now, to prove that this "Relativistic Pauli" Energy-Momentum equation can lead to the Dirac equation
Ps·Ps = [σ⁰(E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (ps⁰)² - (ps)² = (m_oc)² = (E_o/c)²
Ps·Ps = [I(E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (ps⁰)² - (ps)² = (m_oc)² = (E_o/c)²Ps·Ps = (ps⁰)² - (ps)² = (m_oc)²
(ps⁰ + ps) (ps⁰ - ps) = (m_oc)²

Multiply both sides by any arbitrary function, ψχ
(ps⁰ + ps) (ps⁰ - ps)ψχ = (m_oc)²ψχ

We can also split the arbitrary function into two parts, and it still solves the original equation
let (ps⁰ + ps)ψ = (m_oc)χ
and (ps⁰ - ps)χ = (m_oc)ψ

now add and subtract these...
(ps⁰ + ps)ψ + (ps⁰ - ps)χ = (m_oc)χ + (m_oc)ψ
(ps⁰ + ps)ψ - (ps⁰ - ps)χ = (m_oc)χ - (m_oc)ψ

gather terms...
ps⁰ (ψ+χ) + ps (ψ-χ) = (m_oc)(χ+ψ)
ps⁰ (ψ-χ) + ps (ψ+χ) = (m_oc)(χ-ψ)

rearrange a bit...
ps⁰ (ψ+χ) + ps (ψ-χ) = (m_oc)(ψ+χ)
ps (ψ+χ) + ps⁰ (ψ-χ) = -(m_oc)(ψ-χ)

we are free to relabel our variables
let (ψ+χ) = a and (ψ-χ) = -b
ps⁰ a - ps b = (m_oc)a
ps a - ps⁰ b = (m_oc)b

put in matrix form...
[ps⁰ - ps ][a] = (m_oc)[a]
[ps - ps⁰][b]=           [b]

put in suggestive matrix form...
([1 0 ]ps⁰ + [0 -1]ps)[a] = (m_oc)[a]
([0 -1]   [1 0 ]    )[b]=           [b]

put in even more suggestive matrix form...
([1 0 ]σ⁰p⁰ + [0 -1]σ·p)[a] = (m_oc)[a]
([0 -1]      [1 0 ]   )[b]=           [b]

put in highly suggestive matrix form...
([σ⁰ 0 ]p⁰ - [0    σ]·p)[a] = (m_oc)[a]
([0 -σ⁰]   [-σ 0 ] )[b]=         [b]

let Spinor Ψ = [a]
                       [b]

and note that σ⁰ = I

this is equivalent to Dirac Gamma Matrices...
([I 0 ]p⁰ - [0    σ]·p)Ψ = (m_oc)Ψ
([0 -I]   [-σ 0 ]   )

(γ⁰p⁰ - γ·p)Ψ = (m_oc)Ψ

(Γ·P)Ψ = (m_oc)Ψ

(Γ·P) = (m_oc)

(Γ^μP_μ)Ψ = (m_oc)Ψ

iћ(Γ^μ∂_μ)Ψ = (m_oc)Ψ

>Spin	>Statistics	>Relativistic Eqn. Mass = 0	>Relativistic Eqn. Mass <> 0	>Non-Relativistic Eqn. Newtonian Limit √[1+x] ~ (1+x/2) for x<<1 v<<c	>Field Representation	>Polarizations Mass = 0 Mass <> 0
0	Boson: Bose-Einstein	FreeWave N-G bosons (∂·∂) Ψ = 0	Klein-Gordon (-Fock) Higgs bosons (∂·∂ + (m_oc / ћ)² ) Ψ = 0 (ћ∂_slash+im_oc)* (ћ∂_slash-im_oc) Ψ = 0 with minimal EM coupling (D·D + (m_oc / ћ)² ) Ψ = 0 (∂+iqA)·(∂+iqA) + (m_oc / ћ)² ) Ψ = 0	Schrödinger ( iћ∂_t-V+[ћ²∇²/2m])Ψ = 0 with minimal EM coupling (iћ∂_t-qφ-[(p-qa)²/2m])Ψ = 0 (iћ∂_t-qφ-[(iћ∇-qa)²/2m])Ψ = 0 (iћ∂_t-qφ+[(ћ∇+iqa)²/2m])Ψ = 0 Note that qφ = V is approx. when a~0	Ψ scalar = 0-tensor	2? 1
1/2	Fermion Fermi-Dirac	Weyl Idealized Matter Neutrinos (neutrinos now thought to be very low but non-zero mass) (σ·∂) Ψ = 0 factored to Left and Right spinors (σ·∂) Ψ_R = 0 (σ ·∂) Ψ_L = 0	Dirac Matter Leptons/Quarks (i ћγ·∂ - m_oc ) Ψ = 0 (i ћ∂_slash - m_oc) Ψ = 0 with minimal EM coupling (i ћγ·(∂+iqA) - m_oc ) Ψ = 0	Pauli, (Schrödinger-Pauli) with minimal EM coupling (i ћ∂_t - qφ -[(σ·(p-qa)²)/2m])Ψ = 0	Ψ spinor	2 2
1	Boson Bose-Einstein	Maxwell Force/Gauge Fields Photons/Gluons (∂·∂) A = 0 (w Current Source) (∂·∂) A = μ_oJ	Proca Force(ex. W and Z Boson) (∂·∂ + (m_oc / ћ)² ) A = 0		A 4-vector = 1-tensor	2 (= 2 transverse) 3 (= 2 transverse + 1 longitudinal)
3/2	Fermion Fermi-Dirac	Gravitino? ( ε^μνρσγ₅γ_ν∂_ρ) ψ_σ = 0	Rarita-Schwinger (ε^μνρσγ₅γ_ν∂_ρ+m_oσ^μσ)ψ_σ=0		ψ_σspinor-vector	2 4
2	Boson Bose-Einstein	Einstein Graviton?	? ?		tensor = 2-tensor	2 5

Duffin-Kemmer-Petiau Equation = Complex Proca Equation
Duffin-Kemmer Equation: ( β^μ p_μ - M ) ψ = 0 : for a free spin-0 or spin-1 particle

Dim	Type		Hodge Dual
0	scalar
1	vector
2	tensor
3	pseudovector	magnetic field, spin, torque, vorticity, angular momentum
4	pseudoscalar	magnetic charge, magnetic flux, helicity

In Minkowski space (4-dimensions), the { 1 4 6 4 1} Hodge dual of an n-rank (n<=2) tensor will be an (4-n) rank skew-symmetric pseudotensor
Hodge duals
*dt = dx ^ dy ^ dz
*dx = dt ^ dy ^ dz
*dy = - dt ^ dx ^ dz
*dz = dt ^ dx ^ dy

*(dt ^ dx) = -dy ^ dz
*(dt ^ dy) = dx ^ dz
*(dt ^ dz) = -dx ^ dy
*(dx ^ dy) = dt ^ dz
*(dx ^ dz) = -dt ^ dy
*(dy ^ dz) = dt ^ dx

More Research Needed:

Einstein made as stronger statement about EP, known as EPP: In small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments.

SR --> QM, what assumptions necessary & where does it break down
Relational QM
General Continuity of WorldLines
Spin vs. Accel, time component correlation
Relativistic Thermodynamics & SM
Poincare Group & Casimir operators & Casimir Invariants (mass & spin of Poincare field)
Generalized Uncertainty
Points - Waves - Potentials - Fields
Relation between single point and density 4-vectors
Poisson Eqn. / Laplace Eqn.
Continuity eqn --> 4-Vector
Adding Spin to Klein-Gordon
Relativistic Lagrangian & Hamiltonian
Covariant Form Relativistic Equations
Umov-Poynting examples
Dirac - Kemmer generalized eqn.
Hodge Dual examples
Pressure Diffusion Wave/Eqn.
Potential Flow Theory
Schroedinger Eqn as a diffusion equation

Schrödinger equation for a free particle

Main article: Schrödinger equation

With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:

$\psi_t = \frac{i \hbar}{2m} \Delta \psi$ , where i is the unit imaginary number, and $\hbar$ is Planck's constant divided by 2π, and ψ is the wavefunction of the particle.

This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:

$c(\vec R,t) \to \psi(\vec R,t)$

$D \to \frac{i \hbar}{2m}.$

Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wavefunction at any time through an integral on the wavefunction at t=0:

$\psi(\vec R, t) = \int \psi(\vec R^0,t=0) G(\vec R - \vec R^0,t) dR_x^0\,dR_y^0\,dR_z^0$ , with

$G(\vec R,t) = \bigg( \frac{m}{2 \pi i \hbar t} \bigg)^{3/2} e^{-\frac {\vec R^2 m}{2 i \hbar t}}.$

Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wavefunction satisfying Schrödinger's equation might have an origin other than diffusion.

Some examples of equivalent electrical and hydraulic equations:

type	hydraulic	electric	thermal
quantity	volume V [m³]	charge q [C]	heatQ [J]
potential	pressure p [Pa=J/m³]	potential φ [V=J/C]	temperature T [K=J/k_B]
flux	current Φ_V [m³/s]	current I [A=C/s]	heat transfer rate [J/s]
flux density	velocity v [m/s]	j [C/(m²·s) = A/m²]	heat flux [W/m²]
linear model	Poiseuille's law	Ohm's law	Fourier's law

References (on 4-Vectors in SR & QM in SR)

Classical Dynamics of Particles & Systems, 3rd Ed., Jerry B. Marion & Stephen T. Thornton (Chap14)
Classical Electrodynamics, 2nd Ed., J.D. Jackson (Chap11,12)
Classical Mechanics, 2nd Ed., Herbert Goldstein (Chap7,12)
Electromagnetic Field, The, Albert Shadowitz (Chap13-15)
First Course in General Relativity, A, Bernard F. Schutz (Chap1-4)
Fundamental Formulas of Physics, by Donald Howard Menzel (Chap6)
Introduction to Electrodynamics, 2nd Ed., David J. Griffiths (Chap10)
Introduction to Modern Optics, 2nd Ed., Grant R. Fowles (var)
Introduction to Special Relativity, 2nd Ed., Wolfgang Rindler (All) (**pg60-65,82-86**)
Lectures on Quantum Mechanics, Gordon Baym (Chap22,23)
Modern Elementary Particle Physics: The Fundamental Particles and Forces?, Gordon Kane (Chap2+)
Path Integrals and Quantum Processes, Mark Swanson (var)
Quantum Electrodynamics, Richard P. Feynman (Lec7-rest)
Quantum Mechanics, Albert Messiah (Chap20)
Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians, Anthony Sudbery (Chap7)
Spacetime and Geometry: An Introduction to General Relativity, Sean M. Carroll (var)
Statistical Mechanics, by R. K. Pathria (Chap6.5)
Theory of Spinors, The, E'lie Cartan (var)
Topics in Advanced Quantum Mechanics, Barry R. Holstein (Chap3,6,7)

Relativistic Quantum Fields, Mark Hindmarsh, Sussex, UK
Relativity and electromagnetism, Richard Fitzpatrick, Associate Professor of Physics, The University of Texas at Austin
http://farside.ph.utexas.edu/teaching/em/lectures/node106.html

The Relativistic Boltzmann Equation: Theory and Applications, Carlo Cercignani, Gilberto Medeiros Kremer
Essential Relativity: Special, General, and Cosmological, by Wolfgang Rindler
Compendium of Theoretical Physics, by Armin Wachter, Henning Hoeber
Relativistic Quantum Mechanics of Leptons and Fields, by Walter T. Grandy

This remains a work in progress.
Email me, especially if you notice errors (which I will fix ASAP) or have interesting comments.
Please, send comments to John Wilson

quantum
relativity

See SRQM: QM from SR - The 4-Vector RoadMap (.html)
See SRQM: QM from SR - The 4-Vector RoadMap (.pdf)

SRQM Physics Diagramming Method

SRQM 4-Vector : Four-Vector and Lorentz Scalar Diagram

SRQM + EM 4-Vector : Four-Vector and Lorentz Scalar Diagram

SRQM + EM 4-Vector : Four-Vector and Lorentz Scalar Diagram With Tensor Invariants

SRQM 4-Vector : Four-Vector Stress-Energy & Projection Tensors Diagram

SRQM 4-Vector : Four-Vector SR Quantum RoadMap

SRQM + EM 4-Vector : Four-Vector SR Quantum RoadMap

SRQM + EM 4-Vector : Four-Vector SR Quantum RoadMap - Simple

SRQM 4-Vector : Four-Vector New Relativistic Quantum Paradigm

SRQM + EM 4-Vector : Four-Vector New Relativistic Quantum Paradigm (with EM)

SRQM 4-Vector : Four-Vector New Relativistic Quantum Paradigm - Venn Diagram

SRQM 4-Vector : Four-Vector SpaceTime is 4D

SRQM 4-Vector : Four-Vector SpaceTime Orthogonality

SRQM 4-Vector : Four-Vector 4-Position, 4-Velocity, 4-Acceleration Diagram

SRQM 4-Vector : Four-Vector 4-Displacement, 4-Velocity, Relativity of Simultaneity Diagram

SRQM 4-Vector : Four-Vector 4-Velocity, 4-Gradient, Time Dilation Diagram

SRQM 4-Vector : Four-Vector 4-Vector, 4-Velocity, 4-Momentum, E=mc^2 Diagram

SRQM 4-Vector : Four-Vector 4-Velocity, 4-WaveVector, Relativistic Doppler Effect Diagram

SRQM 4-Vector : Four-Vector Wave-Particle Diagram

SRQM 4-Vector : Four-Vector Compton Effect Diagram

SRQM 4-Vector : Four-Vector Aharonov-Bohm Effect Diagram

SRQM 4-Vector : Four-Vector Josephson Junction Effect Diagram

SRQM 4-Vector : Four-Vector Hamilton-Jacobi vs Action, Josephson vs Aharonov-Bohm Diagram

SRQM 4-Vector : Four-Vector Motion of Lorentz Scalar Invariants

SRQM 4-Vector : Four-Vector Motion of Lorentz Scalar Invariants, Conservation Laws & Continuity Equations

SRQM 4-Vector : Four-Vector Speed of Light (c)

SRQM 4-Vector : Four-Vector Minimal Coupling Conservation of 4-TotalMomentum)

SRQM 4-Vector : Four-Vector Relativistic Action (S) Diagram

SRQM 4-Vector : Four-Vector Relativistic Lagrangian Hamiltonian Diagram

SRQM 4-Vector : Four-Vector Relativistic Euler-Lagrange Equation

SRQM 4-Vector : Four-Vector Relativistic EM Equations of Motion

SRQM 4-Vector : Four-Vector Einstein-de Broglie Relation hbar

SRQM 4-Vector : Four-Vector Quantum Canonical Commutation Relation

SRQM 4-Vector : Four-Vector QM Schroedinger Relation

SRQM 4-Vector : Four-Vector Quantum Probability

SRQM 4-Vector : Four-Vector CPT Theorem

SRQM 4-Vector : Four-Vector Lorentz Transforms Connection Map

SRQM 4-Vector : Four-Vector Lorentz Discrete Transforms

SRQM 4-Vector : Four-Vector Lorentz Transforms - Trace Identification

SRQM 4-Vector : Four-Vector Lorentz Lorentz Transforms-Interpretations
CPT Symmetry, Baryon Asymmetry Problem Solution, Matter-Antimatter Symmetry Solution, Arrow-of-Time Problem Solution, Big-Bang!

SRQM 4-Vector : Four-Vector Lorentz Transforms-Interpretations, CPT Symmetry, Baryon Asymmetry Problem Solution, Matter-Antimatter Symmetry Solution, Arrow-of-Time Problem Solution, Big-Bang!

See SRQM: QM from SR - The 4-Vector RoadMap (.html)
See SRQM: QM from SR - The 4-Vector RoadMap (.pdf)

Quantum Mechanics is derivable from Special Relativity
See SRQM - QM from SR - Simple RoadMap (.html)
See SRQM - QM from SR - Simple RoadMap (.pdf)

length/time	[m] meter <*> [s] second	Count of the quantity of separation or distance; Location of <event>'s in spacetime
mass	[kg] kilogram	Count of the quantity of matter; (the "stuff" at an <event>)
EMcharge	[C] Coulomb	Count of the quantity of electric charge; the Coulomb is more fundamental than the Ampere Amperes are just moving Coulombs
temperature	[ºK] Kelvin	Count of the quantity of heat (statistical)

γ	-β_xγ	-β_yγ	-β_zγ
-β_xγ	1+(γ-1)(β_x/β)²	( γ-1)(β_xβ_y)/(β)²	( γ-1)(β_xβ_z)/(β)²
-β_yγ	( γ-1)(β_yβ_x)/(β)²	1+( γ-1)(β_y/β)²	( γ-1)(β_yβ_z)/(β)²
-β_zγ	( γ-1)(β_zβ_x)/(β)²	( γ-1)(β_zβ_y)/(β)²	1+( γ-1)(β_z/β)²

1	0	0	0
0	Cos[φ]	-Sin[φ]	0
0	Sin[φ]	Cos[φ]	0
0	0	0	1

4-Vector Name	4-Vector Components (temporal, spatial) Magnitude & Relation to other 4-vectors	Units (mksC) - Description

		*Calculus*
4-Displacement = 4-Delta = 4-Difference	ΔR = (cΔt, Δr) ΔX = (cΔt, Δx)	[m], Δt = Temporal Displacement, Δr or Δx = Spatial Displacement, (Finite Differences) The 4-vector prototype, the "arrow" linking two <event>'s
4-Differential	dR = (cdt, dr) dX = (cdt, dx)	[m], dt = Temporal Differential, dr or dx = Spatial Differential, (Infinitesimals)
4-Gradient = 4-Del or 4-Partial or 4-∇ (a vector operator)	The tensor gradient (one-form) is technically defined as ∂_μ = ,_μ = ∂/∂x^μ = (∂_t/c,∇) = (∂_t/c,del) => (∂_t/c, ∂_x, ∂_y, ∂_z) using the lower index But the convention with 4-Vectors is to use upper indices The 4-Gradient is the rare example with the upper tensor index having a negative spatial component ∂ = ∂^μ = ∂/∂x_μ = (∂^t/c, -∇) = (∂^t/c, -del) = (∂₀/c, -∇) = (∂_t/c, -∇) => (∂_t/c, -∂_x, -∂_y, -∂_z) => (∂/c∂t, -∂/∂x, -∂/∂y, -∂/∂z) Since (∂^t = ∂_t), we typically write as ∂_t ∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = (∂_t/c,-∇)·(∂_t/c,-∇) = ∂²/c²∂t²-∇·∇ = ∂_t²/c²-∇·∇ = (im_oc / ћ)² = -(m_oc / ћ)² ∂[V·R] = V, where V is not a function of 4-position R from QM connection: ∂ = (∂_t/c, -∇) =?= (∂_t/c, β∂_t/c) =?= (v_phase\|-∇\|/c, -∇) The unit-temporal maybe/maybe not applies here ∂ = ∂_X, as opposed to the 4-MomentumGradient ∂_P	[m^-1], ∂ is the partial derivative, ∇ => (∂/∂x i + ∂/∂y j + ∂/∂z k) is the gradient operator This is a very important 4-vector operator, often used to generate continuity equations Used to obtain field-kinematics It is technically a one-form, a dual of a regular 4-vector ∂^μ = ∂/∂x_u = (∂/c∂t, -∇) and ∂_μ = ∂/∂x^u = (∂/c∂t, ∇) ∂·∂ is also known as the D'alembertian (Wave Operator Δ) I sometimes write out (del) because the nabla/del symbol (∇) is quite often displayed incorrectly in various browsers. It should look like an inverted triangle when displayed correctly. Let g_μ = ∂_μ f = ∂f / ∂x^μ Using the chain rule, one can show: g'_ν = ∂f '/∂x'^ν = Σ ( ∂f / ∂x^μ )( ∂x^μ / ∂x'^ν ) = ∂'_ν f ' = ( ∂_μ f )( ∂'_ν x^μ ) = (g_μ)( ∂'_ν x^μ ) However, this appears to be a standard Lorentz transform ∂'^μ = Λ^μ_ν ∂^ν[ function argument ] = ∂^ν[ function argument ] Λ^μ_ν importantly: U·∂ = γ(∂/∂t + u·∇) = γ d/dt = d/dτ The total derivative wrt. proper time is a Scalar Invariant ∂·X = (∂_t/c,-∇)·(ct,x) = ((∂_t/c)ct - (-∇·x)) = (∂_tt) + (∇·x) = (1) + (3) = 4 The dimensionality of spacetime is 4. ∂[X] = ∂^μ[X^ν] = (∂_t/c,-∇)[(ct,x)] = [(∂_t/c)ct, -∂_xct, -∂_yct, -∂_zct,] [(∂_t/c)x, -∂_xx, -∂_yx, -∂_zx,] [(∂_t/c)y, -∂_xy, -∂_yy, -∂_zy,] [(∂_t/c)z, -∂_xz, -∂_yz, -∂_zz,]= = Diag[(∂_t/c)ct, -∂_xx, -∂_yy, -∂_zz,] = Diag[1,-1,-1,-1] = η^μ^ν ∂[X] = η^μ^ν The 4-Gradient acting on the 4-Position gives the SR Metric ∂·J = (∂_t/c,-∇)·(cρ, j) = ((∂_t/c)cρ)- (-∇·j) = ∂_tρ + ∇·j = 0 for a conserved ElectricCurrent The 4-Gradient leads to various continuity eqns, which lead to various conservation eqns P = (E/c,p) = iћ ∂ = iћ (∂_t/c, -∇) Taking components: Temporal: E = iћ ∂_t Spatial: p = -iћ ∇ The Schrödinger QM relations (∂·∂)G(X\|X') = δ⁴(X-X') The d'Alembertian on the 4D Green's function The 4-Gradient of a Lorentz scalar is a 4-Vector ∂[V·R] = V, where V is not a function of R K = (ω/c, k) R = (ct, r) K·R = -φ_EM = (ωt-k·r) ∂ = (∂_t/c, -∇) ∂[K·R] = (∂_t/c, -∇)[ωt-k·r] = (ω/c, k) = K
4-MomentumGradient used in QM as momentum-space vector operator	∂_P = ∂/∂P_μ = (c∂_E, -∇_p) => (c∂_E, -∂p_x, -∂p_y, -∂p_z) => (c∂/∂E, -∂/∂p_x, -∂/∂p_y, -∂/∂p_z)	[kg^-1 m^-1 s], ∂_E is the momentum-space partial derivative, ∇_p = momentum-space gradient ∂_X[P·X] = P ∂_P[P·X] = X
4-WaveGradient used in QM as wave-space vector operator	∂_K = ∂/∂K_μ = (c∂_ω, -∇_k) => (c∂_ω, -∂k_x, -∂k_y, -∂k_z) => (c∂/∂ω, -∂/∂k_x, -∂/∂k_y, -∂/∂k_z)	[m], ∂_ω is the wave-space partial derivative, ∇_k = wave-space gradient ∂_X[K·X] = K ∂_K[K·X] = X from the de Broglie relations, we get the Fourier transforms ∂_X = -i K ∂_K = i X Example of crystal momentum velocity definition: U = (1/ћ)∂_K[P·U] First, prove that this is true: (1/ћ)∂_K[P·U] = ∂_K[K·U] = ∂_K[(ω/c,k)·γ(c,u)] = ∂_K[γ(ω - k·u)] = γ∂_K[(ω - k·u)] = γ(c∂_ω, -∇_k)[(ω - k·u)] = γ(c∂_ω[(ω - k·u)], -∇_k[(ω - k·u)]) = γ(c∂_ω[ω], -∇_k [- k·u]) = γ(c∂_ω[ω], +∇_k[k·u]) = γ(c,u) = U Next, the part in the function P₁·U₂ = γ[u₂](E-p₁·u₂) = E_rel U = (1/ћ)∂_K[P·U] = (1/ћ)∂_K[E_rel] U = γ(c,u) = (1/ћ)(c∂_ω, -∇_k)[E_rel] Take the spatial part: γ(c,u) = (1/ћ)(c∂_ω, -∇_k)[E_rel] γu = (1/ћ)(-∇_k)[E_rel] Take the Newtonian limit: u = (1/ћ)(-∇_k)[E_rel]

		*Particle/<Event> Dynamics*
4-Position 4-<Event>	R = R^μ = (ct, r), eg. radial coords => (ct,r,θ,z) X = X^μ = (ct, x), eg. Cartesian coords => (ct,x,y,z) Q = Q^μ = (ct, q), eg. Hamiltonian coords => (ct,q₁,q₂,q₃) Note These coordinate choices do not affect the spacetime, they are simply choices of convenience, where mathematical expressions may take on a simpler looking form. R·R = (Δs)² = (ct)²-r·r = (ct)²-\|r\|² {R·R = 0 for photonic path/null path} {ct = \|r\| for photonic} dR/dτ = U	[m], t = Time (temporal), r or x = 3-Position (spatial) Location of an <Event>, the most basic 4-vector (when, where) This is just a 4-Displacement with one of the <event>'s at the origin (0,0,0,0) of the chosen coordinate system c = Speed-of-Light often seen as X when choosing a Cartesian representation often seen as R when choosing a radial/cylindrical/spherical representation interesting derivation: U = dR/dτ dR = dτU R = ∫dτU let U be a constant, then R = τU = t_oU = t_oγ(c, u) = t(c, u) = (ct, ut) = (ct, x) where τ = t_o and t = γt_o time dilation τ = t / γ Position is essentially the ProperTime * 4-Velocity, like some of the other flux 4-vectors
4-Velocity	U = dR/dτ = γ(c, u) = dR/dτ = γdR/dt = γ(c, u_r) = γ(c, u) = γ(c, cβ) = γc(1, β) => γ(c,u_x,u_y,u_z) {U = γ_cc(1,n), for light-like/photonic} {γ_c is infinite, so basically U is not well-defined for photonic} U_o = (c,0) in rest frame U·U = c² U₁·U₂ = γ[u₁]γ[u₂](c²-u₁·u₂) = γ[u₁₂]c² = γ[u_rel]c² dU/dτ = A 4-Velocity is c*UnitTemporal 4-Vector U = cT	[m s^-1], γ = relativistic factor, u_r = Relativistic 3-Velocity, u = dr/dt = Newtonian 3-Velocity u_r = (r)' = r' u = dr/dt = r' thus, u_r = u "U is historically used instead of V" U_o = (c,0), 4-Velocity is always future-pointing time-like usually seen as U, sometimes as V only 3 independent components since U·U = c² = constant the temporal component is determined by the spatial components this actually gives a reason why several 4-vectors are a (Lorentz Scalar)*(4-Velocity): It allows these new 4-vectors to have 4 independent components interesting derivation: Let general 4-vector A = k U = (a_o,a) = kγ(c, u),where k is a constant Then A·U = k U·U = kc² So, a_o = kγc = γ A·U / c And, a = kγu = (γ A·U / c²)u = (a_o/c)u Try 4-momentum P P = (E/c,p) = m_o U E/c = γ P·U /c E = γ P·U or, in Hamiltonian formalism, H = E = γ P·U
4-Acceleration	A = dU/dτ A = γ(c dγ/dt, dγ/dt u+γ a) = dU/dτ = γ dU/dt =d²R/dτ² = γ(c dγ/dt, a_r) = γ(c γ', a_r), where γ' = dγ/dt = γ(c γ', γ' u+γ a) = γ(a_r·u/c, a_r), because A·U = 0 = γ²[(γ²/c)(u·a), a + (γ²/c²)(u·a) u] A_o = (0,a_o) in rest frame A·A = -a² = - γ⁴[a² + ( γ/c)²(u·a)²] = - γ⁶[a² - (u x a)²/c²] dA/dτ = J U·dU/dτ = U·A = 0 : 4-Acceleration orthogonal to it's worldline	[m s^-2], a_r = Relativistic 3-Acceleration, a = du/dt = Newtonian 3-Acceleration a_r = (γu_r)' = γ' u_r + γ u_r' = γ' u + γ a = (γ³/c²)(u·a) u + γ a a = du/dt = u' γ' = dγ/dt = (γ³/c²)(u·a) = (u·a_r)/c² Interesting note: The temporal component has units of frequency, before the c factor, and is given by γ(dγ/dt)=γ(γ') γ(c γ')=γ(u·a_r)/c γ'=(u·a_r)/c² 4-Spin also has a temporal component in this form, given by u·s/c I now wonder if all 4-vectors which are tangent to the worldline possess this "cyclic" feature...
4-Jerk	J = dA/dτ = γ dA/dt =d³R/dτ³ = γ( c(dγ/dt)² + cγ(d²γ/dt²), dγ/dt a_r+γ da_r/dt ) = γ( c γ'² + c γ γ'', γ' a_r + γ a_r' ) = γ( c γ'² + c γ γ'', j_r ) where γ' = dγ/dt, γ'' = d²γ/dt², a_r' = da_r/dt dJ/dτ = S	[m s^-3], j_r = Relativistic 3-Jerk, j = da/dt = Newtonian 3-Jerk j_r = (γa_r)' = γ' a_r + γ a_r' j = da/dt = a' γ' = dγ/dt = (γ³/c²)(u·a) γ'' = dγ'/dt = d²γ/dt² = (γ³/c²)[(3γ²/c²)(u·a)² + (u'·a) + (u·a'*)]
4-Snap = 4-Jounce	S = dJ/dτ = γ dJ/dt =d⁴R/dτ⁴	[m s^-4], s_r = Relativistic 3-Snap, s = dj/dt = Newtonian 3-Snap s_r = (γj_r)' = γ' j_r + γ j_r' s = dj/dt = j'

		*Kinematics*
4-Momentum	P = m_oU = (E_o/c²)U = ћK P = (mc, p) = γm_o(c,u) = (γm_oc,γm_ou) = (E/c, p) = (γE_o/c²)(c,u) = (E/c,βE/c) = (E/c,uE/c²) = (E/c)(1,β) = (E_o/c)γ(1,β) = (\|p\|/\|β\|, p) = (\|p\|c/\|u\|, p) = (v_phase\|p\|/c, p) {P= (ћω/c)(1, n) = (E/c)(1, n) = (\|p\|, p) = \|p\|(1, n) , for light-like/photonic} P_o = (E_o/c, 0) = (m_oc, 0) in rest frame P = ((E_o + p_oV_o)/c²)U, taking into account pressurevolume terms where pressure p = p_o volume V = V_o/ γ P·P* = (m_oc)² = (E_o/c)² {P·P = 0 for photonic} dP/dτ = F Lorentz Force - Covariant eqn. of motion for a particle in an EM field: dP^μ / dτ = q F^μν dX_ν / dτ dP^μ / dτ = q F^μν U_ν dP^μ / dτ = q (∂^uA^ν-∂^νA^u)U_ν 4-Momentum is (m_oc)UnitTemporal 4-Vector P = (m_oc)T 4-Momentum inc. Spin Ps = Σ·P* = Σ^μ_ν P^ν where Σ^μ_ν is the Pauli Spin Matrix Tensor Ps = (σ⁰ E/c,σ·p) where σ⁰ is an identity matrix of appropriate spin dimension and σ is the Pauli Spin Matrix Vector Ps·Ps = (σ⁰ E/c)² - (σ·p)² = (m_oc)² = (E_o/c)² 4-Momentum inc. Spin in External Field where: H = E_T = Hamiltonian = TotalEnergyOfSystem p_T = Total3-MomentumOfSystem Ps = Σ·P P_T = P + qA P = P_T - qA Ps = [σ⁰(E_T/c-qφ/c) , σ·(p_T-qa)] = (σ⁰ E/c,σ·p) Ps·Ps = [σ⁰ (E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)² = (E_o/c)² Note that the intrinsic spin is not something that must come from QM, the spin is an artifact of the Poincare Group, where mass and spin are the Casimir Invariants of the SR Poincare Group	[kg m s^-1], E = Energy, p_r = Relativistic 3-Momentum, p = mdr/dt = Newtonian 3-Momentum p_r = p m_o = RestMass( 0 for photons, + for massive ) 4-Momentum used with single whole particles E = γm_oc² = \|p\|v_phase =√[(\|p\|c)²+(m_oc²)²], in general {E = pc, for photons} 4-Momentum is used with single whole particles Note It is only the 4-Momentum of a closed system that transforms as a 4-vector, not the 4-momenta of its open sub-systems. For example, for a charged capacitor, one must sum both the mechanical and EM momenta together to get an overall 4-vector for the system. Lorentz Force - Covariant eqn. of motion for a particle in an EM field: dP^μ / dτ = q F^μν dX_ν / dτ dP^μ / dτ = q F^μν U_ν F^uv = ( ∂^uA^ν-∂^νA^u) Electromagnetic Field Tensor (F⁰ⁱ = -Eⁱ,F^ij = e^ijkB^k) taking just the μ=1 component dP¹ / dτ = q F^1ν U_ν = q[ F¹⁰ U₀ + F¹¹ U₁ + F¹² U₂ + F¹³ U₃ ] = q[ E_x/c γc + 0 γu_x + B_z γu_y + (-B_y) γu_z ] = q[ E_x γ + 0 γu_x + γ(u x B)_x ] = γq[ E_x + (u x B)_x ] taking 2 & 3, and then gathering the 3-vector components dp/dτ = γq[ E + (u x B) ] dp/dt = q[ E + (u x B) ] taking just the μ=0 component dP⁰ / dτ = q F⁰^ν U_ν = q[ F⁰⁰ U₀ + F⁰¹ U₁ + F⁰² U₂ + F⁰³ U₃ ] = q[ 0 γc + E_x/c γu_x +E_y/c γu_y + E_z/c γu_z ] dP⁰ / dτ = γq E·u dE / dτ = γq E·u, where E=Energy and E=ElectricField dE / dt = q E·u, where E=Energy and E=ElectricField so, [ dP^μ / dτ = q F^μν U_ν ] gives [dE/dt = q E·u] and [dp/dt = q[ E+(u x B)] ] F^μ = q F^μν U_ν
4-MomentumDensity = 4-MassFlux	G = (u/c, g) = (p_mc, g) = p_{o_m} γ(c, u) = p_{o_m}U =(u_o/c²)U = (1/V_o)P = (1/c²)S p_{o_m} = m_on_o?? Starting with 4-Momentum inc. pressure terms... P = ((E_o + p_oV_o)/c²)U P = (m_o + p_oV_o/c²)U P/V_o = (m_o/V_o + p_oV_o/V_oc²)U P/V_o = G = (m_o/V_o + p_o/c²)U G = (p_{o_m} + p_o/c²)U G = ((u_o + p_o)/c²)U, pressure p = p_o	[kg m^-2 s^-1], u = EnergyDen = ne, p_m = MassDen = u/c² g = MomentumDen = (u/c²)u = (e_o)ExB, f = g·u = MomentumFlux u = 3-velocity, n = ParticleDen, e = EnergyPerParticle Proper Density p_{o_m} = m_o / V_o Must be careful here though - Note p_{_m} = m/V = (γm_o)/(V_o/γ) = γ²m_o/V_o = γ²p_{o_m} The mass density p_{_m} goes as the square of γ So this p is actually the (0,0) component of the EM Stress Tensor ∂·G = 0 for a conserved Momentum Density, (for ex. a perfect fluid, which is characterized by density, pressure, and worldline velocity) 4-MomentumDensity is used with mass distributions Note It is only the 4-Momentum of a closed system that transforms as a 4-vector, not the 4-momenta of its open sub-systems. For example, for a charged capacitor, one must sum both the mechanical and EM momenta together to get an overall 4-vector for the system. Start with Energy-Momentum Stress Tensor: T^μν = (p_{o_m}+p/c²)u^μu^ν - pη^μν Contract with the 4-Velocity T^μνu_ν = (p_{o_m}+p/c²)u^μu^νu_ν - pη^μνu_ν T^μνu_ν = (p_{o_m}+p/c²)u^μc² - pu^μ T^μνu_ν = (c²p_{o_m}+p)u^μ - pu^μ T^μνu_ν = c²p_{o_m}u^μ T^μνu_ν = c²G
4-Force or 4-Minkowski Force	F = dP/dτ F = γ(dE/cdt, f_r) = γ(dmc/dt, f_r) = dP/dτ = m_o(dU/dτ) + (dm_o/dτ)U, generally = m_o(A) + (dm_o/dτ)U, generally = γ[(f_r·u + W)/c, f_r] generally where W = ( c² - v² )(dm_o/dτ) F = m_oA , if m_o is stable = γ(f_r·u/c, f_r), if rest-mass preserving = γm_o(c dγ/dt, a_r) = γm_o(c γ', a_r)	[kg m s^-2], dE/dt = Power, f_r = Relativistic 3-Force, f = Newtonian 3- Force f_r = dp/dt = m_o d(γu)/dt = m_o(γ' u + γ u') = m_o(γ' u + γ a) = γm_o[a + (γ²/c²)(u·a) u] = γ[f + (γ²/c²)(u·f) u] f = m_ou' = m_oa a = du/dt = u' Sometimes known as the Minkowski Force 4-Force is used with single whole particles F·U = (m_oA+(dm_o/dτ)U)·U = c²(dm_o/dτ) = γc²(dm_o/dt) U·F = γ²(dE/dt-u·f_r) = γ dm_o/dt c² (pure force if dm_o/dt = 0) solving for dE/dt, the modified rate-of-work equation dE/dt = u·f_r + c²/γ²(dm_o/dτ) dE/dt = u·f_r + ( c² - v² )(dm_o/dτ) (Rate of particle energy change) = (rate of work done by applied force) + (rate of work done by other mechanisms) Let W = ( c² - v² )(dm_o/dτ), the work done by "non-mechanical" effects, ex. heat In the co-moving frame, ω_o = (c²)(dm_o/dτ) Then F = γ[(f_r·u + W)/c, f_r]
4-Force Density	F_d = F/V_o or F/δV_o F_d = γ(du/cdt, f_dr) = dG/dτ??	[kg m^-2 s^-2], 4-Force divided by rest volume element 4-ForceDensity is used with mass distributions

		*Connection to Waves*
4-WaveVector or 4-AngWaveVector or 4-deBroglieWaveVector	K = (ω_o/c²)U = (m_o/ћ)U = (1/ћ)P K = (ω/c, k) = (1/cT,n/λ) = (ω/c, n ω/v_phase) = (v_phase\|k\|/c,k) = (ω/c, β ω/c) = (ω/c)(1,β) = (1/cT)(1,β) = (ω/c, u ω/c²) = (1/ћ)(E/c,p) = (ω_o/c²) γ(c,u) = (\|k\|/\|β\|,k) {K = (ω/c)(1,n) , for light-like/photonic} K·K = (m_oc / ћ)² = (ω_o/c)² = (1/cT_o)² K = (ω_o/c)T = (ω_o/c²)U where T is the unit-temporal 4-vector [ T = γ(1,β) ]	[rad m^-1], ω = AngularFrequency [rad/s], k = WaveNumber or WaveVector [rad/m] n = UnitWaveNormalVector ω = 1/T = 2π/T AngularFreq is 2π rad/Period k = 1/λ = 2π/λ AngularWaveNumber is 2π rad/Wavelength T = Period T = T/2π = reduced Period λ = Wavelength λ = λ/2π = reduced Wavelength h = Planck's Const ћ = h/2π = reduced Planck's Const v_phase = ω/k = ωλ = 1/kT = (E/ћ)/(p/ћ) = E/p = phase_velocity = celerity = velocity of simultaneity K = (ω_o/c²)U k = (ω_o/c²)γu = (γω_o/c²)u Rearranging: c² = ωu/k = ωλu = νλu = v_phaseu: True for all particles If we let u=c, then c = ωk = ωλ = νλ = v_phase: True only for photonic particles ω² = k²c² + ω_o² ω²/k² = c² + ω_o²/k² v_phase² = c² + ω_o²/k² v_phase = √[c² + ω_o²/k²] v_group = dω/dk = u = <event> velocity for SR waves ω_o = RestAngularFrequency( 0 for photons, + for massive ) ω = 2πν, k = 2π/λ ω = γω_o k everywhere points in the direction orthogonal to planes of constant phase φ where phase φ = -K·R= - ( ωt - k·r* ) = ( k·r - ωt ) ∂[K·R] = K, assuming K ≠ f [X] (K·∂)R = (K·∂[R]) = (ω_o/c²)(U·∂)R = (ω_o/c²)(d/dτ)R = (ω_o/c²)U = K (d/dτ) = (U·∂) differential along 4-Velocity direction (i.e. along proper time) (d/dθ) = (K·∂) differential along 4-Wave/Ray direction (null for photonic) ( v_phase * u = c² ) from K = (ω_o/c²)U Since (0 <= u <= c), then (c <= v_phase <=Infinity) such that ( v_phaseu = c² ) The variable u can either be taken to be the velocity of the particle/<event> or the group velocity of the corresponding matter wave. ω = √[k²c² + ω_o²] from K·K* dω/dk = (1/2)(1/√[k² c² + ω_o²])2kc² = (1/ω)kc² = kc²/ω = c²/v_phase = u Thus v_group = u = v_event v_group = dω/dk = (∂ω/∂k_x,∂ω/∂k_y,∂ω/∂k_z) = u v_phase = ω/k = (ω/k_x,ω/k_y,ω/k_z) Relativistic Doppler Effect for Photons ω_obs= ω_emit / γ(1 - β cos[θ]) K = i ∂ from de Broglie relations
4-Frequency *Break with standard notation* better to use the 4-WaveVector	Ν = (ν, c/λ n) =(c/2π)K = (ν, νn) = ν(1,n) , for light-like/photonic	[cyc s^-1], ν = ω/2π, λ = 2π/k ω = 2πν, k = 2π/λ νλ = c, for photonic *this is bad notation based on our 4-vector naming convention* the c-factor should be in the temporal component the 4-vector name should reference the spatial component I simply include it here because it can sometimes be found in the literature
4-CycWaveVector or 4-InverseWaveLength	Kcyc = (ν/c,1/λ n) = (1/cT,1/λ n) = (1/λ)(v_phase/c, n) = (ν)(1/c, n/v_phase) =(1/2π)K = (1/h)P = (ν_o/c²)U = (1/h)P = (m_o/h)U = (ν/c)(1,n) , for light-like/photonic sometimes called L	[cyc m^-1], ν = CyclicalFrequency [cyc/s], λ = WaveLength [m/cyc] n = UnitWaveNormalVector ν = ω/2π = Frequency T = Period T = T/2π = reduced Period λ = Wavelength λ = λ/2π = reduced Wavelength 1/λ = k/2π = Inverse WaveLength [cyc/m] ћ = h/2π = Dirac's Const v_phase = λν = (Phase) Velocity of Wave (sometimes called celerity)

		*Flux 4-Vectors*
	Flux 4-Vectors all in form of : V = {rest_charge_density} U V = {rest_charge}n_o U where n = γn_oalternately, V = (cs,f) where s = source, f = flux vector and ∂·V = 0 for a conserved flux	Flux 4-Vectors all have units of [{charge} m^-2 s^-1] = [{charge_density} m s^-1] Flux is the amount of {charge} that flows through a unit area in a unit time Flux can also be thought of as {charge_density_velocity} = {current_density} {charge} [{charge_unit}] {charge_density} [{charge_units}/m³] {flux} = {charge_density_velocity} = {current_density} [({charge_units}/m³)*(m/s)] = {charge per area per second}
4-NumberFlux "SR Dust"	N = (cn, n_f) = n_o γ(c, u) = n(c, u) = n_oU	[(#) m^-2 s^-1], n_o = RestNumberDensity [#/m³], n = γn_o = NumberDensity [#/m³] n_f = nu = NumberFlux [(#/m³)(m/s)] # of stable particles N = n_oV_o = nV This is the SR "Dust" 4-Vector, which is valid for a perfect gas, i.e. non-interacting particles, no shear stresses, no heat conduction N = Σ_a [∫dτ δ⁴(x-x_a(τ))(dX_a/dτ)] = Σ_a [∫dτ δ⁴(x-x_a(τ))(U_a)] ∂·N* = 0 for a conserved NumberFlux This is equivalent to 4-ProbabilityCurrentDensity [# m^-2 s^-1], 4-Probability Current Density is proportional to the 4-Momentum Equivalent to SR Dust = Number-Flux 4-vector obeys the continuity equation [#/m³], ρ = ProbChargeDensity j = ρu =ProbCurrentDensity = ProbChargeFlux [(#/m³)(m/s)] ρ = (iћ/2m_oc²)(ψ ∂_t[ψ]-∂_t[ψ] ψ) j = (-iћ/2m_o)(ψ ∇[ψ]-∇[ψ] ψ) Conservation of ProbabilityCurrentDensity ∂·J_prob* = ∂ρ/∂t +∇·j = 0
4-VolumetricFlux	V = V_oU??	[(m³) m^-2 s^-1], V_o = RestVolume
4-ElectricCurrentDensity =4-CurrentDensity = 4-ElectricChargeFlux	J = (cρ, j) = ρ_o γ(c, u) = ρ(c, u) = ρ_oU = qn_oU = qN = J_elec	[(C) m^-2 s^-1], ρ_o = RestElecChargeDensity [C/m³], ρ = γρ_o = ElecChargeDensity j = γ ρ_o u = ρu =ElecCurrentDensity = ElecChargeFlux [(C/m³)(m/s)] j = α(E+uxB), α = Conductivity q = Electric Charge [C] ρ_o = qn_o [C/m³] ∂·J* = 0 for a conserved ElectricCurrent Maxwell Equations: (∂·∂)A_EM = μ_o J+∂(∂·A_EM) Inhomogeneous Maxwell Equation (∂·∂)A_EM = μ_o J Homogeneous Maxwell/Lorentz Equation (if ∂·A_EM = 0 Lorenz Gauge)
4-MagneticCurrentDensity = 4-MagneticChargeFlux = Zero (so far..)	J_mag = (cρ_mag, j_mag) = ρ_{o_mag} γ(c, u) = ρ_{o_mag}U = q_{o_mag}n_oU = Zero (so far...)	[(MagCharge) m^-2 s^-1], ρ_{o_mag} = RestMagChargeDensity, ρ_mag = γρ_{o_mag =}MagChargeDensity j_mag = MagCurrentDensity = MagChargeFlux q_{o_mag} = Magnetic Charge to date: ρ_{o_mag} = 0 and j_mag = 0 -- no magnetic (monopole) charges yet discovered
4-ChemicalFlux		[(mol) m^-2 s^-1],
4-MassFlux = 4-MomentumDensity	G = (u/c, g) = (cρ_m, g) = ρ_{o_m} γ(c, u) = ρ_{o_m}U ρ_{o_m} = m_on_o?? = (1/c²)S	[(kg) m^-2 s^-1], u = EnergyDen = ne, p_m = MassDen = u/c² g = MomentumDen = (u/c²)u = (e_o)ExB, f = g·u = MomentumFlux u = 3-velocity, n = ParticleDen, e = EnergyPerParticle Poincare' made the observation that, since the EM momentum of radiation is 1/c² times the Poynting flux of energy, radiation seems to possess a mass density 1/c² times its energy density
4-PoyntingVector = 4-EnergyFlux = 4-RadiativeFlux = 4-MomentumDensity? Technically not a 4-vector Just as the (e) and (b) fields of EM are not the spatial parts of 4-vectors, the Poynting Vector (s) is not the spatial part of a 4-vector. (s) is part of the EM Stress-Energy Tensor. (s) is the T⁰ⁱ component of the symmetric EM Stress-Energy Tensor in the same way that (e) is the F⁰ⁱ component of the antisymmetric EM Faraday Tensor	S = (cu, s) = u_o γ(c, u) = u_oU = c²G ????u_o = E_on_o??	[(J) m^-2 s^-1], u = EnergyDen = ne, s = EnergyFlux = PoyntingVector = uu = c²g = Ne u_e = (ε_oE·E+B·B/μ_o)/2 = (E·D+B·H)/2 = EM energy density technically should be one row of the Energy-stress tensor typically see ∂u/∂t = - ∇·s + J_f · E which in 4-vector notation would be ∂·S = j_f · E, where j_f is the current density of free charges, so not conserved generally however, make the following observation: ∂[u_m]/∂t = j(t,x)·E(t,x), where this is rate of change of kinetic energy of a charge then let u = u_e+u_m, s = s_e+s_m, ∂·S = ∂[u_e+u_m]/∂t +∇·(s_e+s_m) = 0 we have conservation/continuity again, by allowing energy to transform into different types. In the example, energy is passing back & forth between the physical charges and the EM field itself. Energy as a whole is still conserved. ε_o = Permittivity, μ_o = Permeability ε_oμ_o = 1/c² s = (E x B)/μ_o = EnergyCurrentDensity u = 3-velocity, n = ParticleDen, e = EnergyPerParticle, N = ParticleFlux = nu see also Umov-Poynting Vector for generalization to mechanical systems S = (cu, s) = (c(u_e+u_m),s_e+s_m) ∂·S = ∂[u_e+u_m]/∂t +∇·(s_e+s_m) = 0 u_m = mechanical kinetic energy density s_m = mechanical Poynting vector, the flux of their energies The sum of mechanical and EM energies, as well as the sum of mechanical and EM momenta are conserved inside a closed system of fields and charges. Another way to say this is that only the four-momentum of a closed system transforms as a 4-vector, not the four-momentums of its open sub-systems. Since only the microscopic fields E and B are needed in the derivation of S = (1/μ_o)(ExB), assumptions about any material possibly present can be completely avoided, and Poynting's vector as well as the theorem in this definition are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the case above Energy types: electrical, magnetic, thermal, chemical, mechanical, nuclear The density of supplied energy is restricted by the physical properties through which it flows. In a material medium, the power of energy flux U is restricted, (U < vF), where v is the deformation propagation velocity, usually the speed of sound, F may be any elastic or thermal energy, U is a vector. div U determines the amount of energy transformation into a different form. For a gaseous medium, U = a √[T] p, where A is a coefficient which depends on molecular composition, T is temperature, p is pressure. not be be confused with the 4-EntropyFlux Vector S
4-EntropyFlux	S = (cs, s_f) = s_o γ(c, u) = s_oU = sρ_oU more correct term including heat/temp S = s_oU + Q/T_o = sρ_oU + Q/T_o ∂·S >= 0	[(J ºK^-1) m^-2 s^-1], s_o = RestEntropyDensity = q_o/T, s_f = EntropyFlux Entropy S = ∫s_odV = k_B ln Ω, where Ω = # of microstates for a given macrostate ∂·S >= 0 S = s_oU + Q/T_o where Q is the Thermal Heat Flux 4-vector 1st term is entropy carried convectively with mass 2nd term is entropy transported by flow of heat (generalization of dS = dQ/T) If Q·Q = const, then the Jacobian says that dq/q_o ~ dq/T is a Scalar Invariant Q = (cq, q_f) = q γ(c, u) = qU = ( c (k/a)T , q_f) = ( c (k/a)T , -k ∇ [T]) not be be confused with the 4-Poynting Vector S

		*Thermodynamic*
4-InverseTempFlux	β = β_o U = (1/k_BT_o) U where β_o = 1/k_BT_o dS = β·dP, differential entropy * Note* This β is not the relativistic β factor	Considered on Thermodynamic principles also known as a Killing vector "The proper relativistic temperature is not agreed upon by Einstein, Ott, and Landsberg, who respectively think that moving objects are colder, hotter and invariant. You can try reading these and seeing what each do, how they differ in their assumptions and why they disagree with each other. However, given the fact that there does seem to exist genuine disagreement, it is suspected that the matter has not been settled. Also since neither SR nor thermodynamics are complicated in their mathematical settings, the problem is likely to be that of a foundational nature --- i.e. what does temperature mean for a moving object."
4-MomentumTemperature	P_T = P/k_B = (p_t⁰/c, p_T) = (T/c, p_T) = ((E/k_B)/c, p/k_B) p_t⁰ = T = Temperature (in ºK) simply dividing 4-Momentum by Boltzmann's const. k_B which gives E/c = k_BT/c, or E = k_BT	[ºK m^-1s], ºK = degree Kelvin Temperature k_B = 1.380 6504(24)×10⁻²³ [J/ºK] Boltzmann's constant Not sure if this is valid, but perhaps useful as a gauge of photon temperature based entirely on dimensional considerations of k_B [J/ºK] energy/temperature similarly to c [m/s] being a fundamental constant relating length/time

		*Diffusion/Continuity based* see Atomic Diffusion Brownian Motion Electron Diffusion Momentum Diffusion Osmosis Photon Diffusion Reverse Diffusion Thermal Diffusion
4-Potential Flux??	V = (cq, q_f) = q γ(c, u) = qU?? where q = [1]?? = ( c (k/a)φ , q_f)?? = ( c (k/a)φ , -k ∇ [φ])?? needs work	Potential Flow for Velocity?? Velocity Potential "Velocity" Conduction Equation: v = -k ∇ [φ] "Velocity" Diffusion Equation: a ∇·∇ [φ] = ∂φ/∂t where ∇·v ~ ∂φ/∂t Continuity gives ∂·V = ∂[c(k/a)φ]/∂t +∇·v = 0 Thus, [ (k/a)φ ] and [ v ] are components of a 4-vector In fluid dynamics, a potential flow is described by means of a velocity potential , φ being a function of space and time. The flow velocity v is a vector field equal to the negative gradient, ∇, of the velocity potential φ: Incompressible flow In case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence: ∇·v = 0 with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation ∇·∇ φ = 0 where Δ = ∇·∇ is the Laplace operator. In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of the flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of Bernoulli's principle. The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flow Main article: Steady flow The flow of a fluid is said to be steady if v does not vary with time. That is if dv/dt = 0 Incompressible flow Main article: Incompressible flow A fluid is incompressible if the divergence of v is zero: ∇·v = 0 That is, if v is a solenoidal vector field. Irrotational flow Main article: Irrotational flow A flow is irrotational if the curl of v is zero: ∇ x v = 0 That is, if v is an irrotational vector field. Vorticity Main article: Vorticity The vorticity, ω, of a flow can be defined in terms of its flow velocity by ω = ∇ x u Thus in irrotational flow the vorticity is zero. The velocity potential Main article: Potential flow If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field φ such that v = -k ∇ [φ] The scalar field φ is called the velocity potential for the flow. (See Irrotational vector field.) An lamellar vector field is a synonym for an irrotational vector field.[1] The adjective "lamellar" derives from the noun "lamella", which means a thin layer. In Latin, lamella is the diminutive of lamina (but do not confuse with laminar flow). The lamellae to which "lamellar flow" refers are the surfaces of constant potential. An irrotational vector field which is also solenoidal is called a Laplacian vector field. The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v with divergence zero: ∇·v = 0 The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: v = ∇ x A automatically results in the identity (as can be shown, for example, using Cartesian coordinates): ∇·v = ∇·(∇ x A) = 0 The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ x A. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.) In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations: ∇ x v = 0 ∇·v = 0 Since the curl of v is zero, it follows that v can be expressed as the gradient of a scalar potential (see irrotational field) φ : v = ∇ φ (1) Then, since the divergence of v is also zero, it follows from equation (1) that ∇·∇ φ = 0 which is equivalent to ∇² φ = 0 Therefore, the potential of a Laplacian field satisfies Laplace's equation. In fluid dynamics, a potential flow is a velocity field which is described as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of a gradient always being equal to zero (since the curl of a gradient is equivalent to take the cross product of two parallel vectors, which is zero). In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows. Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. A velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region and is irrotational. In such a case, ∇ x u = 0 where u denotes the flow velocity of the fluid. As a result, u can be represented as the gradient of a scalar function Φ: u = ∇ Φ Φ is known as a velocity potential for u. A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant. Unlike a stream function, a velocity potential can exist in three-dimensional flow.
4-HeatFlux 4-ThermalHeatFlux 4-ThermalEnergyFlux	Q = qU = q γ(c, u) = (cq, q) = ( c (k/α)T , q) = ( c (k/α)T , -k ∇ [T]) that q_f = -k ∇ [T] is an additional constraint Alternate Def:??? Q = k ∂[T] = k(1/c ∂T/∂t , - ∇ [T]) = (k/c ∂T/∂t , -k ∇ [T]) which would imply k/c ∂T/∂t = c (k/α) T ???? ∂T/∂t = (c²/α) T ???? I don't think this is right yet another... Q^α = -K P^αβ(T,_β + T A_β/c²) Heat flux 4-vector balances the EM energy flux vector if expansion, rotation, and shear of fluid vanish	[(J) m^-2 s^-1] = [(W) m^-2], Potential Flow for Heat k = Thermal Conductivity [W/(m °K)] Note - this is not Boltzmann's const k_B α = k/(ρc_p) = Thermal Diffusivity [m²/s] ρ = Density [1/m³] c_p = Specific heat capacity [J/ °K] ρc_p = Volumetric heat capacity [J/m³ °K] T = Absolute Temperature [°K] (k/α)T = Thermal energy density [W s / m³] = [J/m³], q = Heat Flux [W/m²] = Thermal Energy Flux Heat Conduction Equation: q = -k ∇ [T] Thermal Diffusion Equation: α ∇·∇ [T] = ∂T/∂t where ∇·q ~ ∂T/∂t Continuity gives ∂·Q = ∂[(k/α)T]/∂t +∇·q = 0 Thus, [c (k/α)T ] and [ q ] are components of a 4-vector Fourier's Law: q = -k ∇ [T] The minus sign ensures that heat flows down the temperature gradient It should be noted that T is a scalar temperature field T[X] see: A Proposed Relativistic, Thermodynamic Four-Vector see Hydraulic Analogies quantity: Heat Q [J] potential: Temperature T [°K] = [J/k_B] flux: Heat Xfer Rate Qdot [J/s] flux density: Heat Flux Qdot'' [W/m²] linear model: Fourier's Law Qdot'' = -k ∇ [T]
4-DarcyFlux 4-HydraulicFlux	Q = (cq, q_f) = q γ(c, u) = qU = ( c (βφ)P , q_f) = ( c (βφ)P , -κ/μ ∇ [P])	[(m³) m^-2 s^-1] = [m s^-1], Potential Flow for Volume/Hydraulics κ = Permeability [m²] μ = Dynamic Viscosity [Pa s] P = Pressure [N m^-2] β = Compressibility Coefficient φ = Porosity [1] q = Flux [m s^-1] = Volumetric Flux <> particle velocity v = q/φ = pore velocity Darcy's Law Equation: q = -κ/μ ∇ [P] Pressure Diffusion Equation: ∇·q = -κ/μ ∇·∇ [P] = βφ∂P/∂t, by imposing incompressibility giving Pressure Diffusion Waves Continuity gives ∂·Q = ∂[βφP]/∂t +∇·q = 0 Thus, [ βφP ] and [ q ] are components of a 4-vector Darcy's Law: q = -κ/μ ∇ [P] The minus sign ensures that flux flows down the pressure gradient Darcy's Law - derivable from Navier-Stokes see Fourier's law for heat conduction see Ohm's law for electrical conduction see Fick's law for diffusion see Hydraulic Analogies quantity: Volume V [m³] potential: Pressure P [Pa] = [J/m³] flux: Current φ_V [m³/s] flux density: Velocity [m/s] linear model: Poiseuille's Law φ_V = ...
4-ElectricChargeFlux	Q = (cq, q_f) = q γ(c, u) = qU = ( c ρ, j)	[(C) m^-2 s^-1], Potential Flow for Charge acts differently, presumably because this is a "charged" field, where the particle interacts with the field. μ = mobility σ = specific conductivity = q n μ where n = concentration of carriers Conservative Potential: E = (- ∇ [φ]) Ohm's Law: j = σ E = -σ ∇ [φ] Gauss Law: ∇·E = ∇·(- ∇ [φ]) = - ∇·∇ [φ] = ρ/ε_o Fick's 1st Law Diffusion: j = - D ∇ ρ where D = μ k T / e = Einstein-Smoluchowski Relation Continuity independently gives ∂·J = ∂[ρ]/∂t +∇·j = 0 Thus, [ ρ ] and [ j ] are components of a 4-vector Continuity independently gives ∂·A = ∂[φ]/∂t +∇·a_EM = 0 in the Lorenz Gauge Thus, [ [φ ] and [ a_EM ] are components of a 4-vector Ohm's Law: j = -σ ∇ [φ] The minus sign ensures that current flows down the potential gradient see Hydraulic Analogies quantity: Charge Q [C] potential: Potential φ [V] = [J/C] flux: Current I [A] = [C/s] flux density: Current Density j [A/m²] linear model: Ohm's Law j = - σ ∇ [φ]

		*Angular Momentum/Spin/Polarization*
4-SpinMomentum or Pauli-Lubanski 4-vector	W = (w₀,w) = (u·w/c,w) because W·U = 0 W = (w⁰,w) = (p·Σ , P⁰Σ + p x k) where Σ is the spin part of angular momentum j W = m_o S where S is the 4-Spin	[spin-momentum], W·W = (u·w/c,w)·(u·w/c,w) = (u·w/c)² - w·w) = - w·w = - m² s(s+1) W² = ( w₀² - w·w ) = - (w·w) = - (P₀²Σ²) = - m²c² ћ² s(s+1) where Σ is the spin part of angular momentum j (P₀²) = (m²c²) (Σ²) = ћ² s(s+1) W·W = 0 for photonic I am suspecting that the s(s+1) value can be derived from the Laplacian acting on a pure radial function. From mathworld.wolfram.com, ∇²[g(r)] = ∇·∇[g(r)] = (2/r)(dg/dr)+d²g/dr² Thus, for a radial power law... ∇²[r^s] = ∇·∇[r^s] = (2/r)(s r^s-1)+(s(s-1) r^s-2) = s(s+1)r^s-2 Consider that the spin 3-vector is of unit length we get s(s+1)(1)r^s-2 = s(s+1) plays the role of covariant angular momentum see Bargmann-Michel-Telegdi (BMT) dynamical eqn
4-Spin	S = (s⁰,s) = (u·s/c,s) because S·U = 0, the spin is orthogonal to world velocity S_o = (0,s_o) in rest frame S = (γ β·s_o , s + [γ²/(γ+1)](β·s_o) β) in moving frame S = (1/m_o) W= (U·U/P·U) W where m_o = √[P·P/U·U] = P·U/U·U Magnetic moment μ = -(g/2)(e/mc) s U·S = 0 : 4-Spin orthogonal to worldline dS/dτ = (-A·S/c²)U : Fermi-Walker Transport	[ J s], = [spin] Spin = IntrinsicAngMomentum, u·s/c = component such that U·S = 0 4-Spin is orthogonal to 4-Velocity, so time component is zero in rest frame S_o=(0,s_o) This is an axial vector, or pseudovector 4-Spin has only 3 independent components, not 4, due to U·S = 0 S_o=(0,s_o), 4-Spin is always space-like S·S = (u·s/c,s)·(u·s/c,s) = ((u·s/c)² - s·s) = - s_o·s_o = - ћ² s_o(s_o+1) Since U·S = 0 then d/dτ [U·S] = 0 = d/dτ[U]·S + U·d/dτ[S] = A·S + U· d/dτ[S] U· d/dτ[S] = - A·S if we assume d/dτ[S] = (k)U then U·d/dτ[S] = kU·U = kc² = -A·S* k = -A·S/c² then d/dτ[S] = (-A·S/c²)U, which is Fermi-Walker Transport of the 4-Spin, and leads to Thomas Precession. Fermi-Walker Transport is the way of transporting a purely spatial vector along the worldline of the particle in such a way that it is as "rotationless" as possible, given that it must remain orthogonal to the worldline. This choice also implies that d/dτ [S·S] = 0, since d/dτ [S·S] ~ [U·S] = 0, which means that the magnitude of the 4-Spin is constant ===== Thomas Precession example: Have a particle in a circular orbit of const radius = r (equivalent to γ'=0 and u·a = 0) and carrying 4-Spin S Also a chance to use cylindrical coords to simplify math Orthonormal basis (e_t,e_r,e_θ,e_z) d/dτ[(e_t; e_r; e_θ; e_z)] ==> (0e_t; γωe_θ; -γωe_r; 0e_z) This gives: R = (ct, r e_r) U = γ(c, rω e_θ) A = γ²(0, -rω² e_r) : The 4-Acceleration is highly simplified by the const r assumption! S = (s_t e_t, s_r e_r + s_θ e_θ + s_z e_z) U·S = 0 so U·S = γ(c, rω e_θ)·(s_t, s_r e_r + s_θ e_θ + s_z e_z) = γ(cs_t - rωs_θ) = 0 s_t = rωs_θ/c A·S = γ²(0, - rω² e_r)·(s_te_t, s_r e_r + s_θ e_θ + s_z e_z) = γ²rω²s_r d/dτ[S] = (-A·S/c²)U = (-γ² rω²s_r/c²)U = (-γ² rω²s_r/c²)γ(c, rω e_θ) = (-γ³rω²s_r/c²)(c, rω e_θ) d/dτ[S] = (-γ³ rω²s_r/c e_t, -γ³ r²ω³s_r/c² e_θ) d/dτ[S] = (-γ³ v²s_r/rc e_t, -γ³ v²ωs_r/c² e_θ) d/dτ[S] = (-γ³ cβ²s_r/r e_t, -γ³ β²ωs_r e_θ) d/dτ[S] = d/dτ[(s_t e_t, s_r e_r + s_θ e_θ + s_z e_z)] = (d/dτ[s_t]e_t + s_td/dτ[e_t] , d/dτ[s_r]e_r + s_rd/dτ[e_r] + d/dτ[s_θ]e_θ + s_θd/dτ[e_θ]+ d/dτ[s_z]e_z + s_zd/dτ[e_z]) = (d/dτ[s_t]e_t, d/dτ[ s_r]e_r + s_rd/dτ[e_r] + d/dτ[ s_θ]e_θ + s_θd/dτ[e_θ]+ d/dτ[s_z]e_z) = (d/dτ[s_t]e_t, d/dτ[s_r]e_r + s_rγωe_θ + d/dτ[ s_θ]e_θ - s_θγωe_r+ d/dτ[ s_z]e_z) Now, match component terms e_t :d/dτ[s_t] = -γ³rω²s_r/c e_r :d/dτ[s_r] - s_θγω = 0 e_θ :d/dτ[s_θ] + s_rγω = -γ³ r²ω³s_r/c² e_z :d/dτ[s_z] = 0 e_t :d/dτ[s_t] = -γ³ rω²s_r/c e_r :d/dτ[s_r] = s_θγω e_θ :d/dτ[s_θ] = -γ³ r²ω³s_r/c²-s_rγω = -s_rγω(γ² r²ω²/c²+1) = - s_r γω(γ²β²+1) = - s_r γω(γ²-1+1) = -s_rγ³ω e_z :d/dτ[s_z] = 0 Can combine the e_r and e_θ into harmonic equation d²/dτ²[s_r] + γ⁴ω²s_r = 0 s_r ~ cos(Ωτ) where Ω = γ²ω whereas orbital frequency is just ~ γω from U = γ(c, rω e_θ) ===== s·s \|s,m> = ћ² s(s+1) \|s,m> s_z \|s,m> = ћ m \|s,m> for s = {0 , 1/2 , 1 , 3/2 , 2 , 5/2 , ...} for m = {-s, -s+1, ..., s-1, s} SpinMultiplicity[m] = (2s+1) denotes the # of possible quantum states of a system with given principal s for s = 0, {m} = {0} singlet for s = 1/2, {m} = {-1/2 , 1/2} doublet for s = 1, {m} = {-1 , 0 , 1} triplet ... \|s\| = ћ √[s(s+1)], [ S_x ,S_y ] = i ћ ε_xyzS^z Spin raising/lowering operators: S_± \|s,m> = ћ √[s(s+1) - m(m ± 1)] \|s,m> where S_± = S_x ± i S_y Bargmann-Michel-Telegdi (BMT) dynamical eqn. for spin dS/dτ = (e/mc)[ (g/2) F^μβS_β + (1/c²)(g/2-1))v^μS_αF^αβv_β ] which leads to Thomas precession in the rest frame
4-SpinDensity
4-Rotation	Omega
4-Polarization or 4-JonesVector	Ε = (ε⁰, ε) = (ε·u/c,ε) for a massive particle = (ε⁰, ε) = ((c/v_phase) ε·n,ε) for a wave {E= (ε⁰, ε) = (ε·n,ε) ,for light-like/photonic} for photon travelling in z-direction using the Jones Vector formalism n = z / \|z\| E = (0,1,0,0) : x-polarized linear E = (0,0,1,0) : y-polarized linear E = √[1/2] (0,1,1,0) : 45 deg from x-polarized linear E = √[1/2] (0,1,i,0) : right-polarized circular = spin 1 E = √[1/2] (0,1,-i,0) : left-polarized circular = spin -1 General-polarized E = (0,Cos[θ]Exp[iα_x],Sin[θ]Exp[iα_y]),0) E* = (0,Cos[θ]Exp[-iα_x],Sin[θ]Exp[-iα_y]),0) Angle θ describes the relation between the amplitudes of the electric fields in the x and y directions Angles α_x and α_y describe the phase relationship between the wave polarized in x and the wave polarized in y Ε·E* = = (+0² - Cos[θ]Exp[iα_x]Cos[θ]Exp[-iα_x] - Sin[θ]Exp[iα_y]Sin[θ]Exp[-iα_y] -0² ) = = (+0-Cos[θ]²-Sin[θ]²-0) = - (Cos[θ]²+Sin[θ]²) = -1 Ε·E* = -1	[1], ε = PolarizationVector This 4-vector has complex components in QM Called helicity for massless particles Helicity is spin component along the direction of motion. "Helicity is the only truly measurable component of spin for a moving particle, but at low enough velocities (non-relativistic), the spin component m along an external axis becomes an alternative observable." Like the 4-Spin, Ε orthogonal to U, or K, so time component = 0 in rest frame This is cancellation of the "scalar" polarization This would again give only 3 independent components Ε·U = 0, Ε·K = 0, Additionally, Ε·E* = -1 (normalized to unity along a spatial direction) Normalization combined with \|u\| = c for photons is enough to reduce it to 2 independent components for photons. Ε·E* = -1 imposes (ε·u/c)² -ε·ε* = -1 (ε·u/c)² -1 = -1 (ε·u/c)² = 0 ε·u = 0, so that the spatial components must be orthogonal. For a massive particle, there is always a rest frame where u = 0, so ε can have 3 independent components. For a photonic particle, there is no rest frame.( ε·u/c = ε·n = 0 ) is therefore an additional constraint, limiting ε to 2 independent components, with polarization ε orthogonal to direction of photon motion n. This is cancellation of the longitudinal polarization. This seems to be related to the Ward-Takahaski Identity. According to Wikipedia-Gauge Fixing, Many of the differences between classical electrodynamics and QED can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances. see Field Quantization, by Walter Greiner, Joachim Reinhardt
4-SpinPolarization	In the rest frame, where K = (m,0), choose a unit 3-vector n as the quantization axis. In a frame in which the momentum is K = (k⁰,k) the spin polarization of a massive particle N = N^μ = ( k·n/m , n + (k·n) k / (m(m + k⁰)) ) N·N = -1, Normalized to spatial unity K·N = 0, Orthogonal to Wave Vector alternately, S = S^μ = ( p·s/m , s + (p·s) p / (m(m + p⁰)) ) which makes more sense, called the covariant spin vector W = (w⁰,w) = (p·Σ , P⁰Σ + p x k) where Σ is the spin part of angular momentum j W² = -m² Σ² = -m² s(s+1) Σ = 0 represents a spin-0 particle Σ = Dirac spinor represents a spin-1/2 particle, with Σ² = 3/4 the unit matrix I W·N = -m Σ·n = - m s, the component of spin along n as measured in the rest frame. s is the spin component in the direction n that would be measured by an observer in the particle's rest frame Apparently this only works for massive particles, so the N 4-vector is undefined for massless Instead, helicity h = Σ·k / \|k\|, h = +1/2 or -1/2 for spin 1/2 particle helicity is component of spin parallel to the 3-vector momentum k w⁰ = (k·Σ) and k⁰ = \|k\| for a massless particle alternately N^μ = ( T^μ - P^μ(pt/m²))(m/\|p\|) where T^μ is the unit time-like 4-vector	I am suspecting that the s(s+1) value can be derived from the Laplacian acting on a pure radial function. From mathworld.wolfram.com, ∇²[g(r)] = ∇·∇[g(r)] = (2/r)(dg/dr)+d²g/dr² Thus, for a radial power law... ∇²[r^s] = ∇·∇[r^s] = (2/r)(s r^s-1)+(s(s-1) r^s-2) = s(s+1)r^s-2 Consider that the spin 3-vector is of unit length we get s(s+1)(1)r^s-2 = s(s+1)
4-PauliMatrix	σ_μ = (σ₀ ,σ₁ ,σ₂ ,σ₃) = (σ₀ ,σ) = (I ,σ) Σ = Σ^μν = Diag[σ⁰,-σ]?? where Σ·Σ = -2σ⁰??	The components of this 4-vector are actually the Pauli Spin Matrices (a·σ)(b·σ) = (a·b)σ⁰ + i(axb)·σ = (a·b)I + i(axb)·σ σ⁰ = I = 2x2 Identity Matrix σ_μ = (σ₀ ,σ₁ ,σ₂ ,σ₃) = (σ₀ ,σ) = (I ,σ) There is 1-1 correspondence between real 4-vectors and 2x2 Hermetian matrices given by: [X] = x^μσ_μ = [x⁰ + x³ x¹ - ix²] [x¹ + ix² x⁰ - x³ ] x^μ = (1/2) Tr[σ_μ X] So, this may actually be a 2-index tensor?? Σ = Σ^μν = Diag[σ⁰,-σ]?? 4-Momentum inc. Spin Ps = Σ·P = Σ^μ_ν P^ν = η_αβ Σ^μα P^β = Ps^μ Σ^μ_ν is a Pauli Spin Matrix Tensor = Diag[σ⁰,σ] Σ^μν is a Pauli Spin Matrix Tensor = Diag[σ⁰,-σ] Ps = Diag[σ⁰,-σ]·P = Diag[σ⁰,-σ]·(E/c,p) = (σ⁰E/c,σ·p) Ps = (ps⁰,ps) = (σ⁰E/c,σ·p) 3-Spin S = (ћ/2)σ [S_x,S_y]=iћS_z The Pauli Matrices form an orthogonal basis for the complex Hilbert space of all 2x2 matrices, meaning that any matrix M = a⁰σ⁰ + a·σ, where A=(a⁰, a) is a 4-vector Relativistic version Ps·Ps = [σ⁰ (E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)² = (E_o/c)² [σ⁰ (E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)² [I (E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)² [I (E_T/c-qφ/c)]² - [I·(p_T-qa)]² - i[σ·((p_T x -qa) + (-qa x p_T))] = (m_oc)² [I (E_T/c-qφ/c)]² - [I·(p_T-qa)]² - i[σ·iћq((-∇_T x -a) + (-a x -∇_T))] = (m_oc)² [I (E_T/c-qφ/c)]² - [I·(p_T-qa)]² - i[σ·iћq(B)] = (m_oc)² [I (E_T/c-qφ/c)]² - [I·(p_T-qa)]² + ћq[σ·B] = (m_oc)² [I (E_T/c-qφ/c)]² = [I·(p_T-qa)]² - ћq[σ·B] + (m_oc)² or, for non-relativistic version [I (E_T/c-qφ/c)]² - [σ·(p_T-qa)]² = (m_oc)² [I (E_T/c-qφ/c)]² = [σ·(p_T-qa)]² + (m_oc)² [I (E_T/c-qφ/c)] = (+/-) Sqrt [ [σ·(p_T-qa)]² + (m_oc)² ] [I (E_T/c-qφ/c)] = (+/-) Sqrt [ [σ·(p_T-qa)]² + (m_oc)² ] [I (E_T/c-qφ/c)] ~ (+/-) [ [σ·(p_T-qa)]²/(2m_oc)+ (m_oc) ] [I (E_T/c-qφ/c)] ~ (+/-) [ ([(p_T-qa)]² - ћq[σ·B])/(2m_oc)+ (m_oc) ] [I (E_T-qφ)] ~ (+/-) [ ([(p_T-qa)]² - ћq[σ·B])/(2m_o)+ (m_oc²) ] E_T ~ qφ (+/-) [ ([(p_T-qa)]² - ћq[σ·B])/(2m_o)+ (m_oc²) ]
4-DiracGamma	Γ = Γ^μ = (γ⁰,γ)	According to some books, not strictly a 4-vector, the Dirac Gamma Matrices are actually matrices to represent intrinsic spin. However, I have seen index raising/lowering Γ_μ = η_μνΓ^μ Γ^μ Γ^ν = η^μν + (1/2)σ^μν σ^μν = [Γ^μ,Γ^ν] Not sure about a Lorentz Boost

		*Electromagnetic Field Potentials*
4-VectorPotential or 4-VectorPotential_EM	A = (φ/c, a) 4-VectorPotential of an arbitrary field A = (φ_EM/c, a_EM) 4-VectorPotential_EM of an EM field technically A[R] = (φ[ct,r]/c, a[ct,r]) or A[X] = (φ[ct,x]/c, a[ct,x]) since this is field defined for all <event>'s in spacetime I believe that in some (possibly all physical) circumstances the following is true: A = (φ_o/c²) U A = (φ/c, a) = (φ_o/c²) γ(c, u) = (γφ_o/c, γφ_o/c²u) giving φ = γφ_o a = (γφ_o/c²)u see the relativistic Lagrangian and Hamiltonian for more reasoning for this... For EM point charges... A_EM = (q/4πε_oc) U / [R·U]_ret [..]_ret implies (R·R = 0, the definition of a light signal) φ_EM = (γφ_o) = (γq/4πε_or) a_EM = (γφ_o/c²)u = (γqμ_o/4πr)u	[kg m <chargetype>^-1 s^-1], for arbitrary field [kg m C^-1 s^-1] for EM field φ_EM = ScalarPotential_EM a_EM = VectorPotential_EM Electric Field E = -∇[φ_EM]-∂a_EM/∂t = -∇[φ_EM]-∂_t[a_EM] Magnetic Field B = ∇ x a_EM Electric Field E [N/C = kg·m·A⁻¹·s⁻³] Magnetic Field B [Wb/m² = kg·s⁻²·A⁻¹ = N·A⁻¹·m⁻¹] ==== For 4-VectorPotenial of a moving point charge (Lienard-Wiechert potential) A_EM = (q/4πε_oc) U / [R·U]_ret = (qμ_oc/4π) U / [R·U]_ret where [..]_ret implies (R·R = 0, the definition of a light signal) If we use the A_EM = (φ_o/c²) U definition and compare terms with above: (φ_o/c²) = (q/4πε_oc) / [R·U]_ret (φ_o) = (qc/4πε_o) / [R·U]_ret (φ_o) = (qc/4πε_o) / [c²τ]_ret (φ_o) = (q/4πε_o) / [ cτ]_ret (φ_o) = (q/4πε_o) / r {which is the correct potential for a point charge in its rest frame} since R·R = (ct)²-\|r\|² = 0 --> ct = r And likewise, the scalar and vector potential of a moving point charge φ = (γφ_o) = (γq/4πε_o) / r a = (φ_o/c²)γu = (γφ_o/c²)u = (γφ_o/c²)u = (φ/c²)u = ((γq/4πc²ε_o) / r)u = ((γqμ_o/4π) / r)u Inhomogeneous Maxwell Equation (∂·∂)A_EM = μ_o J+∂(∂·A_EM) Homogeneous Maxwell/Lorentz Equation (if ∂·A_EM = 0 Lorenz Gauge) (∂·∂)A_EM = μ_o J (∂·∂) = (∂_t²/c²-∇·∇) (∇·∇)[1/r] = Δ[1/r] = -4πδ³(r) if [1/r] is not time dependent then (∂·∂)[1/r] = -(∇·∇)[1/r] = -Δ[1/r] = 4πδ³(r) Let's examine the 4-VectorPotential of a point charge φ = (γΦ_o) = (γq/4πε_o) / r (∂·∂)φ = (∂·∂) (γq/4πε_o) [1/ r] = (γq/4πε_o)(∂·∂)[1/ r] = (γq/4πε_o)4πδ³(r) = (γq/ε_o)δ³(r) = (γq)δ³(r)/ε_o = ρ/ε_o where ρ = (γq)δ³(r) and thus ρ_o = (q)δ³(r) (∂·∂)φ is the temporal part of (∂·∂)A_EM (∂·∂)A_EM = (∂·∂)(Φ/c,a) = (ρ/cε_o,...) = (ρcμ_o,...) = μ_o(ρc,...) = μ_oJ And, of course, once you get it in covariant form, this is general, (∂·∂)A_EM = μ_oJ (∂_t²/c²-∇· 711;)(Φ/c, a) = μ_o (cρ, j) (∂_t²/c²-∇·∇)(Φ) = μ_o( c²ρ ) = ρ/ε_o (∂_t²/c²-∇·∇)(a) = μ_o( j ) Here ρ> and j> are the charge density and current for the matter field (∇·∇)[1/r] = Δ[1/r] = -4πδ³(r) (∇·∇)[1/\|r-r'\|] = Δ[1/\|r-r'\|] = -4πδ³(r-r') If matter field is describing interaction of EM fields with Dirac electron, then one obtains the Maxwell eqns for QED ρ = q ψ^†ψ j = q ψ^†αψ (∂_t²/c²-∇·∇)(Φ) = μ_o( c²ρ ) = ( q ψ^†ψ )/ε_o (∂_t²/c²-∇·∇)(a) = μ_o( q ψ^†αψ ) If there are no sources, i.e. J = 0, then (∂·∂)A_EM = 0 A solution to this are superpositions of polarized EM Plane-Waves A_EM = Σ_n [a_n Ε e^{(i K·R)}] where E is the 4-Polarization Complex phase dθ = (q/ћ) A[X]·dX units?(SI,guassian?) NOTE: Neither the phase nor the components of the phase connection are physically observable although differences in phase connection may be observed via interference experiments. (Ref: Aharanov-Bohm effect.)
4-Potential *Break with standard notation* better to use the 4-VectorPotential	Φ = (φ,c a)	Φ = (φ,c a) where Φ = cA *this is bad notation based on our 4-vector naming convention* the c-factor should be in the temporal component the 4-vector name should reference the spatial component I simply include it here because it can sometimes be found in the literature
4-VectorPotential_Grav 4-VectorPotential_GEM Note This is an approximation only For more accurate results the full GR theory must be used This is just to illustrate a formal analogy between EM and Gravitational formula's in a semi-classical limit.	A_grav = (Φ_grav/c, a_grav) 4-VectorPotential of an arbitrary gravitational field Note This is an approximation only For more accurate results the full GR theory must be used This is just to illustrate a formal analogy between EM and Gravitational formula's in a semi-classical limit. Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation I believe that in some (possibly all physical) circumstances the following is true: A = (Φ_{o_grav}/c²) U A = (Φ_grav/c, a_grav) = (Φ_{o_grav}/c²) γ(c, u) = (γΦ_{o_grav}/c, γΦ_{o_grav}/c²u) giving Φ_grav = γΦ_{o_grav} a = (γΦ_{o_grav}/c²)u see the relativistic Lagrangian and Hamiltonian for more reasoning for this... For point mass-charges... A_grav = (-Gm/c) U / [R·U]_ret [..]_ret implies (R·R = 0, the definition of a light signal) Φ_grav = (γΦ_{o_grav}) = (-γGm/r) a_grav = (γΦ_{o_grav}/c²)u = -(γGm/c²r)u	[kg m <chargetype>^-1 s^-1], for arbitrary field [kg m <kg>^-1 s^-1] for GEM field [m s^-1] for GEM field Φ_GEM = ScalarPotential_GEM a_GEM = VectorPotential_GEM GravitoElectric Field E_grav = -∇[Φ_GEM]-∂a_GEM/∂t = -∇[Φ_GEM]-∂_t[a_GEM] GravitoMagnetic Field B_grav = ∇ x a_GEM GravitoElectric Field E_grav [m·s⁻²] GravitoMagnetic Field B_grav [s⁻¹ ] J_mass = (cρ_mass, j_mass) = ρ_{o_mass}U where ρ_{o_mass} is a mass density basically the conversion from EM to Grav is: q --> m ε_o --> -1/4πG ==== For 4-VectorPotenial of a moving point charge (Lienard-Wiechert potential) A_Grav = -(Gm/c) U / [R·U]_ret where [..]_ret implies (R·R = 0, the definition of a light signal) If we use the A_Grav = (Φ_{o_grav}/c²) U definition and compare terms with above: (Φ_{o_grav}/c²) = -(Gm/c) / [R·U]_ret (Φ_{o_grav}) = -(Gmc) / [R·U]_ret (Φ_{o_grav}) = -(Gmc) / [c²τ]_ret (Φ_{o_grav}) = -(Gm) / [ cτ]_ret (Φ_{o_grav}) = -(Gm) / r {which is the correct potential for a point mass-charge in its rest frame} since R·R = (ct)²-\|r\|² = 0 --> ct = r And likewise, the scalar and vector potential of a moving point mass-charge Φ_grav = (γΦ_{o_grav}) = -(γGm) / r a = (Φ_{o_grav}/c²)γu = (γΦ_{o_grav}/c²)u = (γΦ_{o_grav}/c²)u = (Φ_grav/c²)u = -((γGm/c²) / r)u (∂·∂) = (∂_t²/c²-∇·∇) (∇·∇)[1/r] = Δ[1/r] = -4πδ³(r) if [1/r] is not time dependent then (∂·∂)[1/r] = -(∇·∇)[1/r] = -Δ[1/r] = 4πδ³(r) Let's examine the 4-VectorPotential of a point mass-charge Φ_grav = (γΦ_{o_grav}) = -(γGm) / r (∂·∂)Φ_grav = (∂·∂) (-γGm) [1/ r] = (-γGm)(∂·∂)[1/ r] = (-γGm)4πδ³(r) = (-γ4πGm)δ³(r) = (-γ4πGm)δ³(r) = -4πGρ_mass where ρ_mass = (γm)δ³(r) and thus ρ_{o_mass} = (m)δ³(r) (∂·∂)Φ_grav is the temporal part of (∂·∂)A_grav (∂·∂)A_grav = (∂·∂)(Φ_grav/c,a) = (-4πGρ_mass/c,...) = (-4πGρ_mass/c²)(ρ_massc,...) = (-4πG/c²)J_mass And, of course, once you get it in covariant form, this is general, (∂·∂)A_grav= μ_oJ (∂_t²/c²-∇·∇)(Φ_grav/c, a_grav) = (-4πG/c²) (cρ_mass, j_mass) (∂_t²/c²-∇·∇)(Φ_grav) = (-4πG/c²)( c²ρ_mass ) = (-4πG)ρ_mass (∂_t²/c²-∇·∇)(a) = (-4πG/c²)( j ) (∇·∇)[1/r] = Δ[1/r] = -4πδ³(r) (∇·∇)[1/\|r-r'\|] = Δ[1/\|r-r'\|] = -4πδ³(r-r')
4-VectorPotentialMomentum or 4-PotentialMomentum	Q_EM = (E_EM/c, p_EM) = q A_EM = q (Φ_EM/c, a_EM)	[kg m s^-1], E_EM = ScalarPotenialEnergy p_EM = VectorPotenialMomentum Energy and Momentum of the EM field itself, for a single charge Q_EM = (E_EM/c, p_EM) = q_o A_EM P = ћK K = (ω/c, k) = (ω/c,ω/v_phasen) Q_EM = ћK_EM (for an EM field) q_o (Φ_EM/c, a_EM) = ћ(ω/c, k) q_o Φ_EM = ћω ; q_o a_EM = ћ k
4-Potential or 4-Potential_EM*break with standard notation* better to use the 4-VectorPotential_EM	Φ_EM = (Φ_EM,c a_EM) = c A_EM	[kg m² C^-1 s^-2], Φ_EM = ScalarPotenial_EM ,a_EM = VectorPotenial_EM *this is bad notation based on our 4-vector naming convention* the c-factor should be in the time component the 4-vector name should reference the space component I simply include it here because it can sometimes be found in the literature
4-Momentum_EM 4-CanonicalMomentum 4-TotalMomentum	P_T = (E_T/c + qΦ_EM/c, p_T + qa_EM) P_T = Π = P + qA_EM = m_oU + qA_EM =(H/c,p_T) = (γm_oc+q Φ_EM/c,γm_ou+q a_EM₎	[kg m s^-1], Momentum including effects of potentials also known as Canonical Momentum where P is the Kinetic Momentum term where qA is the Potential Momentum term Total Momentum = Kinetic Part + Potential Part H = γm_oc2 + q φ_EM p_T = γm_ou + q a_EM Can then write dynamic momentum as P = P_T - qA and all the usual relations still hold: P = m_oU = (E_o/c²)U = ћK
4-Gradient_EM Gauge Covariant Derivative	D_EM = (∂/c∂t + iq/ћ Φ_EM/c, -∇ + iq/ћ a_EM) = ∂ + (iq/ћ)A_EM for electrons, commonly seen as D_EM = ∂ - (ie/ћ)A_EM where e is the electric charge	[m^-1], Gradient including effects of EM potentials Minimal coupling based on principle of local gauge invariance

		*Position space & Momentum Space Differentials*
4-Differential	dX = (cdt, dx)	[m], dt = Temporal Differential, dx = Spatial Differential
4-Volume Element (Flux?) 4-SpaceTime Surface (Flux) 4-SpaceTime Surface Element	dV = (dv⁰,dv) = (dV_o) T = (dV_o/c)U = (dV_o) γ(1, β) = γ(dV_o) (1, β) = (dV)(1, β) = (dV,dV β) so that, in a rest frame dV_o = (1)(dV_o,dV_o 0) = (dV_o,0) dV = γdV_o ???? V should be as follows V = V_o/γ Perhaps acts a little differently since this is from a "vector-valued volume element", and not a straight volume. Sometimes denoted as dΣ_μ(x)	[m³], A vector-valued volume element is just a 4-vector that is perpendicular to all spatial vectors in the volume element, and has a magnitude that's proportional to the volume. Using Clifford Algebra one can represent an oriented volume element by a three-form. In a 4d spacetime, a 3-form has a dual representation (Hodges Dual) which is a 1-form, which is basically a vector. Basically this means that you define a volume element by the space-time vector that's perpendicular to it, and you make the length of this space-time vector proportional to the proper volume you wish to represent. Hodge Duality in SR n=4 Minkowski spacetime with metric signature (+,-,-,-) and coordinates (t,x,y,z) gives dt = dx^dy^dz ( is the Hodge star operator, ^ is the wedge product operator) alternately, dV = √[-g]d⁴x is an invariant volume element scalar?? c dt dV = dx⁰ dx¹ dx² dx³ = dx'⁰ dx'¹ dx'² dx'³ c dt = dx⁰ dV = dx¹ dx² dx³ = d³x (c dt)(dV) = (dx⁰ )(dx¹ dx² dx³) = d⁴x d³x = d³x'/γ d³p/p_o = d³p'/p'_o p'_o = p_o/γ d³x d³p = d³x' d³p' dV_o·dX = (dV_o,0)·(cdt,dx) = (dV_o cdt) = dxdydz cdt = d⁴x dV·dX = d⁴x Hence, the differential 4-element is an invariant dq = ρ d³x = j⁰/c d³x Interesting derivation: dq = ρ dV differential charge element (dq)dX = (ρ dV)dX = ρ dV (dt/dt) dX = ρ (dV dt) dX/dt = ρ (dV dt) (1/γ) dX/dτ = ρ (dV dt) (1/γ)U = γρ_o (dV dt) (1/γ) U = ρ_o (dV dt) U = (dV dt) ρ_oU = (dV dt) J = ( dV(c/c) dt)J = ( dV c dt/c) J = (d⁴x/c) J dq dX = (d⁴x/c) J Apparently, (dx¹ dx² dx³)/x⁰ is also an invariant, based on Jacobian also, d³x Δt is an invariant (dV·ΔR) / (U·U) = d³x Δt
4-Momentum Differential	dP = (dE/c, dp)	[kg m s^-1], dE = Temporal Momentum Differential, dp = Spatial Momentum Differential
4-MomentumSpace Volume Element (Flux?)	dV_p = (dv_p0,dv_p) = (dV_po) T = (dV_o/c)U = (dV_po) γ(1, β) = γ(dV_po) (1, β) = (dV_p)(1, β) = (dV_p,dV_p β) so that, in a rest frame dV_po = (1)(dV_po,dV_po 0) = (dV_po,0) Sometimes denoted as dΣ_μ(p) dΣ_μ(p) =(1/3!)ε_{μ ν α β} dp^ν x dp^α x dp^β dΣ_μ(p) =P_μ(d³p/p₀)	[kg³ m³ s^-3], A vector-valued MomentumSpace volume element is just a 4-vector that is perpendicular to all spatial vectors in the MomentumSpace volume element, and has a magnitude that's proportional to the MomentumSpace volume. Using the same Clifford Algebra idea from position-space, I think this can be done (dE/c)(dV_p) = dp⁰ dp¹ dp² dp³ = dp'⁰ dp'¹ dp'² dp'³ dE/ c = dp⁰ dV_p = dp¹ dp² dp³ = d³p (dE/ c)(dV_p) = (dp⁰ )(dp¹ dp² dp³) = d⁴p d³x = d³x'/γ d³p/p_o = d³p'/p'_o p'_o = p_o/γ d³x d³p = d³x' d³p' dV_po·dP = (dV_po,0)·(dE/c,dp) = (dV_po dE/c) = dp_xdp_ydp_z dE/c = d⁴p dV_p·dP = d⁴p Hence, the differential momentum 4-element is an invariant also, dV_o·dV_po = (dV_o,0)·(dV_po,0) = (dV_odV_po) = (dx¹ dx² dx³)(dp¹ dp² dp³) = d³xd³p in the rest frame Thus dV·dV_p = d³x d³p generally, so (d³x d³p) is a Lorentz scalar invariant Apparently, (dp¹ dp² dp³)/p⁰ is also an invariant, based on Jacobian

		*Special 4-Vectors*
4-Zero	Zero = (0,0) = (0,0,0,0)	[*], All components are 0 in all reference frames, the only vector with this property Square Magnitude = 0, Length = \|0\| = 0
4-Null	Null = (a,a) = (a,an) = a(1,n) where n is the unit 3-vector Null·Null = a(1,n)·a(1,n) = a²(11 - n·n) = a²(0) = 0 Null·Null* = 0	[*], Any 4-vector for which the temporal component magnitude equals the spatial component magnitude \|a⁰\| = \|a\| which leads to the magnitude being 0, or LightLike/Photonic ex. The 4-Velocity of a Photon, the 4-Momentum of a Photon
4-Unit Temporal 4-Space Normal	T = U/c = γ(c, u)/c = γ(1, β) = (γ, γβ) T·T = γ(1, β)·γ(1, β) = γ²(11 - β·β) = γ²(1 - β·β) =1 T_o* = (1,0) in rest frame T·T = T_o·T_o = (1,0)·(1,0) = (11 - 0·0) = 1 Temporal Units always temporally separated T₁·T₂ = γ₁γ₂(1²-β₁·β₂) = γ₁₂ = γ_rel dT/dτ = γ(γ', γ'β+γβ') = A/c where γ' = dγ/dt d[T·T*]/dτ = 0 = d[T]/dτ ·T + T·d[T]/dτ d[T]/dτ ·T = 0 Thus d[T]/dτ is orthogonal to T	[1] = dimensionless, The Unit Temporal 4-Vector Square Magnitude = 1, Length = \|1\| = 1 Interesting Formulations Let A=(a⁰,a) be a general 4-Vector (A·T) = (a⁰,a)·γ(1, β) = γ(a⁰-a·β) = 1(a⁰-a·0) = a⁰ which is a Lorentz Invariant way to get (a⁰), the temporal component of A (A·T)² - (A·A) = (a⁰)² - (a⁰a⁰-a·a) = (a⁰)² - (a⁰)² + (a·a) = (a·a) Sqrt[(A·T)² - (A·A)] = Sqrt[(a·a)] = \|a\| which is the Lorentz Invariant way to get the magnitude \|a\|, the spatial component magnitude of A
4-Unit Null	N = (1,n) N·N = (1,n)·(1,n) = (11 - n·n) = 0 N₁·N₂* = (1,n₁)·(1,n₂) = (11 - n₁·n₂) = 1 - n₁·n₂* Photons n₁·n₂ = [1] null-separated Photons n₁·n₂ = [-1 ... 0 ... +0.999...) temporally-separated	[1] = dimensionless, A Unit Null 4-Vector Square Magnitude = 0, Length = \|0\| = 0 Dependent on spatial direction n n = unit 3-vector, \|n\| = 1 The Null Vector is "perpendicular" to itself. Actually not sure if "unit" null really applies here, but in this case at least the spatial components are unit magnitude
4-Unit Spatial 4-Time Normal	S = γ[β_n] (n·β,n) = (γ[β_n]n·β,γ[β_n] n) S = (s⁰,s) = (s·β,s) : due to orthogonality S·T = 0 s = γ[β_n] n : due to unit magnitude S·S = -1 S = γ[β_n](n·β,n) : final form Unit Spatial orthogonal to Unit Temporal S·T = γ[β_n](n·β,n)·γ[β](1, β) = γ[β_n]γ[β](n·β - n·β) = 0 Unit Spatial Magnitude = -1 S·S = γ[β_n](n·β,n)·γ[β_n](n·β,n) = γ[β_n]²(n·β,n)·(n·β,n) = γ[β_n]²[(n·β)² - (n·n)] = γ[β_n]²[(n·β)² - 1] = - γ[β_n]²[1 - (n·β)²] = - γ[β_n]²[1 - (β_n)²] = -1 S_o = (0,n) in rest frame S·S = S_o·S_o = (0,n)·(0,n) = (00 - n·n) = -1 S₁·S₂ = γ[β_n1] (n₁·β₁,n₁)·γ[β_n2] (n₂·β₂,n₂) = γ[β_n1]γ[β_n2](n₁·β₁,n₁)·(n₂·β₂,n₂) = γ[β_n1] γ[β_n2] (n₁·β₁ n₂·β₂ - n₁·n₂) = γ[β_n1](n₁·β₁ - n₁·n₂) = γ[β_n2] (n₂·β₂ - n₁·n₂**)	[1] = dimensionless, A Unit Spatial 4-Vector Square Magnitude = -1, Length = \|-1\| = 1 Dependent on spatial direction n n = unit 3-vector, \|n\| = 1 s = γ[β_n] n The gamma factor γ[β_n] is set to only apply for motion along the direction of s β_n is the part of β along n Spatial units can be temporally-, null-, or spatially-separated depending on direction and relative motion
4-Basis Vectors (1 time + 3 space)	B_t = (1,0,0,0) B_x = (0,1,0,0) B_y = (0,0,1,0) B_z = (0,0,0,1)	A tetrad of 4 mutually orthogonal, unit-length, linearly-independent, basis vectors This is simply one basis, there are others
4-Basis Vectors (complex null tetrad)	B_n1 = √[1/2] (1,0,0,1) B_n2 = √[1/2] (1,0,0,-1) B_n3 = √[1/2] (0,1,i,0) B_n4 = √[1/2] (0,1,-i,0)	A tetrad of complex, linearly-independent, null basis vectors Some, but not all, are orthogonal to one another This is simply one basis, there are others Need to double-check these * see null tetrad, Sachs tetrad, Newman-Penrose tetrad
4-Basis Vectors (real null tetrad)	B_n1 = (1,1,0,0) B_n2 = (1,0,1,0) B_n3 = (1,0,0,1) B_n4 = [1/2](±√3,±1,±1,±1)	A tetrad of real, linearly-independent, null basis vectors This is a non-orthogonal basis
4-ProbabilityCurrentDensity = 4-ProbabilityFlux = 4-QM_ProbabilityFlux = 4-NumberFlux N = SR Dust	J_prob = (cρ_prob, j_prob) = (iћ/2m_o)(ψ<-∂->ψ) J_prob = (iћ/2m_o)( ψ ∂[ψ] - ∂[ψ] ψ ) J_prob* = (cρ, j) = n_o(ψψ) γ(c, u) = ρ(c, u)? J_prob = ρ_{o_prob}U = (ψψ)U = n_oU J_prob,Dirac = (cρ, j) = ic(ψ γ_μ ψ) where ψ = ψ^† γ₄ J_prob = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) - (q/m_o)A (ψψ) for particle in external potential Note: A<-∂->B = A(∂B) - (∂A)B	[# m^-2 s^-1], 4-Probability Current Density is proportional to the 4-Momentum Equivalent to SR Dust = Number-Flux 4-vector obeys the continuity equation [#/m³], ρ = ProbChargeDensity j = ρu =ProbCurrentDensity = ProbChargeFlux [(#/m³)(m/s)] ρ = (iћ/2m_oc²)(ψ ∂_t[ψ]-∂_t[ψ] ψ) j = (-iћ/2m_o)(ψ ∇[ψ]-∇[ψ] ψ) Conservation of ProbabilityCurrentDensity ∂·J_prob* = ∂ρ/∂t +∇·j = 0 ρ_prob = γ(ψψ) = γρ_o See derivation of Newtonian limit below... This is the 4-vector which relates SR to the probabilistic formulation of QM Derived from Klein Gordon eqn. ∂·∂* = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂²/c²∂t²-∇·∇ = -(m_oc / ћ)² ∂·∂ + (m_oc / ћ)² = 0 Wave eqn. on ψ, then pre-mult by ψ* ψ{∂·∂* + (m_oc / ћ)² = 0}ψ ψ*∂·∂ψ + ψ(m_oc / ћ)²ψ = 0 ψ(∂·∂ψ) + (m_oc / ћ)²ψψ = 0 Wave eqn. on ψ, then post-mult by ψ {∂·∂ + (m_oc / ћ)² = 0}ψψ (∂·∂ψ)ψ + (m_oc / ћ)²ψψ = 0 Subtract the two ψ(∂·∂ψ) + (m_oc / ћ)²ψψ -(∂·∂ψ)ψ - (m_oc / ћ)²ψψ = 0 ψ(∂·∂ψ) - (∂·∂ψ)ψ = 0 Note that this is equivalent to: ∂·(ψ∂[ψ]-∂[ψ]ψ) = 0 So, let J_prob* ~ (ψ∂[ψ]-∂[ψ]ψ) Then continuity holds ∂·J_prob = ∂ρ/∂t +∇·j = 0 Conservation of Probability Consider the following purely mathematical argument (based on Green's Vector Identity): ∂·( f ∂[g] - ∂[f] g ), where f and g are SR Lorentz Scalar functions ∂·( f ∂[g] - ∂[f] g ) = ∂·( f ∂[g] ) - ∂·( ∂[f] g ) = (f ∂·∂[g] + ∂[f]·∂[g]) - (∂[f]·∂[g] + ∂·∂[f] g) = f ∂·∂[g] - ∂·∂[f] g So, f ∂·∂[g] - ∂·∂[f] g = ∂·( f ∂[g] - ∂[f] g ) I can also multiply this by a Lorentz Invariant Scalar Constant s s f ∂·∂[g] - ∂·∂[f] g = s ∂·( f ∂[g] - ∂[f] g ) = ∂·s( f ∂[g] - ∂[f] g ) Ok, so we have the math that we need... Now, on to the physics... ∂·∂ = (-im_oc/ћ)² = -(m_oc/ћ)²: The Klein-Gordon Eqn ∂·∂ + (m_oc/ћ)² = 0 Let it act on Lorentz Invariant function g ∂·∂ [g]+ (m_oc/ћ)² [g] = 0 [g] Then pre-multiply by f [f] ∂·∂ [g]+ [f] (m_oc/ћ)² [g] = [f] 0 [g] Resulting in [f] ∂·∂ [g]+ (m_oc/ћ)²[f][g] = 0 Now, do similarly with Lorentz Invariant function f ∂·∂ [f]+ (m_oc/ћ)² [f] = 0 [f] Then post-multiply by g ∂·∂ [f][g]+ (m_oc/ћ)² [f][g] = 0 [f][g] Resulting in ∂·∂ [f][g]+ (m_oc/ћ)² [f][g] = 0 Now, subtract the two equations {[f] ∂·∂ [g]+ (m_oc/ћ)²[f][g] = 0} - {∂·∂ [f][g]+ (m_oc/ћ)² [f][g] = 0} Resulting in [f] ∂·∂ [g]+ (m_oc/ћ)²[f][g] - ∂·∂ [f][g]- (m_oc/ћ)² [f][g] = 0 [f] ∂·∂ [g] - ∂·∂ [f][g] = 0 And as we noted from the mathematical identity at the start... [f] ∂·∂ [g] - ∂·∂ [f][g] = ∂·( f ∂[g] - ∂[f] g ) = 0 s [f] ∂·∂ [g] - ∂·∂ [f][g] = s ∂·( f ∂[g] - ∂[f] g ) = s 0 = 0 Therefore, s ∂·( f ∂[g] - ∂[f] g ) = 0 ∂·s( f ∂[g] - ∂[f] g ) = 0 There is a conserved current 4-vector, J_prob = s ( f ∂[g] - ∂[f] g ), for which ∂·J_prob = 0, and which also solves the Klein-Gordon equation. Let's choose as before (∂ = -i K) with a plane wave function f = ae^-i(K∙X), and choose g = f* = ae^i(K∙X) as it's complex conjugate. At this point, I am going to choose s = (iћ/2m_o), which is Lorentz Scalar Invariant, in order to make the probability have dimensionless units and be normalized to unity in the rest case. Then J_prob = s ( f ∂[g] - ∂[f] g ) = s{f (-i) K g} - {i K f g} = s (-2i) K fg J_prob = -2is K fg J_prob = (ρ_probc,j_prob) = -2is K fg = -2is (ω/c,k) fg Now, take the temporal component ρ_probc = -2is (ω/c) fg ρ_prob = -2is (ω/c²) fg ρ_prob = -2i (iћ/2m_o)(γω_o/c²) fg ρ_prob = (-i² )(2/2)(ћω_o/m_oc²)(γ) fg ρ_prob = (γ) fg Now, to put it in a more obvious form... f = ψ, g = ψ I have put off using psi until now to point out that f and g could be any kind of SR Lorentz Invariant functions. ρ_prob = (γ)(& amp; #968;ψ) = (γ)(ρ_o) (ρ_{o_prob}) = (ψψ) = fg The probability density ρ_prob = γρ_{o_prob} is only equal to (ψψ) in the Newtonian low-energy limiting case where γ --> 1 Thus, the Born Probability Interpretation of (ψψ) only applies in the Newtonian low-energy limiting case J_prob* = s ( f ∂[g] - ∂[f] g ) = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) J_prob* = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) Also, we know that J = ρ_oU and K = P/ћ = (m_o/ћ)U J_prob* = (ρ_{o_prob})(ћ/m_o)K J_prob = (fg)(-2is) K So, everything matches correctly J_prob& lt; /b> = ρ_{o_prob}U = (ψψ)U J_charge* = qJ_prob J_charge = qρ_{o_prob}U = q(ψψ)U If we do the calculation including an EM potential, then the 4-ProbabilityCurrentDensity gains an extra component From before: J = -2is K fg Minimal Coupling is just observation that the 4-TotalMomentum is the sum of the particle 4-Momentum and the field 4-PotentialMomentum We have conservation of the Total 4-Momentum P_T* P_T = P + qA P = P_T-qA K = K_T - (q/ћ)A J_prob = -2is( K )fg J_prob = -2is( K_T - (q/ћ)A )fg J_prob = -2is( K_T )fg + 2is(q/ћ)A )fg J_prob = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) + 2is(q/ћ)A fg J_prob* = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) + 2i(iћ/2m_o)(q/ћ)A fg J_prob* = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) - (q/m_o)A fg J_prob* = (iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) - (q/m_o)A (ψψ) J_charge = qJ_prob J_charge = q[(iћ/2m_o)( ψ* ∂[ψ] - ∂[ψ] ψ ) - (q/m_o)A (ψψ)] J_charge = (iћq/2m_o)( ψ* ∂[ψ] - ∂[ψ*] ψ ) - (q²/m_o)A (ψψ) Next to show Gordon Decomposition of Spin stuff...

<Event> R	Mass m_o = ρ_{o_m}V_o Energy E_o = m_oc²	WaveAngFreq ω_o	ElecCharge q = ρ_oV_o	MassDensity ρ_{o_m} Mass m_o = ρ_{o_m}V_o EnergyDensity u_o = ρ_{o_m}c²	ChargeDensity ρ_o Charge q = ρ_oV_o	NumberDensity n_o ParticleNumber N_o = n_oV_o
<event>	particle mass	wave	elec. charge	mass density	charge density	number density
pos: R = (ct, r)	m_o at R	ω_o at R	q at R	ρ_{o_m} at R	ρ_o at R	n_o at R
vel: U = dR/dτ	P = m_oU = (E_o/c²)U	K = (ω_o/c²)U = (1/ћ)P	J_q = qU	G = ρ_{o_m}U = (u_o/c²)U	J = ρ_oU	N = n_oU
accel: A = dU/dτ	F = dP/dτ			F_d = dG/dτ
jerk: J = dA/dτ
snap: S = dJ/dτ


SiteMap of SciRealm \| About John \| John's Resume/C.V./Bio \| Send email to John John's Science, Math, Philosophy Stuff: The Science Realm: John's Virtual Sci-Tech Universe \| 4-Vectors \| Ambigrams \| Antipodes \| Covert Ops Fragments \| Cyrillic Projector \| Forced Induction (Sums Of Powers Of Integers) \| Fractal Tree \| Frontiers \| IFS Fractals (Assembly) \| IFS Fractals (Javascript) \| JavaScript Graphics \| Kid Science \| Kryptos \| Philosophy \| Photography \| Physics-Planck Constant Measurement Via LEDs \| Prime Sieve \| QM from SR \| QM from SR-Simple RoadMap \| QM from SR-Simple RoadMap (PDF) \| SR 4-Vector & Tensor Calculator \|Quantum Phase \| Quotes \| RuneQuest Cipher Challenge \| Secret Codes & Ciphers \| Scientific Calculator (complex capable) \| Science+Math \| Sci-Pan Pantheist Poems \| Stereograms \| SuperMagicSqr4x4 \|Turkish Grammar \|Melike Wilson Art Site last modified: 2021-Aug	Welcome to Relativistic Quantum Reality: Virtual worlds of imaginary particles: The dreams stuff is made of: Life, the eternal ghost in the machine... This site is dedicated to the quest for knowledge and wisdom, through science, mathematics, philosophy, invention, and technology. May the treasures buried herein spark the fires of imagination like the twinkling jewels of celestial light crowning the midnight sky... Quantum Mechanics is derivable from Special Relativity See SRQM - QM from SR - Simple RoadMap (.html) See SRQM - QM from SR - Simple RoadMap (.pdf) See SRQM - QM from SR - Summary (.pdf) See SRQM - Fundamental Constants (.pdf) See SRQM - Online SR 4-Vector & Tensor Calculator See Online Complex-capable RPN Scientific Calculator *Learn, Discover, Explore*

"Flat" SpaceTime η_μν = g_μν{SR}	t	x	y	z
t	1	0	0	0
x	0	-1	0	0
y	0	0	-1	0
z	0	0	0	-1

<Event>(SR)	<Event>Movement	MassEnergy	Particle-WaveDuality	QuantumMechanics(QM)	SpaceTimeVariations
R = (ct, r)	dR/dτ = U = γ(c,u)	U = P/m_o	P = ћ K	* K = i ∂ *	∂ = (∂/c∂t,-∇)
			or K = (ω_o/c²)U

R = (ct,r)	Location of an <event> (i.e. a particle) within spacetime
U = dR/dτ	Velocity of the <event> is the derivative of <event> position wrt. Proper Time
P = m_oU	Momentum is just the Rest Mass of the particle/<event> times its velocity
K = (1 / ћ )P	A particle's wave vector is just the momentum divided by Dirac's constant, but uncertain by a phase factor
∂ = -i K	The change in spacetime corresponds to (-i) times the wave vector, whatever that means...

(∂·∂ + (m_oc/ћ)² ) Ψ = 0,	Ψ is a scalar Klein-Gordon eqn for massive spin-0 field
(∂·∂ + (m_oc/ћ)² ) A = 0	A is a 4-vector Proca eqn for massive spin-1 field
(∂·∂) Ψ = 0	Ψ is a scalar Free-wave eqn for massless (m_o = 0) spin-0 field
(∂·∂) A = 0	A is a 4-vector Maxwell eqn for massless (m_o = 0) spin-1 field, no current sources

P = i ћ ∂ = -∂(S_act)	∂ = (∂/c∂t,-∇)	A_EM = (0,0) special case
P_EM = P+qA_EM = i ћ D_EM	D_EM = ∂+iq/ћ A_EM	A_EM = (V_EM/c,a_EM)

	Relations involving the 4-Position or 4-Displacment:
R·R = (Δs)² = (ct)²-r·r = (ct)²-\|r\|² R·R = 0 for photonic signal w source at origin	Spacetime position of an <event> wrt. an origin <event>
dR·dR = (ds)² = (c dt)²-dr·dr = (c dt)²-\|dr\|² = (c dτ)²	Differential interval magnitude - the fundamental invariant differential form This is supposed to be locally valid for all <event>'s, even in GR If we let this occur in a "rest frame", then dRo·dRo = (c dt_o)² = (c dτ)²
ΔR·ΔR = (Δs)² = (c Δt)²-Δr·Δr = (c Δt)²-\|Δr\|²	Spacetime displacement interval magnitude - used to derive SR (Δs)² >0 TimeLike Interval(Δs)² = 0 LightLike Interval(Δs)² <0 SpaceLike Interval
∂·R = 4	The divergence of open spacetime is equal to the number of independent dimensions (t,x,y,z)
K·R = -Φ = (ωt-k·r)	Phase of a SR wave; Ψ = a E e^-iK·R Photon Wave Equation (Solution to Maxwell Equation)

R·U = (ct,r)·γ(c,u) = γ(c²t - r·u) = cγ(ct - r·u/c) = c²t_o = c²τ	Part of expression used in Liénard-Wiechert potential t_ret = t - \|x-x'\|/c: (retarded time) r_u = r - r u/c = the virtual present radius vector; i.e., the radius vector directed from the position the charge would occupy at time t' if it had continued with uniform velocity from its retarded position to the field point.
ΔX·U = (cΔt,Δx)·γ(c,u) = γ(c²Δt - Δx·u) = (c²Δt_o) = (c²Δτ) Let T = U/c = γ(c, u)/c = γ(1, β) = (γ, γβ) be the unit-temporal 4-Vector Then... (ΔX·T) = (cΔt,Δx)·γ(1, β) = γ(cΔt-Δx·β) = (cΔt_o) = (cΔτ)	ΔX·U = 0 is an interesting 4-vector condition/definition for simultaneity (The displacement of an external <event> is normal to either <event>'s worldline) ΔX is a displacement vector from <Event> A to <Event> B U is an observer's 4-Velocity wrt. one of the <event>'s The standard definition of simultaneity is when Δt = 0. This gives ΔX·U = -γ(Δx·u). But since we can always choose an observer rest frame we get ΔX·U = 0, which is thus the Lorentz Invariant condition/definition for simultaneity. However, since Δt is one of the components of a 4-vector, it is only true for certain classes of observers. Let's examine the cases when ΔX·U = γ(c²Δt - Δx·u) = 0 then (c²Δt - Δx·u) = 0, since γ is always >=1 then (c²Δt = Δx·u) For simultaneity, Δt = 0, therefore Δx·u = 0 if Δx = 0, then <Event> A and <Event> B are co-local, ie. at the same spatial point if u = 0, then the observer is at rest wrt. the <Event>'s A and B if Δx·u = 0, but Δx > 0 and u > 0, then the observer's motion must be perpendicular to a spatial line from <Event> A to <Event> B

	Relations involving the 4-Velocity:
U·U = c²	The magnitude of 4-Velocity is always c²
U₁·U₂ = γ[u₁]γ[u₂](c²-u₁·u₂) = γ[u_rel]c² U₁·U₁ = U·U = c²	Relative Gamma Factor (The scalar product of two uniformly moving <event>/particle 4-velocities is proportional to the γ factor of their relative velocities) Also get the relativistic addition of velocities with this: γ[u₁]γ[u₂](c²-u₁·u₂) = γ[u_rel]c² γ₁γ₂(c²-u₁·u₂) = γ_relc² γ₁γ₂(1-β₁·β₂) = γ_rel γ₁^-2γ₂^-2(1-β₁·β₂)^-2 = γ_rel^-2 (1-β₁·β₁)(1-β₂·β₂)(1-β₁·β₂)^-2 = (1-β_rel·β_rel) (β_rel·β_rel) = 1-(1-β₁·β₁)(1-β₂·β₂)(1-β₁·β₂)^-2 (β_rel)² = [(1-β₁·β₂)²-(1-β₁·β₁)(1-β₂·β₂)](1-β₁·β₂)^-2 (β_rel)² = [(β₁)²-2(β₁·β₂)+(β₂)²](1-β₁·β₂)^-2 (β_rel)² = [(β₁)-(β₂)]²(1-β₁·β₂)^-2 β_rel = [β₁-β₂]/(1-β₁·β₂) u_rel = [u₁-u₂]/(1-β₁·β₂) or, by letting u₂ --> -u₂ (β₂ --> -β₂) u_rel = [u₁+u₂]/(1+β₁·β₂) = [u₁+u₂]/(1+u₁·u₂/c²)
A₁·U₁ = 0, where A is dU/dτ	The 4-Acceleration of a given particle is always normal to its own worldline
P₁·U₂ = γ[u₂](E-p₁·u₂) = E_rel P₁·U₁ = m_oU₁·U₁ = m_oc² = E_o	Relative Energy The relative energy between <event>'s 1 and 2 is a Lorentz scalar - any observer (<event> 3) must measure the same relative energy between <event>'s 1 and 2 If measuring the 4-Momentum and 4-Velocity of the same <event>, then this simplifies to the rest energy of that <event>
K₁·U₂ = γ[u₂](ω-k₁·u₂) = ω_rel K₁·U₁ = m_o/ћU₁·U₁ = m_oc²/ћ = E_o/ћ = ω_o	Relative Ang. Frequency The relative ang. frequency between <event>'s 1 and 2 is a Lorentz scalar - any observer (<event> 3) must measure the same relative ang. frequency between <event>'s 1 and 2 If measuring the 4-WaveVector and 4-Velocity of the same <event>, then this simplifies to the rest ang. frequency of that <event>
F·U = (m_oA+(dm_o/dτ)U)·U = c²(dm_o/dτ) = γc²(dm_o/dt) U·F = γ²(dE/dt-u·f_r) = γ dm_o/dt c² (pure force if dm_o/dt = 0)	Power Law
U·∂ = γ(∂/∂t + u·∇) = γ d/dt = d/dτ	Relativistic Convective (Time) Derivative, Intrinsic Derivative U·∂ = d/dτ = γ(∂/∂t + u·∇) = γ d/dt let u << c, then γ ~ 1 then U·∂ = d/dτ = γ(∂/∂t + u·∇) ~ (∂/∂t + u·∇) = d/dt ∇_t(v) = (∂/∂t + u·∇) v is the gauge covariant derivative of a fluid where v is a velocity vector field of a fluid
∂·U = 0 (always??)	The General Continuity Equation, one might say the conservation of <event> flux. I believe this is true generally but it needs checking... If true, then it is the source of all the particle / charge / probability continuity equations It can be shown that a scalar (s) and vector (v) which are related through a continuity equation in all frames of reference (∂s/∂t + ∇·v = 0) transform according to the Lorentz transformations and therefore comprise the components of a 4-vector V=(cs,v), where ∂ ·V = 0. Relativistic four-vectors may be identified from the continuity equations of physics. See A Proposed Relativistic, Thermodynamic Four-Vector.

	Relations involving the 4-Acceleration:
A·A = -a² = -γ⁴[a² + (γ/c)²(u·a)²]	Magnitude squared of acceleration
U·A = 0, where A is dU/dτ	The 4-Acceleration of a given particle is always normal to its own worldline U·U = c² d/dτ(U·U) = d/dτ(c²) = 0 d/dτ(U·U) = 2(U·dU/dτ) = 2(U·A) = 0
A·S	Part of the proportionality factor of a 4-Spin Vector Fermi-Walker Transported in time Would also apply to any constant spatial 3-vector that is "attached" to a particle Since U·S = 0 then d/dτ [U·S] = 0 = d/dτ[U]·S + U·d/dτ[S] = A·S + U· d/dτ[S] U· d/dτ[S] = - A·S if we assume d/dτ[S] = (k)U then U·d/dτ[S] = kU·U = kc² = -A·S* k = -A·S/c² then d/dτ[S] = (-A·S/c²)U, which is Fermi-Walker Transport of the 4-Spin and leads to Thomas Precession. Fermi-Walker Transport is the way of transporting a purely spatial vector along the worldline of the particle in such a way that it is as "rotationless" as possible, given that it must remain orthogonal to the worldline. This choice also implies that d/dτ [S·S] = 0, since d/dτ [S·S] ~ [U·S] = 0, which means that the magnitude of the 4-Spin is constant

	Others:
P·P = (m_oc)² = (E_o/c)² = m²(c²-u·u) = (m²c²-p·p) = (E²/c²-p·p) = 0 for light-like/photonic	Square Magnitude of the 4-Momentum Gives the Einstein Energy-Mass Relation
P₁·P₂ = γ[u₁]γ[u₂]m_o1 m_o2(c²-u₁·u₂) = γ[u_r12]m_o1 m_o2c²	Relativistic Billiards... P + Q = P' + Q': Momenta before and after collision generally - Conservation of 4-Momentum P·Q = P'·Q': Momenta in an elastic (rest-mass preserving) collision - Relative velocities conserved
Γ·P= (m_oc) ( Σ·P )·( Σ·P ) = (Ps)·(Ps) = (m_oc)² ( Σ₁·P₁ )·( Σ₂·P₂ ) = (Ps₁)·(Ps₂) = γ[u_r12](m_o1)(m_o2)c² ??	Momentum representation of Dirac Equation, the 4-DiracGamma Γ with 4-Momentum P (Γ·P)Ψ= (m_oc)Ψ (Γ^μP_μ)Ψ= (m_oc)Ψ or, in operator form iћ(Γ^μ∂_μ)Ψ= (m_oc)Ψ (γ⁰p⁰ - γ·p )Ψ= (m_oc)Ψ One can get the equivalent result using the Pauli Spin Matrix Tensor Σ as well 4-Momentum (inc. spin) Ps = Σ·P
N·N = (n_oc)²	Square Magnitude of the 4-NumberCurrentDensity
J·J = (p_oc)² = (qn_oc)²	Square Magnitude of the 4-ElectricCurrentDensity
K·K = (m_oc / ћ)² = (ω_o/c)² = 0 for light-like/photonic	Square Magnitude of the 4-WaveVector
∂·∂ = (∂/c∂t,-∇)·(∂/c∂t,-∇) = ∂²/c²∂t²-∇·∇ = -(m_oc / ћ)²	Klein-Gordon Relativistic Wave Eqn. ∂·∂ is also known as the D'alembertian (Wave Operator) Note: Solutions to the wave equation are linear and allow superposition! This is a general feature of the wave equation, not something that is only from QM
∂·J = ∂ρ/∂t +∇·j = 0	Continuity Equation - Conservation of Electric Charge No sources or sinks Charge is neither created nor destroyed
E·K = 0 E·E* = -1	The Polarization of a photon is orthogonal to direction of wave motion (E·K = 0) (cancellation of "scalar" polarization) The Polarization of a photon is always unit magnitude and space-like (E·E* = -1)(cancellation of the longitudinal polarization) 4-WaveVector K of photon has 4 independent components, points in null direction 4-Polariztion of photon would normally have 4 independent components also, since it is a 4 vector However, the ( E·K = 0 ) Polarization is orthogonal to direction of motion, places 1 constraint, reducing to 3 independent components Ε = (ε⁰, ε) K = (ω/c)(1,n) , for light-like/photonic E·K = 0 = (ε⁰, ε)·(ω/c)(1,n) = (ω/c)(ε⁰ - ε·n), so either ω = 0 {no photon} or (ε⁰ = ε·n) Thus, Ε = (ε⁰, ε) = (ε·n, ε), 3 independent components ε Photon Polarization may have complex components In addition, ( E·E* = -1 ) plus the fact that photons don't have a rest frame, places 1 more constraint, reducing to 2 independent components for photon E·E* = (ε·u/c)(ε·u/c)* -ε·ε* = -1 (ε·u/c)² -1 = -1 because (*ε·ε = 1) (ε·u/c)² = 0 ε·u = 0, so that the spatial components must be orthogonal. For a massive particle, there is always a rest frame where u = 0, so ε can have 3 independent components. For a photonic particle, there is no rest frame.( ε·u/c = ε·n** = 0 ) is therefore an additional constraint, limiting ε to 2 independent components, with polarization ε orthogonal to direction of photon motion n.
A_EM·A_EM = (V_EM/c,a_EM)·(V_EM/c,a_EM) = (V_EM/c)²-a_EM·a_EM = ????	Square Magnitude of the Electromagnetic field
P_T·R = -S = (-∂S/c∂t,∇[S])·R	Action S

∂·J = ∂p/∂t +∇·j = 0	Conservation of 4-CurrentDensity (EM charge): p & j change in ChargeDen wrt. time balanced by flow of CurrentDen
∂·N = ∂/∂t(γ n_o)+∇·(γ n_ou) = ∂n/∂t+∇·n_f = 0	Conservation of 4-NumberFlux (Particle NumberDen, NumFlux): n_&n_f change in NumberDen wrt. time balanced by flow of NumFlux
∂·P = (1/c²)∂E/∂t +∇·p = 0 Sum[P_f-P_i] = Zero	Conservation of 4-Momentum (Energy~Mass, Momentum): E & p change in Energy wrt. time balanced by flow of Momentum Alternately, the Sum[(Final 4-Momenta) - (Initial 4-Momenta)] = Zero 4-Vector Note: this conservation equation, while rarely used, is perfectly acceptable for SR single particles. It is only when a group of particles is treated as a continuous fluid, or motion in GR curved space, that the Energy-Momentum (2,0)Tensor is required. Then, the diagonal pressure terms and off-diagonal shear terms are necessary, basically allowing statistical particle interaction.
∂·K = ∂/c∂t(w/c)+∇·k = (1/c²)∂w/∂t +∇·k = 0.	Conservation of 4-WaveVec (AngFreq, WaveNum): w & k change in AngFreq wrt. time balanced by flow of WaveNum
∂·A_EM = (1/c²)∂V_EM/∂t +∇·a_EM = 0	Conservation of 4-VectPotential_EM (applies in the Lorenz Gauge): V_EM & a_EM change in ScalarPotential wrt. time balanced by flow of VectorPotential

∂·U = ∂/∂t(γ[u])+∇·(γ[u] u) = γ³ (u/c² ∂u/∂t + ∇·u) = ∂·Uo? = 0 if <event> is in a conservative field or space	Conservation of 4-Velocity: (Flux-Gauss' Law)??: γ & γ u change in (γ) wrt. time balanced by flow of (γ u) If this quantity equals zero, then any physical quantity that is just a (constant* 4-velocity) is conserved. For example ∂·P = ∂·(m_oU) = m_o(∂·U) = 0 Also from d/dτ (∂·R) = ∂·U = 0 see also Noether's Theorem

4-Vector(s)	Type	Relativistic Law	Newtonian Limit Low Velocity (v<<c) or Low Energy (E<<m_oc²) Basically, β --> 0, γ --> 1
R = (ct,r)	4-Position	(ct,r) is single 4-vector entity t and r related by Lorentz transform	t independent from r t is independent scalar, r is independent 3-vector
ΔR = (cΔt,Δr)	4-Displacement	Relative Simultaneity Δt' = γ(Δt - β·Δr/c)	Absolute Simultaneity Δt' = Δt
U = dR/dτ	4-Velocity	Relativistic Composition of Velocities u_rel = =[u₁+u₂]/(1+β₁·β₂) =[u₁+u₂]/(1+u₁·u₂/c²) Imposes Universal Speed Limit of c	Additive Velocities u₁₂ = u₁ + u₂ Unlimited Speed
A = dT/dτ	4-Acceleration	Relativistic Larmor Formula Power radiated by moving charge P = = ( q²/ 6πε_oc³)(A·A) = (μ_oq²/6πc)(A·A) = (μ_oq²/6πc) γ⁶/ (a² - (\|u x a\|)²/c²)	Newtonian Larmor Formula Power radiated by a non-relativistic moving charge P = (μ_oq²/6πc)(a²)
P = m_oU	4-Momentum	Einstein Energy-Mass Relation E = γ m_oc² = Sqrt[ m_o²c⁴ + p·p c² ]	Total Energy = Rest Energy + Kinetic Energy E = m_oc² + (p²/2m_o)
∂·P	Divergence of 4-Momentum	Local? Conservation of 4-Momentum	Conservation of Energy, Conservation of Momentum
P₁·P₂	Particle Interaction	Conservation of 4-Momentum	Conservation of Energy, Conservation of Momentum, sometimes Conservation of Kinetic Energy
K =(ω/c,k) =(1/ћ)P = (m_o/ћ)U = (ω_o/c²)U	4-WaveVector and 4-Velocity	Relativistic Doppler Effect, inc. Transverse Doppler Effect a_{o_obs} = = a_{o_emit} / γ(1 - (n·v/c)) = a_{o_emit} / γ(1 - (n·β)) = a_{o_emit} √[1+\|β\|]√[1-\|β\|] / (1 - (n·β)) Relativistic Aberration Effect cos(ø_{_obs}) = [cos(ø_{_emit})-β]/[1-βcos(ø_{_emit})] Relativistic Wave Speed, all elementary particles, matter or photonic λf = c/β = v_phase	Regular Doppler Effect a_{o_obs} = a_{o_emit} √[1+\|β\|]√[1-\|β\|] Newtonian Aberration = None cos(ø_{_obs})= cos(ø_{_emit}) Newtonian Wave Speed, only photonic particles (a rare case when the lightspeed case is chosen for Newtonian description) λf = c
P and K	4-Momentum and 4-WaveVector	Compton Scattering (λ'-λ) = (h/m_oc)(1-cos[ø]) (m_oc²)(1/E'-1/E) = (1-cos[ø]) Ratio of photon energy after/before collision P[E,ø] = 1/[1+(E/m_oc²)(1-cos[ø])] see also Klein-Nishina formula	Thompson Scattering Ratio of photon energy after/before collision: E<<m_oc² P[E,ø] --> 1
∂ = -iK	4-Gradient	D'Alembertian & Klein-Gordon Equation ∂_t²/c² = ∇·∇-(m_oc/ћ)²	Schroedinger Equation (i ћ)( ∂_t ) = - (ћ)²(∇)²/2m_o
∂·J	Divergence of 4-Current	Conservation of 4-EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0	Conservation of 4-EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0
J_prob	Probability CurrentDensity	Conservation of ProbabilityCurrentDensity ρ = (iћ/2m_oc²)(ψ* ∂_t[ψ]-∂_t[ψ] ψ) j = (-iћ/2m_o)(ψ ∇[ψ]-∇[ψ] ψ) ∂·J_prob* = ∂ρ/∂t +∇·j = 0 ρ = γ(ψ*ψ) for time separable wave functions Relativistically, this is conservation of the number of worldlines thru a given spacetime <event>	Conservation of Probability ∂·J_prob = ∂ρ/∂t +∇·j = 0 ρ = (ψ*ψ) for time separable wave functions Typically set so that the sum over all quantum states in space = 1 At low energies/velocities, this appears as the conservation of probability of a given wavefunction at a given spacetime <event> - In other words, the probability interpretation of a wavefunction is just a Newtonian approximation to the more correctly stated conservation of relativistic worldlines. This is why the problem of positive definite probabilities and probabilities >1 vanishes once you consider anti-particles and conservation of charged currents.
A_EM = (Φ_EM/c, a_EM)	4-VectorPotential	4-VectorPotenial of a moving point charge (Lienard-Wiechert potential) A_EM = (q/4πε_oc) U / [R·U]_ret [..]_ret implies (R·R = 0, the definition of a light signal) Φ_EM = (γΦ_o) = (γq/4πε_or) a_EM = (γΦ_o/c²)u = (γqμ_o/4πr)u	Scalar Potential and Vector Potential of a stationary point charge Φ_EM = (q/4πε_or) a_EM = 0 Scalar Potential and Vector Potential of a slowly moving point charge (\|v\|<<c implies γ-->1) Φ_EM = (Φ_o) = (q/4πε_or) a_EM = (Φ_o/c²)u = (qμ_o/4π r)u
Q_EM = (E_EM/c, p_EM) = q A_EM = q (Φ_EM/c, a_EM)	4-VectorPotentialMomentum
P_EM = (E/c + qΦ_EM/c, p + qa_EM) = γ m_o(c,u) P_EM = Π = P + qA_EM = m_oU + qA_EM =(H/c,p_EM) = (γm_oc+q Φ_EM/c,γm_ou+q a_EM)	4-Momentum_EM 4-CanonicalMomentum 4-TotalMomentum	Minimal Coupling ============= Total 4-Momentum = Particle 4-Momentum + Potential(Field) 4-Momentum
D = ∂ + iq/ћ A_EM	Minimal Coupling Prescription	KG equation, with minimal coupling to an EM potential D·D = = -(m_oc/ћ)² (∂ + iq/ћ A_EM)·(∂ + iq/ћ A_EM) + (m_oc/ћ)² = 0	Schroedinger Equation (with external potential) (i ћ)( ∂_t ) = V[x] - (ћ)²(∇)²/2m_o

Invariants	P = Pressure = P_o	N = ParticleNum = N_o	S = Entropy = S_o
Variables	V = Volume = (1/γ)Vol_o	μ = ChemPoten = (1/γ)μ_o	T = Temperature = (1/γ)Temp_o

γ	-(v_x/c)γ	0	0
-(v_x/c)γ	γ	0	0
0	0	1	0
0	0	0	1

Cosh[φ]	-Sinh[φ]	0	0
-Sinh[φ]	Cosh[φ]	0	0
0	0	1	0
0	0	0	1

L = -(P_T·U)/γ	H = γ(P_T·U)	H + L = p_T·u = γ(P_T·U) - (P_T·U)/γ
L = -(P_T·U)/γ L = -((P + Q)·U)/γ L = -(P·U + Q·U)/γ L = -P·U/γ - Q·U/γ L = -m_oU·U/γ - qA·U/γ L = -m_oc²/γ - qA·U/γ L = -m_oc²/γ - q(φ/c, a)·γ(c, u)/γ L = -m_oc²/γ - q(φ/c, a)·(c,u) L = -m_oc²/γ - q(φ - a·u) L = -m_oc²/γ - qφ + qa·u L = -m_oc²/γ - qφ_o/γ L = -(m_oc² + qφ_o)/γ	H = γ(P_T·U) H = γ((P + Q)·U) H = γ(P·U + Q·U) H = γP·U + γQ·U H = γm_oU·U + γqA·U H = γm_oc² + qγφ_o H = γm_oc² + qφ H = ( γβ² + 1/γ )m_oc² + qφ H = ( γm_oβ²c² + m_oc²/γ) + qφ H = ( γm_ou² + m_oc²/γ) + qφ H = p·u + m_oc²/γ + qφ H = E + qφ H = ± c√[m_o²c²+p²] + qφ H = ± c√[m_o²c²+(p_T-qa)²] + qφ	H + L = γ(P_T·U) - (P_T·U)/γ H + L = (γ - 1/γ)(P_T·U) H + L = ( γβ² )(P_T·U) H + L = ( γβ² )((P + Q)·U) H + L = ( γβ² )(P·U + Q·U) H + L = ( γβ² )(m_oc² + qφ_o) H + L = (γm_oβ²c² + qγφ_oβ²) H + L = (γm_ou·uc²/c² + qφ_oγu·u/c²) H + L = (γm_ou·u + qa·u) H + L = (p·u + qa·u) H + L = p_T·u

Four-Vectors (4-Vectors) & Lorentz Invariants of Special Relativity: A Study of Elegant Physics The Ultimate Four-Vector & Lorentz Invariant Reference

Introduction