Four-Vectors (4-Vectors) of Special Relativity:
A Study of Elegant Physics

The 4-vectors (four-vectors) of Special Relativistic (SR) theory are fundamental entities that accurately, precisely, and beautifully describe the physical properties of the world around us. While it is known that SR is not the "deepest" theory, it is valid for the majority of the known universe. It is believed to apply to all forms of interaction, including that of fundamental particles, with the only exception being that of large-scale gravitational phenomena, where spacetime itself is significantly curved, for which General Relativity (GR) is required. The SR 4-vector notation is one of the most powerful tools in understanding the physics of the universe, as it simplifies a great many of the physical relations.

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Introduction

A vector is a mathematical object which has both magnitude and direction.  A common 3-vector is the velocity vector, which tells you in what direction and how fast something is moving.  An important point is that a given vector may be described by more than one coordinate system. 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 image 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 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 events".  I say it this way because all of experimental physics ultimately boils down to taking a measurement.
  (2) The speed of light (c) in vacu 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 Galilean invariance, which also obeys the first postulate.

4-vectors are tensorial entities which display Poincare' 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 Poincare' 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 Poincare' 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 special subset of Poincare' Invariance.

The reason that I really like this notation is that it beautifully and elegantly displays the relations between lots of different physical properties. It also devolves 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-Potential , EM-VectorPotential), (Time Differential, 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.

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

Abbreviations

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

Units of Measure - (SI variant, mksC)

length/time [m] meter <*> [s] second Count of the quantity of separation; Location of events 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
temperature [ºK] Kelvin Count of the quantity of heat (statistical)

Useful SR Quantities

Minkowski Flat (Pseudo-Euclidian) Spacetime Metric:
ημν = gμν = gμν = Diag[+1,-1,-1,-1]

Dimensionless SR Factors:
β = (v/c): Beta factor, the fraction of the speed of light c
β = (u/c): Vector form of Beta factor
γ(u) = dt/dτ: Lorentz Gamma Scaling 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)
φ = Ln[γ(1+ β)]: BoostParameter/Rapidity (which remains additive in SR, unlike v)

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 Functions:
V·V = Vo·Vo : Invariant interval is often easier to calculate in rest frame coordinates
√[1+x] ~ (1+x/2) for x<<1 : Math relation often used to simplify Relativistic eqns. to Newtonian eqns.
δuv = Delta function = (1 if u = v, 0 if u≠v)
γ = c/√[c2-v2] = c/√[c2-u·u]
γ2 = c2/(c2-v2) = c2/(c2-u·u)
v γ = c √[γ2-1]
β γ = √[γ2-1]
c2 dγ = γ3 v dv
d(γ v) = γ3 dv
dγ = γ3 v dv/c2
γ' = dγ/dt = (γ3 v dv/dt)/c2 = (γ3 u·a)/c2 = ar·u/c2
γ'' = dγ'/dt = d2γ/dt2 = (γ3/c2)*[(3γ2/c2)(u·a)2 + (u'·a) + (u·a')]
c = 1/√[εoµo] ~ 2.99729x108 [m/s]
u2 = u2
u·u' = uu' = ua
(u x a)2 = u2a2 - (u·a)2

SR Notation Used

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.
Always check notation conventions in SR & 4-Vector references, they are all relative ;-)

Minkowski SR Metric (time 0-component positive), for which ημν = gμν = gμν = Diag[+1,-1,-1,-1] = Diag[+1,-1]
Signature[ημν] = -2

A = (c at,ax,ay,az) = (c a0,a1,a2,a3) = (c a0,a)   time (a0) in the 0th coord. ( some alternate notations use time as a4 )

Intervals: Time/Temporal (+ interval) = 0 coordinate  ( some alternate notations use time as - interval and space as the + interval)
               Light/Null (0 interval)
               Space/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 Symbols: A = Aμ = (c a0,a) = (ca0,a1+a2+a3) = (c a0,a1,a2,a3), where the raised index indicates dimension, not exponent
4-Vector c-Factor: always applied to "Temporal" scalar component, as necessary to give consistent dimensional units for all vector components (ct,x,y,z) = (ct,x)
  *Note* c-Factor is usually on the top, as ( ct , x , y , z ) = [m], but is sometimes on bottom, as ( E/c , px , py , pz ) = [kg m s-1]

4-Vector Computer HTML Representation:
    SR 4-vector = {BOLD UPPERCASE} = A
    time component = {regular lowercase} = a0
    space 3-vector component = {bold lowercase} = a

Relativistic Component: v --> vo in a rest-frame, typically v = γ vo (dilation) or v = (1/γ) vo (contraction)
eg. t = γ to (time dilation), L = (1/γ) Lo (length contraction)

Imaginary unit: ( i ) used only for QM phenomena, not for SR frame transformations or metric.  To follow up on a quote " 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 separation of SR vs Newtonian concepts, with the Newtonian 3-vector (a) extending naturally into the SR 4-vector (A)
   easy recovery of Newtonian cases by allowing (γ-->1, dγ-->0) when (v<<c)
   easy separation of SR vs QM concepts, no ict's -- ( i ) only enters into QM concepts, such as Photon Polarization
   reduction in number of minus signs (-), eg. U·U = c2, P·P = (moc)2: the square magnitudes of velocity and momentum 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 gμν.  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 ημν = gμν = gμν = Diag[+1,-1,-1,-1].
"Flat" SpaceTime
ημν = gμν{SR} = 
t
x
y
z
+1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1

gμα gμβ = δαβ = (4 if α = β for Minkowski)
g = - Det[gμν] = -1 (for Minkowski) not a scalar invariant
Sqrt[-g]ρ: Scalar density

There are other ways of defining the metrics and 4-vectors available in SR which lead to the same results, but this particular notation has some nice qualities which place it above the others. First, it shows the difference between time and space in the metric. We perceive time differently than space, despite there being only spacetime. Also, this metric gives all of the SR relations (frame transformations) without using the imaginary unit ( i ) in the transforms. This is important, as ( i ) is absolutely essential for the complex wave functions once we get to QM. It is not needed, and would only complicate and confuse matters in SR. This metric will allow us to separate the "real" SR stuff from the "complex/imaginary" QM stuff easily. It also allows for the possibility of complex components in SR 4-vectors.  The choice of +1 for the time component simplifies the derived equations later on, as it usually allows rest frame square magnitudes to be positive.

SR 4-Vector Basics

ημν = gμν = gμν = DiagnolMatrix[1,-1,-1,-1]: Minkowski Spacetime Metric-the "flat" spacetime of SR

A = Aμ = (at,ax,ay,az) = (a0,a1,a2,a3) = (a0,a): Typical SR 4-vector
       Aμ = (at,-ax,-ay,-az) = (a0,-a1,-a2,-a3) = (a0,-a): Typical SR 4-covector, we can always get the 4-vector form with Aμ = gμνAμ
       Basically, this has the effect of putting a minus sign on the space component

B = Bμ = (bt,bx,by,bz) = (b0,b1,b2,b3) = (b0,b): Another typical SR 4-vector

A·B = gμν Aμ Bν = Aν Bν = Aμ Bμ = +a0b0-a1b1-a2b2-a3b3 = (+a0b0-a·b): The Scalar 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·BA·C distributive
d(A·B) = d(AB + A·d(B)  differentiation
B = d(A)/dθ, where θ is a scalar invariant


Aproj = (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||
AA - (A·B)/(B·B) B    Component of A perpendicular to B

ημν  Λμα  Λνβ = ηαβ

if Aμ dXμ = invariant for any dXμ, then Aμ is a 4-vector

A'μ = Λμν Aν: Lorentz Transform (Transformation tensor which gives relations between alternate boosted inertial reference frames)

Λμν = (for x-boost)
γ -(vx/c)γ 0 0
-(vx/c)γ γ 0 0
0 0 1 0
0 0 0 1
γxγ 0 0
xγ γ 0 0
0 0 1 0
0 0 0 1

 
General Lorentz Transformation
Λμν = (for n-boost)
γxγyγzγ
xγ 1+(γ-1)(βx/β)2 ( γ-1)(βxβy/β)2 ( γ-1)(βxβz/β)2
yγ ( γ-1)(βyβx/β)2 1+( γ-1)(βy/β)2 ( γ-1)(βyβz/β)2
zγ ( γ-1)(βzβx/β)2 ( γ-1)(βzβy/β)2 1+( γ-1)(βz/β)2

General Lorentz Boost Transform using just vectors & components-Thank you Jackson, Master of Vectors! Chap. 11
β = v/c, β = |β|, γ = 1/√[1-β2]
a0' = γ(a0-β·a)
a' = a+(γ-1)/β2(β·a)ββ a0

a0' = γ(a0-β·a)  Temporal component
a||' = γ(a||-βa0)   Spatial parallel component
a
' = a   Spatial perpendicular components


We are also able to use the Rapidity
φ = Ln[γ(1+ β)]
eφ = γ(1+β) = √[(1+β)/(1-β)]
β = Tanh[φ], γ = Cosh[φ], γβ = Sinh[φ], φ = Rapidity (which remains additive in SR, unlike v)
Formally, this is like a rotation in 3-space, but becomes a hyperbolic rotation through spacetime for a Lorentz boost
Λuv = (for x-boost)
Cosh[φ] -Sinh[φ] 0 0
-Sinh[φ] Cosh[φ] 0 0
0 0 1 0
0 0 0 1


Time t = γ to  --> Time Dilation (e.g. decay times of unstable particles increase in a cyclotron)
Length L = Lo/γ  --> Length Contraction

Complex SR 4-Vectors

A few 4-vectors are known to have complex components. The Polarization 4-vector is one 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 = (a0c + a1c e1+ a2c e2+ a3c e3): Complex 4-vector has complex components, 1 along time and 3 along space
Scalar[A] = a0c: Just the time component
Vector[A] = a1c e1 + a2c e2 + a3c e3: Just the spatial components
A = Scalar[A] + Vector[A]

A = ( (a0r + a0i ) + (a1r + a1i ) e1 + (a2r + a2i ) e2 + (a3r + a3i ) e3 ): Complex 4-vector has real + imaginary components, 1 each along time and 3 each along space
Re[A] = ( (a0r ) + (a1r ) e1 + (a2r ) e2 + (a3r ) e3 ): Only the real components
Im[A] = ( (a0i ) + (a1i ) e1 + (a2i ) e2 + (a3i ) e3 ): Only the imaginary components
A = Re[A] + i Im[A]

A = (a0r + i a0i,ar + i ai) : Standard 4-vector
A* = (a0r - i a0i,ar - i ai): Complex conjugate 4-vector, just changes the sign of the imaginary component

A = (a0r + i a0i,ar + i ai) : A* = (a0r - i a0i,ar - i ai)
B = (b0r + i b0i,br + i bi) : B* = (b0r - i b0i,br - i bi)

A·B = [( a0r b0r - ar·br ) - ( a0i b0i - ai·bi )] + i [( a0r b0i - ar·bi ) + ( a0i b0r - ai·br )] : General scalar product
A·A = [( a0r2 - ar·ar ) - ( a0i2 - ai·ai )] + 2i [( a0r a0i - ar·ai )] = |A|2 : Scalar product of 4-vector with itself gives the magnitude squared
A·A* = [( a0r2 + a0i2 ) - ( ar·ar + ai·ai )] = Re[A·A*]: Scalar product of 4-vector with its complex conjugate is Real, thus Im[A·A*] = 0

·B = [( ∂/c∂tr b0r + delr·br ) - ( ∂/c∂ti b0i + deli·bi )] + i [( ∂/c∂tr b0i + delr·bi ) + ( ∂/c∂ti b0r + deli·br )]
     = [( ∂/c∂tr b0r + delr·br ) - ( ∂/c∂ti b0i + deli·bi )]
     = 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

i = √[-1] :Imaginary Unit
π = 3.14159265358979... :Circular Const

Fundamental/Universal Physical Constants (Lorentz Scalars)

c = Speed of Light Const ~ 2.99729x108 [m/s]
h = Planck's Constant - relates particle to wave - Action constant
hbar = (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)
kB = Boltzmann's Const ~ 1.3806504(24)×10−23 [J/ ºK]  relates temperature to energy
mo = Rest Mass Const (varies with particle type)
qo = Electric Charge Const (varies with particle type)

Note:
I do not set various fundamental physical constants to dimensionless unity, (i.e. c = h = G = kB = 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.


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

4-Vector 4-Vector = (temporal comp, spatial comp) Units - Description
     
   
***Calculus***
4-Displacement
= 4-Delta
ΔR = (cΔt, Δr) [m]  Δt = Temporal Displacement, Δr = Spatial Displacement, (Finite Differences)
The 4-vector prototype, the "arrow" linking two events
4-Differential dR = (cdt, dr) [m]  dt = Temporal Differential, dr = Spatial Differential, (Infinitesimals)
4-Gradient
= 4-Del or 4-Partial
= ∂/∂xμ = (∂/c∂t, -del) = (∂/c∂t, -)
= (∂/c∂t, -∂/∂x, -∂/∂y, -∂/∂z)
[m-1]  ∂ is the partial derivative, -del = -(∂/∂x i + ∂/∂y j + ∂/∂z k)
This is a very important 4-vector operator, often used to generate continuity equations
∂/∂xu = (∂/c∂t, -del)   and   ∂/∂xu = (∂/c∂t, del)
· is also known as the D'alembertian (Wave Operator Δ)

I usually write out (del) because the nabla/del symbol ()
is quite often not displayed correctly in various browsers.
It should look like an inverted triangle when displayed correctly.

Let gμ = ∂μ f = ∂f / ∂xμ
Using the chain rule, one can show:
g'ν  = ∂f '/∂x'ν = Σ ( ∂f / ∂xμ )( ∂xμ / ∂x'ν ) = ∂'ν f ' = ( ∂μ f )( ∂'ν xμ ) = (gμ)( ∂'ν xμ )
However, this appears to be a standard Lorentz transform
∂'μ = Λμνν[function argument] =  ∂ν[function argument] Λμν  
     
   
***Particle Dynamics***
4-Position
R = Rμ = (ct, r), eg. radial coords
= (ct,r,θ,z)

X = Xμ = (ct, x), eg. cartesian coords
= (ct,x,y,z)
[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 events at the origin (0,0,0,0) of the chosen coordinate systsem
c=Speed-of-Light
sometimes seen as X, other times as R
4-Velocity U = γ(c, u) = dR/dτ = γdR/dt
= γ(c, ur) = γ(c, u)
= γ(c,ux,uy,uz)

= γcc(1,n), for light-like/photonic

Uo = (c,0) in rest frame
[m s-1ur = Relativistic 3-Velocity,  u = dr/dt = Newtonian 3-Velocity
ur = (r)' = r'
u = dr/dt = r'
thus, ur = u
"U is historically used instead of V"
Uo = (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
4-Acceleration
A = γ(c dγ/dt, dγ/dt ua) = dU/dτ = γ dU/dt
=d2R/dτ2
= γ(c dγ/dt, ar)
= γ(c γ', ar), where γ' = dγ/dt
= γ(c γ', γ' ua)
= γ(ar·u/c, ar), because A·U = 0

Ao = (0,ao) in rest frame
[m s-2ar = Relativistic 3-Acceleration,  a = du/dt  = Newtonian 3-Acceleration
ar = (γur)' =  γ' ur + γ ur' =  γ' u + γ a = (γ3/c2)(u·a) ua
a = du/dt = u'
γ' = dγ/dt = (γ3/c2)(u·a) = ar·u/c2
4-Jerk
J = dA/dτ = γ dA/dt
=d3R/dτ3
= γ( c(dγ/dt)2 + cγ(d2γ/dt2), dγ/dt ar+γ dar/dt )
= γ( c γ'2 + c γ γ'', γ' ar + γ ar' )
= γ( c γ'2 + c γ γ'', jr )
where γ' = dγ/dt, γ'' = d2γ/dt2, ar' = dar/dt
[m s-3jr = Relativistic 3-Jerk,  j = da/dt  = Newtonian 3-Jerk
jr = (γar)' = γ' ar + γ ar'
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')]
     
   
***Kinematics***
4-Momentum P = (E/c, p) = (mc, p) = mo γ(c,u)
= moU = hbarK
=
(Eo/c2)U

= (hbarω/c)(1, n) = (E/c)(1, n), for light-like/photonic
Po = (Eo/c,0) = (moc,0) in rest frame

P = ((Eo + poVo)/c2)U, taking into account pressure*volume terms where
pressure p = po
volume V = Vo/ γ
[kg m s-1]   E = Energy, p = Relativistic 3-Momentum
mo = RestMass( 0 for photons, + for massive )

4-Momentum used with single whole particles
P·P = (moc)2 = (Eo/c)2 generally
P·P = 0 for photonic



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.
4-MomentumDensity
= 4-MassFlux
G = (u/c, g) = (pmc, g) = po_m γ(c, u)
= po_mU
=(uo/c2)U
= (1/Vo)P = (1/c2)S
po_m = mono??

G = ((uo + po)/c2)U,
pressure p = po
[kg m-2 s-1]   u = EnergyDen = ne, pm = MassDen = u/c2
g = MomentumDen = (u/c2)u = (eo)ExB,   f = g·u = MomentumFlux
u
= 3-velocity, n = ParticleDen, e = EnergyPerParticle



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.
4-Force
 or
4-Minkowski Force
F = γ(dE/cdt, fr) = dP/dτ
= (modU/dτ) + (dmo/dτ)U, generally
= ( moA) + (dmo/dτ)U, generally
=
moA , if mo is stable
=
γ(fr·u/c, fr), if rest-mass preserving
= γmo(c dγ/dt, ar) = γmo(c γ', ar)
[kg m s-2]   dE/dt = Power, fr = Relativistic 3-Force,  f = Newtonian 3- Force
                  fr = dp/dt = mo d(γu)/dt = mo(γ' u + γ u') = mo(γ' u + γ a)
                   f = mou' = moa
                  a
= du/dt = u'

Sometimes known as the Minkowski Force

4-Force is used with single whole particles
4-Force Density Fd = γ(du/cdt, fdr) = dG/dτ?? [kg m-2 s-2] 4-Force divided by volume

4-ForceDensity is used with mass distributions
     
   
***Connection to Waves***
4-WaveVector
 or
4-AngWaveVector
K = (ω/c, k) = (ω/c,ω/vphase n)
= (1/hbar)(E/c,p) = (ωo/c2) γ(c,u)
= (1/hbar)P = (ωo/c2)U
=
(1/hbar)P = (mo/hbar)U

= (ω/c)(1, n) , for light-like/photonic
[rad m-1]
ω = AngularFrequency [rad/s], k = WaveNumber or WaveVector [rad/m]
n = UnitWaveNormalVector, vphase = phase_velocity
ωo = RestAngularFrequency( 0 for photons, + for massive )
ω = 2πν, k = 2π/λ
k everywhere points in the direction orthogonal to planes of constant phase φ
where phase φ = -K·R= -(ωt - k·r) = (k·r - ωt)
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
ν
λ = c, for photonic

***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 is common in the literature
4-CycWaveVector
 or
4-InverseWaveLength
Kcyc = (ν/c,1/λ n)
= (1/λ)(w/c, n) = (ν)(1/c, n/w)
=(1/2π)K
= (1/h)P = (νo/c2)U
= (1/h)P = (mo/h)U

= (ν/c)(1, n) , for light-like/photonic

sometimes called
L
[cyc m-1ν = CyclicalFrequency [cyc/s], λ = WaveLength [m/cyc]

ν = ω/2π = Frequency
1/λ = k/2π = Inverse WaveLength [cyc/m]
hbar = h/2π = Dirac's Const
w = λν = (Phase) Velocity of Wave
n = UnitWaveNormalVector
               
     
   
***Flux 4-Vectors***
 
Flux 4-Vectors all in form of :
 V = {rest_charge_density} U
 V
= {rest_charge}no U
where n = γno

alternately,

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, nf) = no γ(c, u) = n(c, u)
= noU
[# m-2 s-1]  no = RestNumberDensity [#/m3], n = γno = NumberDensity [#/m3]
nf = nu = NumberFlux [(#/m3)*(m/s)]
# of stable particles N = noVo = 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 = Σa [∫dτ δ4(x-xa(τ))(dXa/dτ)] = = Σa [∫dτ δ4(x-xa(τ))(Ua)]
4-VolumetricFlux V  = VoU?? [(m3) m-2 s-1] Vo = RestVolume
4-ElectricCurrentDensity
=4-CurrentDensity
= 4-ElectricChargeFlux
J = (cρ, j) = ρo γ(c, u) = ρ(c, u)
= ρoU = qonoU = qoN
= Jelec
[C m-2 s-1]   ρo = RestElecChargeDensity [C/m3], ρ = γρo = ElecChargeDensity
                   j = γ ρo u = ρu =ElecCurrentDensity = ElecChargeFlux [(C/m3)*(m/s)]
                   j = α(E+uxB), α = Conductivity
                  qo = Electric Charge [C]
                  ρo = qono [C/m3]
4-MagneticCurrentDensity
= 4-MagneticChargeFlux
= Zero (so far..)
Jmag = (cρmag, jmag) = ρo_mag γ(c, u)
= ρo_magU = qo_magnoU
= Zero (so far...)
[MagCharge m-2 s-1]  ρo_mag = RestMagChargeDensity, ρmag = γρo_mag = MagChargeDensity
                                  jmag = MagCurrentDensity = MagChargeFlux
                                  qo_mag = Magnetic Charge  
to date: ρo_mag = 0 and jmag = 0  -- no magnetic (monopole) charges yet discovered
4-ChemicalFlux   [(mol) m-2 s-1]
4-MassFlux
= 4-MomentumDensity
G = (u/c, g) = (cρm, g) = ρo_m γ(c, u) = ρo_mU

ρo_m = mono?? = (1/c2)S
[(kg) m-2 s-1]   u = EnergyDen = ne, pm = MassDen = u/c2
g = MomentumDen = (u/c2)u = (eo)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?
S = (cu, s) = uo γ(c, u) = uoU = c2G

????uo = Eono ??
[(J) m-2 s-1]   u = EnergyDen = ne, s = EnergyFlux = PoyntingVector = uu = c2g = Ne
ue = (εoE·E+B·Bo)/2 = (E·D+B·H)/2 = EM energy density

typically see ∂u/∂t = - del·s + Jf · E
which in 4-vector notation would be ·S = jf · E, where jf is the current density of free charges, so not conserved generally
however, make the following observation
∂(um)/∂t = j(t,xE(t,x), where this is rate of change of kinetic energy of a charge
then let
u = ue+um, s = se+sm, ·S = ∂(ue+um)/∂t +del·(se+sm) = 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.


εo = Permittivity, µo = Permeability
εoµo = 1/c2
s = (E x B)/µo = 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(ue+um),se+sm)
·S = ∂(ue+um)/∂t +del·(se+sm) = 0
um = mechanical kinetic energy density
sm = 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/µo)(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, sf) = so γ(c, u) = soU [(J ºK-1) m-2 s-1]  so = RestEntropyDensity = qo/T, sf = EntropyFlux
Entropy S = ∫sodV = kB ln  Ω, where Ω = # of microstates for a given macrostate
·S >= 0

alternate def:
S = soU + Q/To
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)

not be be confused with the 4-Poynting Vector S
     
   
***Thermodynamic***
4-InverseTempFlux β = βo U = (1/kBTo) U
where βo = 1/kBTo
dS = β·dP,  differential entropy
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 PT = P/kB = (pT0/c, pT) = (T/c, pT) = ((E/kB)/c, p/kB)
pT0 = T = Temperature (in ºK)

simply dividing 4-Momentum by Boltzmann's const. kB
which gives E/c = kBT/c, or E = kBT
[ºK m-1 s]
ºK = degree Kelvin Temperature
kB = 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 kB [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, qf) = qo γ(c, u) = qoU??

where qo = [1]??


= ( c (k/a)
φ , qf)??
= ( c (k/a)
φ , -k del [φ])??

needs work
Potential Flow for Velocity??
Velocity Potential

"Velocity" Conduction Equation:  v = -k del [
φ]
"Velocity" Diffusion Equation:  a del·del [
φ] = ∂φ/∂t
where del·v ~ ∂
φ/∂t

Continuity gives ·V = ∂[c(k/a)
φ]/∂t +del·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, del, of the velocity potential φ:[1]

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:[1]

  del·v = 0

with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation[1]

del·del 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

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

A fluid is incompressible if the divergence of v is zero:

  del·v = 0

That is, if v is a solenoidal vector field.

 

 Irrotational flow

A flow is irrotational if the curl of v is zero:

  del x v = 0

That is, if v is an irrotational vector field.

 

Vorticity

The vorticity, ω, of a flow can be defined in terms of its flow velocity by

 \omega=\nabla\times\mathbf{u}.

Thus in irrotational flow the vorticity is zero.

 

The velocity potential

If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field φ such that

v = -k del [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:

   del·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 = del x A

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

     del·v = del·(del x A) = 0

The converse also holds: for any solenoidal v there exists a vector potential A such that v = del 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:

      del x v = 0
      del
·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 = del φ      (1)

Then, since the divergence of v is also zero, it follows from equation (1) that

     del·del φ = 0

which is equivalent to

     del2 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,

     del 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 = del 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 = (cq, qf) = qo γ(c, u) = qoU
= ( c (k/a)T , qf)
= ( c (k/a)T , -k del [T])
[(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 kB*
a = k/(ρcp) = Thermal Diffusivity [m2/s]
ρ = Density [1/m3]
cp = Specific heat capacity [J/ °K]
ρcp = Volumetric heat capacity [J/m3 °K]
T = Absolute Temperature [°K]
(k/a)T = Thermal energy density [W s / m3] = [J/m3], q = Heat Flux [W/m2] = Thermal Energy Flux

Heat Conduction Equation:  q = -k del [T]
Thermal Diffusion Equation:  a del·del [T] = ∂T/∂t
where del·q ~ ∂T/∂t

Continuity gives ·Q = ∂[c(k/a)T]/∂t +del·q = 0
Thus, [ (k/a)T ] and [ q ] are components of a 4-vector

Fourier's Law: q = -k del [T]
The minus sign ensures that heat flows down the temperature gradient

see Hydraulic Analogies
quantity: Heat Q [J]
potential: Temperature T [°K] = [J/kB]
flux: Heat Xfer Rate Qdot [J/s]
flux density: Heat Flux Qdot'' [W/m2]
linear model: Fourier's Law Qdot'' = -k del [T]
4-DarcyFlux
4-HydraulicFlux
Q = (cq, qf) = qo γ(c, u) = qoU
= ( c (βφ)P , qf)
= ( c (βφ)P , -κ/μ del [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 del [P]
Pressure Diffusion Equation: del·q  =  -κ/μ del·del [P]  = βφ∂P/∂t, by imposing incompressibility
giving Pressure Diffusion Waves

Continuity gives ·Q = ∂[βφP]/∂t +del·q = 0
Thus, [ βφP ] and [ q ] are components of a 4-vector

Darcy's Law: q = -κ/μ del [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 φV [m3/s]
flux density: Velocity [m/s]
linear model: Poiseuille's Law φV = ...
4-ElectricChargeFlux Q = (cq, qf) = qo γ(c, u) = qoU
= ( 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 = (- del [φ])
Ohm's Law: j = σ E = -σ del [φ]
Gauss Law: del·E = del·(- del [φ])  = - del·del [φ] = ρ/ε0

Fick's 1st Law Diffusion: j = - D del ρ
where D = μ k T / e = Einstein-Smoluchowski Relation



Continuity independently gives ·J = ∂[ρ]/∂t +del·j = 0
Thus, [ ρ ] and [ j ] are components of a 4-vector

Continuity independently gives ·A = ∂[φ]/∂t +del·aEM = 0 in the Lorenz Gauge
Thus, [ [φ ] and [ aEM ] are components of a 4-vector

Ohm's Law: j = -σ del [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 del [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 =
mo 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 hbar2 s(s+1)
where Σ is the spin part of angular momentum j
(P02) = (m2c2)
(Σ2) = hbar2 s(s+1)

W·W = 0 for photonic

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

S = (γ β·so , s + [γ2/(γ+1)](β·so) β) in moving frame

S = (1/mo) W= (U·U/P·U) W
where mo = √[P·P/U·U] = P·U/U·U

Magnetic moment
μ = -(g/2)(e/mc) s

So = (0,so) in rest frame
[ 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 So=(0,so)
This is an axial vector, or pseudovector
4-Spin has only 3 independent components, not 4, due to U·S = 0
So=(0,so), 4-Spin is always space-like
S·S = (u·s/c,s)·(u·s/c,s) = ((u·s/c)2 - s·s) = - so·so = - hbar2 so(so+1)

s·s |s,m> = hbar2 s(s+1) |s,m>
sz |s,m> = hbar 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| = hbar √[s(s+1)], 
[ Sx ,Sy ] = i hbar εxyz Sz

Spin raising/lowering operators:
S± |s,m> = hbar √[s(s+1) - m(m ± 1)] |s,m> where S± = Sx ± i Sy

Bargmann-Michel-Telegdi (BMT) dynamical eqn. for spin
dS/dτ = (e/mc)[ (g/2) FμβSβ + (1/c2)(g/2-1))vμSαFαβvβ ]
which leads to Thomas precession in the rest frame
4-Polarization
or
4-JonesVector
Ε = (ε0, ε) = (ε·u/c,ε) for a massive particle
= (ε0, ε) = ((c/vphase) ε·n,ε) for a wave

= (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[iαx],Sin[θ]Exp[iαy]),0)
E* = (0,Cos[θ]Exp[-iαx],Sin[θ]Exp[-iαy]),0)

Angle θ describes the relation between the amplitudes of the electric fields in the x and y directions
Angles αx and αy describe the phase relationship between the wave polarized in x and the wave polarized in y

Ε·E*
= (+02 
- Cos[θ]Exp[iαx]Cos[θ]Exp[-iαx]
- Sin[θ]Exp[iαy]Sin[θ]Exp[-iαy]
-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.
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 = Nμ = ( 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 = Sμ = ( 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
Nμ = ( Tμ - Pμ(pt/m2))(m/|p|) where Tμ is the unit time-like 4-vector
 
4-PauliMatrix Σ = Σμ = (σ0, σ)

where Σ·Σ = -2σ0
The components of this 4-vector are actually the Pauli Matrices
     
   
***Electromagnetic Field Potentials***
4-VectorPotential
 or
4-VectorPotentialEM
AEM = (ΦEM/c, aEM) [kg m C-1 s-1]   ΦEM = ScalarPotenialEM   aEM = VectorPotenialEM
Electric Field E = -delEM]-∂aEM/∂t, Magnetic Field B = del x aEM
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 (Liénard-Wiechert potential)
AEM = (q/c4πε0) U / (R·U) where (R·R = 0, the definition of a light signal)

4-VectorPotentail of an Ε polarized plane-wave
AEM ~ Ε Exp[i K·R]

Homogeneous Maxwell/Lorentz Equation  (if ·AEM = 0 Lorenz Gauge)
(·)AEM = µ0 J  
4-VectorPotentialMomentum
 or 
4-PotentialMomentum
QEM = (EEM/c, pEM) = qo AEM [kg m s-1]   EEM = ScalarPotenialEnergy   pEM = VectorPotenialMomentum
Energy and Momentum of the EM field itself, for a single charge
4-Potential
 or
4-PotentialEM

***break with standard notation***

better to use the 4-VectorPotentialEM
ΦEM = (ΦEM,c aEM)
= c AEM
[kg m2 C-1 s-2]   ΦEM = ScalarPotenialEM   aEM = VectorPotenialEM

***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 is common in the literature
4-MomentumEM
4-CanonicalMomentum
4-TotalMomentum
PEM = (E/c + qΦEM/c, p + qaEM) = γ mo(c,u)

PEM = Π = P + q AEM
[kg m s-1] **Momentum including effects of EM 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
4-GradientEM
Gauge Covariant Derivative
DEM = (∂/c∂t + iq/hbar ΦEM/c, -del + iq/hbar aEM)
= + (iq/hbar)AEM
for electrons, commonly seen as
DEM = - (ie/hbar)AEM
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 dV = (c dv0,dv)
= (dVo) U
= (dVo) γ(c, u)
= γ(dVo) (c, u)
= (dV)(c, u)
= (c dV,dV u)

so that, in a rest frame
dVo = (1)(c dVo,dVo 0) = (c dVo,0)

dV =  γdVo ????

V should be as follows
V = Vo

Perhaps acts a little differently since this is from a "vector-valued volume element", and not a straight volume.
[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/po = d3p'/p'o
p'o = po
d3x d3p = d3x' d3p'

dVo·dX = (dVo,0)·(cdt,dx) = (dVo 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  = γρo (dV dt) (1/γ) U = ρo (dV dt)  U = (dV dt)  ρoU = (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
dVp = (c dvp0,dvp)
= (dVpo) U
= (dVpo) γ(c, u)
= γ(dVpo) (c, u)
= (dVp)(c, u)
= (c dVp,dVp u)

so that, in a rest frame
dVpo = (1)(c dVpo,dVpo 0) = (c dVpo,0)
[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)(dVp) = dp0 dp1 dp2 dp3 =  dp'0 dp'1 dp'2 dp'3
dE/ c = dp0
dVp = dp1 dp2 dp3 =  d3p
(dE/ c)(dVp) = (dp0 )(dp1 dp2 dp3) =  d4p

d3x = d3x'/γ
d3p/po = d3p'/p'o
p'o = po
d3x d3p = d3x' d3p'

dVpo·dP = (dVpo,0)·(dE/c,dp) = (dVpo dE/c) = dpxdpydpz dE/c = d4p
dVp·dP = d4p
Hence, the differential momentum 4-element is an invariant


also,

dVo·dVpo = (dVo,0)·(dVpo,0) = (dVodVpo) = (dx1 dx2 dx3)(dp1 dp2 dp3)
= d3xd3p in the rest frame
Thus dV·dVp = 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) [*] 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(1*1 - 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 T = (1,0)
T·T = (1,0)·(1,0) = (1*1 - 0·0) = 1
[*] The Unit Temporal 4-Vector
Square Magnitude = 1, Length = |1| = 1
4-Unit Null N = (1,n)
N·N = (1,n)·(1,n) = (1*1 - n·n) = 0
[*] The Unit Null 4-Vector
n = unit 3-vector, |n| = 1
Square Magnitude = 0, Length = |0| = 0
The Null Vector is "perpendicular" to itself.
4-Unit Spatial S = (0,n)
S·S = (0,n)·(0,n) = (0*0 - n·n) = -1
[*] The Unit Spatial 4-Vector
n = unit 3-vector, |n| = 1
Square Magnitude = -1, Length = |-1| = 1
4-Basis Vectors 
(1 time + 3 space)
Bt = (1,0,0,0)
Bx = (0,1,0,0)
By = (0,0,1,0)
Bz = (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 
(null tetrad)
Bn1 = √[1/2] (1,0,0,1)
Bn2 = √[1/2] (1,0,0,-1)
Bn3 = √[1/2] (0,1,i,0)
Bn4 = √[1/2] (0,1,-i,0)
A tetrad of unit-length, linearly-independent, null basis vectors
Note the complex components, since there can be only 2 real linearly independent null vectors
This is simply one basis, there are others
** Need to double-check these ***
see null tetrad, Sachs tetrad, Newman-Penrose tetrad


Fundamental/Universal Relations


Event Tracking Relations
Event R Mass mo


Energy Eo = moc2
WaveAngFreq ωo


MassDensity ρo_m
Mass mo = ρo_mVo

EnergyDensity uo = ρo_mc2
ChargeDensity ρo
Charge Qo = ρoVo

NumberDensity no
ParticleNumber No = noVo

event particle wave density density density
pos: R = (ct, r) mo at R ωo at R ρo_m at R ρo at R no at R
vel: U = dR/dτ P = moU = (Eo/c2)U K = (ωo/c2)U = (1/hbar)P G = ρo_mU = (uo/c2)U J = ρoU N = noU
accel: A = dU/dτ F = dP/dτ   Fd = dG/dτ    
jerk: J = dA/dτ          


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


·R = (∂/c∂t,-del)·(ct,r) = (∂/c∂t[ct]+del·r) = (∂/∂t[t]+del·r) = (1+3) = 4
·R = 4  The divergence of open spacetime is equal to the number of independent dimensions (1 time + 3 space)

d/dτ (·R) = d/dτ (4) = 0
d/dτ (·R) = d/dτ () · R + ∂·d/dτ (R) = d/dτ () · R + ∂·U = γ d/dt () · R + ∂·U = γ (d/dt(∂/c∂t), -d/dt(del))·(ct,r) + ∂·U = γ (d/dt(∂/c∂t)(ct)+d/dt(del))·r + ∂·U = γ (d/dt(∂/∂t)(t)+d/dt(del))·r + ∂·U = γ (d/dt(1)+d/dt(3)) + ∂·U = 0 + ∂·U = ·U
thus, 
·U = 0, which is the General SR Continuity Equation, one might say the conservation of event flux or continuity of worldlines.
Due to this property, any Lorentz scalar constant times 4-Velocity U is a conserved quantity.
For example, let N = noU, so ∂·N = ·noU = no·U = no(0) = 0.  The quantity no is conserved.

Any "charge" constant becomes a 4-vector when multiplied by the 4-Velocity, and obeys the Conservation of Charge/Continuity equation ·J = ∂ρ/∂t +del·j = 0 where J = ρoU
let Charge Qo = ρoVo, where ρo is the "rest charge density", ρ = γρo is the relativistic "charge density", Vo is the rest volume, and j = γρou = ρu is the "ChargeDensity-Flux or Current Density"
then ChargeFlux 4-Vector = CurrentDensity 4-Vector J = ρoU = ρo γ(c, u) = ρ(c, u) = (cρ, j)
In the case of "electric" charge, ρo is the "rest electric-charge density", and j is the ElectricChargedensity-flux = electric current density
In the case of "number" charge, ρo is the "rest number-charge density"
In the case of "mass" charge, ρo is the "rest mass density", and j is the mass-flux = mass current density = momentum density

Poincaré transformation or an inhomogeneous Lorentz transformation:
ημν Λμα Λ νβ = ηαβ

Chain rule on the 4-gradient:
Let gμ = ∂μ f = ∂f / ∂xμ
Using the chain rule, one can show:
g'ν  = ∂f '/∂x'ν = Σ ( ∂f / ∂xμ )( ∂xμ / ∂x'ν ) = ∂'ν f ' = ( ∂μ f )( ∂'ν xμ ) = (gμ)( ∂'ν xμ )
where the brackets indicate that the gradient acts only on the function inside the given bracket
However, this appears to be a standard Lorentz transform
∂'μ = Λμνν[function argument] =  ∂ν[function argument] Λμν

Let ∂'·J' = ∂ρ'/∂t +del·j' = 0 be an arbitrary 4-vector continuity equation.
∂'·J' = ημν ∂'μ J'ν = ημν Λμαα J'ν = ημν Λμαα Λνβ Jβ = ημν Λμα Λνβα Jβ = ηαβ α Jβ = ·J
Assuming that the 4-gradient acts only on the 4-vector J, and not on the metric and Lorentz transforms,
which appears to be the case based on the chain rule
So, ∂'·J' = ·J, the continuity equations holds in arbitrary inertial reference frames

Consider a scalar ( s ) and a vector ( v ) related by a continuity equation, ∂s/∂t +del·v = 0.
If this equation holds in all inertial reference frames, then s and v must be components of a 4-vector (cs, v).

see http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/proposed_relativistic.htm



SR Path to QM

Event(SR) EventMovement MassEnergy Particle-WaveDuality QuantumMechanics(QM) SpaceTimeVariations
R = (ct, r) dR/dτ = U = γ(c,u) U = P/mo P = hbar K ***  K = i   *** = (∂/c∂t,-del)
      or K =  (ωo/c2)U    

d/dτ[R] = (i hbar / mo)       Event motion ~ spacetime structure  - depends on i hbar / mo

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

R = (ct,r) Location of an event (i.e. a particle) within spacetime
U = dR/dτ Velocity of the event is the derivative of position wrt. Proper Time
P = moU Momentum is just the Rest Mass of the particle times its velocity
K = 1/hbar 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...

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 = (moc)2
K·K = (moc / hbar)2 = (ωo/c)2
· = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂2/c2∂t2-del·del = -(moc / hbar)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:
· + (moc / hbar)2 = 0
(· + (moc / hbar)2 ) Ψ = 0, where Ψ is a scalar   Klein-Gordon eqn for massive spin-0 field
(· + (moc / hbar)2 ) A = 0, where A is a 4-vector  Proca eqn for massive spin-1 field
and let mo --> 0
(·) Ψ = 0, where Ψ is a scalar   Free-wave eqn for massless spin-0 field
(·) A = 0, where A is a 4-vector  Maxwell eqn for massless spin-1 field, no current sources


Interesting Note about Proca eqn.
"Massive charged vector field - represent with complex four-vector field φμ(X) and impose "Lorenz condition" (∂μφμ) = 0 so that φ0(X) -the scalar polarization- , can be discarded and the Klein-Gordon equations emerge for the other three components φi(X

(· + (moc / hbar)2 ) A = 0, where A is a 4-vector  Proca eqn for massive spin-1 field
rewrite in index notation
(∂μμ + (moc / hbar)2 ) Aν = 0 and combine with the Lorenz gauge condition (∂μAμ = 0)
apparently, this conjunction is equivalent to
μ( ∂μ Aν - ∂ν Aμ )+ (moc / hbar)2  Aν = 0
which is the Euler-Lagrange equation for the Proca Action

see Conceptual Foundations of Modern Particle Physics, ~ pg. 100


Momentum/Gradient Relations(Correspondences)
P = i hbar = -(Sact) = (∂/c∂t,-del) AEM = (0,0) *special case*
PEM = P+qAEM = i hbar DEM DEM = +iq/hbar AEM AEM = (VEM/c,aEM)

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

  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
Spacetime position of an event wrt. an origin event
dR·dR = (ds)2 = (c dt)2-dr·dr = (c dt)2-|dr|2 Differential interval magnitude - the fundamental invariant differential form
ΔR·ΔR = (ds)2 = (c Δt)2r·Δr = (c Δt)2-|Δr|2 Spacetime displacement interval magnitude - used to derive SR
·R = 4 The divergence of open spacetime is equal to the number of independent dimensions (t,x,y,z)
K·R = -φEM = (ωt-k·r) Phase of a SR wave; Psi = a E e -iK·R Photon Wave Equation (Solution to Maxwell Equation)
R·U = (ct,r)·γ(c,u ) = γ(c2t - r·u) Part of expression used in Liénard-Wiechert potential
   
Relations involving the 4-Velocity:
U·U = c2 The magnitude of 4-Velocity is always c2
U·A = 0 The 4-Acceleration is always normal to a particles worldline
P·U = moc2 = Eo Rest Energy
K·U = moc2/hbar = Eo/hbar = ωo Rest Ang. Frequency
F·U =  (moA+(dmo/dτ)UU = c2(dmo/dτ) = γc2(dmo/dt)

U
·F = γ2(dE/dt-u·f) = γ dmo/dt c2  
(pure force if dmo/dt = 0)
Power Law
U1·U2 = γ[u1]γ[u2](c2-u1·u2) = γ[ur]c2 (The scalar product of two uniformly moving particles is proportional to the γ factor of their relative velocities)
U· = d/dτ = γ(∂/∂t + u·del) = γ d/dt Relativistic Convective (Time) Derivative
U· = d/dτ = γ(∂/∂t + u·del) = γ d/dt

let u << c, then γ ~ 1
then U· = d/dτ = γ(∂/∂t + u·del) ~ (∂/∂t + u·del) = d/dt
delt(v) = (∂/∂t + u·del) v
is the gauge covariant derivative of a fluid where v is a velocity vector field of a fluid
·U = 0 The General Continuity Equation, one might say the conservation of event flux.
I believe this is true generally but it needs checking...
   
Relations involving the 4-Acceleration:
A·A = -a2 Magnitude squared of acceleration
U·A = 0 The 4-Acceleration is always normal to a particles worldline
U·U = c2
d/dτ(U·U) = d/dτ(c2) = 0
d/dτ(U·U) = 2*(U·dU/dτ) = 2*(U·A) = 0
   
Others:
P·P = (moc)2 = (Eo/c)2 
= m2(c2-u·u)
= (m2c2-p·p) = (E2/c2-p·p)
= 0 for light-like/photonic
Square Magnitude of the 4-Momentum
P1·P2 = γ[u1]γ[u2]mo1 mo2(c2-u1·u2)
= γ[ur12]mo1 mo2c2
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
N·N = (noc)2 Square Magnitude of the 4-NumberCurrentDensity
J·J = (poc)2 = (qonoc)2 Square Magnitude of the 4-ElectricCurrentDensity
K·K = (moc / hbar)2 = (ωo/c)2 
= 0 for light-like/photonic
Square Magnitude of the 4-WaveVector
· = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂2/c2∂t2-del·del = -(moc / hbar)2 Klein-Gordon Relativistic Wave Eqn.
** · is also known as the D'alembertian (Wave Operator) **
·J = dp/∂t +del·j = 0 Continuity Equation - Conservation of Electric Charge
No sources or sinks
Charge is neither created nor destroyed
E·K = 0 The Polarization of a photon is orthogonal to direction of wave motion (cancellation of "scalar" polarization)
E·E* = -1 The Polarization of a photon is always unit magnitude and space-like (cancellation of the longitudinal polarization)
AEM·AEM = (VEM/c,aEM)·(VEM/c,aEM) = (VEM/c)2-aEM·aEM = ???? Square Magnitude of the Electromagnetic field


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 change upon physical interaction (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.


Invariant Quantities (Lorentz Scalars~A·B)

c = √[U·U] Speed of Light: c (in vacuum)    E ~ cp
h = √[P·P/L·L] = P·L / L·L
hbar = √[P·P/K·K] = P·K / K·K
Planck's const: h                    E ~ hν
Dirac's const: hbar                  E ~ hbarω
kB = √[P·P/PT·PT]  = P·PT / PT·PT Boltzmann's const: kB             E ~ kBT
   
γrel = U·Vo/U·V Relative Relativistic Gamma Factor
   
Δs = √[ΔR·ΔR] = √[c2Δt2 - Δx2 - Δy2 - Δz2] Displacement
       < 0 Outside LightCone, Acausal SpaceLike Separation
Δs2 = 0 "On The LightCone", LightLike Signal Separation
       > 0 Inside LightCone, Causally TimeLike Separation
ds = √[dR·dR] = √[c2dt2 - dx2 - dy2 - dz2] Differential Length of World Line Element
ds2 = c22
dτ = √[dR·dR/U·U] Differential Proper Time, aka. the Eigentime differential
d/dτ = U· = γ(∂/∂t + u·del)
       = γ d/dt
Derivative wrt Proper Time  d/dτ
 d/dt = total time derivative,  ∂/∂t = partial time derivative
Δ = · = (∂/c∂t,-del)·(∂/c∂t,-del
= ∂2/c2∂t2-del·del  
D'Alembertian/wave operator
d4x = dV·dX Spacetime position-space differential "4-volume" element
Note: may need a correction factor if not using the Minkowski Metric
d4p = dVp·dP Spacetime momentum-space differential "4-volume" element
d3xd3p = dV·dVp Spacetime position-momentum differential 3-volume element
δ4(x-y) 4-D Dirac Delta Function
   
mo = √[P·P/U·U] = P·U/U·U RestMass of a Particle mo  ( 0 for photons, + for massive )
qo = √[J·J/N·N] =  J·N/N·N RestElectricCharge of a Particle qo
3-vector
so
= √[-S·S] = hbar √[s(s+1)]
Spin so
S·S = - so·so = - hbar2 so(so+1)
magnetic moment  
   
Eo = P·U = moc2 RestEnergy of a Particle ( 0 for photons, + for massive )
ωo = K·U = moc2/hbar RestAngFrequency of a Particle ( 0 for photons, + for massive )
   
φ = -K·R *** Phase of a wave ***, e.g. an EM wave, a plane wave
However, could also be a de Broglie matter wave...
Sact = -P·R = Integral[dt L;ti,tf] Action Variable S of Action Integral
γ L =  ?? Relativistic Lagrangian 
Action Integral Action Integral
   
no = √[N·N/U·U] = N·U/U·U Particle RestNumberDensity (for stat mech)
so = √[S·S/U·U] = S·U/U·U RestEntropyDensity (for stat mech)
   
Ωo = Ω Ω = # of microstates = (N!) / (n0!n1!n2!...)
No = N (Stable) Particle Number: N = nV = (n/γ)(γ V) = noVo = No
Po = P Pressure of system: P = Po
So = S = kB ln Ω Entropy: S = sV = (s/γ)(γ V) = soVo = So
To = γ T RestTemperature (according to Einstein/Planck def.)
Qo = γ Q RestHeat
Vo = γ V RestVolume
dS = kB d(ln Ω) = δQ / T Change in Entropy
   
Π (pα,xα) =

= dN/(d3x d3P)

=       (2j+1)/h3
  ---------------------
  exp[-(P·u)/kT - θ] - ε
Invariant equilibrium distribution function for relativistic gas
 j = particle spin
h = Planck's const
u = mean 4-velocity

   = 1 Bose-Einstein statistics
ε = 0 Maxwell-Boltzmann statistics
   = -1 Fermi-Dirac statistics

kTθ = Chemical potential μ=(ρ+p)/n - Ts
   
Fuv Fuv = 2(B2 - E2/c2) EM invariant
Gcd Fcd = εabcdFabFcd =(2/c)(B·E) EM invariant

 

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

·J = ∂p/∂t +del·j = 0 Conservation of 4-CurrentDensity (EM charge): p & j
change in ChargeDen wrt. time balanced by flow of CurrentDen
·N = ∂/∂t(γ no)+del·(γ nou)
     = ∂n/∂t+del·nf = 0
Conservation of 4-NumberFlux (Particle NumberDen, NumFlux): n & nf
change in NumberDen wrt. time balanced by flow of NumFlux
·P = (1/c2)∂E/∂t +del·p = 0

Sum[Pf-Pi] = 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 single particles. It is only when a group of particles is treated as a continuous fluid 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)+del·k
     =
(1/c2)∂w/∂t +del·k = 0.
Conservation of 4-WaveVec (AngFreq, WaveNum): w & k
change in AngFreq wrt. time balanced by flow of WaveNum
·AEM = (1/c2)∂VEM/∂t +del·aEM = 0 Conservation of 4-VectPotentialEM (applies in the Lorenz Gauge): VEM & aEM
change in ScalarPotential wrt. time balanced by flow of VectorPotential
 
·U = ∂/∂t(γ[u])+del·(γ[u] u)
       = γ3 (u/c2 ∂u/∂t + del·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 = ·(moU) = mo(·U) = 0
Also from d/dτ (·R) = ·U = 0
see also Noether's Theorem





Lorentz 4-Tensors

ημν =  ημν =  Diag[1,-1,-1,-1]
  = +1 if μ = ν = 0
  = -1 if μ = ν = 1,2,3
  = 0 if μ ≠ ν
Minkowski Metric (flat spacetime)
(pseudo-Euclidean)
           1 if a=b, 
δab =  
           0 if a≠b
Kronecker Delta
                       = +1 if {abcd} is an even permutation of {0123}
εabcd = -εabcd = -1 if {abcd} is an odd permutation of {0123}
                       = 0 otherwise
Levi-Civita symbol
technically a pseudotensor
εabcd = g εabcd = - εabcd = since g = -1 for Minkowski
Fuv =
0 -Ex/c -Ey/c -Ez/c
Ex/c 0 Bz -By
Ey/c -Bz 0 Bx
Ez/c By -Bx 0
EM Field Tensor
Fuv = ∂uAv-∂vAu  Electromagnetic Field Tensor (F0i = -Ei,Fij = eijkBk)
Gcd = (1/2)εabcdFab
0 -Bx -By -Bz
Bx 0 Ez/c -Ey/c
By -Ez/c 0 Ex/c
Bz Ey/c -Ex/c 0
Dual EM Field Tensor
Tab
W Sx/c Sy/c Sz/c
Sx/c xx xy xz
Sy/c yx yy yz
Sz/c zx zy zz
Energy-Momentum Stress Tensor

W = Energy Density
S = Energy Flux Density?

The stress-energy tensor of a relativistic fluid can be written in the form

Tab = μ ua ub +  p hab + (ua qb + qa ub)  + πab

Here

  • the world lines of the fluid elements are the integral curves of the velocity vector ua,
  • the projection tensor hab = gab + ua ub projects other tensors onto hyperplane elements orthogonal to ua,
  • the matter density is given by the scalar function μ,
  • the pressure is given by the scalar function p,
  • the heat flux vector is given by qa,
  • the viscous shear tensor is given by πab.

The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that

qa ub = 0, πab ub = 0, 

This means that they are effectively three-dimensional quantities, and since the viscous stress tensor is symmetric and traceless, they have respectively 3 and 5 linearly independent components. Together with the density and pressure, this makes a total of 10 linearly independent components, which is the number of linearly independent components in a four-dimensional symmetric rank two tensor.





Derived Equations

E2 = p·p c2 + mo2c4: Energy of a particle has a Momentum component and a RestMass component

Total Energy:    E = mc2 = γ[u] moc2 = hbarω
Kinetic Energy: T = mc2-moc2 = (γ[u]-1) moc2 = (γ-1) moc2
Rest Energy:     Eo = moc2


|
|___
|      |
|      |γ[u]  
| m  |
|      |
|___|____
 mo
Relativistic (apparent) mass m = AreaLike = γ[u] * mo = hbar w/c2 = E/c2
Theoretically, this would scale like a δ-function for photons{mo -->0,u -->c,γ-->Infinity}
Thus, the relativistic mass of a photon is proportional to w, the angular frequency
There is also a rest frequency wo = moc2/hbar, even when the massive particle is at rest. Mass is always "spinning" about the time dimension.

U·U = c2  , d/dτ(U·U) = d/dτ(c2) = 0 , d/dτ(U·U) = 2*(U·dU/dτ) = 2*(U·A) = 0
U·A = 0: The 4-Acceleration is orthogonal to its own 4-Velocity (Any acceleration is orthogonal to its own world-line, i.e. you don't accelerate in time).  
U
plays the part of the tangent vector of the world-line, and A plays the part of the normal vector of the world-line.
The curvature of a world-line is given by a/c2.

U1·U2 = γ[u1]γ[u2](c2-u1·u2) = γ[ur]c2 (The scalar product of two uniformly moving particles is proportional to the γ factor of their relative velocities)

·R = (∂/c∂t,-del)·(ct,r) = (∂/c∂t[ct]+del·r) = (∂/∂t[t]+del·r) = (1+3) = 4
·R = 4  The divergence of open space is equal to the number of independent dimensions

d/dτ (·R) = d/dτ (4) = 0
d/dτ (·R) = d/dτ () · R + ∂·d/dτ (R) = d/dτ () · R + ∂·U = γ d/dt () · R + ∂·U = γ d/dt (R + ∂·U = γ (d/dt(∂/c∂t), -d/dt(del))·(ct,r) + ∂·U = γ (d/dt(∂/c∂t)(ct)+d/dt(del))·r + ∂·U = γ (d/dt(∂/∂t)(t)+d/dt(del))·r + ∂·U = γ (d/dt(1)+d/dt(3))+ ∂·U = ·U 
thus, 
·U = 0, which is the general SR continuity equation, one might say the conservation of event flux.  Due to this property, any Lorentz scalar constant times 4-Velocity U is a conserved quantity.
For example, let N = noU, so ∂·N = ·noU = no·U = no(0) = 0.  The quantity no is conserved.
Alternately, ·U = (∂/c∂t, -del)·γ(c, u) = ·Uo = (∂/c∂t, -del)·(c, 0) = ∂/c∂t (c) = ∂/∂t (1) = 0


Compton Scattering
P·P = (moc)2  = = >0 for photons
Pphot1·Pphot2 = hbar2K1·K2 = (hbar2ω1ω2/c2)(1-n1·n2) = (hbar2ω1ω2/c2)(1-cos[ø])
Pphot·Pmass = hbarK·P = (hbarω/c)(1,n)·(E/c,p) = (hbarω/c)(E/c-n·p) = (hbarωEo/c2) = (hbarωmo)
Pphot + Pmass = P'phot + P'mass   Conservation of 4-Momentum in Photon-Massive Interaction
Pphot + Pmass - P'phot = P'mass   rearrange
(Pphot + Pmass - P'phot)2 = (P'mass)2  square to get scalars
(Pphot·Pphot + 2 Pphot·Pmass - 2 Pphot·P'phoot + Pmass·Pmass - 2 Pmass·P'phoot + P'phot·P'phot) = (P'mass)2
(0 + 2 Pphot·Pmass - 2 Pphot·P'phot + (moc)2 - 2 Pmass·P'phot + 0) = ((moc)2)2
Pphot·Pmass - Pmass·P'phot = Pphot·P'phot
(hbarωmo)-(hbarω'mo) = (hbar2ωω'/c2)(1-cos[ø])
(ω-ω')/(ωω') = (hbar/moc2)(1-cos[ø])
(1/ω'-1/ω) = (hbar/moc2)(1-cos[ø])
(1/(2pi v')-1/(2pi v)) = (hbar/moc2)(1-cos[ø])
(1/(v')-1/(v)) = (h/moc2)(1-cos[ø])
(λ'/c-λ/c) = (h/moc2)(1-cos[ø])
(λ'-λ) = (h/moc)(1-cos[ø])
(λ'-λ) = (h/moc)(2sin2[ø/2])  Compton scattering with Compton wavelength (h/moc)


Relativistic Doppler Effect
A = (c at, a), a generic SR 4-vector under observation, relative to observer
A·U = a Lorentz invariant, upon which all observers agree
take A·U --> A·Uo =  (c at, a)·(c,0) = c2at = the value of the temporal component of A as seen by observer U
now, let there be an observer Uobs at rest and an emitter Uemit moving with respect to Uobs
Uobs = (c,0): observer at rest
Uemit = γ(c,v): velocity of emitter relative to observer
*NOTE: This could be v already past obs and pointing away from obs, or v not yet reached obs and pointing toward obs.*
A·Uobs =  (c at, a)·(c,0) = c2at = c2atobs
A·Uemit =  (c at, a)·γ(c,v) = γ(c2at - a·v)= c2atemit
A·Uobs / A·Uemit = c2atobs / c2atemit = atobs / atemit = atobs / atemit = c2at / γ(c2at - a·v) = 1 / γ(1 - a·v/atc2) = 1 / γ[1 - (|a|/atc)*(n·v/c)]
atobs / atemit = 1 / γ(1 - (|a|/atc)*(n·v/c)) = 1 / γ[1 - (|a|/atc)*(v Cos[θ]/c)], 
atobs = atemit / γ(1 - (|a|/atc)*(n·v/c)) = atemit / γ[1 - (|a|/atc)*(v Cos[θ]/c)]

if A is photonic, then  (|a|/atc) = 1, then atobs = atemit / γ(1 - (n·v/c)) = atemit / γ[1 - (v Cos[θ]/c)] = atemit / γ[1 - (β Cos[θ])]
if θ = 0, then atobs = atemit / γ[1 - (v Cos[0]/c)] = atemit / γ[1 - (v/c)] = atemit / γ[1 - (v/c)] = atemit √[(1 + v/c)/(1 - v/c)] = atemit √[(1+β)/(1-β)]
thus, atobs = atemit √[(1 + v/c)/(1 - v/c)] for direct line
Now, experiment decides whether the v is towards or away from observer.
Experiments show blueshift when v towards obs., and redshift when v away from obs.
so, atobs = atemit √[(1 + v/c)/(1 - v/c)] for v toward obs.

Note that at could be any temporal component, ie. (E/c) for 4-Momentum P, (ω/c) for 4-Wavevector K, (ρc) for 4-CurrentDensity J, etc.

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) = (To dSo - Po dVo + µo dNo) = 0 ??
U·d(P) = γ(dE-u·dp) = (TodSo - PodVo + µodNo) = const = ? 0 ?

U·d(P) = γ(dE-u·dp) = γ(T dS - P dV + Sum[µi dNi] + w·dL + E·dP + B·dM) ???
E = Energy, [Total energy of system]
u
= Velocity, p = Momentum, [Translational/Kinetic energy]
T = Temperature, S = Entropy [Heat energy]
P = Pressure, V = Volume [Mechanical compression energy?]
µ = Chemical Potential, N = Particle Number, ["Chemical" energy = energy per particle]  (Sum over different particle types)
w
= Angular Velocity, L = Angular Momentum, [Rotational energy]
E = Electric Field, P = Polarization, [Electrical energy]
B = Magnetic Field, M = Magnetization, [Magnetic energy]
Always have (intensive var * differential extensive var), intensive = sys size independent, extensive = sys size proportional

U = γ(c, u), P = (E/c,p), U·U = c2 , P·P = (moc)2
U·P = γ(c E/c-u·p) = γ(E-u·p) = γ(T S - P V + µ N) = (To So - Po Vo + µo No)  ?
U·P = γ(E-u·p) = (To So - Po Vo + µo No) = moc2 ? for a spatially homogeneous system: relativistic Gibbs-Duhem eqn.

Invariants P = Pressure = Po N = ParticleNum = No S = Entropy = So
Variables V = Volume = (1/γ)Volo µ = ChemPoten = (1/γ)µo T = Temperature = (1/γ)Tempo

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

Time t = γ to
Length L = Lo

Heat Q = Qo
dQo = TodSo
InertialMassDen(of radiation field)  q = P/vV = γ qo

Total Particle Number N = No is an invariant, because the NumberDensity n varies as n = γ no, but this is balanced by Volume V = Vo
NumberDenstiy n = γ no   where NumberFlux 4-Vector N = (cn,nf) = no γ(c, u) = noU, no = No/(Δ_xo*Δ_yo*Δ_zo)
N = n * V = (γ no)*(Vo/γ) = no* Vo = No
N·N = (noc)2

Total Entropy S = So is an invariant, because the EntropyDensity s varies as s = γ so, but this is balanced by Volume V = Vo
EntropyDensity s = γ so    where EntropyFlux 4-Vector S = (cs,sf) = so γ(c, u) = soU, so = So/(Δ_xo*Δ_yo*Δ_zo)
S = s * V = (γ so)*(Vo/γ) = so* Vo = So
S·S = (soc)2

Relativistic Analytical-Mech

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

Relativistic Quantum Mechanics

· = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂2/c2∂t2-del·del = -(moc / hbar)2: Klein-Gordon Relativistic Wave eqn.
DEM = (∂/c∂t + iq/hbar VEM/c, -del + iq/hbar aEM)
      = + (iq/hbar)AEM

DEM·DEM = -(moc / hbar)2: Klein-Gordon Relativistic Wave eqn. in electromagnetic potentials
( + (iq/hbar)AEM)·( + (iq/hbar)AEM) = -(moc / hbar)2: Klein-Gordon Relativistic Wave eqn. w/ electromagnetic potentials
(·) + (iq/hbar)(·AEM + AEM·) + (iq/hbar)2(AEM·AEM) = -(moc / hbar)2: Klein-Gordon Relativistic Wave eqn. w/ electromagnetic potentials
if (·AEM + AEM·) = 0
then (·) + (iq/hbar)2(AEM·AEM) = -(moc / hbar)2
(∂/c∂t,-del)·(∂/c∂t,-del) + (iq/hbar)2((VEM/c, aEM)·(VEM/c, aEM)) = -(moc / hbar)2
(∂2/c2∂t2-del·del) + (iq/hbar)2(VEM/c)2-(aEM·aEM) = -(moc / hbar)2
(∂2/c2∂t2+(iq/chbar)2(VEM2)-(del·del+(iq/hbar)2(aEM·aEM) = -(moc / hbar)2
if ·AEM & AEM· = 0
then (∂2/c2∂t2+(iq/chbar)2(VEM2)-(  del·del+(iq/hbar)2(aEM·aEM) = -(moc / hbar)2
...not finished...


Newtonian Approximations

E2 = p·p c2 + mo2c4: Relativistic Energy of a particle
E2 = mo2c4 + p·p c2
E2 = mo2c4 * [1 + p·p / mo2c2]
E = ± moc2 * √[ 1 + p·p / mo2c2 ]
E ~ ± moc2 * [ 1 + p·p / 2mo2c2 + ...] for p2<<mo2c2
E ~ ± [moc2 + p·p / 2mo + ...]
choosing the positive root and discarding higher order terms...
E ~ Eo +|p|2 / 2mo
Total Energy = Rest Energy + Newtonian Momentum term


· = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂2/c2∂t2-del·del = -(moc / hbar)2: Klein-Gordon Relativistic Wave eqn.
2/c2∂t2 = del·del-(moc / hbar)2
2/c2∂t2 = (imoc / hbar)2+del·del
(i hbar)22/c2∂t2 = (i hbar)2(imoc / hbar)2+(i hbar)2del·del
(i hbar)22/c2∂t2 = (moc)2+(i hbar)2del·del
(i hbar)22/∂t2 = (moc2)2*[1 + (i hbar/moc)2del·del]
(i hbar)∂/∂t = ± (moc2)*√[1 + (i hbar/moc)2del·del]
(i hbar)∂/∂t ~ ± (moc2)*[1 + (1/2)*(i hbar/moc)2del·del + ...] for (hbar)2*del·del<<(moc)2 ,generally a very good approx. for non-relativistic systems
(i hbar)∂/∂t ~ ± [(moc2) + (i2 hbar2/2mo)del·del + ...]
choosing the positive root and discarding higher order terms...
(i hbar)∂/∂t ~ (moc2) - (hbar2/2mo)|del|2
(i hbar)∂/∂t ~ V(t,r) - ( hbar2/2mo)|del|2  becomes the time dependent Schrödinger eqn. by equating rest energy with the potential energy of the particle

Interesting Relations

(K = mo/hbarU = ωo/c2 U) gives (c2/vphase 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 (vphase * u = c2). In the case of photons, the phase velocity = particle velocity = c.  In the case of matter particles, the phase velocity vphase = 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 vphase 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 vphase is infinite, which makes sense since they all appear to flash simultaneously.

(·)AEM = µ0 J+(·AEM) Inhomogeneous Maxwell Equation
(·)AEM = µ0 J  Homogeneous Maxwell/Lorentz Equation  (if ·AEM = 0 Lorenz Gauge)
·J = dp/∂t +del·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

P·Uobs = E/c γ[uobs]c-p·γ[uobs] uobs = γ[uobs](E-p·uobs)
P·Uobs[uobs = 0] = E (RestFrame Invariant expression for energy)
K·Uobs = w/c γ[uobs]c-k·γ[uobs] uobs = γ[uobs](w-k·uobs)
K·Uobs[uobs = 0] = w (RestFrame Invariant expression for angular frequency)
R·Uobs = ct γ[uobs]c-r·γ[uobs] uobs = γ[uobs](c2t-p·uobs)
R·Uobs[uobs = 0]/c2 = t (RestFrame Invariant expression for time)
J
·Uobs = cp γ[uobs]c-j·γ[uobs] uobs = γ[uobs](pc2-j·uobs)
J·Uobs[uobs = 0]/c2 = p (RestFrame Invariant expression for ElecChargeDensity)

Fuv = ∂uAv-∂vAu  Electromagnetic Field Tensor (F0i = -Ei,Fij = eijkBk)
L = -1/4 Fuv Fuv - Ju Au : Lagrangian Density for EM field

L = -moco/γ -V: Relativistic Lagrangian function of a Particle in a Conservative Potential
VEM = q U·AEM/γ: Potential of EM field
LEM = -moc2/γ - q U·AEM/γ = - (P·P/m0 + q0U·AEM)/γ = - (m0U·U + q0U·AEM)/γ

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

Larmor formula can be written in Lorentz invariant form
P = -( q2/ 6πε0c3)(A·A)
= ( q2/ 6πε0c3) γ6/ (u'2 - (u x u')2/c2)

Relativistic Power radiated by moving charge by Abraham-Lorentz-Dirac force
P = (μoq2a2γ6)/(6πc)  


Liénard-Wiechert potentials - potential due to a moving charge
Aμ(x) = (q/c4πε0) Uμ / ( Rν Uν ) where Rν is a null vector (Rν Rν = 0)

AEM
= (q/c4πε0) U / (R·U) where (R·R = 0, the definition of a light signal)
= (q/c4πε0) U / ( cγ ( |r|-r·u/c ) )
= (q/c24π ε0)(c,u)/( |r|-r·u/c )
and therefore
φEM = (q / 4 π ε0 ) 1/[ r - r·u/c]
aEM = (µ0 q / 4 π) [u]/[ r - r·u/c]
where terms in square brackets [] indicate retarded quantities
(R·U) = (ct,r)·γ(c,u) = γ(c2t - r·u) = cγ(ct - r·u/c)
tret = t - |x-x'|/c: (retarded time)
ru = 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-moc2 = (γ[u]-1) moc2 = (γ-1) moc2 Relativistic Kinetic Energy: 
F·dX/dt = dT/dt: Also holds in Relativistic Mechanics
F·U =  (moA+(dmo/dτ)UU = c2(dmo/dτ) = γc2(dmo/dt)




Things to look-up


It can be shown that an scalar (s) and vector (v) which are related through a continuity equation in all frames of reference (∂s/∂t + del·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

Also, the diffusion equation can be derived from the continuity equation, which states that a change in density in any part of a system is due to inflow/outflow of material into/out-of that part of the system.  Essentially, no material is created/destroyed.  ·J = dp/∂t +del·j = 0

If j is the flux of diffusing material, then the diffusion equation is obtained by combining continuity with the assumption that the flux of diffusing material in any part of the system is proportional to the local density gradient.  j = - D(p) del  p.  see Fick's law of diffusion


Not every vector field has a scalar potential; those which do are called conservative, corresponding to the notion of conservative force in physics. Among velocity fields, any lamellar field has a scalar potential, whereas a solenoidal field only has a scalar potential in the special case when it is a Laplacian field.

In vector calculus a conservative vector field is a vector field which is the gradient of a scalar potential. There are two closely related concepts: path independence and irrotational vector fields. Every conservative vector field has zero curl (and is thus irrotational), and every conservative vector field has the path independence property. In fact, these three properties are equivalent in many 'real-world' applications.

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:

   del·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 = del x A

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

     del·v = del·(del x A) = 0

The converse also holds: for any solenoidal v there exists a vector potential A such that v = del 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:

      del x v = 0
      del
·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 = del φ      (1)

Then, since the divergence of v is also zero, it follows from equation (1) that

     del·del φ = 0

which is equivalent to

     del2 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,

     del 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 = del 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.

see Cosmological Physics
Relativistic Euler Equations:
dv/dt = - 1/[γ2(ρ + p/c2)](del p + p'v/c2): Conservation of Momentum 
d/dt[γ2(ρ + p/c2)] = p'/c2 - γ2(ρ + p/c2)del·v: Conservation of Energy
where p' = ∂ p/∂ t
·J = 0 where J = noU (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](del p): Conservation of Momentum 
d/dt[(ρ)] = - (ρ)del·v: Conservation of Mass
p = Pressure
ΔE = - p ΔV
E = ρ c2 V
ΔV / V = - Δρoo


Relativisitic Bernoulli's eqn.
γ w / ρo = const



Examples of Covariant equations

Equation of motion for a free particle
d2R/dτ2 = A = 0

Klein-Gordon Relativistic Wave eqn. = Wave equation for a scalar field (no spins)
(·)φ + (moc / hbar)2φ = 0

Maxwell Equations: 
(·)AEM = µ0 J+(·AEM) Inhomogeneous Maxwell Equation
(·)AEM = µ0 J  Homogeneous Maxwell/Lorentz Equation  (if ·AEM = 0 Lorenz Gauge)

where AEM = µ0 Integral[  [ J ]/r dV] ??? is a solution to the Maxwell Equation

 Lorentz Force - Covariant eqn. of motion for a particle in an EM field:
dPμ / dτ = (q/c) Fμν dXν / dτ




Quantum Commutation & SR Uncertainty Relations

[ ∂u , Rv ] = ∂u Rv - Rv u = gμν   quantum commutator with pure 4-gradient      [since ( Rv u = 0) generally???]
but K = i d
[ Ku , Rv ] = i gμν   quantum commutator with 4-wave vector
but P = hbar K
[ Pu , Rv ] = i hbar gμν   quantum commutator with 4-momentum

[ Ru , Pv ] = Ru Pv - Pv Ru = (- i hbar gμν)   SR quantum commutator with 4-momentum

this gives
[ x , px ] = [ y , py ] = [ z , pz ] = (i hbar)
[ ct , E/c ] = [ t , E ] = (- i hbar) :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 px > = hbar / 2) and (Δ t * Δ E > = hbar / 2)
or more generally
(Δ Ru * Δ Pv > = hbar duv / 2)
or
(Δ Ru * Δ Kv > = duv / 2)
(Δ x * Δ kx > = 1/2) and (Δ t * Δ w > = 1/2)

[ Ru , Rv ] = Ru Rv - Rv Ru = 0 : All position coordinates commute
[ Pu , Pv ] = Pu Pv - Pv Pu = 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 gμν 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 "events" 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?
--------
need to derive:
(Δ phix * Δ Lx > = hbar / 2)
where phix is angle about x, and Lx is angular momentum about x

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


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 = Λμν Xν + Aμ

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)

Λμν (a Lorentz Transform - maps spacetime onto itself and therefore preserves the inner product)
Λμν Λμλ =  gνλ (the Minkowski Metric)
Aμ = (Space-time Translation)

Unitary Operators representing these transforms:
U(A,1) = Exp[ i P·A ]
U(0,Λ) = Exp[ i Mμν Λμν ]

Poincare group has 10 generators (spacetime 4-generators)
Pμ    (4 generators of space-time translation = Conservation of 4-Momentum)
Mμν (6 generators of Lorentz group = 3 orbital angular momenta + 3 Lorentz boosts)

[ Pμ, Pν ] = 0  (Energy/Momentum commutes with itself)

[ Mμν, Pσ ] = - i ( Pμ gνσ - Pν gμσ )
or
[ Mμν, Pσ ] = i ( gνσ Pμ - gμσ Pν )    {one of these has a sign error I think}

[ Mμν, Mρσ ] = -i ( Mνσ gμρ - Mμσ gνρ + Mρν gμσ - Mρμ gνσ )

Then, define the spatial 3-generators:
"Spatial Rotation" generators Ji = -(1/2) εijk Mjk (for i=1,2,3), are Hermetian, (Mjk) = Mjk
"Lorentz Boost" generators Ki = Mi0 (for i=1,2,3), are anti-Hermetian, (Mi0) = - Mi0

[ Ji , Pk ] = i εikl Pl
[ Ji , P0 ] = 0  (Spin commutes Energy)
[ Ki , Pk ] = i P0 gik
[ Ki , P0 ] = - i Pi
[ Jm , Jn ] = i εmnk Jk
[ Jm , Kn ] = i εmnk Kk
[ Km , Kn ] = - i εmnk Jk

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 = ημν Pμ Pν = Pμ Pμ 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 = ημν Wμ Wν = Wμ Wμ 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 Wμ as the Pauli-Lubanski (mixed) Spin-Momentum four vector
With dimensional units this would be W2 = - m2c2hbar2 s(s+1)

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


Wσ = (1/2) εσμνρ Mμν Pρ
or
Wσ = - (1/2) εμνρσ Mμν Pρ
such that 

[ Wσ , Pμ ] = 0
[ Mμν , Wσ ] = -i ( Wμ gνσ - Wν gμσ )
[ Wλ , Wσ ] = i ελσαβ Wα Pβ

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

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
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 Wμ

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> = at(k)|0> is characterized by the eigenvalues
p0|k> = hbarω(k)|k>, p|k> = hbark|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) Aμ vector field
(1,0)  (+)  (0,1) Fuv = ∂uAv-∂vAu   EM field tensor


Consider an arbitrary spacetime vector xμ
Construct the 2 x 2 Hermitian matrix X = X

X = xμσμ ( x0 +   x3 x1 - i x2 )
( x1 + i x2 x0 -   x3 )

then Det[X] = x2 = x·x = ημν xμ xν  




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


Spin Statistics Eqn. Mass=0 Eqn. Mass<>0 Representation Polarizations
Mass=0
Mass<>0
0 Boson:
Bose-Einstein
FreeWave

N-G bosons
Klein-Gordon

Higgs bosons
scalar = 0-tensor 1?
1
1/2 Fermion
Fermi-Dirac
Weyl

Matter Neutrinos
Dirac

Matter Leptons/Quarks
spinor 2
2
1 Boson
Bose-Einstein
Maxwell

Force/Gauge Fields Photons/Gluons
Proca

Force
vector = 1-tensor 2 (= 2 transverse)
3 (= 2 transverse + 1 longitutinal)
3/2 Fermion
Fermi-Dirac
Gravitino?  Rarita-Schwinger spinor-vector 2
4
2 Boson
Bose-Einstein
Einstein
Graviton
tensor = 2-tensor 2
5



Dim Type   Hodge Dual
0 scalar    
1 vector    
2 tensor    
3 pseudovector magnietic 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:

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

[edit] Schrödinger equation for a free particle

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 , 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/kB]
flux current ΦV [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.
Please, send comments/corrections to John

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