The FourVectors (4Vectors) and Lorentz Invariants 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 and quantum effects, with the only exception being that of largescale gravitational phenomena, where spacetime itself is significantly curved, for which General Relativity (GR) is required. The SR 4vector 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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A vector is a mathematical object which has both magnitude and direction.
It is a powerful tool for describing physical phenomena. A common 3vector
is the velocity vector (v_{x}, v_{y}, v_{z}),
which tells you in what direction and how fast something is moving.
One might use the (x, y, z) coordinates to write down the velocity vector
of some object in the laboratory. That would be an example of a
rectilinear coordinate system. Another person might use a coordinate
system that is rotated wrt. the first observer, with components (y', x',
z). The same vector might also be described by the (r, θ, φ) spherical
coordinate system. Within a given coordinate system, each component is
typically orthogonal to each other component. While these different
coordinate systems will usually have different numbers in the vector
3tuple, 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 pointofview of the given vector.
The extension of 3vectors to that of 4vectors is a simple idea. Let's
imagine some event in spacetime. The location of the event in the
Newtonian world would be it's 3position (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 4position (ct,x,y,z). All that we have done is to
insert the time (t) into the vector as another component. The factor of
(c) is put with it to make the dimensional units work out right. (
[m/s]*[s] = [m]). So, each component now has overall units of [m] for this
4vector. 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 SRSpecial 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.
4vectors are tensorial entities which display Poincare' Invariance,
meaning they leave invariant the differential squared interval (ds)^{2}
= (cdt)^{2}dx^{2}dy^{2}dz^{2}. 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 4vectors 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 4vector itself is an invariant, meaning that it
gives valid physical information for all inertial observers. Likewise, the
scalar products of Lorentz Invariant 4vectors are themselves invariant
quantities, known as Lorentz Scalars. Lorentz Invariance is a
subset of the more general Poincare' Invariance.
The reason that I really like 4vectors and their notation is that they beautifully and
elegantly display the relations between lots of different physical
properties. They also devolve very nicely into the limiting/approximate
Newtonian cases of {v<<c} by letting {γ >1 and dγ/dt >0}. SR
tells us that several different physical properties are actually dual
aspects of the same thing, with the only real difference being one's point
of view, or reference frame. Examples include: (Time , Space), (Energy ,
Momentum), (Power , Force), (Frequency , WaveNumber), (ChargeDensity ,
CurrentDensity), (EMScalarPotential , EMVectorPotential), (Time Differential,
Spatial Gradient), etc. Also, things are even more related than that. The
4Momentum is just a constant times 4Velocity. The 4WaveVector is just a
constant times 4Momentum. 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.
QM = Quantum Mechanics SR = Special Relativity
SM = Statistical Mechanics GR = General Relativity
length/time 
[m] meter <*> [s] second 
Count of the quantity of separation or distance; 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) 
Velocity
v_{group} or v or u: v
= cβ = c^{2}/v_{p} = group velocity = event velocity= [0..c], {u is historically used in SR
notation}
v_{phase} or v_{p}: v_{p} = c/β = c^{2}/v = phase velocity = celerity
= [c..Infinity]
Minkowski Flat (PseudoEuclidian) Spacetime Metric:
η_{μν} = η^{μν} = SR g^{μν} = Diag[+1,1,1,1]
Dimensionless SR Factors:
β = (v/c) = (v_{group}/c) = (c/v_{phase})=
[0..1]: Relativistic Beta factor, the fraction of the speed of light c
β = (u/c): Vector form of Beta factor, u is the velocity
γ[u] = dt/dτ: Lorentz Gamma Scaling Factor (Relativistic Gamma factor)
γ = (1 / √[1(v/c)^{2}] ) = (1 / √[1(u·u/c^{2})]
): Lorentz Gamma Scaling Factor (~1 for v<<c), (>>1 for v~c)
γ = (1 / √[1β^{2}] ) = (1 / √[1β·β] ): Lorentz Gamma
Scaling Factor (~1 for β<<1), (>>1 for β~1)
γ = (1 / √[1β^{2}] ) = 1/√[(1+β)/(1β)] : Useful for Doppler Shift Eqns
φ = Ln[γ(1+ β)] ~ Atanh[β]: BoostParameter/Rapidity (which remains
additive in SR, unlike v)
e^{φ} = γ(1+β) = √[(1+β)/(1β)]
β = Tanh[φ], γ = Cosh[φ], γβ = Sinh[φ], φ = Rapidity (which remains
strictly additive in SR, unlike v)
D = 1 / [γ(1  β Cos[θ] )] = 1 / [γ(1  β·n )]: Relativistic
Doppler Factor (sometimes called a relativistic beaming factor)
D_{+} = γ(1 + β Cos[θ] ): Forward jet Doppler shift
D_{} = γ(1  β Cos[θ] ): Counterjet Doppler shift
Temporal Factors:
τ = t / γ : Proper Time = Rest Time (time as measured in a frame at rest)
dτ = dt / γ : Differential of Proper Time
d/dτ = γ d/dt = U·∂ : Differential wrt Proper Time
Useful SR Formulas:
V·V = Vo·Vo : Invariant interval is often easier to
calculate in rest frame coordinates
√[1+x] ~ (1+x/2) for x ~ 0 : Math relation often used to simplify
Relativistic eqns. to Newtonian eqns.
1/√[1+x] ~ (1x/2) for x ~ 0 : Math relation sometimes used to simplify
Relativistic eqns. to Newtonian eqns.
δ^{uv} = Delta function = (1 if u = v, 0 if u ≠ v)
γ = (1 / √[1(u·u/c^{2})]) = c/√[c^{2}v^{2}]
= c/√[c^{2}u·u]
γ^{2} = c^{2}/(c^{2}v^{2}) = c^{2}/(c^{2}u·u)
= 1/(1 β^{2})
c^{2}/γ^{2} = (c^{2}v^{2})
v γ = c √[γ^{2}1]
β γ = √[γ^{2}1]
(1β^{2})γ^{2} = 1
(1β^{2}) = 1/γ^{2}
β^{2}γ^{2} = γ^{2}1
β^{2}γ^{2} +1 = γ^{2}
β^{2}γ = (γ1/γ)
c^{2} dγ = γ^{3} v dv
d(γ v) = c^{2} dγ / v = γ^{3} dv
dγ = γ^{3} v dv / c^{2} = γ^{3} β dβ
dβ = dv / c
dγ/dv = γ^{3} v / c^{2}
d(γ^{1})/dv =  γ v / c^{2}
γ' = dγ/dt = (γ^{3} v dv/dt)/c^{2} = (γ^{3} u·a)/c^{2}
= (u·a_{r})/c^{2}
γ'' = dγ'/dt = d^{2}γ/dt^{2} = (γ^{3}/c^{2})*[(3γ^{2}/c^{2})(u·a)^{2}
+ (u'·a) + (u·a')]
u^{2} = u^{2}
u·u' = uu' = ua
(u x a)^{2} + (u·a)^{2} = u^{2}a^{2}
sin^{2} + cos^{2}
= 1
(∇·∇)[1/r]
= Δ[1/r] = 4πδ^{3}(r)
(∇·∇)[1/rr']
= Δ[1/rr'] = 4πδ^{3}(rr') Green's function for Poisson's Eqn
**NOTE**
All results below use the timepositive SR Minkowski Metric η_{μν} =
Diag[+1,1,1,1].
If you wish to do GR, with other metrics g_{μν}, then some results
below may need GR modification, such as the GR √[g] for whichever metric
you are using...
You have been warned.
There are several different SR notations available that are,
mathematically speaking, equivalent.
However, some are easier to employ than others. I have used that one which
seems the most practical and least errorprone.
If you mix notations, you will get errors! Always check notation
conventions in SR & 4Vector references, they are all relative ;)
Minkowski SR Metric (time 0component positive), for which η_{μν}
= η^{μν} = SR g_{μν} = SR g^{μν} = Diag[+1,1] =
Diag[+1,1,1,1]
Signature[η^{μν}] = 2
Generic 4Vector Definition:
A = A^{u}
= (a^{0},a) = (a^{0},a^{1},a^{2},a^{3}) time (a^{0}) in the 0th coord. (
some alternate notations use time as a^{4} )
Specific coordinate system representations:
A = A^{u}
= (a^{0},a) = (a^{0},a^{1},a^{2},a^{3}) => (a^{t},a^{x},a^{y},a^{z}) {for rectangular/Cartesian coords}
A = A^{u}
= (a^{0},a) = (a^{0},a^{1},a^{2},a^{3}) => (a^{t},a^{r},a^{θ},a^{z}) {for cylindrical coords}
A = A^{u}
= (a^{0},a) = (a^{0},a^{1},a^{2},a^{3}) => (a^{t},a^{r},a^{θ},a^{φ}) {for spherical coords}
Note that the superscripted variables are not exponents, they are upper tensor indices
Intervals:
TimeLike/Temporal (+ interval) = 0 coordinate ( some alternate notations
use time as  interval and space as the + interval)
LightLike/Null (0 interval)
SpaceLike/Spatial ( interval) = 1,2,3 coordinates
Temporal Components: Future(+), Now(0), Past()
4Vector Name: always references the "Spatial" 3vector component
(basically trying to extend the Newtonian 3vector to SR 4vector)
4Vector Magnitude: usually references the "temporal" scalar
component (because many vectors in the rest frame only have a temporal
component)
4Vector Tensor Indices: I use the convention of [Greek symbols dim{0..3} = time+space], [Latin symbols dim{1..3} = only space]
4Vector Symbols: A = A^{μ} = (a^{0},a)
= (a^{0},a^{i}) = (a^{0},a^{1}+a^{2}+a^{3}) = (a^{0},a^{1},a^{2},a^{3}),
where the raised index indicates dimension, not exponent
4Vector Definition: A = A^{μ} , always references the upper tensor index unless otherwise noted
4Vector cFactor: almost always applied to "Temporal" scalar component, as
necessary to give consistent dimensional units for all vector components
(a^{0},a^{1},a^{2},a^{3}) <==>
(ct,x,y,z) = (ct,x)
*Note* cFactor can be on the top, as ( ct , x , y , z ) = [m], or on
bottom, as ( E/c , p_{x} , p_{y} , p_{z} ) = [kg m
s^{1}]
*Note* P = (E/c, p) = (mc, p);
the 4Momentum is a good case showing top or bottom, with E = mc^{2}
4Vector Computer HTML Representation:
SR 4vector = {BOLD UPPERCASE} = A
time scalar component = {regular lowercase} = a^{0}
space 3vector component = {bold lowercase} = a
Contraction & Dilation Relativistic Component: v > v_{o}
in a restframe, typically v = γ v_{o} (dilation) or v = (1/γ) v_{o}
(contraction)
eg.
t = γ t_{o} (time dilation)  pertains to temporal separation
between two events
L = (1/γ) L_{o} (length contraction)  pertains to the spatial
separation between two parallel world lines
This is somewhat confusing due
Generally, timelike quantities get dilated, spacelike quantities get
contracted by motion
Also, I typically denote "atrest" invariant quantities with a "naught",
or "_{o}", i.e.:
L_{o} (invariant rest length = proper length), relativistic length
L = (1/γ) L_{o}
V_{o} (invariant rest volume), relativistic volume V = (1/γ) V_{o}
m_{o} (invariant rest mass), relativistic mass m = γ m_{o}
E_{o} (invariant rest energy), relativistic energy E = γ E_{o}
ω_{o} (invariant rest angfrequency), relativistic angfrequency ω
= γ ω_{o}
ρ_{o} (invariant rest chargedensity), relativistic chargedensity
ρ = γ ρ_{o}
n_{o} (invariant rest numberdensity), relativistic numberdensity
n = γ n_{o
}t_{o} (invariant rest time = proper time), relativistic time
t = γ t_{o} = γ τ
etc.
This avoids the confusion of some texts which use just "m" as invariant
mass, or just "ρ" as invariant chargedensity.
It also helps to avoid confusion such as:
If the mass m of an object increases with velocity, wouldn't it
have be a black hole in some reference frames (near c), since the mass
increases with velocity.
Answer  no. The rest mass m_{o} does not
change. The relativistic mass is simply an "apparent" mass, how the
object is velocityrelated to an observer, not how much "stuff" is in
it...
The apparent increase is fully due to the gamma factor( γ ), which is
simply a reflection of relative motion.
Imaginary unit: ( i ) used only for QM phenomena, not for SR frame
transformations or metric. To follow up on a quote from MTW " ict was put to
the sword ".
This allows all the purely SR stuff to use only real numbers.
Imaginary/complex stuff apparently only enters the scene via QM.
( some alternate notations use the imaginary unit ( i ) in the
components/frame transformations/metric )
So, in summary, this notation allows:
easy recovery of Newtonian cases by allowing (γ>1, dγ>0) when
(v<<c)
easy separation of SR vs Newtonian concepts, with the Newtonian 3vector (a)
extending naturally into the SR 4vector (A)
easy separation of SR vs QM concepts, no ict's  ( i ) only enters into
QM concepts, such as Photon Polarization, Quantum Probability Current,
etc.
easy separation of relativistic quantities vs. invariant quantities, E = γ
E_{o}
reduction in number of minus signs (), eg. U·U = c^{2}, P·P
= (m_{o}c)^{2}: the square magnitudes of velocity,
momentum, wavevector, and other velocitybased vectors are positive
The main assumption of SR, or GR for that matter, is that the structure of spacetime is described by a metric g_{μν}. A metric tells how the spacetime is put together, or how distances are measured within the spacetime. These distances are known as intervals. In GR, the metric may take a number of different values, depending on various circumstances which determine its curvature. We are interested in the flat/pseudoEuclidean spacetime of SR, also known as the Minkowski Metric, for which g_{μν} => η_{μν} = η^{μν} = Diag[+1,1,1,1].
"Flat" SpaceTime 




t 
1 
0 
0 
0 
x 
0 
1 
0 
0 
y 
0 
0 
1 
0 
z 
0 
0 
0 
1 
η_{μα} η^{μβ} = δ_{α}^{β} = (4
if α = β for Minkowski)
g =  Det[g_{μν}] = 1 (for Minkowski) not a scalar invariant
Sqrt[g]ρ: Scalar density
There are other ways of defining the metrics and 4vectors 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 4vectors. The choice of +1 for the time component simplifies
the derived equations later on, as it allows restframe squaremagnitudes
to be positive for most quantities of interest.
η_{μν} = SR g_{μν} = SR g^{μν} =
DiagonalMatrix[1,1,1,1]: Minkowski Spacetime Metricthe "flat"
spacetime of SR
A = A^{μ} = (a^{0},a^{i}) = (a^{0},a^{1},a^{2},a^{3}) => (a^{t},a^{x},a^{y},a^{z})
:
Typical
SR 4vector (using all upper indices)
A_{μ} = (a_{0},a_{i}) = (a_{0},a_{1},a_{2},a_{3}) => (a_{t},a_{x},a_{y},a_{z}) :
Typical SR 4covector (using all lower indices)
We can always get the alternate form by applying the Minkowski Metric Tensor: A^{μ}
= η^{μν}A_{ν} and A_{μ} = η_{μν} A^{ν}
Basically, this has the effect of putting a
minus sign on the space component(s)
A = A^{μ} = (a^{0},a^{i}) = (a^{0},a^{1},a^{2},a^{3}) = (a^{0}, a^{1}, a^{2}, a^{3}) = (a^{0},a)
:Typical
SR 4vector (all upper indices)
A_{μ} = (a_{0},a_{i}) = (a_{0},a_{1},a_{2},a_{3}) = (a^{0},a^{1},a^{2},a^{3}) = (a^{0},a):
Typical SR 4covector (converted using Metric Tensor)
It is occasionally convenient to choose a particular basis to simplify component calculations
Typical bases are rectangular, cylindrical, spherical
A = A^{μ} = (a^{0},a^{i}) = (a^{0},a^{1},a^{2},a^{3}) = (a^{0},a) = (a^{0},a^{1},a^{2},a^{3}) = (a^{0},a) => (a^{t},a^{x},a^{y},a^{z})
:Typical SR 4vector (choosing the rectangular basis)
A_{μ} = (a_{0},a_{i}) = (a_{0},a_{1},a_{2},a_{3}) = (a_{0},a) = (a^{0},a^{1},a^{2},a^{3}) = (a^{0},a) => (a_{t},a_{x},a_{y},a_{z}) = (a^{t},a^{x},a^{y},a^{z})
:
Typical SR 4covector (choosing the rectangular basis)
B = B^{μ} = (b^{0},b^{1},b^{2},b^{3}) = (b^{0},b) = (b^{t},b^{x},b^{y},b^{z})
:
Another
typical SR 4vector
A·B = η_{μν} A^{μ} B^{ν} = A_{ν} B^{ν}
= A^{μ} B_{μ} = +a^{0}b^{0}a^{1}b^{1}a^{2}b^{2}a^{3}b^{3}
= (+a^{0}b^{0}a·b): The Scalar Product or Invariant Product relation,
used to make SR invariants
c(A + B) = (cA + cB) scalar multiplication
A·A = A^{2} = (+a_{0}^{2}  a_{1}^{2}
 a_{2}^{2}  a_{3}^{2}) = (+a_{0}^{2}
 a·a) magnitude squared, which can be {  , 0 , + }
A = A = √A^{2} >= 0 absolute magnitude or
length, which can be { 0 , + }
A·B = B·A commutative, with the exception of the (∂)
operator, since it only acts to the right
A·(B + C) = A·B + A·C distributive
d(A·B) = d(A)·B + A·d(B)
differentiation
B = d(A)/dθ, where θ is a Lorentz Scalar Invariant
A_{proj} = (A·B)/(B·B) B =
Projection of A along B
A_{} = (A·B)/(B·B) B =
Component of A parallel to B
A_{⊥} = A  A_{}
A_{⊥} =A  (A·B)/(B·B) B =
Component of A perpendicular to B
Let A = (a^{0},a) be a general 4Vector and T = U/c = γ(c, u)/c = γ(1, β) =
(γ, γβ) be the unittemporal 4Vector
Then...
(A·T) = (a^{0},a)·γ(1, β) = γ(a^{0}a·β) = 1(a^{0}_{o}a·0) = a^{0}_{o}
which is a Lorentz Invariant way to get (a^{0}_{o}), the rest temporal component of A
Let A = (a^{0},a) be a general 4Vector and S = γ_{βn}(β·n, n) be the unitspatial 4Vector
Then...
(A·S) = (a^{0},a)·γ_{βn}(β·n, n) = γ_{βn}(a^{0}β·n, a·n) = 1(a^{0}a·0  a·n) = a·n
which is a Lorentz Invariant way to get (a·n), the rest spatial component of A along the n direction
If A is a timelike vector, then you can also do the following:
(A·T)^{2}  (A·A) = (a^{0}_{o})^{2}  (a^{0}a^{0}a·a) = (a^{0}_{o})^{2}  (a^{0})^{2} + (a·a) = (a·a)
Sqrt[(A·T)^{2}  (A·A)] = Sqrt[(a·a)] = a
which is the Lorentz Invariant way to get the magnitude a, the spatial component magnitude of a timelike A
Also:
(A·A) = (a^{0})^{2}  (a·a) = invariant = (a^{0}_{o})^{2} ,where (a^{0}_{o}) is the Lorentz Scalar Invariant "temporal rest value" for those vectors that can be at rest, and just the invariant for others
(a^{0})^{2} = (a^{0}_{o})^{2} + (a·a)
Consider the following identity:
γ = 1/Sqrt[1β^{2}] : γ^{2} = 1 + γ^{2}β^{2}
Multiply by the square of a Lorentz Scalar Invariant (a^{0}_{o}):
γ^{2}(a^{0}_{o})^{2} = (a^{0}_{o})^{2} + γ^{2}(a^{0}_{o})^{2}β^{2}
Compare Terms:
(a^{0})^{2} = (a^{0}_{o})^{2} + (a·a)
γ^{2}(a^{0}_{o})^{2} = (a^{0}_{o})^{2} + γ^{2}(a^{0}_{o})^{2}β^{2}
Notice the following correspondences:
a^{0} = γa^{0}_{o}, The temporal component is the Gamma Factor times the rest value
a = γa^{0}_{o}β = a^{0}β, The spatial component is temporal component times the Beta Factor
Try it with the 4Momentum P = (E/c,p)
(E/c) = γ(E_{o}/c) or E = γE_{o}
p = γ(E_{o}/c)β = γ(E_{o}/c)(v/c) = γ(E_{o}v/c^{2}) = γ(m_{o}v) = γm_{o}v
p = (E/c)β = (mc)β = (mv) = γm_{o}v
If β=1 then p = E/c or E = pc, which is correct for photons
It also shows that as β > 1: γ > Infinity, (a^{0}_{o}) > 0, a^{0} = (γa^{0}_{o}) > some finite value
Special Relativity is interesting in that it is one area of physics where {Infinity*Zero = Finite Value} for certain variables.
eg. E = γE_{o}
For photons, the rest energy E_{o} = 0, the gamma factor γ = Infinity, but the overall energy of photon E = finite value for a given observer.
One can do this with any SR rest value variable. Always pair γ={1..Infinity} with a_{o}={large..0} to get a finite value a = γa_{o}={something finite}
These correspondences can also be generated by letting A = LorentzScalar (a^{0}_{o}) * TemporalUnit 4Vector T
A = (a^{0}_{o})T = (a^{0}_{o})γ(1,β) = γ(a^{0}_{o})(1,β) = (γa^{0}_{o})(1,β) = (a^{0})(1,β) = (a^{0},a^{0}β) = (a^{0},a)
If β>0 , then A > (a^{0},a^{0}β) > (a^{0}_{o},0), which has (A·A) = (a^{0}_{o})^{2} as expected
If β>1 , then A > (a^{0},a^{0}n) > a^{0}(1,n), which has (A·A) = (a^{0})^{2} (1^{2}  n·n) = 0 as expected
==========
if A·A = const
then
dA^{1}dA^{2}dA^{3} / A_{0}
dA^{0}dA^{2}dA^{3} / A_{1}
dA^{0}dA^{1}dA^{3} / A_{2}
dA^{0}dA^{1}dA^{2} / A_{3}
are all scalar invariants
from Jacobian derivation
============
if A_{μ} dX^{μ} = invariant for any dX^{μ}, then A_{μ}
is a 4vector
η_{μν} Λ^{μ}_{α} Λ^{ν}_{β} = η_{αβ} This is basically the reason why the Scalar Product relation gives invariants
η_{μν} (A'^{μ})(B'^{ν}) = η_{μν} (Λ^{μ}_{α}A^{α})(Λ^{ν}_{β}B^{β}) = η_{μν} (Λ^{μ}_{α}Λ^{ν}_{β})(A^{α})(B^{β}) = η_{αβ} (A^{α})(B^{β}) = η_{αβ} (A^{α}B^{β})
Thus, A'·B' = A·B, where the primed 4vectors are just Lorentz Transformed versions of the unprimed ones
Λ^{α}_{μ} Λ^{μ}_{β} = d^{α}_{β}
A'^{μ} = Λ^{μ}_{ν} A^{ν}: Lorentz
Transform (Transformation tensor which gives relations between alternate
boosted inertial reference frames)
Λ^{μ}'_{ν} = (∂X^{μ}'/∂X^{ν})
Λ^{μ}_{ν} = (for xboost)
γ 
(v_{x}/c)γ 
0 
0 
(v_{x}/c)γ 
γ 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
or
γ 
β_{x}γ 
0 
0 
β_{x}γ 
γ 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
General Lorentz Transformation
Λ^{μ}_{ν} = (for nboost)
γ 
β_{x}γ 
β_{y}γ 
β_{z}γ 
β_{x}γ 
1+(γ1)(β_{x}/β)^{2} 
( γ1)(β_{x}β_{y})/(β)^{2} 
( γ1)(β_{x}β_{z})/(β)^{2} 
β_{y}γ 
( γ1)(β_{y}β_{x})/(β)^{2} 
1+( γ1)(β_{y}/β)^{2} 
( γ1)(β_{y}β_{z})/(β)^{2} 
β_{z}γ 
( γ1)(β_{z}β_{x})/(β)^{2} 
( γ1)(β_{z}β_{y})/(β)^{2} 
1+( γ1)(β_{z}/β)^{2} 
General Lorentz Boost Transform using just vectors & componentsThank
you Jackson, Master of Vectors! Chap. 11
β = v/c, β = β, γ = 1/√[1β^{2}]
a^{0}' = γ(a^{0}β·a)
a' = a+(β·a)β(γ1)/β^{2}γ β a^{0}
a^{0}' = γ(a^{0}β·a) Temporal component
a^{}' = γ(a^{}βa^{0}) Spatial parallel
component
a^{⊥}' = a^{⊥} Spatial perpendicular
components
Contraction & Dilation Relativistic Component: v > v_{o}
in a restframe, typically v = γ v_{o} (dilation) or v = (1/γ) v_{o}
(contraction)
eg.
t = γ t_{o} (time dilation)  pertains to temporal separation
between two events
L = (1/γ) L_{o} (length contraction)  pertains to the spatial
separation between two parallel world lines
Since the transformations are symmetric in the temporal and spatial parts
of the 4vector, it is somewhat confusing how the gamma factor is
inversely related for times compared to lengths. The time dilation
compares the separation in proper time between events on the worldline of
a single particle. The length contraction is comparing separations
between differing events however. The length must be measured along
lines of simultaneity, and the events of the endpoints while simultaneous
n the rest frame, are not simultaneous in the moving frame.
We are also able to use the Rapidity
φ = Ln[γ(1+ β)] = Rapidity (which remains strictly additive in SR, unlike
v)
φ = aTanh[pc/E] = (1/2) Ln[(E+pc)/(Epc)]
e^{φ} = γ(1+β) = √[(1+β)/(1β)]
β = Tanh[φ], γ = Cosh[φ], γβ = Sinh[φ]
φ_{AC} = φ_{AB} + φ_{BC}
Rapidity of C wrt. A = Rapidity of B wrt. A + Rapidity of C wrt. B,
provided that A,B,C are colinear
i.e. Rapidity is strictly additive only for colinear points
Λ^{u}_{v} = (for xboost, y & z unchanged)
Cosh[φ] 
Sinh[φ] 
0 
0 
Sinh[φ] 
Cosh[φ] 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
R_{z} = (for xy rotation about zaxis, t & z unchanged)
1 
0 
0 
0 
0 
Cos[φ] 
Sin[φ] 
0 
0 
Sin[φ] 
Cos[φ] 
0 
0 
0 
0 
1 
Time t = γ t_{o} > Time Dilation (e.g. decay times of
unstable particles increase in a cyclotron)
Length L = L_{o}/γ > Length Contraction
A few 4vectors are known to have complex components. The Polarization
4vector and ProbabilityCurrent 4vector are a couple of these.
It will be assumed that all physical 4vectors may potentially be complex,
although, as far as I know, these only come into play via QM...
i = √[1] :Imaginary Unit
e_{0}: Unit vector in the temporal direction
(typically not used since the temporal unit is always considered a scalar)
e_{1}, e_{2}, e_{3}
: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 4vectors, the superscript is the tensor
index, not exponentiation.
A = (a^{0}_{c} + a^{1}_{c} e_{1}+
a^{2}_{c} e_{2}+ a^{3}_{c}
e_{3}): Complex 4vector has complex components, 1
along time and 3 along space
Scalar[A] = a^{0}_{c}: Just the time component
Vector[A] = a^{1}_{c} e_{1}
+ a^{2}_{c} e_{2} + a^{3}_{c}
e_{3}: Just the spatial components
A = Scalar[A] + Vector[A]
A = ( (a^{0}_{r} + a^{0}_{i}
) + (a^{1}_{r} + a^{1}_{i}
) e_{1} + (a^{2}_{r}
+ a^{2}_{i} ) e_{2} +
(a^{3}_{r} + a^{3}_{i} ) e_{3}
): Complex 4vector has real + imaginary
components, 1 each along time and 3 each along space
Re[A] = ( (a^{0}_{r}
) + (a^{1}_{r} ) e_{1}
+ (a^{2}_{r} ) e_{2}
+ (a^{3}_{r} ) e_{3}
): Only the real components
Im[A] = ( (a^{0}_{i}
) + (a^{1}_{i} ) e_{1}
+ (a^{2}_{i} ) e_{2}
+ (a^{3}_{i} ) e_{3}
): Only the imaginary components
A = Re[A] + i Im[A]
A = (a^{0}_{r} + i a^{0}_{i},a_{r}
+ i a_{i}) : Standard 4vector
A^{*} = (a^{0}_{r}  i a^{0}_{i},a_{r}
 i a_{i}): Complex conjugate 4vector, just changes the
sign of the imaginary component
A = (a^{0}_{r} + i a^{0}_{i},a_{r}
+ i a_{i}) : A^{*} = (a^{0}_{r}
 i a^{0}_{i},a_{r}  i a_{i})
B = (b^{0}_{r} + i b^{0}_{i},b_{r
}+ i b_{i}) : B^{*} = (b^{0}_{r}
 i b^{0}_{i},b_{r } i b_{i})
A·B = [( a^{0}_{r} b^{0}_{r}
 a_{r}·b_{r} )  ( a^{0}_{i}
b^{0}_{i}  a_{i}·b_{i} )] + i [( a^{0}_{r}
b^{0}_{i}  a_{r}·b_{i} ) +
( a^{0}_{i} b^{0}_{r}  a_{i}·b_{r}
)] : General scalar product
A·A = [( a^{0}_{r}^{2}
 a_{r}·a_{r} )  ( a^{0}_{i}^{2}
 a_{i}·a_{i} )]
+ 2i [( a^{0}_{r} a^{0}_{i}
 a_{r}·a_{i} )]
= A^{2} : Scalar product of 4vector with itself gives
the magnitude squared
A·A^{*} = [( a^{0}_{r}^{2}
+ a^{0}_{i}^{2} )  ( a_{r}·a_{r}
+ a_{i}·a_{i} )]
= Re[A·A^{*}]: Scalar product of 4vector with its complex
conjugate is Real, thus Im[A·A^{*}] = 0
∂·B = [( ∂/c∂t_{r} b^{0}_{r}
+ ∇_{r}·b_{r} )  ( ∂/c∂t_{i} b^{0}_{i}
+ ∇_{i}·b_{i} )]
+ i [( ∂/c∂t_{r} b^{0}_{i}
+ ∇_{r}·b_{i} ) + ( ∂/c∂t_{i} b^{0}_{r}
+ ∇_{i}·b_{r} )]
= [( ∂/c∂t_{r} b^{0}_{r}
+ ∇_{r}·b_{r} )  ( ∂/c∂t_{i} b^{0}_{i}
+ ∇_{i}·b_{i} )]
= Re[∂·B]
The 4Divergence of a Complex 4Vector 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.
i = √[1] :Imaginary Unit
π = 3.14159265358979... :Circular Const
c = Speed of Light Const = 1/√[ε_{o}μ_{o}] ~ 2.99729x10^{8}
[m/s]
h = Planck's Constant  relates particle to wave  Action constant
ћ = (h/2π) = Planck's Reduced Const , aka. Dirac's Const  same idea as
transforming between cycles and radians for angles
In essence, the reduced Planck constant is a conversion factor between
phase (in radians) and action (in jouleseconds)
k_{B} = Boltzmann's Const ~ 1.3806504(24)×10^{−23} [J/ ºK]
relates temperature to energy
m_{o} = Rest Mass Const (varies with particle type)
q = Electric Charge Const (varies with particle type)
Note:
I do not set various fundamental physical constants
to dimensionless unity, (i.e. c = h = G = k_{B} = 1).
While doing so may make the mathematics/geometry a bit easier, it
ultimately obscures the physics.
While pure 4Vectors may be Math, SR 4Vectors is Physics. I prefer to
keep the dimensional units.
Also, it is much easier to set them to unity in a final formula than to
figure out where they go later if you need them.
4Vector Name 
4Vector Components 
Units (mksC)  Description 






4Displacement 
ΔR = (cΔt, Δr) 
[m], Δt = Temporal Displacement, Δr
or Δx = Spatial Displacement, (Finite Differences) 
4Differential 
dR = (cdt, dr) 
[m], dt = Temporal Differential, dr or dx = Spatial Differential, (Infinitesimals) 
4Gradient 
The tensor gradient is technically defined as 
[m^{1}], ∂ is the partial
derivative, ∇ => (∂/∂x i + ∂/∂y j +
∂/∂z k) is the gradient operator 
4MomentumGradient 
∂_{P} = ∂/∂P_{μ} = (c∂_{E}, ∇_{p}) 
[kg^{1} m^{1} s], ∂_{E} is the
momentumspace partial derivative, ∇_{p} = momentumspace gradient 
4WaveGradient 
∂_{K} = ∂/∂K_{μ} = (c∂_{ω}, ∇_{k}) 
[m], ∂_{ω} is the wavespace partial derivative, ∇_{k} = wavespace gradient (1/ћ)∂_{K}[P·U] 






4Position 
R = R^{μ} = (ct, r), eg. radial coords 
[m], t = Time (temporal), r or x
= 3Position (spatial) 
4Velocity 
U = dR/dτ 
[m s^{1}], γ = relativistic
factor, u_{r} = Relativistic
3Velocity, u = dr/dt = Newtonian 3Velocity 
4Acceleration 
A = dU/dτ 
[m s^{2}], a_{r}
= Relativistic 3Acceleration, a = du/dt =
Newtonian 3Acceleration a_{r} = (γu_{r})' = γ' u_{r} + γ u_{r}' = γ' u + γ a = (γ^{3}/c^{2})(u·a) u + γ a a = du/dt = u' γ' = dγ/dt = (γ^{3}/c^{2})(u·a) = (u·a_{r})/c^{2} Interesting note: The temporal component has units of frequency, before the c factor, and is given by γ(dγ/dt)=γ(γ') γ(c γ')=γ(u·a_{r})/c γ'=(u·a_{r})/c^{2} 4Spin also has a temporal component in this form, given by u·s/c I now wonder if all 4vectors which are tangent to the worldline possess this "cyclic" feature... 
4Jerk 
J = dA/dτ 
[m s^{3}], j_{r}
= Relativistic 3Jerk, j = da/dt = Newtonian
3Jerk 
4Snap 
S = dJ/dτ 
[m s^{4}], s_{r}
= Relativistic 3Snap, s = dj/dt = Newtonian
3Snap 






4Momentum 
P = m_{o}U = (E_{o}/c^{2})U
= ћK 
[kg m s^{1}], E = Energy, p_{r}
= Relativistic 3Momentum, p = mdr/dt = Newtonian
3Momentum 
4MomentumDensity 
G = (u/c, g) = (p_{m}c, g)
= p_{o_m} γ(c, u) 
[kg m^{2} s^{1}], u
= EnergyDen = ne, p_{m} = MassDen = u/c^{2} 
4Force 
F = dP/dτ 
[kg m s^{2}], dE/dt = Power, f_{r}
= Relativistic 3Force, f = Newtonian 3 Force 
4Force Density 
F_{d} = F/V_{o} or F/δV_{o} F_{d} = γ(du/cdt, f_{dr}) = dG/dτ?? 
[kg m^{2} s^{2}], 4Force
divided by rest volume element 






4WaveVector 
K = (ω_{o}/c^{2})U = (m_{o}/ћ)U = (1/ћ)P 
[rad m^{1}], ω = AngularFrequency
[rad/s], k = WaveNumber or WaveVector [rad/m] 
4Frequency 
Ν = (ν, c/λ n) 
[cyc s^{1}], ν = ω/2π, λ = 2π/k 
4CycWaveVector 
Kcyc = (ν/c,1/λ n) = (1/cT,1/λ n) 
[cyc m^{1}], ν = CyclicalFrequency
[cyc/s], λ = WaveLength [m/cyc] 









4NumberFlux 
N = (cn, n_{f}) = n_{o}
γ(c, u) = n(c, u) 
[(#) m^{2} s^{1}], n_{o}
= RestNumberDensity [#/m^{3}], n = γn_{o} =
NumberDensity [#/m^{3}] 
4VolumetricFlux 
V = V_{o}U?? 
[(m^{3}) m^{2} s^{1}], V_{o} = RestVolume 
4ElectricCurrentDensity 
J = (cρ, j) = ρ_{o} γ(c, u) = ρ(c,
u) 
[(C) m^{2} s^{1}], ρ_{o}
= RestElecChargeDensity [C/m^{3}], ρ = γρ_{o} =
ElecChargeDensity 
4MagneticCurrentDensity 
J_{mag} = (cρ_{mag}, j_{mag})
= ρ_{o_mag} γ(c, u) 
[(MagCharge) m^{2} s^{1}],
ρ_{o_mag} = RestMagChargeDensity, ρ_{mag} = γρ_{o_mag
= }MagChargeDensity 
4ChemicalFlux 

[(mol) m^{2} s^{1}], 
4MassFlux 
G = (u/c, g) = (cρ_{m}, g)
= ρ_{o_m} γ(c, u) = ρ_{o_m}U 
[(kg) m^{2} s^{1}], u
= EnergyDen = ne, p_{m} = MassDen = u/c^{2} 
4PoyntingVector 
S = (cu, s) = u_{o} γ(c, u)
= u_{o}U = c^{2}G 
[(J) m^{2} s^{1}], u
= EnergyDen = ne, s = EnergyFlux = PoyntingVector = uu
= c^{2}g = Ne 
4EntropyFlux 
S = (cs, s_{f}) 
[(J ºK^{1}) m^{2} s^{1}], s_{o} = RestEntropyDensity = q_{o}/T, s_{f}
= EntropyFlux 






4InverseTempFlux 
β = β_{o} U = (1/k_{B}T_{o})
U 
Considered on Thermodynamic
principles 
4MomentumTemperature 
P_{T} = P/k_{B} = (p_{t}^{0}/c,
p_{T}) = (T/c, p_{T})
= ((E/k_{B})/c, p/k_{B}) 
[ºK m^{1 }s], 






4Potential Flux?? 
V = (cq, q_{f}) = q γ(c, u)
= qU?? 
Potential Flow for Velocity?? In fluid dynamics, a potential flow is described by means of a velocity potential , φ being a function of space and time. The flow velocity v is a vector field equal to the negative gradient, ∇, of the velocity potential φ: Incompressible flowIn case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence: ∇·v = 0with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation ∇·∇ φ = 0where 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 flowMain article: Steady flow Incompressible flowMain article: Incompressible flow Irrotational flowMain article: Irrotational
flow VorticityMain article: Vorticity
The velocity potentialMain article: Potential flow 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. 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: automatically results in the identity (as can be shown, for
example, using Cartesian coordinates): The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ x A. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.) In vector calculus, a Laplacian vector field is a vector
field which is both irrotational and incompressible. If the field
is denoted as v, then it is described by the following
differential equations: 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 : Then, since the divergence of v is also zero, it follows from equation (1) that ∇·∇ φ = 0 which is equivalent to Therefore, the potential of a Laplacian field satisfies Laplace's
equation. 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 simplyconnected region and is irrotational. In
such a case, where u denotes the flow velocity of the fluid. As a
result, u can be represented as the gradient of a scalar
function 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 threedimensional flow. 
4HeatFlux 
Q = qU 
[(J) m^{2} s^{1}] = [(W)
m^{2}], 
4DarcyFlux 
Q = (cq, q_{f}) = q γ(c, u)
= qU = ( c (βφ)P , q_{f}) 
[(m^{3}) m^{2} s^{1}] = [m s^{1}], 
4ElectricChargeFlux 
Q = (cq, q_{f}) = q γ(c, u) = qU = ( c ρ, j) 
[(C) m^{2} s^{1}], 






4SpinMomentum 
W = (w_{0},w) = (u·w/c,w) 
[spinmomentum], 
4Spin 
S = (s^{0},s) = (u·s/c,s) 
[ J s], = [spin] Spin =
IntrinsicAngMomentum, u·s/c = component such that U·S
= 0 ===== 
4SpinDensity  
4Rotation 
Omega 

4Polarization 
Ε = (ε^{0}, ε) = (ε·u/c,ε)
for a massive particle 
[1], ε = PolarizationVector **This
4vector has complex components in QM** 
4SpinPolarization 
In the rest frame, where K = (m,0), choose a unit
3vector n as the quantization axis. 
I am suspecting that the s(s+1) value can be derived from the Laplacian acting on a pure radial function. From mathworld.wolfram.com, 
4PauliMatrix 

The components of this 4vector are
actually the Pauli Spin Matrices 
4DiracGamma  Γ = Γ^{μ} = (γ^{0},γ)  According to some books, not strictly a 4vector, the Dirac Gamma
Matrices are actually matrices to represent intrinsic spin.
However, I have seen index raising/lowering Γ_{μ} = η_{μν}Γ^{μ} Γ^{μ} Γ^{ν} = η^{μν} + (1/2)σ^{μν} σ^{μν} = [Γ^{μ},Γ^{ν}] Not sure about a Lorentz Boost 






4VectorPotential 
A = (φ/c, a) 
[kg m <chargetype>^{1} s^{1}], for arbitrary field 
4Potential ***Break with standard notation*** better to use the 4VectorPotential  Φ = (φ,c a)  Φ = (φ,c a) where Φ = cA ***this is bad notation based on our 4vector naming convention*** the cfactor should be in the temporal component the 4vector name should reference the spatial component I simply include it here because it can sometimes be found in the literature 

A_{grav} = (Φ_{grav}/c, a_{grav}) 
[kg m <chargetype>^{1} s^{1}], for arbitrary field 
4VectorPotentialMomentum 
Q_{EM} = (E_{EM}/c, p_{EM}) 
[kg m s^{1}], 
4Potential 
Φ_{EM} = (Φ_{EM},c a_{EM}) 
[kg m^{2} C^{1} s^{2}],
Φ_{EM} = ScalarPotenial_{EM} ,a_{EM}
= VectorPotenial_{EM} 
4Momentum_{EM} 
P_{T} = (E_{T}/c + qΦ_{EM}/c,
p_{T} + qa_{EM}) 
[kg m s^{1}], **Momentum including
effects of potentials** H = γm_{o}c2 + q φ_{EM} 
4Gradient_{EM} 
D_{EM} = (∂/c∂t + iq/ћ Φ_{EM}/c, ∇
+ iq/ћ a_{EM}) 
[m^{1}], **Gradient including
effects of EM potentials** 






4Differential 
dX = (cdt, dx) 
[m], dt = Temporal Differential, dx = Spatial Differential 
4Volume Element (Flux?) 
dV = (dv^{0},dv) 
[m^{3}], A vectorvalued volume
element is just a 4vector that is perpendicular to all spatial
vectors in the volume element, and has a magnitude that's
proportional to the volume. 
4Momentum Differential 
dP = (dE/c, dp) 
[kg m s^{1}], dE = Temporal Momentum Differential, dp = Spatial Momentum Differential 
4MomentumSpace 
dV_{p} = (dv_{p0},dv_{p}) 
[kg^{3} m^{3} s^{3}],
A
vectorvalued MomentumSpace volume element is just a 4vector
that is perpendicular to all spatial vectors in the
MomentumSpace volume element, and has a magnitude that's
proportional to the MomentumSpace volume. 






4Zero 
Zero = (0,0) = (0,0,0,0) 
[*], All components are 0 in all reference
frames, the only vector with this property 
4Null 
Null = (a,a) = (a,an) = a(1,n) 
[*], Any 4vector for which the temporal
component magnitude equals the spatial component magnitude 
4Unit Temporal 
T = U/c = γ(c, u)/c = γ(1, β) =
(γ, γβ) 
[1] = dimensionless, The Unit Temporal
4Vector 
4Unit Null 
N = (1,n) 
[1] = dimensionless, A Unit Null 4Vector 
4Unit Spatial 
S = γ[β_{n}] (n·β,n) = (γ[β_{n}]n·β,γ[β_{n}]
n) 
[1] = dimensionless, A Unit Spatial
4Vector 
4Basis Vectors 
B_{t} = (1,0,0,0) 
A tetrad of 4 mutually orthogonal,
unitlength, linearlyindependent, basis vectors 
4Basis Vectors 
B_{n1} = √[1/2] (1,0,0,1) 
A tetrad of complex, linearlyindependent,
null basis vectors 
4Basis Vectors 
B_{n1} = (1,1,0,0) 
A tetrad of real, linearlyindependent,
null basis vectors 
4ProbabilityCurrentDensity 
J_{prob} = (cρ_{prob}, j_{prob}) = (iћ/2m_{o})(ψ*<∂>ψ) 
[# m^{2} s^{1}],
4Probability Current Density is proportional to the 4Momentum 
Event Tracking Relations
Event R 
Mass m_{o} = ρ_{o_m}V_{o} 
WaveAngFreq ω_{o} 
ElecCharge q = ρ_{o}V_{o} 
MassDensity ρ_{o_m} 
ChargeDensity ρ_{o} 
NumberDensity n_{o} 
event 
particle 
wave 
elec. charge 
mass 
charge 
number 
pos: R = (ct, r) 
m_{o} at R 
ω_{o} at R 
q at R 
ρ_{o_m} at R 
ρ_{o} at R 
n_{o} at R 
vel: U = dR/dτ 
P = m_{o}U = (E_{o}/c^{2})U 
K = (ω_{o}/c^{2})U = (1/ћ)P 
J_{q} = qU 
G = ρ_{o_m}U = (u_{o}/c^{2})U 
J = ρ_{o}U 
N = n_{o}U 
accel: A = dU/dτ 
F = dP/dτ 


F_{d} = dG/dτ 


jerk: J = dA/dτ 






snap: S = dJ/dτ 
∂·R = (∂/c∂t,∇)·(ct,r) = (∂/c∂t[ct]+∇·r)
= (∂/∂t[t]+∇·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[∇])·(ct,r) + ∂·U = γ (d/dt[∂/c∂t][ct]+d/dt[∇])·r
+ ∂·U = γ (d/dt[∂/∂t][t]+d/dt[∇])·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 4Velocity U is
a conserved quantity.
For example, let N = n_{o}U, so ∂·N = ∂·n_{o}U
= n_{o}∂·U = n_{o}(0) = 0. The quantity n_{o}
is conserved.
Any "charge" constant becomes a 4vector when multiplied by the
4Velocity, and obeys the Conservation of Charge/Continuity equation
∂·J = ∂ρ/∂t +∇·j = 0 where J = ρ_{o}U
let Charge q = ρ_{o}V_{o}, where ρ_{o} is the
"rest charge density", ρ = γρ_{o} is the relativistic "charge
density", V_{o} is the rest volume, and j = γρ_{o}u
= ρu is the "ChargeDensityFlux or Current Density"
then ChargeFlux 4Vector = CurrentDensity 4Vector J = ρ_{o}U
= ρ_{o} γ(c, u) = ρ(c, u) = (cρ, j)
In the case of "electric" charge, ρ_{o} is the "rest
electriccharge density", and j is the ElectricChargedensityflux
= electric current density
In the case of "number" charge, ρ_{o} is the "rest numbercharge
density"
In the case of "mass" charge, ρ_{o} is the "rest mass density",
and j is the massflux = mass current density = momentum density
It can be shown that a scalar (s) and vector (v) which are related
through a continuity equation in all frames of reference (∂s/∂t + ∇·v
= 0) transform according to the Lorentz transformations and therefore
comprise the components of a 4vector V=(cs,v), where ∂ ·V
= 0. Relativistic fourvectors may be identified from the continuity
equations of physics. See A
Proposed Relativistic, Thermodynamic FourVector.
Poincaré transformation or an inhomogeneous Lorentz
transformation:
η_{μν} Λ^{μ}_{α} Λ ^{ν}_{β} = η_{αβ
}
Chain rule on the 4gradient:
Let g_{μ} = ∂_{μ} f = ∂f / ∂x^{μ}
Using the chain rule, one can show:
g'_{ν} = ∂f '/∂x'^{ν} = Σ ( ∂f / ∂x^{μ} )( ∂x^{μ}
/ ∂x'^{ν} ) = ∂'_{ν} f ' = ( ∂_{μ} f )( ∂'_{ν}
x^{μ} ) = (g_{μ})( ∂'_{ν} x^{μ} )
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} = Λ^{μ}_{ν} ∂^{ν}[function
argument] = ∂^{ν}[function argument] Λ^{μ}_{ν}
Let ∂'·J' = ∂ρ'/∂t +∇·j' = 0 be an arbitrary 4vector
continuity equation.
∂'·J' = η_{μν} ∂'^{μ} J'^{ν} = η_{μν}
Λ^{μ}_{α} ∂^{α} J'^{ν} = η_{μν} Λ^{μ}_{α}
∂^{α} Λ^{ν}_{β} J^{β} = η_{μν} Λ^{μ}_{α}
Λ^{ν}_{β} ∂^{α} J^{β} = η_{αβ }∂^{α}
J^{β} = ∂·J
Assuming that the 4gradient acts only on the 4vector 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 +∇·v = 0.
If this equation holds in all inertial reference frames, then s and v
must be components of a 4vector (cs, v).
see http://www.grc.nasa.gov/WWW/K12/Numbers/Math/Mathematical_Thinking/proposed_relativistic.htm
Event(SR) 
EventMovement 
MassEnergy 
ParticleWaveDuality 
QuantumMechanics(QM) 
SpaceTimeVariations 
R = (ct, r) 
dR/dτ = U = γ(c,u) 
U = P/m_{o} 
P = ћ K 
*** K = i ∂ *** 
∂ = (∂/c∂t,∇) 



or K = (ω_{o}/c^{2})U 


d/dτ[R] = (i ћ / m_{o}) ∂ Event motion ~
spacetime structure  depends on i ћ / m_{o}
So, the following assumptions within SRSpecial Relativity lead to
QMQuantum 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 event position wrt. Proper Time 
P = m_{o}U 
Momentum is just the Rest Mass of the particle/event times its velocity 
K = (1 / ћ )P 
A particle's wave vector is just the momentum divided by Dirac's constant, but uncertain by a phase factor 
∂ = i K 
The change in spacetime corresponds to (i) times the wave vector, whatever that means... 
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 = c^{2}
P·P = (m_{o}c)^{2}
K·K = (m_{o}c / ћ)^{2} = (ω_{o}/c)^{2}
∂·∂ = (∂/c∂t,∇)·(∂/c∂t,∇) = ∂^{2}/c^{2}∂t^{2}∇·∇
= (m_{o}c / ћ)^{2} : KleinGordon Relativistic Wave Eqn.
Each relation may seem simple, but there is a lot of complexity generated
by each level.
*see QM from SR
(Quantum Mechanics derived from Special Relativity)*
This can be further explored:
∂·∂ + (m_{o}c / ћ)^{2} = 0
(∂·∂ + (m_{o}c/ћ)^{2} ) Ψ = 0, 
Ψ is a scalar KleinGordon eqn for massive spin0 field 
(∂·∂ + (m_{o}c/ћ)^{2} ) A = 0 
A is a 4vector Proca eqn for massive spin1 field 
(∂·∂) Ψ = 0 
Ψ is a scalar Freewave eqn for massless (m_{o} = 0) spin0 field 
(∂·∂) A = 0 
A is a 4vector Maxwell eqn for massless (m_{o} = 0) spin1 field, no current sources 
Interesting Note about Proca eqn.
"Massive charged vector field  represent with complex fourvector field φ^{μ}(X)
and impose "Lorenz condition" (∂_{μ}φ^{μ}) = 0 so that φ^{0}(X)
the scalar polarization , can be discarded and the KleinGordon
equations emerge for the other three components φ^{i}(X)
(∂·∂ + (m_{o}c / ћ)^{2} ) A = 0, where A
is a 4vector Proca eqn for massive spin1 field
rewrite in index notation
(∂_{μ}∂^{μ} + (m_{o}c / ћ)^{2} ) A^{ν}
= 0 and combine with the Lorenz gauge condition (∂_{μ}A^{μ}
= 0)
apparently, this conjunction is equivalent to
∂_{μ}( ∂^{μ} A^{ν}  ∂^{ν} A^{μ}
)+ (m_{o}c / ћ)^{2} A^{ν} = 0
which is the EulerLagrange equation for the Proca Action
see Conceptual Foundations of Modern Particle Physics, ~ pg. 100
Momentum/Gradient Relations(Correspondences)
P = i ћ ∂ = ∂(S_{act}) 
∂ = (∂/c∂t,∇) 
A_{EM} = (0,0) *special case* 
P_{EM} = P+qA_{EM} = i ћ D_{EM} 
D_{EM} = ∂+iq/ћ A_{EM} 
A_{EM} = (V_{EM}/c,a_{EM}) 

Relations involving the 4Position or 4Displacment: 
R·R = (Δs)^{2} =
(ct)^{2}r·r = (ct)^{2}r^{2}

Spacetime position of an event wrt. an origin event 
dR·dR = (ds)^{2} 
Differential interval magnitude  the fundamental invariant
differential form 
ΔR·ΔR = (Δs)^{2} = (c Δt)^{2}Δr·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 = Φ = (ωtk·r) 
Phase of a SR wave; Ψ = a E e^{ iK·R} Photon Wave Equation (Solution to Maxwell Equation) 


R·U = (ct,r)·γ(c,u) = γ(c^{2}t  r·u) = cγ(ct  r·u/c) 
Part of expression used in LiénardWiechert potential 
ΔX·U =
(cΔt,Δx)·γ(c,u) = γ(c^{2}Δt  Δx·u) = (c^{2}Δt_{o}) = (c^{2}Δτ) Let T = U/c = γ(c, u)/c = γ(1, β) = (γ, γβ) be the unittemporal 4Vector Then... (ΔX·T) = (cΔt,Δx)·γ(1, β) = γ(cΔtΔx·β) = (cΔt_{o}) = (cΔτ) 
ΔX·U = 0 is an interesting 4vector condition/definition for
simultaneity (The displacement of an external event is normal to either event's worldline) ΔX is a displacement vector from Event A to Event B U is an observer's 4Velocity wrt. one of the events The standard definition of simultaneity is when Δt = 0. This gives ΔX·U = γ(Δx·u). But since we can always choose an observer rest frame we get ΔX·U = 0, which is thus the Lorentz Invariant condition/definition for simultaneity. However, since Δt is one of the components of a 4vector, it is only true for certain classes of observers. Let's examine the cases when ΔX·U = γ(c^{2}Δt  Δx·u) = 0 then (c^{2}Δt  Δx·u) = 0, since γ is always >=1 then (c^{2}Δt = Δx·u) For simultaneity, Δt = 0, therefore Δx·u = 0 if Δx = 0, then Event A and Event B are colocal, ie. at the same spatial point if u = 0, then the observer is at rest wrt. the Events A and B if Δx·u = 0, but Δx > 0 and u > 0, then the observer's motion must be perpendicular to a spatial line from Event A to Event B 



Relations involving the 4Velocity: 
U·U = c^{2} 
The magnitude of 4Velocity is always c^{2} 
U_{1}·U_{2}
= γ[u_{1}]γ[u_{2}](c^{2}u_{1}·u_{2})
= γ[u_{rel}]c^{2} 
Relative Gamma Factor 
A_{1}·U_{1} = 0, where A is dU/dτ 
The 4Acceleration of a given particle is always normal to its own worldline 
P_{1}·U_{2} = γ[u_{2}](Ep_{1}·u_{2})
= E_{rel} 
Relative Energy 
K_{1}·U_{2} = γ[u_{2}](ωk_{1}·u_{2})
= ω_{rel} 
Relative Ang. Frequency 
F·U = (m_{o}A+(dm_{o}/dτ)U)·U
= c^{2}(dm_{o}/dτ) = γc^{2}(dm_{o}/dt) 
Power Law 
U·∂ = γ(∂/∂t + u·∇) = γ d/dt = d/dτ 
Relativistic Convective (Time) Derivative, Intrinsic Derivative 
∂·U = 0 (always??) 
The General Continuity Equation, one might say the conservation
of event flux. 



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



Others: 
P·P = (m_{o}c)^{2} = (E_{o}/c)^{2} 
Square Magnitude of the 4Momentum 
P_{1}·P_{2} = γ[u_{1}]γ[u_{2}]m_{o1}
m_{o2}(c^{2}u_{1}·u_{2}) 
Relativistic Billiards... 
Γ·P= (m_{o}c) ( Σ·P )·( Σ·P ) = (Ps)·(Ps) = (m_{o}c)^{2} ( Σ_{1}·P_{1} )·( Σ_{2}·P_{2} ) = (Ps_{1})·(Ps_{2}) = γ[u_{r12}](m_{o1})(m_{o2})c^{2} ?? 
Momentum representation of Dirac Equation, the 4DiracGamma Γ with
4Momentum P (Γ·P)Ψ= (m_{o}c)Ψ (Γ^{μ}P_{μ})Ψ= (m_{o}c)Ψ or, in operator form iћ(Γ^{μ}∂_{μ})Ψ= (m_{o}c)Ψ (γ^{0}p^{0}  γ·p )Ψ= (m_{o}c)Ψ One can get the equivalent result using the Pauli Spin Matrix Tensor Σ as well 4Momentum (inc. spin) Ps = Σ·P 
N·N = (n_{o}c)^{2} 
Square Magnitude of the 4NumberCurrentDensity 
J·J = (p_{o}c)^{2} = (qn_{o}c)^{2} 
Square Magnitude of the 4ElectricCurrentDensity 
K·K = (m_{o}c / ћ)^{2} = (ω_{o}/c)^{2} 
Square Magnitude of the 4WaveVector 
∂·∂ = (∂/c∂t,∇)·(∂/c∂t,∇) = ∂^{2}/c^{2}∂t^{2}∇·∇ = (m_{o}c / ћ)^{2} 
KleinGordon Relativistic Wave Eqn. 
∂·J = ∂ρ/∂t +∇·j = 0 
Continuity Equation  Conservation of Electric Charge 
E·K = 0 
The Polarization of a photon is orthogonal to direction of wave
motion (E·K = 0) (cancellation of "scalar" polarization) 
A_{EM}·A_{EM} = (V_{EM}/c,a_{EM})·(V_{EM}/c,a_{EM}) = (V_{EM}/c)^{2}a_{EM}·a_{EM} = ???? 
Square Magnitude of the Electromagnetic field 
P_{T}·R = S = (∂S/c∂t,∇[S])·R  Action S 
There is an important distinction between an invariant
quantity and a conserved quantity.
An invariant quantity has the same value wrt. all inertial systems, but
may possibly change upon physical interaction for a system of multiple particles
(e.g. a fission/fusion reaction
"redistributes" the rest masses).
A conserved quantity maintains the same value both before and after an
interaction, although the component values may appear different in
different frames.
In 4vector notation:
An invariant quantity is a Lorentz Scalar, the dot product of two
4Vectors, A·B = invariant = same value for all inertial
observers.
A conserved quantity is a component of a 4Vector that has 4Divergence =
0, ∂·V = 0.
Relativistic Invariant Quantities (Lorentz Scalars~A·B),
although perhaps there might be some that are simply just scalars...
also known as Relativistic Covariance = Relativistic
Invariance = Lorentz Invariance
Lorentz Scalars = World Scalars = Invariant Scalars = Lorentz Invariants
Any quantity involving counting of particles or states  Entropy, Number of particles in a volume, Number of microstates/macrostate, etc.  
c = √[U·U] 
Speed of Light: c (in vacuum) E ~ cp 

h = √[P·P/L·L] = P·L /
L·L 
Planck's const: h, E ~ hν 

k_{B} = √[P·P/P_{T}·P_{T}] = P·P_{T} / P_{T}·P_{T??} 
Boltzmann's const: k_{B}, E ~ k_{B}T 




γ_{rel} = V·U / U·U = V·U / V·V 
Relative Relativistic Gamma Factor, between 4velocities U and V 




Δs = √[ΔR·ΔR] = √[c^{2}Δt^{2}  Δx^{2}  Δy^{2}  Δz^{2}] 
Displacement 

Δσ = √[ΔR·ΔR] = √[(c^{2}Δt^{2}  Δx^{2}  Δy^{2}  Δz^{2})] 
Proper Distance, for SpaceLike Intervals 

ds = √[dR·dR] = √[c^{2}dt^{2}
 dx^{2}  dy^{2}  dz^{2}] 
Differential Length of World Line
Element, the Spacetime Interval ds 

dτ = √[dR·dR/U·U] 
Differential Proper Time, aka. the Eigentime differential 

d/dτ = U·∂ = γ(∂/∂t + u·∇) = γ d/dt 
Derivative wrt Proper Time d/dτ 

Δ = ∂·∂ = (∂/c∂t,∇)·(∂/c∂t,∇) 
D'Alembertian/wave operator 

d^{4}x 
Invariant 4volume 

d^{4}p = dP·dV_{p} 
Spacetime momentumspace differential "4volume" element 

d^{4}k = dK·dV_{k} 
Spacetime wavevecspace differential "4volume" element?? 

d^{3}xd^{3}p = dV_{phase} = dμ[t] = dV·dV_{p} 
Invariant phase volume 

δ^{4}(xy) 
4D Dirac Delta Function 

G(xx') 
Green's Function where
∂·∂ G(xx') = δ^{4}(xx') 

ε_{o} 
The Electric Constant ε_{o}  
μ_{o} = (∂·∂)A /  J  
The Magnetic Constant μ_{o}  
f(t,x,p) 
One Particle Distribution Function 

N[t] = f(t,x,p)
dμ[t] = ∫N·dV 
Number of particles in a volume element  
d^{3}p/(2E) 
Invariant phasespace element 

d^{3}p/(2E) = p^{2}dpdΩ/(2E)  Spacetime
phasespace differential 3volume element, Cartesian vs. spherical
basis dΩ =sin(θ) dθ dφ = Solid Angle Element in direction of Ω 

(2E)δ^{3}(pp'), (2ω)δ^{3}(kk')  Spacetime phasespace
differential 3volume element, Dirac form need to get exact units, etc. corrected δ^{4}(pp') => Lorentz Invariant δ( E  E_{o} )δ^{3}(pp') δ( E  E_{o} )δ^{3}(pp') / δ( F[E] ) Divide by another Lorentz invariant δ( E  E_{o} )δ^{3}(pp') / δ( E  E_{o} )/2E δ( E  E_{o} ) 2E δ^{3}(pp') / δ( E  E_{o} ) 2E δ^{3}(pp') 

d^{3}p/E E d^{3}x d^{3}xd^{3}p I_{ν}/ν^{3} 
d^{3}x = c dt
dA_{⊥} : The photons contained in a cylinder of base area
A traveling a distance dx = c dt d^{3}p = (h/c)^{3} ν^{2} dν d[cos(θ)] dφ : Spherical coords I_{ν} d ν d[ cos( θ)] dφ dA_{⊥} dt = Energy carried by photons in range (ν, ν+dν) 2 ν^{2}/c^{2} dν d[ cos( θ)] d φ dA_{⊥} dt = 2 d^{3}pd^{3}x / h^{3} (c^{2} I_{ν} )/( 2 ν^{2} )= Energy per mode n = (c^{2} I_{ν} )/( 2 h ν^{3} ) = Number of photons per quantum state, itself an invariant thus I_{ν} /ν^{3} is invariant 

u[ε,Ω]/ε^{3}  Specific spectral energy density over dimensionless energy cubed  
I_{ε}[Ω]/ε^{3}  Intensity over dimensionless energy cubed  
j[ε,Ω]/ε^{2}  Emissivity over dimensionless energy squared  
I_{ν}/ν^{3}  Spectral
Intensity / ν^{3} Lorentz Invariant related to Relativistic Beaming or Doppler Beaming 

m_{o} = √[P·P/U·U] = P·U/U·U 
RestMass of a Particle m_{o} ( 0 for photons, + for massive ) 

ρ_{mo} = m_{o}n_{o}

ProperMassDensity ρ_{mo} of
a continuum in the comoving frame of n_{o} 

q = √[J·J/N·N] = J·N/N·N 
RestElectricCharge of a Particle q 

3vector 
Spin s_{o} 

magnetic moment 





E_{o} = P·U = m_{o}c^{2} 
RestEnergy of a Particle ( 0 for photons, + for massive ) 

ω_{o} = K·U = m_{o}c^{2}/ћ 
RestAngFrequency of a Particle ( 0 for photons, + for massive ) 




Φ_{T} =  K_{T}·R 
*** Phase of a wave ***, 

S = S_{action} =  P_{T}·R
= ∫[dt L;t_{i},t_{f}] 
Action Variable S of Action
Integral 

γL =  (P + qA)·U 
Relativistic Lagrangian L * gamma
(γ) 

L 
Lagrangian Density 

L_{1 }H_{1}  Extended
Lagrangian/Hamiltonian formalism L_{1} = (+/?) γL L_{1} + H_{1} = P_{T}·U H_{1} seems to end up being identically 0 in all frames, H_{1} = H  E = 0, so I guess it is a Lorentz scalar as well I'm not totally sure of the sign for L_{1}, it seems to differ in the various papers I have read on this In any case, we have the following: L + H = p_{T}·u {noncovariant, but true relativistically/Newtonian, for the conventional Lagrangian/Hamiltonian} L_{1} + H_{1} = P_{T}·U {covariant generally true, for the extended Lagrangian/Hamiltonian} L = T  V, only in Newtonian 

H / γ = P_{t}·U  Relativistic
Hamiltonian H / gamma (γ) Start with Lagrangian L[q_{i},q_{i}'], a function of coords q_{i} and their time derivatives q_{i}' Conjugate momenta p_{i} = ∂L/∂q_{i}' Then Hamiltonian H = Σ [p_{i}q_{i}';i]  L Then, Eqns. of Motion p' = ∂H/∂q q' = ∂H/∂p ex. Lagrangian of a free particle L = m_{o}c^{2}/γ p_{i} = ∂L/∂q_{i}' = γm_{o}u_{i} (or p = γm_{o}u) H = Σ [p_{i}q_{i}';i]  L = p·u  L = γm_{o}u·u + m_{o}c^{2}/γ = γm_{o}c^{2} where γ^{2} = c^{2}/(c^{2}v^{2}) = c^{2}/(c^{2}u·u) p' = ∂H/∂q = 0, since H = γm_{o}c^{2} has no explicit dependence on q q' = ∂H/∂p = u So, p' ~ a = 0 ie. no acceleration q' = u as we expect Hamiltonian for a free particle: H = γm_{o}c^{2} = E ; H/γ = m_{o}c^{2} = E_{o}  ex. Lagrangian of a charged particle in EM field
L = (P + Q_{EM})·U/γ L = (P·U + Q_{EM}·U)/γ L = P·U/γ  Q_{EM}·U/γ L = m_{o}U·U/γ  qA_{EM}·U/γ L = m_{o}c^{2}/γ  qA_{EM}·U/γ L = m_{o}c^{2}/γ  q(Φ_{EM}/c, a_{EM})·γ(c, u)/γ L = m_{o}c^{2}/γ  q(Φ_{EM}/c_{,} a_{EM})·(c, u) L = m_{o}c^{2}/γ  q(Φ_{EM}  a_{EM}·u) L = m_{o}c^{2}/γ  qΦ_{EM} + qa_{EM}·u L = m_{o}c^{2}/γ  qΦ_{oEM}/γ L = (m_{o}c^{2} + qΦ_{oEM})/γ p_{canonical} = p_{total} = ∂L/∂q' = (γm_{o}u) + (qa_{EM}) = (p_{dynamical}) + (qa_{EM}) = (p_{kinetic}) + (p_{potential}) Hence, γm_{o}u = p_{total}  qa_{EM} Equation of motion: (leading to negative gradient of potential) dp/dt = ∂L/∂x =  q(∂Φ_{EM}/∂x  ∂a_{EM}/∂x·u)
H = p_{T}·u  L H = γm_{o}u·u + qa_{EM}·u  L H = γm_{o}u·u + qa_{EM}·u + m_{o}c^{2}/γ + q(Φ_{EM}  a_{EM}·u) H = γm_{o}u·u + m_{o}c^{2}/γ + q(Φ_{EM}) H = p·u + m_{o}c^{2}/γ + q(Φ_{EM}) H = γm_{o}c^{2} + qΦ_{EM} H = E + V = (rest+kinetic) + (potential) H = m_{o}c^{2} + (γ1)m_{o}c^{2} + qΦ_{EM} H = (rest) + (kinetic) + (potential) also, since E=√[p·p c^{2} + m_{o}^{2}c^{4}] H = √[p_{T}^{2}c^{2} + m_{o}^{2}c^{4}] + qΦ_{EM } H = √[(p_{kinetic}  qa_{EM})^{2} c^{2} + m_{o}^{2}c^{4}] + qΦ_{EM }q' = ∂H/∂p = (p  qa_{EM}) / √[(p  qa_{EM})^{2} /c^{2} + m_{o}^{2}] p' = ∂H/∂q = q(∇ aEM) ·u  q∇Φ_{EM } this leads to the Lorentz force (here E and B are the classical electric and magnetic fields, not 4vectors): p_{T}' = f = q(E + v x B) representing the rate at which the EM field adds relativistic momentum to a charged particle dp/dτ = γq(E + v x B) The nonrelativistic Lagrangian L is an approximation of the relativistic one: L = (m_{o}c^{2} + qΦ_{oEM})/γ L = (m_{o}c^{2} + qΦ_{oEM})/γ = √[1(v/c)^{2}](m_{o}c^{2} + qΦ_{oEM}) ~ (m_{o}c^{2} + qΦ_{oEM})  (1/2)(m_{o}c^{2}v^{2}/c^{2} + qΦ_{oEM}v^{2}/c^{2}) ~ (m_{o}c^{2} + qΦ_{oEM})  (1/2)(m_{o}v^{2} + 0) L ~ (1/2)(m_{o}v^{2})  (m_{o}c^{2} + qΦ_{oEM}) L ~ (Kinetic)  (Rest+Potential) = T  V (for v << c) The large constant coming from the restmass is simply ignored in classical mechanics. The gamma factor in the Lagrangian corresponds to the time dilation of an object moving at v. In QM words: the number of phase changes (ticks) over the trajectory of the particle (the t' axis) is less by a factor gamma. The nonrelativistic Hamiltonian H is an approximation of the relativistic one: H = γ(m_{o}c^{2} + qΦ_{oEM}) H = (1/√[1(v/c)^{2}])(m_{o}c^{2} + qΦ_{oEM}) ~ [1+(v/c)^{2}/2])(m_{o}c^{2} + qΦ_{oEM}) = (m_{o}c^{2} + qΦ_{oEM})+(1/2)(m_{o}c^{2}v^{2}/c^{2} + qΦ_{oEM}v^{2}/c^{2}) ~ (m_{o}c^{2} + qΦ_{oEM}) +(1/2)(m_{o}v^{2} + 0) H ~ (1/2)(m_{o}v^{2}) + (m_{o}c^{2} + qΦ_{oEM}) H ~ (Kinetic) + (Rest+Potential) = T + V (for v << c) In QM words: the number of phase changes (ticks) over the t axis is higher by a factor gamma. Thus, L ~ TV and H ~ T+V only in the nonrelativistic limit (v<<c) 

T = (1/2)m_{o}U·U= (1/2)P·U ?? 
Relativistic Kinetic Energy Term T 

ψ[R] , ψ*[R] 
Scalar Quantum Wave Function 







n_{o} = √[N·N/U·U] = N·U/U·U = n/γ 
Particle RestNumberDensity (for stat mech) 

s_{o} = √[S·S/U·U] = S·U/U·U 
RestEntropyDensity (for stat mech) 




Ω_{o} = Ω 
Ω = # of microstates = (N!) / (n_{0}!n_{1}!n_{2}!...) 

N_{o} = N 
(Stable) Particle Number: N = nV =
(n/γ)(γ V) = n_{o}V_{o} = N_{o} 

P_{o} = P 
Pressure of system (eg. of a tensorial perfect fluid): P = P_{o} 

S_{o} = S = k_{B} ln Ω 
Entropy: S = sV = (s/γ)(γ V) = s_{o}V_{o} = S_{o} , 

T_{o} = γ T 
RestTemperature (according to Einstein/Planck def.) 

q = γ Q 
RestHeat 

V_{o} = γ V 
RestVolume 

dS = k_{B} d(ln Ω) = δQ / T 
Change in Entropy 




Π (p^{α},x^{α})
= 
Invariant equilibrium distribution
function for relativistic gas 




F_{uv}^{ }F^{uv}
= 2(B^{2}  E^{2}/c^{2}) 
EM invariant 

G_{cd}^{ }F^{cd}
= ε_{abcd}F^{ab}F^{cd}
=(2/c)(B·E) 
EM invariant 

P =
(2q^{2}/3c^{3}) γ^{4}(a_{perp}^{2}+
γ^{2}a^{2}) P = (μ_{o}q^{2}a^{2}γ^{6})/(6πc) in parallel Generally P = μ_{o}q^{2}(A·A)/(6πc) 
Radiated Power P, total power is Lorentz invariant for processes with symmetry in the rest frame  
I_{v}/v^{3}  Spectral Intensity / v3  
Helicity  A massless particle moves with the speed of light, so a real observer (who must always travel at less than the speed of light) cannot be in any reference frame where the particle appears to reverse its relative direction, meaning that all real observers see the same chirality. Because of this, the direction of spin of massless particles is not affected by a Lorentz boost (change of viewpoint) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: the helicity is a relativistic invariant 
Conserved Quantities (components of V, such that the 4Divergence ∂·V = 0 )
∂·J = ∂p/∂t +∇·j = 0 
Conservation of 4CurrentDensity (EM
charge): p & j 
∂·N = ∂/∂t(γ n_{o})+∇·(γ
n_{o}u) 
Conservation of 4NumberFlux (Particle
NumberDen, NumFlux): n_{ & }n_{f
} 
∂·P = (1/c^{2})∂E/∂t +∇·p
= 0 
Conservation of 4Momentum (Energy~Mass,
Momentum): E & p 
∂·K = ∂/c∂t(w/c)+∇·k 
Conservation of 4WaveVec (AngFreq,
WaveNum): w & k 
∂·A_{EM} = (1/c^{2})∂V_{EM}/∂t +∇·a_{EM} = 0 
Conservation of 4VectPotential_{EM}
(applies in the Lorenz Gauge): V_{EM} & a_{EM
} 


∂·U = ∂/∂t(γ[u])+∇·(γ[u] u) 
Conservation of 4Velocity: (FluxGauss'
Law)??: γ & γ u 
Lorentz 4Tensors
η_{μν} = η^{μν} =
Diag[1,1,1,1] 
Minkowski Metric (flat spacetime) 

1 if a=b, 
Kronecker Delta 

= +1 if {abcd} is an even permutation of
{0123} 
LeviCivita symbol 

F_{uv}^{ }=

EM Field Tensor 

G_{cd} = (1/2)ε_{abcd}F^{ab} =

Dual EM Field Tensor 

[Generally {including shear forces}]
[For a Perfect Fluid {no viscosity}] T^{μν} = (ρ_{eo}+p)U^{μ}U^{ν}/c^{2}  pη^{μν} : where η^{μν} = Diag[1,1,1,1] T^{μν} =
(both of these are Lorentz Scalars) [For Dust {no particle interaction}] T^{μν} = (ρ_{eo})U^{μ}U^{ν}/c^{2} = (ρ_{mo})U^{μ}U^{ν} T^{μν} =
and p = p_{o} = 0 for dust {particles do not interact} 
EnergyMomentum Stress Tensor The stressenergy tensor of a relativistic
fluid can be written in the form Here
The heat flux vector and viscous shear tensor are transverse to the world lines, in the sense that
This means that they are effectively threedimensional 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 fourdimensional symmetric rank two tensor. 
*Note*
When deriving the Newtonian Limit, always use the Low Velocity
(v<<c) or Low Energy (E<<m_{o}c^{2})
approximations, as these apply to "real" situations
Do not use the (c > Infinitity) approximation  while technically
making the math work, it is however an unphysical situation
4Vector(s)  Type  Relativistic Law  Newtonian Limit Low Velocity (v<<c) or Low Energy (E<<m_{o}c^{2}) Basically, β > 0, γ > 1 
R = (ct,r)  4Position  (ct,r) is single 4vector entity t and r related by Lorentz transform 
t independent from r t is independent scalar, r is independent 3vector 
ΔR = (cΔt,Δr)  4Displacement  Relative Simultaneity Δt' = γ(Δt  β·Δr/c) 
Absolute Simultaneity Δt' = Δt 
U = dR/dτ  4Velocity  Relativistic Composition of Velocities u_{rel} = =[u_{1}+u_{2}]/(1+β_{1}·β_{2}) =[u_{1}+u_{2}]/(1+u_{1}·u_{2}/c^{2}) Imposes Universal Speed Limit of c 
Additive Velocities u_{12} = u_{1} + u_{2} Unlimited Speed 
A = dT/dτ  4Acceleration  Relativistic Larmor Formula Power radiated by moving charge P = = ( q^{2}/ 6πε_{o}c^{3})(A·A) = (μ_{o}q^{2}/6πc)(A·A) = (μ_{o}q^{2}/6πc) γ^{6}/ (a^{2}  (u x a)^{2}/c^{2}) 
Newtonian Larmor Formula Power radiated by a nonrelativistic moving charge P = (μ_{o}q^{2}/6πc)(a^{2}) 
P = m_{o}U  4Momentum  Einstein EnergyMass Relation E = γ m_{o}c^{2} = Sqrt[ m_{o}^{2}c^{4} + p·p c^{2} ] 
Total Energy = Rest Energy + Kinetic Energy E = m_{o}c^{2} + (p^{2}/2m_{o}) 
∂·P  Divergence of 4Momentum  Local? Conservation of 4Momentum  Conservation of Energy, Conservation of Momentum 
P_{1}·P_{2}  Particle Interaction  Conservation of 4Momentum  Conservation of Energy, Conservation of Momentum, sometimes Conservation of Kinetic Energy 
K =(ω/c,k) =(1/ћ)P = (m_{o}/ћ)U = (ω_{o}/c^{2})U 
4WaveVector and 4Velocity 
Relativistic Doppler Effect, inc. Transverse Doppler Effect a_{o_obs} = = a_{o_emit} / γ(1  (n·v/c)) = a_{o_emit} / γ(1  (n·β)) = a_{o_emit} √[1+β]√[1β] / (1  (n·β)) Relativistic Aberration Effect cos(ø_{_obs}) = [cos(ø_{_emit})β]/[1βcos(ø_{_emit})] Relativistic Wave Speed, all elementary particles, matter or photonic λf = c/β = v_{phase} 
Regular Doppler Effect a_{o_obs} = a_{o_emit} √[1+β]√[1β] Newtonian Aberration = None cos(ø_{_obs})= cos(ø_{_emit}) Newtonian Wave Speed, only photonic particles (a rare case when the lightspeed case is chosen for Newtonian description) λf = c 
P and K  4Momentum and 4WaveVector 
Compton Scattering (λ'λ) = (h/m_{o}c)(1cos[ø]) (m_{o}c^{2})(1/E'1/E) = (1cos[ø]) Ratio of photon energy after/before collision P[E,ø] = 1/[1+(E/m_{o}c^{2})(1cos[ø])] see also KleinNishina formula 
Thompson Scattering Ratio of photon energy after/before collision: E<<m_{o}c^{2} P[E,ø] > 1 
∂ = iK  4Gradient  D'Alembertian & KleinGordon Equation ∂_{t}^{2}/c^{2} = ∇·∇(m_{o}c/ћ)^{2} 
Schroedinger Equation (i ћ)( ∂_{t} ) =  (ћ)^{2}(∇)^{2}/2m_{o} 
∂·J  Divergence of 4Current  Conservation of 4EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0 
Conservation of 4EM_CurrentDensity ∂·J = ∂/c∂t(cp)+∇·j = ∂p/∂t +∇·j = 0 
J_{prob}  Probability CurrentDensity  Conservation of ProbabilityCurrentDensity ρ = (iћ/2m_{o}c^{2})(ψ* ∂_{t}[ψ]∂_{t}[ψ*] ψ) j = (iћ/2m_{o})(ψ* ∇[ψ]∇[ψ*] ψ) ∂·J_{prob} = ∂ρ/∂t +∇·j = 0 ρ = γ(ψ*ψ) for time separable wave functions Relativistically, this is conservation of the number of worldlines thru a given spacetime event 
Conservation of Probability ∂·J_{prob} = ∂ρ/∂t +∇·j = 0 ρ = (ψ*ψ) for time separable wave functions Typically set so that the sum over all quantum states in space = 1 At low energies/velocities, this appears as the conservation of probability of a given wavefunction at a given spacetime event  In other words, the probability interpretation of a wavefunction is just a Newtonian approximation to the more correctly stated conservation of relativistic worldlines. This is why the problem of positive definite probabilities and probabilities >1 vanishes once you consider antiparticles and conservation of charged currents. 
A_{EM} = (Φ_{EM}/c, a_{EM})  4VectorPotential  4VectorPotenial of a moving point charge (LienardWiechert potential) A_{EM} = (q/4πε_{o}c) U / [R·U]_{ret} [..]_{ret} implies (R·R = 0, the definition of a light signal) Φ_{EM} = (γΦ_{o}) = (γq/4πε_{o}r) a_{EM} = (γΦ_{o}/c^{2})u = (γqμ_{o}/4πr)u 
Scalar Potential and Vector Potential of a stationary point charge Φ_{EM } = (q/4πε_{o}r) a_{EM } = 0 Scalar Potential and Vector Potential of a slowly moving point charge (v<<c implies γ>1) Φ_{EM} = (Φ_{o}) = (q/4πε_{o}r) a_{EM} = (Φ_{o}/c^{2})u = (qμ_{o}/4π r)u 
Q_{EM} = (E_{EM}/c, p_{EM}) = q A_{EM} = q (Φ_{EM}/c, a_{EM}) 
4VectorPotentialMomentum 


P_{EM} = (E/c + qΦ_{EM}/c, p +
qa_{EM}) = γ m_{o}(c,u) P_{EM} = Π = P + qA_{EM} = m_{o}U + qA_{EM} =(H/c,p_{EM}) = (γm_{o}c+q Φ_{EM}/c,γm_{o}u+q a_{EM}) 
4Momentum_{EM} 4CanonicalMomentum 4TotalMomentum 
Minimal Coupling ============= Total 4Momentum = Particle 4Momentum + Potential(Field) 4Momentum 

D = ∂ + iq/ћ A_{EM}  Minimal Coupling Prescription 
KG equation, with minimal coupling to an EM
potential D·D = = (m_{o}c/ћ)^{2} (∂ + iq/ћ A_{EM})·(∂ + iq/ћ A_{EM}) + (m_{o}c/ћ)^{2} = 0 
Schroedinger Equation (with external potential) (i ћ)( ∂_{t} ) = V[x]  (ћ)^{2}(∇)^{2}/2m_{o} 
E^{2 = }p·p c^{2} + m_{o}^{2}c^{4}:
Energy of a particle has a Momentum component and a RestMass component
Total Energy: E = mc^{2} = γ[u] m_{o}c^{2} = ћω
Kinetic Energy: T = mc^{2}m_{o}c^{2} = (γ[u]1) m_{o}c^{2}
= (γ1) m_{o}c^{2}
Rest Energy: E_{o} = m_{o}c^{2
}

___
 
 γ[u]
 m 
 
_______
m_{o}
Relativistic (apparent) mass m = AreaLike = γ[u] * m_{o} = ћ w/c^{2}
= E/c^{2}
Theoretically, this would scale like a δfunction for photons{m_{o}
>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 ω_{o} = m_{o}c^{2}/ћ,
even when the massive particle is at rest. Mass is always "spinning" about
the time dimension.
U·U = c^{2} , d/dτ(U·U) = d/dτ(c^{2}) = 0 ,
d/dτ(U·U) = 2*(U·dU/dτ) = 2*(U·A) = 0
U·A = 0: The 4Acceleration is orthogonal to its own 4Velocity
(Any acceleration is orthogonal to its own worldline, i.e. you don't
accelerate in time).
U plays the part of the tangent vector of the worldline, and A
plays the part of the normal vector of the worldline.
The curvature of a worldline is given by a/c^{2}.
U_{1}·U_{2} = γ[u_{1}]γ[u_{2}](c^{2}u_{1}·u_{2})
= γ[u_{r}]c^{2} (The scalar product of two uniformly
moving particles is proportional to the γ factor of their relative
velocities)
∂·R = (∂/c∂t,∇)·(ct,r) = (∂/c∂t[ct]+∇·r)
= (∂/∂t[t]+∇·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(∇))·(ct,r) +
∂·U = γ (d/dt(∂/c∂t)(ct)+d/dt(∇))·r + ∂·U= γ (d/dt(∂/∂t)(t)+d/dt(∇))·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 4Velocity U is
a conserved quantity.
For example, let N = n_{o}U, so ∂·N =
∂·n_{o}U = n_{o}∂·U = n_{o}(0)
= 0. The quantity n_{o} is conserved.
Alternately, ∂·U = (∂/c∂t, ∇)·γ(c, u) = ∂·U_{o}
= (∂/c∂t, ∇)·(c, 0) = ∂/c∂t (c) = ∂/∂t (1) = 0
Compton Scattering
P·P = (m_{o}c)^{2} ==>0 for photons
P_{phot1}·P_{phot2} = ћ^{2}K_{1}·K_{2}
= (ћ^{2}ω_{1}ω_{2}/c^{2})(1n_{1}·n_{2})
= (ћ^{2}ω_{1}ω_{2}/c^{2})(1cos[ø])
P_{phot}·P_{mass} = ћK·P
= (ћω/c)(1,n)·(E/c,p) = (ћω/c)(E/cn·p) = (ћωE_{o}/c^{2})
= (ћωm_{o})
P_{phot} + P_{mass} = P'_{phot}
+ P'_{mass} Conservation of 4Momentum in
PhotonMassive Interaction
P_{phot} + P_{mass}  P'_{phot}
= P'_{mass} rearrange
(P_{phot} + P_{mass}  P'_{phot})^{2}
= (P'_{mass})^{2} square to get scalars
(P_{phot}·P_{phot} + 2 P_{phot}·P_{mass}
 2 P_{phot}·P'_{phot} + P_{mass}·P_{mass}
 2 P_{mass}·P'_{phoot} + P'_{phot}·P'_{phot})
= (P'_{mass})^{2}
(0 + 2 P_{phot}·P_{mass}  2
P_{phot}·P'_{phot} + (m_{o}c)^{2}
 2 P_{mass}·P'_{phot} + 0) =
(m_{o}c)^{2}
P_{phot}·P_{mass}  P_{mass}·P'_{phot}
= P_{phot}·P'_{phot}
(ћωm_{o})(ћω'm_{o}) = (ћ^{2}ωω'/c^{2})(1cos[ø])
(ωω')/(ωω') = (ћ/m_{o}c^{2})(1cos[ø])
(1/ω'1/ω) = (ћ/m_{o}c^{2})(1cos[ø])
ω = 2 pi v = 2 pi c / λ : 1/ω = λ / 2 pi c
: ћ = h / 2 pi
(λ'λ) = (h/m_{o}c)(1cos[ø])
(λ'λ) = (h/m_{o}c)(2sin^{2}[ø/2]) Compton scattering with
Compton wavelength (h/m_{o}c)
(m_{o}c^{2})(1/E'1/E) = (1cos[ø])
Relativistic Doppler Effect
A = (a^{0}, a), a generic SR 4vector under
observation, relative to observer
A·U = a Lorentz invariant, upon which all observers agree
take A·U > A·U_{o} = (a^{0}, a)·(c,0)
= ca^{0} = the value of the temporal component of A as
seen by observer U
now, let there be an observer U_{obs} at rest and
an emitter U_{emit} moving with respect to U_{obs}
U_{obs} = (c,0): observer at rest
U_{emit} = γ(c,v): velocity of
emitter relative to observer

A·U_{obs} = (a^{0}, a)·(c,0)
= c a^{0} = ca^{0}_{_obs}
A·U_{emit} = (a^{0}, a)·γ(c,v)
= γ(ca^{0}  a·v)= ca^{0}_{_emit}

A·U_{obs} / A·U_{emit} = ca^{0}_{_obs}
/ ca^{0}_{_emit} = a^{0}_{_obs} / a^{0}_{_emit}
A·U_{obs} / A·U_{emit} = ca_{0}
/ γ(ca^{0}  a·v) = 1 / γ(1  a·v/a^{0}c)
= 1 / γ[1  (a/a^{0})*(n·v/c)]

a^{0}_{_obs} / a^{0}_{_emit} = 1 / γ(1  (a/a^{0})*(n·v/c))
a_{0_obs} =
= a^{0}_{_emit} / γ(1  (a/a^{0})*(n·v/c))
= a^{0}_{_emit} / γ(1  (a/a^{0})*(n·β))
= a^{0}_{_emit} / γ(1  (a/a^{0})*(β cos[θ]))
if A is photonic, then (a/a^{0}) = 1, then a^{0}_{_obs}
= a^{0}_{_emit} / γ(1  (n·v/c)) = a^{0}_{_emit} / γ(1 
(n·β))
= a_{0_emit} √[1+β]√[1β] / (1  (n·β))
thus, for photonic Doppler shifts,
if {n toward and β toward obs},
then a^{0}_{_obs} = a^{0}_{_emit} / γ[1  β Cos[0°]] = a^{0}_{_emit}
/ γ[1  β] = a^{0}_{_emit} √[(1+β)/(1β)] >Doppler
BlueShift

if {n toward and β 90° to obs},
then a^{0}_{_obs} = a^{0}_{_emit} / γ[1  (β Cos[90°])] = a^{0}_{_emit}
/ γ[1  0] = a^{0}_{_emit} / γ >the transverse Doppler
effect

if {n toward and β away from obs},
then a^{0}_{_obs} = a^{0}_{_emit} / γ[1  β Cos[180°]] = a^{0}_{_emit}
/ γ[1 + β] = a^{0}_{_emit} √[(1β)/(1+β)] >Doppler
RedShift
Note that a^{0} could be any temporal component, ie. (E/c) for
4Momentum P, (ω/c) for 4Wavevector K, (ρc) for
4CurrentDensity J, etc.
Commonly used notation would be the 4frequency, for which a^{0} =
ν/c, leading to:
thus, for photonic frequency Doppler shifts,
if {n toward and β toward obs}, then ν_{obs} = ν_{emit}
√[(1+β)/(1β)] >Doppler BlueShift
if {n toward and β 90° to obs}, then ν_{obs} = ν_{emit}
/ γ >the transverse Doppler effect
if {n toward and β away from obs}, then ν_{obs} = ν_{emit}
√[(1β)/(1+β)] >Doppler RedShift
U = γ(c, u), P = (E/c,p), d(P) =
(dE/c,dp)
U·d(P) = γ(c dE/cu·dp) = γ(dEu·dp)
= γ(T dS  P dV + μ dN) = (T_{o} dS_{o}  P_{o}
dV_{o} + μ_{o} dN_{o}) = 0 ??
U·d(P) = γ(dEu·dp) = (T_{o}dS_{o}  P_{o}dV_{o} + μ_{o}dN_{o}) = const = ? 0 ? 
U·d(P) = γ(dEu·dp) = γ(T dS  P dV +
Sum[μ_{i} dN_{i}] + 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 = c^{2
}, P·P = (m_{o}c)^{2}
U·P = γ(c E/cu·p) = γ(Eu·p) = γ(T S  P V +
μ N) = (T_{o} S_{o}  P_{o} V_{o}
+ μ_{o} N_{o}) ?
U·P = γ(Eu·p) = (T_{o} S_{o}  P_{o} V_{o} + μ_{o} N_{o}) = m_{o}c^{2 }? for a spatially homogeneous system: relativistic GibbsDuhem eqn. 
Invariants 
P = Pressure = P_{o} 
N = ParticleNum = N_{o} 
S = Entropy = S_{o} 
Variables 
V = Volume = (1/γ)Vol_{o} 
μ = ChemPoten = (1/γ)μ_{o} 
T = Temperature = (1/γ)Temp_{o} 
V*P (particle superstructure = Vol*Press)
μ*N (particle structure = ChemPoten*ParticleNum)
T*S (particle substructure = Temp*Entropy)
Time t = γ t_{o}
Length L = L_{o}/γ
Heat Q = q/γ
dq = T_{o}dS_{o}
InertialMassDen(of radiation field) q = P/vV = γ q
Total Particle Number N = N_{o} is an invariant, because the
NumberDensity n varies as n = γ n_{o}, but this is balanced by
Volume V = V_{o}/γ
NumberDenstiy n = γ n_{o} where NumberFlux 4Vector N =
(cn,n_{f}) = n_{o} γ(c, u) = n_{o}U,n_{o}
= N_{o}/(Δ_x_{o}*Δ_y_{o}*Δ_z_{o})
N = n * V = (γ n_{o})*(V_{o}/γ) = n_{o}* V_{o}
= N_{o}
N·N = (n_{o}c)^{2}
Total Entropy S = S_{o} is an invariant, because the
EntropyDensity s varies as s = γ s_{o}, but this is balanced by
Volume V = V_{o}/γ
EntropyDensity s = γ s_{o} where EntropyFlux 4Vector S =
(cs,s_{f}) = s_{o} γ(c, u) = s_{o}U,s_{o}
= S_{o}/(Δ_x_{o}*Δ_y_{o}*Δ_z_{o})
S = s * V = (γ s_{o})*(V_{o}/γ) = s_{o}* V_{o}
= S_{o}
S·S = (s_{o}c)^{2}
Action S = S(ct,x,y,z)
dS/dτ = 0
dS/dτ = U·∂(S) = γ(∂S/∂t + u·∇(S)) = 0
see Menzel pg.172
√[1+x] ~ (1+x/2) for x<<1 This mathematical formula is used
to derive the Newtonian limit of the various relativistic entities
γ = (1 / √[1(v/c)^{2}] )
γ > 1 for v<<c
All of the formulas below can also be generated from the 4Velocity
Relation and multiplying by the appropriate Lorentz scalar:
U·U = γ[u]^{2}(c^{2}u·u) = c^{2}
γ^{2}(1β·β) = 1
γ^{2} = 1 + γ^{2}β^{2}
γ = ±√[1 + γ^{2}β^{2}]
We choose the positive root since γ is always positive
γ = √[1 + γ^{2}β^{2}]
γ ~ [1 + γ^{2}β^{2}/2] for (γ^{2}β^{2}
<< 1)
4Momentum
P = (E/c, p)
P·P = (E_{o}/c)^{2} = (m_{o}c)^{2}
E^{2 } = E_{o}^{2} + p·p c^{2}
E = E_{o}√[ 1 + p·p c^{2} / E_{o}^{2}]
E ~ E_{o}( 1 + p·p c^{2} / 2 E_{o}^{2}
+ ...) for  p·p c^{2}  <<  E_{o}^{2}
 discarding higher order terms...
E ~ ( E_{o} + p·p c^{2} / 2 E_{o} ) for 
p·p c^{2}  <<  E_{o}^{2} 
E ~ ( E_{o} + p·p / 2 m_{o} ) for  p·p c^{2}
 <<  E_{o}^{2}  where E_{o} = m_{o}c^{2}
E ~ ( E_{o} + p^{2} / 2 m_{o} ) for  p
c  <<  E_{o} 
Total Energy = Rest Energy + Newtonian Momentum term
alternately:
γ ~ [1 + γ^{2}β^{2}/2]
γE_{o} ~ E_{o}[1 + γ^{2}β^{2}/2]
E ~ [E_{o} + γ^{2}E_{o}β^{2}/2]
E ~ [E_{o} + γ^{2}m_{o}c^{2}β^{2}/2]
E ~ [E_{o} + γ^{2}m_{o}^{2}c^{2}β^{2}/2m_{o}]
E ~ [E_{o} + γ^{2}m_{o}^{2}v^{2}/2m_{o}]
E ~ [E_{o} + p^{2}/2m_{o}]
4WaveVector
K = (ω/c, k)
K·K = (ω_{o}/c)^{2} = (E_{o}/ћc)^{2}
= (m_{o}c / ћ)^{2}
ω^{2 } = ω_{o}^{2} + k·k c^{2}
ω = ω_{o} √[ 1 + k·k c^{2} / ω_{o}^{2}
]
ω ~ ω_{o}( 1 + k·k c^{2} / 2 ω_{o}^{2}
+ ...) for  k·k c^{2}  <<  ω_{o}^{2}
 and choosing the positive root and discarding higher order terms...
ω ~ ( ω_{o} + k·k c^{2} / 2 ω_{o} )
for  k·k c^{2}  <<  ω_{o}^{2}

ω ~ ( ω_{o} + ћ k·k / 2 m_{o} ) for  k·k
c^{2}  <<  ω_{o}^{2}  where ω_{o}
= m_{o}c^{2} / ћ
ω ~ ( ω_{o} + ћ k^{2} / 2 m_{o} ) for  k
c  <<  ω_{o} 
Total Angular Frequency = Rest Angular Frequency + Newtonian Wave Number
term
4Gradient (Wave equation)
∂ = ∂/∂x_{μ} = (∂/c∂t, ∇) =
(∂_{t}/c,
∇)
∂·∂ = (∂_{to}/c)^{2} = ( i m_{o}c / ћ )^{2}:
KleinGordon Relativistic Wave eqn.
∂_{t}^{2 } = ∂_{to}^{2} + ∇·∇
c^{2}
∂_{t} = ±∂_{to}√[ 1 + ∇·∇ c^{2}
/ ∂_{to}^{2}]
∂_{t} ~ ∂_{to}( 1 + ∇·∇ c^{2} /
2 ∂_{to}^{2} + ...) for  ∇·∇ c^{2}
 <<  ∂_{to}^{2}  and choosing the positive root
and discarding higher order terms...
∂_{t} ~ ( ∂_{to} + ∇·∇ c^{2} /
2 ∂_{to} ) for  ∇·∇ c^{2}  << 
∂_{to}^{2} 
∂_{t} ~ ( ∂_{to}  ћ ∇·∇ / i 2 m_{o}
) for  ∇·∇ c^{2}  <<  ∂_{to}^{2}
 where ∂_{to} =  i m_{o}c^{2} / ћ
∂_{t} ~ ( ∂_{to}  ћ ∇^{2} / i 2
m_{o} ) for  ∇ c  <<  ∂_{to} 
or, in more standard form
i ћ ∂_{t} ~ ( i ћ ∂_{to}  ћ^{2} ∇^{2}
/ 2 m_{o} ) for  ∇ c  <<  ∂_{to} 
where i ћ ∂_{to} = E_{o}, the rest energy of the potential
V
i ћ ∂_{t} ~ ( V(x,t)  ћ^{2} ∇^{2}
/ 2 m_{o} ) for  ∇ c  <<  ∂_{to} 
Time dependent Schroedinger equation is just the Newtonian approximation
of the KleinGordon Relativistic Wave eqn.
4ProbabilityCurrentDensity (change in form of Probability Density)
J = (cρ, j) = (iћ/2m_{o})(ψ*∂[ψ]∂[ψ*]ψ)
taking the temporal component, the relativistic probability density
ρ = (iћ/2m_{o}c^{2})(ψ* ∂_{t}[ψ]∂_{t}[ψ*]
ψ)
assume wave solution in following general form:
ψ = A f [k] e^{(iωt)} and ψ* = A* f [k]* e^{(+iωt)}
then
∂_{t}[ψ] = (iω)A f [k] e^{(iωt)} = (iω)ψ and ∂_{t}[ψ*]
= (+iω)A* f [k]* e^{(+iωt)} = (+iω)ψ*
then
ρ = (iћ/2m_{o}c^{2})(ψ* ∂_{t}[ψ]∂_{t}[ψ*]
ψ)
ρ = (iћ/2m_{o}c^{2})((iω)ψ*ψ(+iω)ψ*ψ)
ρ = (iћ/2m_{o}c^{2})((2iω)ψ*ψ)
ρ = (ћω/m_{o}c^{2})(ψ*ψ)
now use the Newtonian form of ω from above
ρ ~ [ћ( ω_{o} + ћ k^{2} / 2 m_{o} )/m_{o}c^{2}](ψ*ψ)
ρ ~ [(ћω_{o}/m_{o}c^{2}) + (ћћ
k^{2} / 2 m_{o}m_{o}c^{2})](ψ*ψ)
ρ ~ [(ћω_{o}/m_{o}c^{2}) + (ћω_{o}ћω_{o}
k^{2}c^{2} / 2 ω_{o}ω_{o}m_{o}c^{2}m_{o}c^{2}
)](ψ*ψ)
ρ ~ [(1) + ( k^{2}c^{2} / 2 ω_{o}^{2})](ψ*ψ),
but
 k c  <<  ω_{o} 
ρ ~ [(1) + (~0)](ψ*ψ) because 2nd term is very small in
nonrelativistic limit
ρ ~ (ψ*ψ)
The standard probability density (ψ*ψ) is the Newtonian
approximation of the temporal component of the 4ProbabilityCurrent
Alternately, use ω = γω_{o}
ρ = (ћω/m_{o}c^{2})(ψ*ψ)
ρ = (ћγω_{o}/m_{o}c^{2})(ψ*ψ)
ρ = (γ)(ψ*ψ)
ρ ~ (ψ*ψ) where γ>1 in the Newtonian limit
∂·∂ = (∂/c∂t,∇)·(∂/c∂t,∇) = ∂^{2}/c^{2}∂t^{2}∇·∇
= (m_{o}c / ћ)^{2}: KleinGordon Relativistic Wave eqn.
D_{EM} = (∂/c∂t + iq/ћ V_{EM}/c,
∇ + iq/ћ a_{EM}) = ∂ + (iq/ћ)A_{EM}
D_{EM}·D_{EM} = (m_{o}c
/ ћ)^{2}: KleinGordon Relativistic Wave eqn. in electromagnetic
potentials
(∂ + (iq/ћ)A_{EM})·(∂ + (iq/ћ)A_{EM})
= (m_{o}c / ћ)^{2}: KleinGordon Relativistic Wave eqn.
w/ electromagnetic potentials
(∂·∂) + (iq/ћ)(∂·A_{EM} + A_{EM}·∂)
+ (iq/ћ)^{2}(A_{EM}·A_{EM})
= (m_{o}c / ћ)^{2}: KleinGordon Relativistic Wave eqn.
w/ electromagnetic potentials
if (∂·A_{EM} + A_{EM}·∂)
= 0
then (∂·∂) + (iq/ћ)^{2}(A_{EM}·A_{EM})
= (m_{o}c / ћ)^{2}
(∂/c∂t,∇)·(∂/c∂t,∇) + (iq/ћ)^{2}((V_{EM}/c,
a_{EM})·(V_{EM}/c,
a_{EM})) = (m_{o}c / ћ)^{2}
(∂^{2}/c^{2}∂t^{2}∇·∇) + (iq/ћ)^{2}(
(V_{EM}/c)^{2}(a_{EM}·a_{EM}) ) = (m_{o}c
/ ћ)^{2}
(∂^{2}/c^{2}∂t^{2}+(iq/cћ)^{2}(V_{EM}^{2})(∇·∇+(iq/ћ)^{2}(a_{EM}·a_{EM})
) = (m_{o}c / ћ)^{2}
The KleinGordon equation is more general than the Schrödinger equation,
but simplifies to the Schrödinger equation in the (φ/c)<<1 limit.
∂·∂ = (∂/c∂t,∇)·(∂/c∂t,∇) = ∂^{2}/c^{2}∂t^{2}∇·∇
= (m_{o}c / ћ)^{2}: KleinGordon Relativistic Wave eqn.
∂^{2}/c^{2}∂t^{2 } = ∇·∇(m_{o}c
/ ћ)^{2}
∂^{2}/c^{2}∂t^{2 } = (im_{o}c / ћ)^{2}+∇·∇
(i ћ)^{2}∂^{2}/c^{2}∂t^{2 } = (i ћ)^{2}(im_{o}c
/ ћ)^{2}+(i ћ)^{2}∇·∇
(i ћ)^{2}∂^{2}/c^{2}∂t^{2 } = (m_{o}c)^{2}+(i
ћ)^{2}∇·∇
(i ћ)^{2}∂^{2}/∂t^{2 } = (m_{o}c^{2})^{2}*[1
+
(i ћ/m_{o}c)^{2}∇·∇]
(i ћ)∂/∂t = ± (m_{o}c^{2})*Sqrt[1 + (i ћ/m_{o}c)^{2}∇·∇]
(i ћ)∂/∂t ~ ± (m_{o}c^{2})*[1 + (1/2)*(i ћ/m_{o}c)^{2}∇·∇
+ ...] for ( ћ)^{2}*∇·∇<<(m_{o}c)^{2}
,generally a very good approx. for nonrelativistic systems
(i ћ)∂/∂t ~ ± [(m_{o}c^{2}) + (i^{2} ћ^{2}/2m_{o})∇·∇
+ ...]
choosing the positive root and discarding higher order terms...
(i ћ)∂/∂t ~ (m_{o}c^{2})  ( ћ^{2}/2m_{o})∇^{2}
(i ћ)∂/∂t ~  ( ћ^{2}/2m_{o})∇^{2}
becomes
the time dependent Schrödinger eqn. for a free particle
Also, extensions into EM fields (or other types of relativistic
potentials) can be made using D = ∂ + iq/ћ A_{EM}
where A_{EM} is the EM vector potential and q is the EM
charge,
and allowing D·D = (m_{o}c/ћ)^{2} to be the more
correct EM quantum wave equation.
D·D = (m_{o}c/ћ)^{2}
(∂ + iq/ћ A_{EM})·(∂ + iq/ћ A_{EM})
+ (m_{o}c/ћ)^{2} = 0
let A'_{EM} = iq/ћ A_{EM}
let M = m_{o}c/ћ_{
}then (∂ + A'_{EM})·(∂ + A'_{EM})
+ (M)^{2} = 0
∂·∂ + ∂·A'_{EM} + 2 A'_{EM}·∂ + A'_{EM}·A'_{EM}
+ (M)^{2} = 0
now the trick is that factor of 2, it comes about by keeping track of
tensor notation...
a weakness of strick 4vector notation
let the 4Vector potential be a conservative field, then ∂·A_{EM}
=
0
(∂·∂) + 2(A'_{EM}·∂) + (A'_{EM}·A'_{EM})
+
(M)^{2} = 0
expanding to temporal/spatial components...
( ∂_{t}^{2}/c^{2}∇·∇ ) + 2(φ'/c
∂_{t}/c  a'·∇ ) + ( φ'^{2}/c^{2} a'·a'
) + (M)^{2} = 0
gathering like components
( ∂_{t}^{2}/c^{2} + 2φ'/c ∂_{t}/c
+ φ'^{2}/c^{2} )  (∇·∇ + 2
a'·∇ + a'·a' ) + (M)^{2} = 0
( ∂_{t}^{2} + 2φ'∂_{t} + φ'^{2}
)  c^{2}(∇·∇ + 2 a'·∇
+ a'·a'
) + c^{2}(M)^{2} = 0
( ∂_{t} + φ' )^{2}  c^{2}(∇ + a'
)^{2} + c^{2}(M)^{2} = 0
multiply everything by (i ћ)^{2}
(i ћ)^{2}( ∂_{t} + φ' )^{2}  c^{2}(i ћ)^{2}(∇
+ a' )^{2} + c^{2}(i
ћ)^{2}(M)^{2} = 0
put into suggestive form
(i ћ)^{2}( ∂_{t} + φ' )^{2} =  c^{2}(i ћ)^{2}(M)^{2}
+ c^{2}(i ћ)^{2}(∇ + a'
)^{2}
(i ћ)^{2}( ∂_{t} + φ' )^{2} = i^{2}c^{2}(i
ћ)^{2}(M)^{2} + c^{2}(i ћ)^{2}(∇
+ a' )^{2}
(i ћ)^{2}( ∂_{t} + φ' )^{2} = i^{2}c^{2}(i
ћ)^{2}(M)^{2} [1 + c^{2}(i ћ)^{2}(∇
+ a' )^{2}/ i^{2}c^{2}(i
ћ)^{2}(M)^{2} ]
(i ћ)^{2}( ∂_{t} + φ' )^{2} = i^{2}c^{2}(i
ћ)^{2}(M)^{2} [1 + (∇ + a'
)^{2}/ i^{2}(M)^{2} ]
take Sqrt of both sides
(i ћ)( ∂_{t} + φ' ) = ic(i ћ)(M) Sqrt[1 + (∇ + a' )^{2}/ i^{2}(M)^{2}
]
use Newtonian approx Sqrt[1+x] ~ ±[1+x/2] for x<<1
(i ћ)( ∂_{t} + φ' ) ~ ic(i ћ)(M) ±[1 + (∇ + a' )^{2}/2 i^{2}(M)^{2}
]
(i ћ)( ∂_{t} + φ' ) ~ ±[ic(i ћ)(M) + ic(i ћ)(M)(∇ +
a' )^{2}/2 i^{2}(M)^{2}
]
(i ћ)( ∂_{t} + φ' ) ~ ±[c(i^{2} ћ)(M) + c( ћ)(∇
+ a' )^{2}/2(M) ]
remember M = m_{o}c/ћ
(i ћ)( ∂_{t} + φ' ) ~ ±[c(i^{2} ћ)(m_{o}c/ћ)
+ c( ћ)(∇ + a' )^{2}/2(m_{o}c/ћ)
]
(i ћ)( ∂_{t} + φ' ) ~ ±[c(i^{2})(m_{o}c) +
(ћ)^{2}(∇ + a' )^{2}/2(m_{o})
]
(i ћ)( ∂_{t} + φ' ) ~ ±[(m_{o}c^{2}) + (ћ)^{2}(∇
+ a' )^{2}/(2m_{o}) ]
remember A'_{EM} = iq/ћ A_{EM}
(i ћ)( ∂_{t} + iq/ћφ ) ~ ±[(m_{o}c^{2}) +
(ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
]
(i ћ)( ∂_{t} ) + (i ћ)(iq/ћ)(φ) ~ ±[(m_{o}c^{2})
+ (ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
]
(i ћ)( ∂_{t} ) + (i^{2})(qφ ) ~ ±[(m_{o}c^{2})
+ (ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
]
(i ћ)( ∂_{t} ) (qφ ) ~ ±[(m_{o}c^{2}) +
(ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
]
(i ћ)( ∂_{t} ) ~ (qφ )±[(m_{o}c^{2})
+ (ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
]
take the negative root
(i ћ)( ∂_{t} ) ~ (qφ ) + [(m_{o}c^{2})
 (ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
]
Here is the general Newtonian result
(i ћ)( ∂_{t} ) ~ (qφ ) + (m_{o}c^{2})
 (ћ)^{2}(∇ + iq/ћa )^{2}/2m_{o}
or
(i ћ)( ∂_{t} ) ~ (qφ ) + (m_{o}c^{2})
+ [( ћ / i )∇ + qa ]^{2}/2m_{o}
call (qφ ) + (m_{o}c^{2}) = V[x]
(i ћ)( ∂_{t} ) ~ V[x]  (ћ)^{2}(∇
+ iq/ћa )^{2}/2m_{o
}
typically the vector potential a is zero in most nonrelativistic settings
(i ћ)( ∂_{t} ) ~ V[x]  (ћ)^{2}(∇)^{2}/2m_{o}
And there you have it, the Schrodinger Equation with a potential
The assumptions for nonrelativistic equation were:
Conservative field A_{EM}, then ∂·A_{EM}
=
0
(∇ + a' )^{2}/ i^{2}(M)^{2}
= (∇ + a' )^{2}/ i^{2}(m_{o}c/ћ)^{2}
= (ћ)^{2}(∇ + a' )^{2}/
i^{2}(m_{o}c)^{2} is near zero
i.e. (ћ)^{2}(∇ + a'
)^{2} << (m_{o}c)^{2}, a good approximation
for lowenergy systems
Arbitrarily chose vector potential a=0
Or keep it around for a nearPauli equation (we would just have to track
spins, not included in this derivation)
(K = m_{o}/ћU = ω_{o}/c^{2 }U)
gives (c^{2}/v_{phase} n = u) Both the
wave vector and particle velocity point in the same direction; along the
worldline. The product of the phase velocity and the particle velocity
always equals c^{2}. ( v_{phase} * u = c^{2} ). In
the case of photons, the phase velocity = particle velocity = c. In the
case of matter particles, the phase velocity v_{phase} = c^{2}/u
> c and particle velocity u<c. What does this mean? Suppose that you
have a collection of particles traveling at identical velocities that all
flash at the same time. The v_{phase} is the speed at which the
flash moves in other reference frames, and can be considered the speed of
propagation of simultaneity. For particles which are at rest, the v_{phase}
is infinite, which makes sense since they all appear to flash
simultaneously. v_{phase} (the phase velocity) is sometimes
known as the celerity.
(∂·∂)A_{EM} = μ_{o} J+∂(∂·A_{EM})
Inhomogeneous Maxwell Equation
(∂·∂)A_{EM} = μ_{o} J
Homogeneous Maxwell/Lorentz Equation (if ∂·A_{EM}
= 0 Lorenz Gauge)
∂·J = ∂ρ/∂t +∇·j = 0 Conservation of
EMcurrent
Psi = a E e^{ iK·R} Photon
Wave Equation (Solution to Maxwell Equation)
E·K = 0 The Polarization of a photon is orthogonal to the
WaveVector of that photon
V·U_{obs} = γ_{v}(c,v)·γ[u_{obs}]
u_{obs} = γ_{v}γ[u_{obs}](c^{2}v·u_{obs})
V·U_{obs}[u_{obs} = 0]/c^{2}
= γ_{v} (RestFrame Invariant expression for relative gamma
factor)
P·U_{obs} = E/c γ[u_{obs}]cp·γ[u_{obs}]
u_{obs} = γ[u_{obs}](Ep·u_{obs})
P·U_{obs}[u_{obs} = 0]
= E (RestFrame Invariant expression for energy)
K·U_{obs} = w/c γ[u_{obs}]ck·γ[u_{obs}]
u_{obs} = γ[u_{obs}](wk·u_{obs})
K·U_{obs}[u_{obs} = 0]
= w (RestFrame Invariant expression for angular frequency)
R·U_{obs} = ct γ[u_{obs}]cr·γ[u_{obs}]
u_{obs} = γ[u_{obs}](c^{2}tp·u_{obs})
R·U_{obs}[u_{obs} = 0]/c^{2}
= t (RestFrame Invariant expression for time)
J·U_{obs} = cp γ[u_{obs}]cj·γ[u_{obs}]
u_{obs} = γ[u_{obs}](pc^{2}j·u_{obs})
J·U_{obs}[u_{obs} = 0]/c^{2}
= ρ (RestFrame Invariant expression for ElecChargeDensity)
F^{uv} = ∂^{u}A^{v}∂^{v}A^{u}
Electromagnetic Field Tensor (F^{0i} = E^{i},F^{ij}
= e^{ijk}B^{k})
L = 1/4 F_{uv}^{ }F^{uv} 
J_{u} A^{u} : Lagrangian Density for EM field
L = m_{o}c^{2}/γ V: Relativistic Lagrangian function of
a Particle in a Conservative Potential
V_{EM} = q U·A_{EM}/γ: Potential of
EM field
L_{EM} = m_{o}c^{2}/γ  q U·A_{EM}/γ
=  (P·P/m_{o} + qU·A_{EM})/γ
=  (m_{o}U·U + qU·A_{EM})/γ
d/dτ = U·∂ = γ d/dt
U·∂/γ = ∂/∂t + u·∇ = d/dt : Convective Derivative
Larmor formula can be written in Lorentz invariant form
P = ( q^{2}/ 6πε_{o}c^{3})(A·A)
=
(μ_{o}q^{2})/(6πc)(A·A) Guassian units?
= ( q^{2}/ 6πε_{o}c^{3}) γ^{6}/ (u'^{2}
 (u x u')^{2}/c^{2})
= ( q^{2}/ 6πε_{o}c^{3}) γ^{6}/ (β'^{2}
 (β x β')^{2})
= (2q^{2}/ 3c(1β'^{2})^{3}) γ^{6}/ ( β'^{2}
 ( β x β')^{2}) SI Units?
alternate Larmor formula:
P = (2/3)(q^{2}/ m_{o}^{2}c^{3})(F·F)
SI
units?
P = ( q^{2}/ 6πε_{o}m_{o}^{2}c^{3})(F·F)
Guassian
units?
Relativistic Power radiated by moving charge by AbrahamLorentzDirac
force
P = (μ_{o}q^{2}a^{2}γ^{6})/(6πc)
====
LiénardWiechert potentials  potential due to a moving charge
A^{μ}(x) = (q/c4πε_{o}) U^{μ }/ ( R_{ν }U^{ν}
) where R_{ν} is a null vector (R_{ν }R^{ν} = 0)
A_{EM} = (q/c4πε_{o}) U / (R·U)
where (R·R = 0, the definition of a light signal)
= (q/c4πε_{o}) U / ( cγ ( rr·u/c )
)
= (q/c^{2}4π ε_{o})(c,u)/( rr·u/c )
and therefore
φ_{EM} = (q / 4 π ε_{o} ) 1/[ r  r·u/c]_{ret}
a_{EM} = (μ_{o} q / 4 π) [u]/[ r  r·u/c]_{ret}
where terms in square brackets [] indicate retarded quantities
(R·U) = (ct,r)·γ(c,u) = γ(c^{2}t  r·u)
= cγ(ct  r·u/c)
t_{ret} = t  xx'/c: (retarded time)
r_{u} = r  r u/c = the virtual
present radius vector; i.e., the radius vector directed
from the position the charge would occupy at time t' if it had continued
with uniform velocity from its retarded position to the field point.
=====
F =  grad V(x): Particle moving in conservative
force field
mc^{2} + V(x) = E = const: Relativistic energy conservation in
conservative force fields
T = mc^{2}m_{o}c^{2} = (γ[u]1) m_{o}c^{2}
= (γ1) m_{o}c^{2} Relativistic Kinetic Energy:
F·dX/dt = dT/dt: Also holds in Relativistic Mechanics
F·U = (m_{o}A+(dm_{o}/dτ)U)·U
= c^{2}(dm_{o}/dτ) = γc^{2}(dm_{o}/dt)
Relativistic Perfect Fluids, where dissipative effects (viscosity,
heat conduction, etc.) are neglected.
Particle 4Flow N is a conservative quantity whose balance eqn. is ∂·N
= 0
N = (cn, n_{f}) = n_{o}
γ(c, u) = n(c, u) = n_{o}U
∂·N = ∂n/∂t +∇·(nu) = 0, where n=n_{o}
γ
∂·N = ∂n_{o} γ/∂t +∇·(n_{o} γu) = 0
In nonrelativistic limit this becomes ∂n_{o}/∂t
+∇·( n_{o}u) = 0
T^{αβ} = ((ne+p)/c^{2})U^{α} U^{β}  p η^{αβ}
∂_{β}T^{αβ} = 0,
Consevation of EnergyMomentum Tensor
Since the speed of light is so large, it is difficult to come up with some
ordinary type phenomena that rely on SR. There are a few, however.
Relativistic quantum chemistry:
The yellowish color of the elements gold and cesium, which would otherwise
be silvery/white : http://www.fourmilab.ch/documents/golden_glow/
The corrosion resistance of the element gold
Low melting point of element Mercury
About 10 of the 12 volts of a car's leadacid battery due to relativistic
effects, tinacid batteries (similar outer orbitals) too weak
Navigation:
GPS Satellite system  Would go out of synch within minutes without the
relativistic corrections, up to about 10 km difference /day
Very slight time differences of atomic clocks carried on airliners
EM:
Homopolar/Unipolar generator/motor  solution to Faraday's Paradox : http://www.physics.umd.edu/lecdem/outreach/QOTW/arch11/q218unipolar.pdf
Relativistic electron diffraction, or any other high speed electron
experiments
Faraday's Law, where a moving magnet generates an electric field, for
instance current along a wire
A plain old electromagnet, where the magnetic field is generated by moving
electrons
Various:
Fizeau's experiment of light in moving water
Cyclotron frequency of electron in magnetic field, increased effective
mass from relativistic gamma factor
Muon travel time thru atmosphere
Magnet moving past wire loop vs. wire loop moving past magnet
In atomic physics, the fine structure describes the splitting of the
spectral lines of atoms due to 1st order relativistic corrections.
Spinorbit splitting.
Cerenkov Radiation: Matter particles moving thru a medium at greater than
the local speed of light thru the medium emit Cerenkov radiation.
Relativity gives us fermions and FermiDirac statistics and the whole
structure of matter relies on the nature of fermions.
Relativity explains low energy aspects of the microscopic structure of
matter, such as atomic spectra.
It is a general property for any interacting fermion to show spinorbit
behavior, a consequence of Lorentz Invariance.
Magnetism as the dynamic effect of moving electrical charges.
Cathode Ray Tubes, CRT's, in old style television sets and computer
monitors, had electrons moving at up to 30% c, and the magnets controlling
the beam had to be shaped with relativistic effects accounted for.
Nuclear power: One gets the massive power amounts from fission and fusion
reactions based on the relativity, which gives magnitudes more power than
regular chemical reactions.
Sunlight: It's based on nuclear fusion, which requires relativity.
Slow moving clocks:
The measurement of time dilation at everyday speeds has been accomplished
as well. Chou et al. (2010) created two clocks each holding a single 27Al^{+}
ion in a Paul trap. In one clock, the Al^{+} ion was accompanied
by a 9Be^{+} ion as a "logic" ion, while in the other, it was
accompanied by a 25Mg^{+} ion. The two clocks were situated in
separate laboratories and connected with a 75 m long, phasestabilized
optical fiber for exchange of clock signals. These optical atomic clocks
emitted frequencies in the petahertz (1 PHz = 10^{15} Hz) range
and had frequency uncertainties in the 10^{17} range. With these
clocks, it was possible to measure a frequency shift due to time dilation
of ~10^{16} at speeds below 36 km/h (< 10 m/s, the speed of a
fast runner) by comparing the rates of moving and resting aluminum ions.
It was also possible to detect gravitational time dilation from a
difference in elevation between the two clocks of 33 cm
It can be shown that a scalar (s) and vector (v) which are related
through a continuity equation in all frames of reference (∂s/∂t + ∇·v
= 0) transform according to the Lorentz transformations and therefore
comprise the components of a 4vector V=(cs,v), where ∂ ·V
= 0. Relativistic fourvectors may be identified from the continuity
equations of physics. See A
Proposed Relativistic, Thermodynamic FourVector.
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/outof that part of the system.
Essentially, no material is created/destroyed. ∂·J = ∂p/∂t +∇·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) ∇ 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 'realworld' 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:
∇·v = 0
The fundamental theorem of vector calculus states that any vector field
can be expressed as the sum of a conservative vector field and a
solenoidal field. The condition of zero divergence is satisfied whenever a
vector field v has only a vector potential component, because the
definition of the vector potential A as:
v = ∇ x A
automatically results in the identity (as can be shown, for example,
using Cartesian coordinates):
∇·v = ∇·(∇ x A) = 0
The converse also holds: for any solenoidal v there exists a vector potential A such that v = ∇ x A. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)
In vector calculus, a Laplacian vector field is a vector field
which is both irrotational and incompressible. If the field is denoted as
v, then it is described by the following differential equations:
∇ x v = 0
∇·v = 0
Since the curl of v is zero, it follows that v can be
expressed as the gradient of a scalar potential (see irrotational field) φ:
v = ∇ φ (1)
Then, since the divergence of v is also zero, it follows from equation (1) that
∇·∇ φ = 0
which is equivalent to
∇^{2} φ = 0
Therefore, the potential of a Laplacian field satisfies Laplace's
equation.
In fluid dynamics, a potential flow is a velocity field which is
described as the gradient of a scalar function: the velocity potential. As
a result, a potential flow is characterized by an irrotational velocity
field, which is a valid approximation for several applications. The
irrotationality of a potential flow is due to the curl of a gradient
always being equal to zero (since the curl of a gradient is equivalent to
take the cross product of two parallel vectors, which is zero).
In case of an incompressible flow the velocity potential satisfies the Laplace's equation. However, potential flows have also been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.
Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, and groundwater flow.
For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable.
A velocity potential is used in fluid dynamics, when a fluid
occupies a simplyconnected region and is irrotational. In such a case,
∇ x u = 0
where u denotes the flow velocity of the fluid. As a result, u
can be represented as the gradient of a scalar function Φ:
u = ∇ Φ
Φ is known as a velocity potential for u.
A velocity potential is not unique. If a is a constant then Φ + a is also a velocity potential for u. Conversely, if Ψ is a velocity potential for u then Ψ = Φ + b for some constant b. In other words, velocity potentials are unique up to a constant.
Unlike a stream function, a velocity potential can exist in threedimensional flow.
see Cosmological Physics
Relativistic Euler Equations:
dv/dt =  1/[γ^{2}(ρ + p/c^{2})](∇ p + p'v/c^{2}):
Conservation of Momentum
d/dt[γ^{2}(ρ + p/c^{2})] = p'/c^{2}  γ^{2}(ρ
+ p/c^{2})∇·v: Conservation of Energy
where p' = ∂ p/∂ t
∂·J = 0 where J = n_{o}U (J is the Number
Flux here)
Relativistic Enthalpy w = (ρ + p/c^{2})
d/dt[γw/n] = p'/γnc^{2}
Thus, in steady flow, γ * (enthalpy/particle) = const.
In nonrelativistic limit these reduce to
dv/dt =  1/[ρ](∇ p): Conservation of Momentum
d/dt[(ρ)] =  (ρ)∇·v: Conservation of Mass
p = Pressure
ΔE =  p ΔV
E = ρ c^{2} V
ΔV / V =  Δρ_{o}/ρ_{o}
Relativistic Bernoulli's eqn.
γ w / ρ_{o} = const
L = (P_{T}·U)/γ  H = γ(P_{T}·U)  H + L = p_{T}·u = γ(P_{T}·U)  (P_{T}·U)/γ 
L = (P_{T}·U)/γ L = ((P + Q)·U)/γ L = (P·U + Q·U)/γ L = P·U/γ  Q·U/γ L = m_{o}U·U/γ  qA·U/γ L = m_{o}c^{2}/γ  qA·U/γ L = m_{o}c^{2}/γ  q(φ/c, a)·γ(c, u)/γ L = m_{o}c^{2}/γ  q(φ/c, a)·(c,u) L = m_{o}c^{2}/γ  q(φ  a·u) L = m_{o}c^{2}/γ  qφ + qa·u L = m_{o}c^{2}/γ  qφ_{o}/γ L = (m_{o}c^{2} + qφ_{o})/γ 
H = γ(P_{T}·U) H = γ((P + Q)·U) H = γ(P·U + Q·U) H = γP·U + γQ·U H = γm_{o}U·U + γqA·U H = γm_{o}c^{2} + qγφ_{o} H = γm_{o}c^{2} + qφ H = ( γβ^{2} + 1/γ )m_{o}c^{2} + qφ H = ( γm_{o}β^{2}c^{2} + m_{o}c^{2}/γ) + qφ H = ( γm_{o}u^{2} + m_{o}c^{2}/γ) + qφ H = p·u + m_{o}c^{2}/γ + qφ H = E + qφ H = ± c√[m_{o}^{2}c^{2}+p^{2}] + qφ H = ± c√[m_{o}^{2}c^{2}+(p_{T}qa)^{2}] + qφ 
H + L = γ(P_{T}·U)  (P_{T}·U)/γ H + L = (γ  1/γ)(P_{T}·U) H + L = ( γβ^{2} )(P_{T}·U) H + L = ( γβ^{2} )((P + Q)·U) H + L = ( γβ^{2} )(P·U + Q·U) H + L = ( γβ^{2} )(m_{o}c^{2} + qφ_{o}) H + L = (γm_{o}β^{2}c^{2} + qγφ_{o}β^{2}) H + L = (γm_{o}u·uc^{2}/c^{2} + qφ_{o}γu·u/c^{2}) H + L = (γm_{o}u·u + qa·u) H + L = (p·u + qa·u) H + L = p_{T}·u 
Next, let's look at Quantum Commutation Relations...
Nonzero Commutation Relation between position and momentum:
[X^{u},P^{v}] =  i ћ η^{uv}
this gives
[ x , p_{x} ] = [ y , p_{y} ] = [ z , p_{z} ] = (i
ћ)
[ ct , E/c ] = [ t , E ] = ( i ћ) :assuming that one can treat the time
as an operator...
both of these yield the familiar uncertainty relations:
Generalized Uncertainty relation: (Δ A) * (Δ B) > = (1/2) < i[A,B]
> see Sudbury pg. 59 for a great derivation
(Δ x * Δ p_{x} > = ћ / 2) and (Δ t * Δ E > = ћ / 2)
or more generally
(Δ R^{u} * Δ P^{v} > = ћ δ^{uv} / 2)
or
(Δ R^{u} * Δ K^{v} > = δ^{uv} / 2)
(Δ x * Δ k_{x} > = 1/2) and (Δ t * Δ w > = 1/2)
[ R^{u} , R^{v} ] = R^{u }R^{v}  R^{v
}R^{u} = 0 : All position coordinates commute
[ P^{u} , P^{v} ] = P^{u }P^{v}  P^{v
}P^{u} = 0 : All momentum coordinates commute
While I'm at it, a small comment about the quantum uncertainty relation. A
great many books state that the quantum uncertainty relations mean that a
"particle" cannot simultaneously have precise properties of position and
momentum. I disagree with that interpretation. The uncertainty relations,
the mathematical structure of the argument, say nothing about
"simultaneous" measurements. They do say something about "sequential"
measurements. A measurement of one variable places the system in a state
such that if the next measurement is that of a noncommuting variable of
the first, then the uncertainty must be of a minimum>0 amount. Also,
note that the uncertainty relations are not necessarily about the size of
h. Nor are they about the factor of ( i ) in the commutation relation. It
would appear that they are about the metric g^{μν} itself, which
has a nonzero result for sequential, noncommuting 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 spacelike 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? The answer is of course that the one measurement does not affect the spacelike separated other measurement.

need to derive:
(Δ phi_{x} * Δ L_{x} > = ћ / 2)
where phi_{x} is angle about x, and L_{x} is angular
momentum about x
 timelike interval(+)
/
lightlike interval(0)
worldline


c
\ future /
\  /
\ 
/  spacelike
interval()
\/now
/\
/
 \
elsewhere
/  \
/ past \

c
(0,0) ZeroNull 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 PhotonicLightLike
(a,b) a<b SpaceLike
Any TimeLike 4Vector (a,b) may be boosted into a Pure TimeLike
(ka,0) state
Any SpaceLike 4Vector (a,b) may be boosted into a Pure SpaceLike
(0,kb) state
R·U = (ct,r)·γ(c,u
) = γ(c^{2}t  r·u) = cγ(ct  r·u/c) = c^{2}t_{o}
= c^{2}τ
d[R·U]/dτ = d[c^{2}τ]/dτ
d[R]/dτ·U+R·d[U]/dτ = c^{2}
U·U + R·A = c^{2}
but U·U = c^{2} so R·A = 0 { Remember, this is a derivation for a single particle with no interactions, so A = 0 }
d[U·U]/dτ = d[c^{2}]/dτ = 0
U·d[U]/dτ +d[U]/dτ·U = 0
U·A +A·U = 0
U·A = 0, a nice general result that a particle's acceleration is perpendicular to its worldline.
A·A = a^{2} = γ^{4}[a^{2}
+ (γ/c)^{2}(u·a)^{2}]
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 (KleinGordon) quantum field  which is mathematically
the simplest QFT that still contains a continuous global [U(1)] internal
symmetry. A real (Hermetian) scalar QFT is mathematically still simpler,
but the absence of "charge" renders it uninteresting for most purposes.
Poincare group (aka inhomogeneous Lorentz group) and its representations
The set of Lorentz transforms and spacetime translations (Λ,A) such that:
X'^{μ} = Λ^{μ}_{ν} X^{ν} + A^{μ}
with conditions:
Det[Λ] = +1 (excludes discrete transforms of space inversion => proper)
Λ^{0}_{0} >= +1 (excluded discrete transforms of time
inversion => orthochronous, preserve direction of time)
Λ^{μ}_{ν} (a Lorentz Transform  maps spacetime onto
itself and therefore preserves the inner product)
Λ^{μ}_{ν} Λ_{μλ} = g_{νλ} (the
Minkowski Metric)
A^{μ} = (Spacetime Translation)
Unitary Operators representing these transforms:
U(A,1) = Exp[ i P·A ]
U(0,Λ) = Exp[ i M^{μν} Λ_{μν} ]
Poincare group has 10 generators (spacetime 4generators)
P^{μ} (4 generators of spacetime translation = Conservation of
4Momentum)
M^{μν} (6 generators of Lorentz group = 3 orbital angular momenta
+ 3 Lorentz boosts)
[ P^{μ}, P^{ν} ] = 0 (Energy/Momentum commutes with
itself)
[ M^{μν}, P^{σ} ] =  i ( P^{μ} g^{νσ}  P^{ν}
g^{μσ} )
or
[ M_{μν}, P_{σ} ] = i ( g_{νσ} P_{μ}  g_{μσ}
P_{ν} ) {one of these has a sign error I think}
[ M^{μν}, M^{ρσ} ] = i ( M^{νσ} g^{μρ} 
M^{μσ} g^{νρ} + M^{ρν} g^{μσ}  M^{ρμ}
g^{νσ} )
Then, define the spatial 3generators:
"Spatial Rotation" generators J_{i} = (1/2) ε_{ijk} M^{jk}
(for i=1,2,3), are Hermetian, (M^{jk})^{†} = M^{jk}
"Lorentz Boost" generators K_{i} = M_{i0} (for i=1,2,3),
are antiHermetian, (M_{i0})^{†} =  M_{i0}
[ J_{i} , P_{k} ] = i ε_{ikl }P^{l}
[ J_{i} , P_{0} ] = 0 (Spin commutes Energy)
[ K_{i} , P_{k} ] = i P_{0 }g_{ik}
[ K_{i} , P_{0} ] =  i P_{i}
[ J_{m} , J_{n} ] = i ε_{mnk }J^{k}
[ J_{m} , K_{n} ] = i ε_{mnk }K^{k}
[ K_{m} , K_{n} ] =  i ε_{mnk }J^{k}
Covariance of physical laws under Poincare trans. imply that all
quantities defined in Minkowski spacetime 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) P^{2} = η_{μν} P^{μ} P^{ν} = P_{μ}
P^{μ} with corresponding eigenvalues P^{2} = m^{2}
which measure the invariant mass of field configurations.
In the real world we observe only timelike or lightlike fourmomenta,
i.e. particles with positive or zero mass. Furthermore, the temporal
components are always positive.
With dimensional units this would be P^{2} = m^{2}c^{2}
(2) W^{2} = η_{μν} W^{μ} W^{ν} = W_{μ}
W^{μ} with corresponding eigenvalues W^{2} = ( w_{0}^{2}
 w·w ) =  (w·w) =  (P_{0}^{2}j^{2})
=  m^{2} s(s+1),
which measure the invariant spin of the particle, where there are (2s+1)
spin states
(or 2 polarization/helicity states for massless fields)
with W_{μ} as the PauliLubanski (mixed) SpinMomentum four vector
With dimensional units this would be W^{2} =  m^{2}c^{2}ћ^{2}
s(s+1)
Note: Massless representation give P^{2} = m^{2} = 0 and W^{2}
=  m^{2} s(s+1) = 0
For instance, for a photonic P^{μ} = E(1,0,0,1), one has W^{μ}
= M_{12} P^{μ}
so that M_{12} takes the possible eigenvalues ± s
W_{σ} = (1/2) ε_{σμνρ} M^{μν} P^{ρ}
or
W_{σ} =  (1/2) ε_{μνρσ} M^{μν} P^{ρ}
such that
[ W_{σ} , P^{μ} ] = 0
[ M_{μν} , W_{σ} ] = i ( W_{μ} g_{νσ}  W_{ν}
g_{μσ} )
[ W_{λ} , W_{σ} ] = i ε_{λσαβ} W^{α} P^{β}
Further,
W = (w^{0},w) = (p·j , P_{0}j
 p x k)
w^{0} = p·j
w = P_{0}j  p x k
where
j = (M_{32},M_{13},M_{21}) are the 3
components of angular momentum, where [J_{1},J_{2}] = i J_{3}
and cyclic permutations
k = (M_{01},M_{02},M_{03}) are boosts in 3
Cartesian directions
Wigner's classification: (nonnegative energy irreducible unitary
representations of the Poincare group)
The irreducible unitary representations of the Poincare' group are
classified according to the eigenvalues of P^{2} and W^{2}
They fall into several classes:
1a) P^{2} = m^{2} > 0 and P_{0} > 0: Massive
particle
1b) P^{2} = m^{2} > 0 and P_{0} < 0: Massive
antiparticle??
2a) P^{2} = 0 and P_{0} > 0: Photonic
2b) P^{2} = 0 and P_{0} < 0: Photonic??
3) P^{2} = 0 and P_{0} = 0: P in the 4Zero,
the vacuum
4) P^{2} = m^{2} < 0: Tachyonic
A complete set of commuting observables is composed of P^{2}, the
3 components of p, W^{2}, and one of the 4 components of W^{μ}
The eigenvalues of P^{2} (mass) and W^{2} (spin)
distinguish (possibly together with other quantum numbers) different
particles. This is the general result for finitemass 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 1particle state k> = a^{t}(k)0> is
characterized by the eigenvalues
p^{0}k> = ћω(k)k>, pk>
= ћkk>, W^{2}k> = 0
thus showing that the quanta of such a quantum field may be identified
with particles of definite energymomentum 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
pointparticles. It is the Poincare invariance properties, the
relativistic covariance of such systems, that also justifies, on account
of Noether's theorem, this physical interpretation.
One has to learn how to extend the above description to more general field
theories whose quanta are particles of nonvanishing spin or helicity. One
then has to consider collections of fields whose components also mix under
Lorentz transforms.
One may list the representations which are invariant under parity and
correspond to the lowest spin/helicity content possible.
(0,0) 
φ 
scalar field 
(1/2,0) (+) (0,1/2) 
ψ 
Dirac spinor 
(1/2,1/2) 
A_{μ} 
vector field 
(1,0) (+) (0,1) 
F^{uv} = ∂^{u}A^{v}∂^{v}A^{u} 
EM field tensor 
Consider an arbitrary spacetime vector x^{μ}
Construct the 2 x 2 Hermitian matrix X = X^{†}
X = x^{μ}σ_{μ} = 
( x^{0} + x^{3} 
x^{1}  i x^{2} ) 











( x^{1} + i x^{2} 
x^{0}  x^{3} ) 
then Det[X] = x^{2} = x·x = η_{μν} x^{μ}
x^{ν}
see Proceedings of the Third International Workshop on Contemporary
Problems in Physics, By Jan Govaerts, M. Norbert Hounkonnou, Alfred
Z. Msezane
see Conceptual Foundations of Modern Particle Physics, Robert Eugene
Marshak
see Fundamentals of Neutrino Physics and Astrophysics, Carlo Giunti
see Kinematical Theory of Spinning Particles, Martin Rivas
All of the relativistic wave equations can be derived from a common
source, the relativistic massenergy relation, inc. spin, in an EM field
4Momentum inc. Spin
Ps = Σ·P = Σ^{μ}_{ν} P^{ν}
= η_{αβ} Σ^{μα} P^{β} = Ps^{μ}
Σ^{μ}_{ν} is a Pauli Spin Matrix Tensor = Diag[σ^{0},σ]
Σ^{μν} is a Pauli Spin Matrix Tensor = Diag[σ^{0},σ]
Ps = Diag[σ^{0},σ]·P = Diag[σ^{0},σ]·(E/c,p)
= (σ^{0}E/c,σ·p)
Ps = (ps^{0},ps) =
(σ^{0}E/c,σ·p)
where σ^{0} is an identity matrix of appropriate spin dimension
and σ is the Pauli Spin Matrix Vector
4Momentum inc. Spin in External Field
where:
H = E_{t} = Hamiltonian = TotalEnergyOfSystem
p_{T} = Total3MomentumOfSystem
P_{T}= P + qA
P = P_{T}  qA
Ps = (ps^{0},ps) =
(σ^{0} E/c,σ·p) = [σ^{0}(E_{t}/cqφ/c) , σ·(p_{t}qa)]
Ps·Ps = (ps^{0})^{2}  (ps)^{2}
= [σ^{0}(E/c)]^{2}  [σ·(p)]^{2} =
[σ^{0}(E_{T}/cqφ/c)]^{2}  [σ·(p_{T}qa)]^{2}
= (m_{o}c)^{2} = (E_{o}/c)^{2}
The 4Total Momentum (inc. External Field MinimalCoupling and Spin)
P_{s} = Σ·P
= Σ·(P_{T}qA)
= [σ^{0}(E_{T}/cqφ/c),σ·(p_{T}qa)]
where Σ = Σ^{μν}
are the Pauli Spin Matrices, and taking the Einstein summation gives the σ^{0}
and σ
P_{s}·P_{s} =( Σ·P
)^{2} = [ Σ·(P_{T}qA)]^{2}
= [σ^{0}(E_{T}/cqφ/c)]^{2}  [σ·(p_{T}qa)]^{2}
= (m_{o}c)^{2}
( Σ·P )^{2}
= (m_{o}c)^{2}
( Σ·∂ )^{2}
= (m_{o}c/ћ)^{2}
( Σ·∂ )^{2}
+ (m_{o}c/ћ)^{2} = 0
( Σ·(D(i/h)qA)
)^{2} + (m_{o}c/ћ)^{2} = 0
Now, to prove that this "Relativistic Pauli" EnergyMomentum equation can
lead to the Dirac equation
Ps·Ps = [σ^{0}(E_{T}/cqφ/c)]^{2}
 [σ·(p_{T}qa)]^{2}
= (ps^{0})^{2}  (ps)^{2}
= (m_{o}c)^{2} = (E_{o}/c)^{2}
Ps·Ps = [I(E_{T}/cqφ/c)]^{2}
 [σ·(p_{T}qa)]^{2}
= (ps^{0})^{2}  (ps)^{2}
= (m_{o}c)^{2} = (E_{o}/c)^{2
}Ps·Ps = (ps^{0})^{2}  (ps)^{2}
= (m_{o}c)^{2}
(ps^{0} + ps) (ps^{0}
 ps) = (m_{o}c)^{2}
Multiply both sides by any arbitrary function, ψχ
(ps^{0} + ps) (ps^{0}
 ps)ψχ = (m_{o}c)^{2}ψχ
We can also split the arbitrary function into two parts, and it still
solves the original equation
let (ps^{0} + ps)ψ
= (m_{o}c)χ
and (ps^{0}  ps)χ
= (m_{o}c)ψ
now add and subtract these...
(ps^{0} + ps)ψ + (ps^{0}
 ps)χ = (m_{o}c)χ + (m_{o}c)ψ
(ps^{0} + ps)ψ  (ps^{0}
 ps)χ = (m_{o}c)χ
 (m_{o}c)ψ
gather terms...
ps^{0} (ψ+χ) + ps (ψχ) =
(m_{o}c)(χ+ψ)
ps^{0} (ψχ) + ps (ψ+χ) =
(m_{o}c)(χψ)
rearrange a bit...
ps^{0} (ψ+χ) + ps (ψχ) =
(m_{o}c)(ψ+χ)
ps (ψ+χ) + ps^{0} (ψχ) =
(m_{o}c)(ψχ)
we are free to relabel our variables
let (ψ+χ) = a and (ψχ) = b
ps^{0} a  ps b = (m_{o}c)a
ps a  ps^{0} b = (m_{o}c)b
put in matrix form...
[ps^{0}  ps ][a]
= (m_{o}c)[a]
[ps  ps^{0}][b]=
[b]
put in suggestive matrix form...
([1 0 ]ps^{0} + [0 1]ps)[a] = (m_{o}c)[a]
([0 1] [1 0 ]
)[b]= [b]
put in even more suggestive matrix form...
([1 0 ]σ^{0}p^{0} + [0 1]σ·p)[a]
= (m_{o}c)[a]
([0 1] [1 0
]
)[b]= [b]
put in highly suggestive matrix form...
([σ^{0} 0 ]p^{0}  [0 σ]·p)[a] = (m_{o}c)[a]
([0 σ^{0}] [σ
0 ] )[b]=
[b]
let Spinor Ψ = [a]
[b]
and note that σ^{0} = I
this is equivalent to Dirac Gamma Matrices...
([I 0 ]p^{0}  [0 σ]·p)Ψ
= (m_{o}c)Ψ
([0 I] [σ
0 ] )
(γ^{0}p^{0}  γ·p)Ψ
= (m_{o}c)Ψ
(Γ·P)Ψ = (m_{o}c)Ψ
(Γ·P) = (m_{o}c)
(Γ^{μ}P_{μ})Ψ = (m_{o}c)Ψ
iћ(Γ^{μ}∂_{μ})Ψ = (m_{o}c)Ψ
>Spin 
>Statistics 
>Relativistic Eqn. 
>Relativistic Eqn. 
>NonRelativistic Eqn. Newtonian Limit √[1+x] ~ (1+x/2) for x<<1 v<<c 
>Field 
>Polarizations 
0 
Boson: 
FreeWave 
KleinGordon (Fock) 
Schrödinger ( iћ∂_{t}V+[ћ^{2}∇^{2}/2m])Ψ = 0 with minimal EM coupling (iћ∂_{t}qφ[(pqa)^{2}/2m])Ψ = 0 (iћ∂_{t}qφ[(iћ∇qa)^{2}/2m])Ψ = 0 (iћ∂_{t}qφ+[(ћ∇+iqa)^{2}/2m])Ψ = 0 Note that qφ = V is approx. when a~0 
Ψ 
2? 
1/2 
Fermion 
Weyl (σ·∂) Ψ = 0 
Dirac 
Pauli, (SchrödingerPauli) with minimal EM coupling (i ћ∂_{t}  qφ [(σ·(pqa)^{2})/2m])Ψ = 0 
Ψ 
2 
1 
Boson 
Maxwell 
Proca 
A 
2 (= 2 transverse) 

3/2 
Fermion 
Gravitino? 
RaritaSchwinger 
ψ_{σ}^{ }spinorvector 
2 

2 
Boson 
Einstein 
? ? 
tensor = 2tensor 
2 
DuffinKemmerPetiau Equation = Complex Proca Equation
DuffinKemmer Equation: ( β^{μ} p_{μ}  M ) ψ = 0 : for a
free spin0 or spin1 particle
Dim 
Type 

Hodge Dual 
0 
scalar 


1 
vector 


2 
tensor 


3 
pseudovector 
magnetic field, spin, torque, vorticity, angular momentum 

4 
pseudoscalar 
magnetic charge, magnetic flux, helicity 

In Minkowski space (4dimensions), the { 1 4 6 4 1} Hodge dual of an
nrank (n<=2) tensor will be an (4n) rank skewsymmetric 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
Einstein made as stronger statement about EP, known as EPP: In small
enough regions of spacetime, the laws of physics reduce to those of
special relativity; it is impossible to detect the existence of a
gravitational field by means of local experiments.
SR > QM, what assumptions necessary & where does it break down
Relational QM
General Continuity of WorldLines
Spin vs. Accel, time component correlation
Relativistic Thermodynamics & SM
Poincare Group & Casimir operators & Casimir Invariants (mass
& spin of Poincare field)
Generalized Uncertainty
Points  Waves  Potentials  Fields
Relation between single point and density 4vectors
Poisson Eqn. / Laplace Eqn.
Continuity eqn > 4Vector
Adding Spin to KleinGordon
Relativistic Lagrangian & Hamiltonian
Covariant Form Relativistic Equations
UmovPoynting examples
Dirac  Kemmer generalized eqn.
Hodge Dual examples
Pressure Diffusion Wave/Eqn.
Potential Flow Theory
Schroedinger Eqn as a diffusion equation
Main article: Schrödinger equation
With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:
This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
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:
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 [m^{3}] 
charge q [C] 
heatQ [J] 
potential 
pressure p [Pa=J/m^{3}] 
potential φ [V=J/C] 
temperature T [K=J/k_{B}] 
flux 
current Φ_{V} [m^{3}/s] 
current I [A=C/s] 
heat transfer rate [J/s] 
flux density 
velocity v [m/s] 
j [C/(m^{2}·s) = A/m²] 
heat flux [W/m^{2}] 
linear model 
Poiseuille's law 
Ohm's law 
Fourier's law 
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 (Chap1315)
First Course in General Relativity, A, Bernard F. Schutz (Chap14)
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)
(**pg6065,8286**)
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 (Lec7rest)
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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Quantum Mechanics is derivable from Special Relativity See QM from SRSimple RoadMap 