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

The Ultimate
Four-Vector & Lorentz Invariant Reference

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

[ŗK] Kelvin

Count of the quantity of heat (statistical)

"Flat" SpaceTime
η_μν = g_μν{SR}

t

x

y

z

t

1

0

0

0

x

0

-1

0

0

y

0

0

-1

0

z

0

0

0

-1

γ

-(v_x/c)γ

0

0

-(v_x/c)γ

γ

0

0

0

0

1

0

0

0

0

1

γ

-β_xγ

0

0

-β_xγ

γ

0

0

0

0

1

0

0

0

0

1

γ

-β_xγ

-β_yγ

-β_zγ

-β_xγ

1+(γ-1)(β_x/β)²

( γ-1)(β_xβ_y/β)²

( γ-1)(β_xβ_z/β)²

-β_yγ

( γ-1)(β_yβ_x/β)²

1+( γ-1)(β_y/β)²

( γ-1)(β_yβ_z/β)²

-β_zγ

( γ-1)(β_zβ_x/β)²

( γ-1)(β_zβ_y/β)²

1+( γ-1)(β_z/β)²

Cosh[φ]

-Sinh[φ]

0

0

-Sinh[φ]

Cosh[φ]

0

0

0

0

1

0

0

0

0

1

1

0

0

0

0

Cos[φ]

-Sin[φ]

0

0

Sin[φ]

Cos[φ]

0

0

0

0

1

4-Vector Name

4-Vector Components
(temporal, spatial)

Magnitude & Relation to other 4-vectors

Units (mksC) - Description

***Calculus***

4-Displacement
= 4-Delta
= 4-Difference

ΔR = (cΔt, Δr)

[m], Δt = Temporal Displacement, Δr = Spatial Displacement, (Finite Differences)

The 4-vector prototype, the "arrow" linking two events

4-Differential

dR = (cdt, dr)

[m], dt = Temporal Differential, dr = Spatial Differential, (Infinitesimals)

4-Gradient
= 4-Del or 4-Partial
(a vector operator)

∂ = ∂/∂x_μ = (∂/c∂t, -del) = (∂/c∂t, -∇)
= (∂/c∂t, -∂/∂x, -∂/∂y, -∂/∂z)
= (∂_t/c, -∂_x, -∂_y, -∂_z)
= (∂₀/c, -∇) = (∂_t/c, -∇)


∂·∂ = (∂/c∂t,-del)·(∂/c∂t,-del) = (∂_t/c,-del)·(∂_t/c,-del)
= ∂²/c²∂t²-del·del = ∂_t²/c²-del·del


∂[V·R] = V, where V is not a function of 4-position R

[m^-1], ∂ is the partial derivative, -del = -(∂/∂x i + ∂/∂y j + ∂/∂z k) is the negative gradient operator

This is a very important 4-vector operator, often used to generate continuity equations
Used to obtain field-kinematics
It is technically a one-form, a dual of a regular 4-vector
∂/∂x_u = (∂/c∂t, -del) and ∂/∂x^u = (∂/c∂t, del)
∂·∂ is also known as the D'alembertian (Wave Operator Δ)

I usually write out (del) because the nabla/del symbol (∇)
is quite often displayed incorrectly in various browsers.
It should look like an inverted triangle when displayed correctly.

Let g_μ = ∂_μ f = ∂f / ∂x^μ
Using the chain rule, one can show:
g'_ν = ∂f '/∂x'^ν = Σ ( ∂f / ∂x^μ )( ∂x^μ / ∂x'^ν ) = ∂'_ν f ' = ( ∂_μ f )( ∂'_ν x^μ ) = (g_μ)( ∂'_ν x^μ )
However, this appears to be a standard Lorentz transform
∂'^μ = Λ^μ_ν ∂^ν[ function argument ] =  ∂^ν[ function argument ] Λ^μ_ν

importantly:
U·∂ = γ(∂/∂t + u·del) = γ d/dt = d/dτ

The 4-Gradient of a Lorentz scalar is a 4-Vector
∂[V·R] = V, where V is not a function of R

K = (ω/c, k)
R = (ct, r)
K·R = -φ_EM = (ωt-k·r)
∂ = (∂_t/c, -del)
∂[K·R] = (∂_t/c, -del)[ωt-k·r] = (ω/c, k) = K

4-MomentumGradient??

used in QM as momentum-space
vector operator

∂_p= ∂/∂p_μ = (c∂/∂E, -del_p)??
= (c∂/∂E, -∂/∂p_x, -∂/∂p_y, -∂/∂p_z)??

[kg^-1 m^-1 s], ∂_p is the momentum-space partial derivative, -del_p = negative momentum-space gradient

∂[P·R] = P
∂_p[P·R] = R

Not sure is this is actually used or not

***Particle/Event Dynamics***

4-Position
4-Event

R = R^μ = (ct, r), eg. radial coords
= (ct,r,θ,z)

X = X^μ = (ct, x), eg. Cartesian coords
= (ct,x,y,z)

Q = Q^μ = (ct, q), eg. Hamiltonian coords
= (ct,q₁,q₂,q₃)


*Note*
These coordinate choices do not affect the spacetime, they are simply choices of convenience, where mathematical expressions may take on a simpler looking form.

R·R = (Δs)² = (ct)²-r·r = (ct)²-|r|² 

{R·R = 0 for photonic path/null path}

[m], t = Time (temporal), r or x = 3-Position (spatial)

Location of an Event, the most basic 4-vector (when, where)
This is just a 4-Displacement with one of the events at the origin (0,0,0,0) of the chosen coordinate system
c = Speed-of-Light
often seen as X when choosing a Cartesian representation
often seen as R when choosing a radial/cylindrical/spherical representation



interesting derivation:
U = dR/dτ
dR = dτU
R = ∫dτU
let U be a constant, then R = τU = t_oU = t_oγ(c, u) = t(c, u) = (ct, ut) = (ct, x)
where τ = t_o and t = γt_o
time dilation τ = t / γ
Position is essentially the ProperTime * 4-Velocity, like some of the other flux 4-vectors

4-Velocity

U = dR/dτ
= γ(c, u) = dR/dτ = γdR/dt
= γ(c, u_r) = γ(c, u)
= γ(c,u_x,u_y,u_z)
= γ(c, cβ) = γc(1, β)

{U = γ_cc(1,n), for light-like/photonic}

U_o = (c,0) in rest frame

U·U = c²
U₁·U₂ = γ[u₁]γ[u₂](c²-u₁·u₂) = γ[u_rel]c²

[m s^-1], γ = relativistic factor,  u_r = Relativistic 3-Velocity, u = dr/dt = Newtonian 3-Velocity

u_r = (r)' = r'
u = dr/dt = r'
thus, u_r = u
"U is historically used instead of V"
U_o = (c,0), 4-Velocity is always future-pointing time-like
usually seen as U, sometimes as V
only 3 independent components since U·U = c² = constant
the temporal component is determined by the spatial components

this actually gives a reason why several 4-vectors are a (Lorentz Scalar)*(4-Velocity):
It allows these new 4-vectors to have 4 independent components

interesting derivation:
Let A = k U = (a_o,a) = kγ(c, u),where k is a constant
Then  A·U = k U·U = kc²
So, a_o = kγc = γ A·U / c
And, a = kγu = (γ A·U / c²)u = (a_o/c)u

Try 4-momentum P
P = (E/c,p) = m_o U

E/c = γ P·U /c
E = γ P·U
or, in Hamiltonian formalism, H = E = γ P·U

4-Acceleration

A = dU/dτ
A = γ(c dγ/dt, dγ/dt u+γ a) = dU/dτ = γ dU/dt
=d²R/dτ²
= γ(c dγ/dt, a_r)
= γ(c γ', a_r), where γ' = dγ/dt
= γ(c γ', γ' u+γ a)
= γ(a_r·u/c, a_r), because A·U = 0
= γ²[(γ²/c)(u·a), a + (γ²/c²)(u·a) u]


A_o = (0,a_o) in rest frame

A·A = -a²  
=  - γ⁴[a² + ( γ/c)²(u·a)²]
=  - γ⁶[a² - (u x a)²/c²]

4-Jerk

J = dA/dτ
= γ dA/dt
=d³R/dτ³
= γ( c(dγ/dt)² + cγ(d²γ/dt²), dγ/dt a_r+γ da_r/dt )
= γ( c γ'² + c γ γ'', γ' a_r + γ a_r' )
= γ( c γ'² + c γ γ'', j_r )
where γ' = dγ/dt, γ'' = d²γ/dt², a_r' = da_r/dt

[m s^-3], j_r = Relativistic 3-Jerk, j = da/dt = Newtonian 3-Jerk

j_r = (γa_r)' = γ' a_r + γ a_r'
j = da/dt = a'
γ' = dγ/dt = (γ³/c²)(u·a)
γ'' = dγ'/dt = d²γ/dt² = (γ³/c²)*[(3γ²/c²)(u·a)² + (u'·a) + (u·a')]

4-Snap
= 4-Jounce

S = dJ/dτ
= γ dJ/dt
=d⁴R/dτ⁴

[m s^-4], s_r = Relativistic 3-Snap, s = dj/dt = Newtonian 3-Snap

s_r = (γj_r)' = γ' j_r + γ j_r'
s = dj/dt = j'

***Kinematics***

4-Momentum

P = m_oU = (E_o/c²)U = h_barK

P = (E/c, p) = (mc, p)
= m_o γ(c,u)
= (v_phase|p|/c, p)



{P= (h_barω/c)(1, n) = (E/c)(1, n) = (|p|, p) = |p|(1, n) , for light-like/photonic}



P_o = (E_o/c, 0) = (m_oc, 0) in rest frame

P = ((E_o + p_oV_o)/c²)U, taking into account pressure*volume terms where
pressure p = p_o
volume V = V_o/ γ

P·P = (m_oc)² = (E_o/c)²
{P·P = 0 for photonic}

[kg m s^-1], E = Energy, p_r = Relativistic 3-Momentum, p = mdr/dt = Newtonian 3-Momentum

p_r = p
m_o = RestMass( 0 for photons, + for massive )

4-Momentum used with single whole particles



E = γm_oc² = |p|v_phase , in general
E = pc, for photons


4-Momentum is used with single whole particles
*Note* It is only the 4-Momentum of a closed system that transforms as a 4-vector, not the 4-momenta of its open sub-systems. For example, for a charged capacitor, one must sum both the mechanical and EM momenta together to get an overall 4-vector for the system.

4-MomentumDensity
= 4-MassFlux

G = (u/c, g) = (p_mc, g) = p_{o_m} γ(c, u)
= p_{o_m}U
=(u_o/c²)U = (1/V_o)P = (1/c²)S p_{o_m} = m_on_o??

Starting with 4-Momentum inc. pressure terms...
P = ((E_o + p_oV_o)/c²)U
P = (m_o + p_oV_o/c²)U
P/V_o =  (m_o/V_o + p_oV_o/V_oc²)U
P/V_o = G = (m_o/V_o + p_o/c²)U
G = (p_{o_m} + p_o/c²)U
G = ((u_o + p_o)/c²)U,
pressure p = p_o

[kg m^-2 s^-1], u = EnergyDen = ne, p_m = MassDen = u/c²

g = MomentumDen = (u/c²)u = (e_o)ExB, f = g·u = MomentumFlux
u = 3-velocity, n = ParticleDen, e = EnergyPerParticle

Proper Density p_{o_m} = m_o / V_o

Must be careful here though - *Note*
p_{_m} = m/V = (γm_o)/(V_o/γ) = γ²m_o/V_o = γ²p_{o_m}
The mass density p_{_m} goes as the square of γ
So this p is actually the (0,0) component of the EM Stress Tensor

∂·G = 0 for a conserved Momentum Density, (for ex. a perfect fluid, which is characterized by density, pressure, and worldline velocity)

4-MomentumDensity is used with mass distributions
*Note*  It is only the 4-Momentum of a closed system that transforms as a 4-vector, not the 4-momenta of its open sub-systems.  For example, for a charged capacitor, one must sum both the mechanical and EM momenta together to get an overall 4-vector for the system.

Start with Energy-Momentum Stress Tensor:
T^μν = (p_{o_m}+p/c²)u^μu^ν - pη^μν
Contract with the 4-Velocity
T^μνu_ν = (p_{o_m}+p/c²)u^μu^νu_ν - pη^μνu_ν
T^μνu_ν = (p_{o_m}+p/c²)u^μc² - pu^μ
T^μνu_ν = (c²p_{o_m}+p)u^μ - pu^μ
T^μνu_ν = c²p_{o_m}u^μ
T^μνu_ν = c²G

4-Force
 or
4-Minkowski Force

F = dP/dτ

F = γ(dE/cdt, f_r) = γ(dmc/dt, f_r) = dP/dτ
= m_o(dU/dτ) + (dm_o/dτ)U, generally
=  m_o(A) + (dm_o/dτ)U, generally
= γ[(f_r·u + W)/c, f_r] generally
where W = ( c² - v² )(dm_o/dτ)

F = m_oA , if m_o is stable
= γ(f_r·u/c, f_r), if rest-mass preserving
= γm_o(c dγ/dt, a_r) = γm_o(c γ', a_r)

[kg m s^-2], dE/dt = Power, f_r = Relativistic 3-Force, f = Newtonian 3- Force

f_r = dp/dt = m_o d(γu)/dt = m_o(γ' u + γ u') = m_o(γ' u + γ a) = γm_o[a + (γ²/c²)(u·a) u]
= γ[f + (γ²/c²)(u·f) u]
f = m_ou' = m_oa
a = du/dt = u'

Sometimes known as the Minkowski Force

4-Force is used with single whole particles

F·U =  (m_oA+(dm_o/dτ)U)·U = c²(dm_o/dτ) = γc²(dm_o/dt)

U·F = γ²(dE/dt-u·f_r) = γ dm_o/dt c²
(pure force if dm_o/dt = 0)

solving for dE/dt, the modified rate-of-work equation
dE/dt = u·f_r + c²/γ²(dm_o/dτ)
dE/dt = u·f_r + ( c² - v² )(dm_o/dτ)
(Rate of particle energy change) = (rate of work done by applied force) + (rate of work done by other mechanisms)

Let W = ( c² - v² )(dm_o/dτ), the work done by "non-mechanical" effects, ex. heat
In the co-moving frame,
ω_o = (c²)(dm_o/dτ)

Then F = γ[(f_r·u + W)/c, f_r]

4-Force Density

F_d = F/V_o or F/δV_o F_d = γ(du/cdt, f_dr) = dG/dτ??

[kg m^-2 s^-2], 4-Force divided by rest volume element

4-ForceDensity is used with mass distributions

***Connection to Waves***

4-WaveVector
 or
4-AngWaveVector

K = (1/h_bar)P = (m_o/h_bar)U = (ω_o/c²)U

K = (ω/c, k)
= (ω/c, n ω/v_phase) = (v_phase|k|/c,k)
= (1/h_bar)(E/c,p)
= (ω_o/c²) γ(c,u)

{K = (ω/c)(1, n) , for light-like/photonic}

K·K = (m_oc / h_bar)² = (ω_o/c)²

[rad m^-1], ω = AngularFrequency [rad/s], k = WaveNumber or WaveVector [rad/m]

n = UnitWaveNormalVector
v_phase = ω/k = (E/h_bar)/(p/h_bar) = E/p = phase_velocity = celerity
ω² = k²c² + ω_o²
ω²/k² = c² + ω_o²/k²
v_phase² = c² + ω_o²/k²
v_phase = √[c² + ω_o²/k²]
v_group = dω/dk = u = event velocity for SR waves
ω_o = RestAngularFrequency( 0 for photons, + for massive )
ω = 2πν, k = 2π/λ
ω = γω_o
k everywhere points in the direction orthogonal to planes of constant phase φ
where phase φ = -K·R= - ( ωt - k·r ) = ( k·r - ωt )

∂[K·R] = K
( v_phase * u = c² ) from K = (ω_o/c²)U
Since (0 <= u <= c), then (c <= v_phase <=Infinity) such that ( v_phase*u = c² )
The variable u can either be taken to be the velocity of the particle/event or the group velocity of the corresponding matter wave.
ω = √[k²c² + ω_o²] from K·K
dω/dk = (1/2)(1/√[k² c² + ω_o²])*2kc² = (1/ω)*kc² =  kc²/ω =  c²/v_phase = u
Thus v_group = u = v_event
v_group = dω/dk = (∂ω/∂k_x,∂ω/∂k_y,∂ω/∂k_z) = u
v_phase = ω/k = (ω/k_x,ω/k_y,ω/k_z)

4-Frequency

***Break with standard notation***

better to use the 4-WaveVector

Ν = (ν, c/λ n)
=(c/2π)K
= (ν, νn) = ν(1,n) , for light-like/photonic

[cyc s^-1], ν = ω/2π, λ = 2π/k

ω = 2πν, k = 2π/λ
νλ = c, for photonic

***this is bad notation based on our 4-vector naming convention***
the c-factor should be in the temporal component
the 4-vector name should reference the spatial component
I simply include it here because it is common in the literature

4-CycWaveVector
or
4-InverseWaveLength

Kcyc = (ν/c,1/λ n)
= (1/λ)(v_phase/c, n) = (ν)(1/c, n/v_phase)
=(1/2π)K = (1/h)P = (ν_o/c²)U = (1/h)P = (m_o/h)U
= (ν/c)(1, n) , for light-like/photonic

sometimes called L

[cyc m^-1], ν = CyclicalFrequency [cyc/s], λ = WaveLength [m/cyc]

n = UnitWaveNormalVector
ν = ω/2π = Frequency
1/λ = k/2π = Inverse WaveLength [cyc/m]
h_bar = h/2π = Dirac's Const
v_phase = λν = (Phase) Velocity of Wave (sometimes called celerity)

***Flux 4-Vectors***

Flux 4-Vectors all in form of : V = {rest_charge_density} U
V = {rest_charge}n_o U
where n = γn_o
alternately,
V = (cs,f)
where s = source, f = flux vector
and ∂·V = 0 for a conserved flux

Flux 4-Vectors all have units of [{charge} m^-2 s^-1] = [{charge_density} m s^-1]
Flux is the amount of {charge} that flows through a unit area in a unit time
Flux can also be thought of as {charge_density_velocity} = {current_density}
{charge} [{charge_unit}]
{charge_density} [{charge_units}/m³]
{flux} = {charge_density_velocity} = {current_density} [({charge_units}/m³)*(m/s)]
= {charge per area per second}

4-NumberFlux
"SR Dust"

N = (cn, n_f) = n_o γ(c, u) = n(c, u)
= n_oU

[(#) m^-2 s^-1], n_o = RestNumberDensity [#/m³], n = γn_o = NumberDensity [#/m³]
n_f = nu = NumberFlux [(#/m³)*(m/s)]
# of stable particles N = n_oV_o = nV
This is the SR "Dust" 4-Vector, which is valid for a perfect gas,
i.e. non-interacting particles, no shear stresses, no heat conduction

N = Σ_a [∫dτ δ⁴(x-x_a(τ))(dX_a/dτ)] = = Σ_a [∫dτ δ⁴(x-x_a(τ))(U_a)]
∂·N = 0 for a conserved NumberFlux

4-VolumetricFlux

V  = V_oU??

[(m³) m^-2 s^-1], V_o = RestVolume

4-ElectricCurrentDensity
=4-CurrentDensity
= 4-ElectricChargeFlux

J = (cρ, j) = ρ_o γ(c, u) = ρ(c, u)
= ρ_oU = qn_oU = qN = J_elec

[(C) m^-2 s^-1], ρ_o = RestElecChargeDensity [C/m³], ρ = γρ_o = ElecChargeDensity
j = γ ρ_o u = ρu =ElecCurrentDensity = ElecChargeFlux [(C/m³)*(m/s)]
j = α(E+uxB), α = Conductivity
q = Electric Charge [C]
ρ_o = qn_o [C/m³]
∂·J = 0 for a conserved ElectricCurrent

4-MagneticCurrentDensity
= 4-MagneticChargeFlux
= Zero (so far..)

J_mag = (cρ_mag, j_mag) = ρ_{o_mag} γ(c, u)
= ρ_{o_mag}U = q_{o_mag}n_oU
= Zero (so far...)

[(MagCharge) m^-2 s^-1], ρ_{o_mag} = RestMagChargeDensity, ρ_mag = γρ_{o_mag =} MagChargeDensity
j_mag = MagCurrentDensity = MagChargeFlux
q_{o_mag} = Magnetic Charge
to date: ρ_{o_mag} = 0 and j_mag = 0 -- no magnetic (monopole) charges yet discovered

4-ChemicalFlux

[(mol) m^-2 s^-1],

4-MassFlux
= 4-MomentumDensity

G = (u/c, g) = (cρ_m, g) = ρ_{o_m} γ(c, u) = ρ_{o_m}U
ρ_{o_m} = m_on_o?? = (1/c²)S

[(kg) m^-2 s^-1], u = EnergyDen = ne, p_m = MassDen = u/c²
g = MomentumDen = (u/c²)u = (e_o)ExB, f = g·u = MomentumFlux
u = 3-velocity, n = ParticleDen, e = EnergyPerParticle

Poincare' made the observation that,
since the EM momentum of radiation is 1/c² times the Poynting flux of energy,
radiation seems to possess a mass density 1/c² times its energy density

4-PoyntingVector
= 4-EnergyFlux
= 4-RadiativeFlux

= 4-MomentumDensity?

S = (cu, s) = u_o γ(c, u) = u_oU = c²G
????u_o = E_on_o ??

[(J) m^-2 s^-1], u = EnergyDen = ne, s = EnergyFlux = PoyntingVector = uu = c²g = Ne
u_e = (ε_oE·E+B·B/µ_o)/2 = (E·D+B·H)/2 = EM energy density

typically see ∂u/∂t = - del·s + J_f · E
which in 4-vector notation would be ∂·S = j_f · E, where j_f is the current density of free charges, so not conserved generally
however, make the following observation:
∂(u_m)/∂t = j(t,x)·E(t,x), where this is rate of change of kinetic energy of a charge
then let
u = u_e+u_m, s = s_e+s_m, ∂·S = ∂(u_e+u_m)/∂t +del·(s_e+s_m) = 0
we have conservation/continuity again, by allowing energy to transform into different types. In the example, energy is passing back & forth between the physical charges and the EM field itself. Energy as a whole is still conserved.


ε_o = Permittivity, µ_o = Permeability
ε_oµ_o = 1/c²
s = (E x B)/µ_o = EnergyCurrentDensity
u = 3-velocity, n = ParticleDen, e = EnergyPerParticle, N = ParticleFlux = nu

see also Umov-Poynting Vector for generalization to mechanical systems
S = (cu, s) = (c(u_e+u_m),s_e+s_m)
∂·S = ∂(u_e+u_m)/∂t +del·(s_e+s_m) = 0
u_m = mechanical kinetic energy density
s_m = mechanical Poynting vector, the flux of their energies
The sum of mechanical and EM energies, as well as the sum of mechanical and EM momenta are conserved inside a closed system of fields and charges.  Another way to say this is that only the four-momentum of a closed system transforms as a 4-vector, not the four-momentums of its open sub-systems.

Since only the microscopic fields E and B are needed in the derivation of S = (1/µ_o)(ExB), assumptions about any material possibly present can be completely avoided, and Poynting's vector as well as the theorem in this definition are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the case above

Energy types: electrical, magnetic, thermal, chemical, mechanical, nuclear
The density of supplied energy is restricted by the physical properties through which it flows. In a material medium, the power of energy flux U is restricted, (U < vF), where v is the deformation propagation velocity, usually the speed of sound, F may be any elastic or thermal energy, U is a vector. div U determines the amount of energy transformation into a different form. For a gaseous medium, U = a √[T] p, where A is a coefficient which depends on molecular composition, T is temperature, p is pressure.

not be be confused with the 4-EntropyFlux Vector S

4-EntropyFlux

S = (cs, s_f) 
= s_o γ(c, u)
= s_oU
= sρ_oU

more correct term including heat/temp
S = s_oU + Q/T_o 
= sρ_oU + Q/T_o
∂·S >= 0

[(J ŗK^-1) m^-2 s^-1],  s_o = RestEntropyDensity = q_o/T, s_f = EntropyFlux
Entropy S = ∫s_odV = k_B ln  Ω, where Ω = # of microstates for a given macrostate
∂·S >= 0

S = s_oU + Q/T_o
where Q is the Thermal Heat Flux 4-vector
1st term is entropy carried convectively with mass
2nd term is entropy transported by flow of heat (generalization of dS = dQ/T)

If Q·Q = const, then the Jacobian says that dq/q_o ~ dq/T is a Scalar Invariant

Q = (cq, q_f) = q γ(c, u) = qU = ( c (k/a)T , q_f)
= ( c (k/a)T , -k del [T])

not be be confused with the 4-Poynting Vector S

***Thermodynamic***

4-InverseTempFlux

β = β_o U = (1/k_BT_o) U
where β_o = 1/k_BT_o
dS = β·dP,  differential entropy

Considered on  Thermodynamic principles
also known as a Killing vector

"The proper relativistic temperature is not agreed upon by Einstein, Ott, and Landsberg, who respectively think that moving objects are colder, hotter and invariant. You can try reading these and seeing what each do, how they differ in their assumptions and why they disagree with each other. However, given the fact that there does seem to exist genuine disagreement, it is suspected that the matter has not been settled. Also since neither SR nor thermodynamics are complicated in their mathematical settings, the problem is likely to be that of a foundational nature --- i.e. what does temperature mean for a moving object."

4-MomentumTemperature

P_T = P/k_B = (p_T⁰/c, p_T) = (T/c, p_T) = ((E/k_B)/c, p/k_B)
p_T⁰ = T = Temperature (in ŗK)

simply dividing 4-Momentum by Boltzmann's const. k_B
which gives E/c = k_BT/c, or E = k_BT

[ŗK m^-1 s],
ŗK = degree Kelvin Temperature
k_B = 1.380 6504(24)×10⁻²³ [J/ŗK] Boltzmann's constant

Not sure if this is valid, but perhaps useful as a gauge of photon temperature
based entirely on dimensional considerations of k_B [J/ŗK] energy/temperature
similarly to c [m/s] being a fundamental constant relating length/time

***Diffusion/Continuity based***
see
Atomic Diffusion
Brownian Motion
Electron Diffusion
Momentum Diffusion
Osmosis
Photon Diffusion
Reverse Diffusion
Thermal Diffusion

4-Potential Flux??

V = (cq, q_f) = q γ(c, u) = qU??
where q = [1]??
= ( c (k/a)φ , q_f)??
= ( c (k/a)φ , -k del [φ])??

needs work

Potential Flow for Velocity??
Velocity Potential

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

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

In fluid dynamics, a potential flow is described by means of a velocity potential , φ being a function of space and time. The flow velocity v is a vector field equal to the negative gradient, del, of the velocity potential φ:

Incompressible flow

In case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence:

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

where Δ = ∇·∇ is the Laplace operator. In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of the flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of Bernoulli's principle.

The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:

Steady flow

Main article: Steady flow
The flow of a fluid is said to be steady if v does not vary with time. That is if
dv/dt = 0

Incompressible flow

Main article: Incompressible flow
A fluid is incompressible if the divergence of v is zero:
del·v = 0
That is, if v is a solenoidal vector field.

Irrotational flow

Main article: Irrotational flow
A flow is irrotational if the curl of v is zero:
del x v = 0
That is, if v is an irrotational vector field.

Vorticity

Main article: Vorticity
The vorticity, ω, of a flow can be defined in terms of its flow velocity by
ω = del x u  
Thus in irrotational flow the vorticity is zero.

The velocity potential

Main article: Potential flow
If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field φ such that
v = -k del [φ]
The scalar field φ is called the velocity potential for the flow. (See Irrotational vector field.)

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

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

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

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

del·v = 0

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

v = del x A

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

del·v = del·(del x A) = 0

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

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

del x v = 0
del·v = 0

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

v = del φ (1)

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

del·del φ = 0

which is equivalent to

del² φ = 0

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

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

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

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

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

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

del x u = 0

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

u = del Φ

Φ is known as a velocity potential for u.

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

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

4-HeatFlux
4-ThermalHeatFlux
4-ThermalEnergyFlux

Q = qU  
= q γ(c, u)
= (cq, q_f) 
= ( c (k/α)T , q_f)
= ( c (k/α)T , -k del [T])

that q_f = -k del [T] is an additional constraint


Alternate Def:???
Q = k ∂[T]
= k(1/c ∂T/∂t , - del [T])
= (k/c ∂T/∂t , -k del [T])

which would imply
k/c ∂T/∂t = c (k/α) T  ????
∂T/∂t = (c² /α) T  ????


yet another...
Q^α = -K P^αβ(T,_β + T A_β/c²)

Heat flux 4-vector balances the EM energy flux vector if expansion, rotation, and shear of fluid vanish

[(J) m^-2 s^-1] = [(W) m^-2],
Potential Flow for Heat
k = Thermal Conductivity [W/(m °K)] *Note - this is not Boltzmann's const k_B*
α = k/(ρc_p) = Thermal Diffusivity [m²/s]
ρ = Density [1/m³]
c_p = Specific heat capacity [J/ °K]
ρc_p = Volumetric heat capacity [J/m³ °K]
T = Absolute Temperature [°K]
(k/α)T = Thermal energy density [W s / m³] = [J/m³], q = Heat Flux [W/m²] = Thermal Energy Flux

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

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

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

It should be noted that T is a scalar temperature field T[X]

see: A Proposed Relativistic, Thermodynamic Four-Vector

see Hydraulic Analogies
quantity: Heat Q [J]
potential: Temperature T [°K] = [J/k_B]
flux: Heat Xfer Rate Qdot [J/s]
flux density: Heat Flux Qdot'' [W/m²]
linear model: Fourier's Law Qdot'' = -k del [T]

4-DarcyFlux
4-HydraulicFlux

Q = (cq, q_f) = q γ(c, u) = qU = ( c (βφ)P , q_f)
= ( c (βφ)P , -κ/μ del [P])

[(m³) m^-2 s^-1] = [m s^-1],
Potential Flow for Volume/Hydraulics
κ = Permeability [m²]
μ = Dynamic Viscosity [Pa s]
P = Pressure [N m^-2]
β = Compressibility Coefficient
φ = Porosity [1] q = Flux [m s^-1] = Volumetric Flux <> particle velocity
v = q/φ = pore velocity

Darcy's Law Equation: q = -κ/μ del [P]
Pressure Diffusion Equation: del·q = -κ/μ del·del [P] = βφ∂P/∂t, by imposing incompressibility
giving Pressure Diffusion Waves

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

Darcy's Law: q = -κ/μ del [P]
The minus sign ensures that flux flows down the pressure gradient

Darcy's Law - derivable from Navier-Stokes
see Fourier's law for heat conduction
see Ohm's law for electrical conduction
see Fick's law for diffusion

see Hydraulic Analogies
quantity: Volume V [m³]
potential: Pressure P [Pa] = [J/m³]
flux: Current φ_V [m³/s]
flux density: Velocity [m/s]
linear model: Poiseuille's Law φ_V = ...

4-ElectricChargeFlux

Q = (cq, q_f) = q γ(c, u) = qU = ( c ρ, j)

[(C) m^-2 s^-1],
Potential Flow for Charge
acts differently, presumably because this is a "charged" field, where the particle interacts with the field.

μ = mobility
σ = specific conductivity = q n μ
where n = concentration of carriers

Conservative Potential: E = (- del [φ])
Ohm's Law: j = σ E = -σ del [φ]
Gauss Law: del·E = del·(- del [φ]) = - del·del [φ] = ρ/ε_o

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



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

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

Ohm's Law: j = -σ del [φ]
The minus sign ensures that current flows down the potential gradient

see Hydraulic Analogies
quantity: Charge Q [C]
potential: Potential φ [V] = [J/C]
flux: Current I [A] = [C/s]
flux density: Current Density j [A/m²]
linear model: Ohm's Law j = - σ del [φ]

***Angular Momentum/Spin/Polarization***

4-SpinMomentum
or
Pauli-Lubanski 4-vector

W = (w₀,w) = (u·w/c,w)
because W·U = 0
W = (w⁰,w) = (p·Σ , P⁰Σ + p x k)
where Σ is the spin part of angular momentum j
W = m_o S
where S is the 4-Spin

[spin-momentum],
W·W = (u·w/c,w)·(u·w/c,w) = (u·w/c)² - w·w) = - w·w = - m² s(s+1) W² = ( w₀² - w·w ) = - (w·w) = - (P₀²Σ²) = - m²c² h_bar² s(s+1)
where Σ is the spin part of angular momentum j
(P₀²) = (m²c²)
(Σ²) = h_bar² s(s+1)

W·W = 0 for photonic

plays the role of covariant angular momentum

see Bargmann-Michel-Telegdi (BMT) dynamical eqn

4-Spin

S = (s⁰,s) = (u·s/c,s)
because S·U = 0
S = (γ β·s_o , s + [γ²/(γ+1)](β·s_o) β) in moving frame
S = (1/m_o) W= (U·U/P·U) W
where m_o = √[P·P/U·U] = P·U/U·U

Magnetic moment μ = -(g/2)(e/mc) s
S_o = (0,s_o) in rest frame

[ J s], = [spin] Spin = IntrinsicAngMomentum, u·s/c = component such that U·S = 0
4-Spin is orthogonal to 4-Velocity, so time component is zero in rest frame S_o=(0,s_o)
This is an axial vector, or pseudovector
4-Spin has only 3 independent components, not 4, due to U·S = 0
S_o=(0,s_o), 4-Spin is always space-like
S·S = (u·s/c,s)·(u·s/c,s) = ((u·s/c)² - s·s) = - s_o·s_o = - h_bar² s_o(s_o+1)

s·s |s,m> = h_bar² s(s+1) |s,m>
s_z |s,m> = h_bar m |s,m>
for s = {0 , 1/2 , 1 , 3/2 , 2 , 5/2 , ...}
for m = {-s, -s+1, ..., s-1, s}
SpinMultiplicity[m] = (2s+1) denotes the # of possible quantum states of a system with given principal s
for s = 0, {m} = {0} singlet
for s = 1/2, {m} = {-1/2 , 1/2} doublet
for s = 1, {m} = {-1 , 0 , 1} triplet

|s| = h_bar √[s(s+1)],
[ S_x ,S_y ] = i h_bar ε_xyz S^z

Spin raising/lowering operators:
S_± |s,m> = h_bar √[s(s+1) - m(m ± 1)] |s,m> where S_± = S_x ± i S_y

Bargmann-Michel-Telegdi (BMT) dynamical eqn. for spin
dS/dτ = (e/mc)[ (g/2) F^μβS_β + (1/c²)(g/2-1))v^μS_αF^αβv_β ]
which leads to Thomas precession in the rest frame

4-SpinDensity

4-Polarization
or
4-JonesVector

Ε = (ε⁰, ε) = (ε·u/c,ε) for a massive particle
= (ε⁰, ε) = ((c/v_phase) ε·n,ε) for a wave

= (ε⁰, ε) = (ε·n,ε) ,for light-like/photonic

for photon travelling in z-direction
using the Jones Vector formalism n = z / |z| E = (0,1,0,0) :
x-polarized linear E = (0,0,1,0) :
y-polarized linear E = √[1/2] (0,1,1,0) :
45 deg from x-polarized linear E = √[1/2] (0,1,i,0) :
right-polarized circular = spin 1
E = √[1/2] (0,1,-i,0) :
left-polarized circular = spin -1

General-polarized E = (0,Cos[θ]Exp[iα_x],Sin[θ]Exp[iα_y]),0) E* = (0,Cos[θ]Exp[-iα_x],Sin[θ]Exp[-iα_y]),0)

Angle θ describes the relation between the amplitudes of the electric fields in the x and y directions
Angles α_x and α_y describe the phase relationship between the wave polarized in x and the wave polarized in y
Ε·E* =
= (+0²
- Cos[θ]Exp[iα_x]Cos[θ]Exp[-iα_x]
- Sin[θ]Exp[iα_y]Sin[θ]Exp[-iα_y]
-0² ) =
= (+0-Cos[θ]²-Sin[θ]²-0)
= - (Cos[θ]²+Sin[θ]²)
= -1 Ε·E* = -1

[1], ε = PolarizationVector **This 4-vector has complex components in QM**
Called helicity for massless particles
Helicity is spin component along the direction of motion. "Helicity is the only truly measurable component of spin for a moving particle, but at low enough velocities (non-relativistic), the spin component m along an external axis becomes an alternative observable."
Like the 4-Spin, Ε orthogonal to U, or K, so time component = 0 in rest frame
This is cancellation of the "scalar" polarization
This would again give only 3 independent components
Ε·U = 0, Ε·K = 0,
Additionally, Ε·E* = -1 (normalized to unity along a spatial direction)
Normalization combined with |u| = c for photons
is enough to reduce it to 2 independent components for photons.
Ε·E* = -1 imposes
(ε·u/c)² -ε·ε* = -1
(ε·u/c)² -1 = -1
(ε·u/c)² = 0
ε·u = 0, so that the spatial components must be orthogonal.
For a massive particle, there is always a rest frame where u = 0, so ε can have 3 independent components.
For a photonic particle, there is no rest frame.( ε·u/c = ε·n = 0 ) is therefore an additional constraint, limiting ε to 2 independent components, with polarization ε orthogonal to direction of photon motion n.
This is cancellation of the longitudinal polarization.
According to Wikipedia-Gauge Fixing, Many of the differences between classical electrodynamics and QED can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances.
see Field Quantization, by Walter Greiner, Joachim Reinhardt

4-SpinPolarization

In the rest frame, where K = (m,0), choose a unit 3-vector n as the quantization axis.
In a frame in which the momentum is K = (k⁰,k)
the spin polarization of a massive particle N = N^μ = ( k·n/m , n + (k·n) k / (m(m + k⁰)) )
N·N = -1, Normalized to spatial unity K·N = 0, Orthogonal to Wave Vector

alternately, S = S^μ = ( p·s/m , s + (p·s) p / (m(m + p⁰)) )
which makes more sense, called the covariant spin vector
W = (w⁰,w) = (p·Σ , P⁰Σ + p x k)
where Σ is the spin part of angular momentum j

W² = -m² Σ² = -m² s(s+1)
Σ = 0 represents a spin-0 particle
Σ = Dirac spinor represents a spin-1/2 particle, with Σ² = 3/4 the unit matrix I
W·N = -m Σ·n = - m s, the component of spin along n as measured in the rest frame.
s is the spin component in the direction n that would be measured by an observer in the particle's rest frame

Apparently this only works for massive particles, so the N 4-vector is undefined for massless

Instead, helicity h = Σ·k / |k|,
h = +1/2 or -1/2 for spin 1/2 particle
helicity is component of spin parallel to the 3-vector momentum k

w⁰ = (k·Σ) and k⁰ = |k| for a massless particle

alternately
N^μ = ( T^μ - P^μ(pt/m²))(m/|p|) where T^μ is the unit time-like 4-vector

4-PauliMatrix

Σ = Σ^μ = (σ⁰, σ)

where Σ·Σ = -2σ⁰

The components of this 4-vector are actually the Pauli Matrices

***Electromagnetic Field Potentials***

4-VectorPotential
 or
4-VectorPotential_EM

A = (Φ/c, a)
4-VectorPotential of an arbitrary field

A_EM = (Φ_EM/c, a_EM)
4-VectorPotential of the EM field


I believe that in some circumstances the following is true:

A = (Φ_o/c²) U

A = (Φ/c, a)
= Φ_o/c²  γ(c, u)
= (γΦ_o/c, γΦ_o/c²u) 

giving
Φ = γΦ_o
a = γΦ_o/c²u

see the relativistic Lagrangian and Hamiltonian for reasoning for this...

[kg m <chargetype>^-1 s^-1], for arbitrary field
[kg m C^-1 s^-1] for EM field
Φ_EM = ScalarPotenial_EM
a_EM = VectorPotenial_EM
Electric Field E = -del[Φ_EM]-∂a_EM/∂t
Magnetic Field B = del x a_EM
Electric Field E [N/C = kg·m·A⁻¹·s⁻³]
Magnetic Field B [Wb/m² = kg·s⁻²·A⁻¹ = N·A⁻¹·m⁻¹]

for 4-VectorPotenial of a moving point charge (Liénard-Wiechert potential)
A_EM = (q/c4πε₀) U / (R·U) where (R·R = 0, the definition of a light signal)

4-VectorPotentail of an Ε polarized plane-wave
A_EM ~ Ε Exp[i K·R]
(∂·∂)A_EM = µ₀ J+∂(∂·A_EM) Inhomogeneous Maxwell Equation

(∂·∂)A_EM = µ₀ J Homogeneous Maxwell/Lorentz Equation (if ∂·A_EM = 0 Lorenz Gauge)

Complex phase  dθ = q A[X]·dX units?(SI,guassian?)
NOTE: Neither the phase nor the components of the phase connection are physically observable although differences in phase connection may be observed via interference experiments. (Ref: Aharanov-Bhom effect.)

4-VectorPotentialMomentum
 or 
4-PotentialMomentum

Q_EM = (E_EM/c, p_EM)
  = q A_EM
  = q (Φ_EM/c, a_EM)

[kg m s^-1],
E_EM = ScalarPotenialEnergy
p_EM = VectorPotenialMomentum
Energy and Momentum of the EM field itself, for a single charge
Q_EM = (E_EM/c, p_EM) = q_o A_EM
P = h_barK
K = (ω/c, k) = (ω/c,ω/v_phase n)
Q_EM = h_barK_EM (for an EM field)
q_o (Φ_EM/c, a_EM) = h_bar(ω/c, k)
q_o Φ_EM = h_barω ; q_o a_EM = h_bar k

4-Potential
 or
4-Potential_EM
***break with standard notation***

better to use the 4-VectorPotential_EM

Φ_EM = (Φ_EM,c a_EM)
= c A_EM

[kg m² C^-1 s^-2], Φ_EM = ScalarPotenial_EM ,a_EM = VectorPotenial_EM

***this is bad notation based on our 4-vector naming convention***
the c-factor should be in the time component
the 4-vector name should reference the space component
I simply include it here because it is common in the literature

4-Momentum_EM
4-CanonicalMomentum
4-TotalMomentum

P_T = (E_T/c + qΦ_EM/c, p_T + qa_EM)
P_T = Π = P + qA_EM = m_oU + qA_EM
=(H/c,p_T)
= (γm_oc+q Φ_EM/c,γm_ou+q a_EM)

[kg m s^-1], **Momentum including effects of potentials**
also known as Canonical Momentum
where P is the Kinetic Momentum term
where qA is the Potential Momentum term
Total Momentum = Kinetic Part + Potential Part

H = γmoc2 + q ΦEM
p_T = γmou + q aEM

Can then write dynamic momentum as P = P_T - qA
and all the usual relations still hold:
P = m_oU = (E_o/c²)U = h_barK

4-Gradient_EM
Gauge Covariant Derivative

D_EM = (∂/c∂t + iq/h_bar Φ_EM/c, -del + iq/h_bar a_EM)
= ∂ + (iq/h_bar)A_EM
for electrons, commonly seen as
D_EM = ∂ - (ie/h_bar)A_EM
where e is the electric charge

[m^-1], **Gradient including effects of EM potentials**

Minimal coupling based on principle of local gauge invariance

***Position space & Momentum Space Differentials***

4-Differential

dX = (cdt, dx)

[m], dt = Temporal Differential, dx = Spatial Differential

4-Volume Element (Flux?)

dV = (c dv₀,dv)
= (dV_o) U
= (dV_o) γ(c, u)
= γ(dV_o) (c, u)
= (dV)(c, u)
= (c dV,dV u)

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

dV =  γdV_o ????

V should be as follows
V = V_o/γ

Perhaps acts a little differently since this is from a "vector-valued volume element", and not a straight volume.

[m³], A vector-valued volume element is just a 4-vector that is perpendicular to all spatial vectors in the volume element, and has a magnitude that's proportional to the volume.

Using Clifford Algebra one can represent an oriented volume element by a three-form. In a 4d spacetime, a 3-form has a dual representation (Hodges Dual) which is a 1-form, which is basically a vector.
Basically this means that you define a volume element by the space-time vector that's perpendicular to it, and you make the length of this space-time vector proportional to the proper volume you wish to represent.

Hodge Duality in SR
n=4 Minkowski spacetime with metric signature (+,-,-,-) and coordinates (t,x,y,z) gives
*dt = dx^dy^dz (* is the Hodge star operator, ^ is the wedge product operator)

alternately, dV = √[-g]d⁴x is an invariant volume element scalar??

c dt dV = dx⁰ dx¹ dx² dx³ = dx'⁰ dx'¹ dx'² dx'³
c dt = dx⁰
dV = dx¹ dx² dx³ = d³x
(c dt)(dV) = (dx⁰ )(dx¹ dx² dx³) = d⁴x

d³x = d³x'/γ
d³p/p_o = d³p'/p'_o
p'_o = p_o/γ
d³x d³p = d³x' d³p'

dV_o·dX = (dV_o,0)·(cdt,dx) = (dV_o cdt) = dxdydz cdt = d⁴x
dV·dX = d⁴x
Hence, the differential 4-element is an invariant

dq = ρ d³x = j⁰/c d³x
Interesting derivation:

dq = ρ dV differential charge element
(dq)dX = (ρ dV)dX = ρ dV (dt/dt) dX = ρ (dV dt) dX/dt = ρ (dV dt) (1/γ) dX/dτ = ρ (dV dt) (1/γ)U = γρ_o (dV dt) (1/γ) U = ρ_o (dV dt) U = (dV dt) ρ_oU = (dV dt) J = ( dV(c/c) dt)J = ( dV c dt/c) J = (d⁴x/c) J
dq dX = (d⁴x/c) J

Apparently, (dx¹ dx² dx³)/x⁰ is also an invariant, based on Jacobian

also, d³x Δt is an invariant
(dV·ΔR) / (U·U) = d³x Δt

4-Momentum Differential

dP = (dE/c, dp)

[kg m s^-1], dE = Temporal Momentum Differential, dp = Spatial Momentum Differential

4-MomentumSpace
   Volume Element (Flux?)

dV_p = (c dv_p0,dv_p)
= (dV_po) U
= (dV_po) γ(c, u)
= γ(dV_po) (c, u)
= (dV_p)(c, u)
= (c dV_p,dV_p u)

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

[kg³ m³ s^-3], A vector-valued MomentumSpace volume element is just a 4-vector that is perpendicular to all spatial vectors in the MomentumSpace volume element, and has a magnitude that's proportional to the MomentumSpace volume.

Using the same Clifford Algebra idea from position-space, I think this can be done

(dE/c)(dV_p) = dp⁰ dp¹ dp² dp³ = dp'⁰ dp'¹ dp'² dp'³
dE/ c = dp⁰
dV_p = dp¹ dp² dp³ = d³p
(dE/ c)(dV_p) = (dp⁰ )(dp¹ dp² dp³) = d⁴p

d³x = d³x'/γ
d³p/p_o = d³p'/p'_o
p'_o = p_o/γ
d³x d³p = d³x' d³p'

dV_po·dP = (dV_po,0)·(dE/c,dp) = (dV_po dE/c) = dp_xdp_ydp_z dE/c = d⁴p
dV_p·dP = d⁴p
Hence, the differential momentum 4-element is an invariant


also,

dV_o·dV_po = (dV_o,0)·(dV_po,0) = (dV_odV_po) = (dx¹ dx² dx³)(dp¹ dp² dp³)
= d³xd³p in the rest frame
Thus dV·dV_p = d³x d³p generally, so (d³x d³p) is a Lorentz scalar invariant

Apparently, (dp¹ dp² dp³)/p⁰ is also an invariant, based on Jacobian

***Special 4-Vectors***

4-Zero

Zero = (0,0) = (0,0,0,0)

[*], All components are 0 in all reference frames, the only vector with this property
Square Magnitude = 0, Length = |0| = 0

4-Null

Null = (a,a) = (a,an) = a(1,n)
where n is the unit 3-vector

Null·Null = a(1,n)·a(1,n) = a²(1*1 - n·n) = a²(0) = 0
Null·Null = 0

[*], Any 4-vector for which the temporal component magnitude equals the spatial component magnitude
|a⁰| = |a|
which leads to the magnitude being 0, or LightLike/Photonic
ex. The 4-Velocity of a Photon, the 4-Momentum of a Photon

4-Unit Temporal

T = (1,0)
T·T = (1,0)·(1,0) = (1*1 - 0·0) = 1

[*], The Unit Temporal 4-Vector
Square Magnitude = 1, Length = |1| = 1

4-Unit Null

N = (1,n)
N·N = (1,n)·(1,n) = (1*1 - n·n) = 0

[*], The Unit Null 4-Vector
n = unit 3-vector, |n| = 1
Square Magnitude = 0, Length = |0| = 0
The Null Vector is "perpendicular" to itself.
Actually not sure if "unit" null applies here...

4-Unit Spatial

S = (0,n)
S·S = (0,n)·(0,n) = (0*0 - n·n) = -1

[*], The Unit Spatial 4-Vector
n = unit 3-vector, |n| = 1
Square Magnitude = -1, Length = |-1| = 1

4-Basis Vectors 
(1 time + 3 space)

B_t = (1,0,0,0)
B_x = (0,1,0,0)
B_y = (0,0,1,0)
B_z = (0,0,0,1)

A tetrad of 4 mutually orthogonal, unit-length, linearly-independent, basis vectors
This is simply one basis, there are others

4-Basis Vectors 
(null tetrad)

B_n1 = √[1/2] (1,0,0,1)
B_n2 = √[1/2] (1,0,0,-1)
B_n3 = √[1/2] (0,1,i,0)
B_n4 = √[1/2] (0,1,-i,0)

A tetrad of unit-length, linearly-independent, null basis vectors
Note the complex components, since there can be only 2 real linearly independent null vectors
This is simply one basis, there are others
** Need to double-check these ***
see null tetrad, Sachs tetrad, Newman-Penrose tetrad

4-ProbabilityCurrentDensity
= 4-ProbabilityFlux
= 4-QM_ProbabilityFlux

J_prob = (cρ, j) = (ih_barρ_o/m_o)(ψ*<-∂->ψ)/2 

J_prob = (cρ, j) = (ih_barρ_o/m_o)(ψ*∂[ψ]-∂[ψ*]ψ)/2
  
J_prob = (cρ, j) = ρ_o(ψ*ψ) γ(c, u) = ρ(c, u)?
= ρ_o(ψ*ψ)U?


J_prob,Dirac = (cρ, j) = ic(ψ γ_μ ψ)

where ψ = ψt γ₄


J_prob = (ih_barρ_o/2m_o)(ψ*<-∂->ψ + 2iqAψ*ψ)
for particle in external potential

Note:
A<-∂->B = A(∂B) - (∂A)B

[# m^-2 s^-1],  4-Probability Current Density is proportional to the 4-Momentum

obeys the continuity equation
[#/m³], ρ = ProbChargeDensity
j = ρu =ProbCurrentDensity = ProbChargeFlux [(#/m³)*(m/s)]

ρ = (ih_barρ_o/2m_oc²)(ψ* ∂_t[ψ]-∂_t[ψ*] ψ) 
j = (-ih_barρ_o/2m_o)(ψ* del[ψ]-del[ψ*] ψ) 

Conservation of ProbabilityCurrentDensity
∂·J_prob = ∂ρ/∂t +del·j = 0

ρ = (ρ_o)γ(ψ*ψ) for time separable wave functions
See derivation of Newtonian limit below...

This is the 4-vector which relates SR to the probabilistic formulation of QM

Derived from Klein Gordon eqn. ∂·∂ = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂²/c²∂t²-del·del = -(m_oc / h_bar)²
∂·∂ + (m_oc / h_bar)² = 0

Wave eqn. on ψ, then pre-mult by ψ*
ψ*{∂·∂ + (m_oc / h_bar)² = 0}ψ
ψ*∂·∂ψ + ψ*(m_oc / h_bar)²ψ = 0
ψ*(∂·∂ψ) + (m_oc / h_bar)²ψ*ψ = 0

Wave eqn. on ψ*, then post-mult by ψ
{∂·∂ + (m_oc / h_bar)² = 0}ψ*ψ
(∂·∂ψ*)ψ + (m_oc / h_bar)²ψ*ψ = 0

Subtract the two
ψ*(∂·∂ψ) + (m_oc / h_bar)²ψ*ψ -(∂·∂ψ*)ψ - (m_oc / h_bar)²ψ*ψ = 0
ψ*(∂·∂ψ) - (∂·∂ψ*)ψ = 0

Note that this is equivalent to:
∂·(ψ*∂[ψ]-∂[ψ*]ψ) = 0

So, let
J_prob ~ (ψ*∂[ψ]-∂[ψ*]ψ)

Then continuity holds
∂·J_prob = ∂ρ/∂t +del·j = 0
Conservation of Probability

Event R

Mass m_o = ρ_{o_m}V_o


Energy E_o = m_oc²

WaveAngFreq ω_o

ElecCharge q = ρ_oV_o

MassDensity ρ_{o_m}
Mass m_o = ρ_{o_m}V_o

EnergyDensity u_o = ρ_{o_m}c²

ChargeDensity ρ_o
Charge q = ρ_oV_o

NumberDensity n_o
ParticleNumber N_o = n_oV_o

event

particle
mass

wave

elec. charge

mass
density

charge
density

number
density

pos: R = (ct, r)

m_o at R

ω_o at R

q at R

ρ_{o_m} at R

ρ_o at R

n_o at R

vel: U = dR/dτ

P = m_oU = (E_o/c²)U

K = (ω_o/c²)U = (1/h_bar)P

Jq = qU

G = ρ_{o_m}U = (u_o/c²)U

J = ρ_oU

N = n_oU

accel: A = dU/dτ

F = dP/dτ

F_d = dG/dτ

jerk: J = dA/dτ

Event(SR)

EventMovement

MassEnergy

Particle-WaveDuality

QuantumMechanics(QM)

SpaceTimeVariations

R = (ct, r)

dR/dτ = U = γ(c,u)

U = P/m_o

P = h_bar K

***  K = i ∂  ***

∂ = (∂/c∂t,-del)

or K =  (ω_o/c²)U

R = (ct,r)

Location of an event (i.e. a particle) within spacetime

U = dR/dτ

Velocity of the event is the derivative of event position wrt. Proper Time

P = m_oU

Momentum is just the Rest Mass of the particle/event times its velocity

K = (1 / h_bar )P

A particle's wave vector is just the momentum divided by Dirac's constant, but uncertain by a phase factor

∂ = -i K

The change in spacetime corresponds to (-i) times the wave vector, whatever that means...

(∂·∂ + (m_oc / h_bar)² ) Ψ = 0,

Ψ is a scalar Klein-Gordon eqn for massive spin-0 field

(∂·∂ + (m_oc / h_bar)² ) A = 0

A is a 4-vector Proca eqn for massive spin-1 field

(∂·∂) Ψ = 0

Ψ is a scalar Free-wave eqn for massless (m_o = 0) spin-0 field

(∂·∂) A = 0

A is a 4-vector Maxwell eqn for massless (m_o = 0) spin-1 field, no current sources

P = i h_bar ∂ = -∂(S_act)

∂ = (∂/c∂t,-del)

A_EM = (0,0) *special case*

P_EM = P+qA_EM = i h_bar D_EM

D_EM = ∂+iq/h_bar A_EM

A_EM = (V_EM/c,a_EM)

Relations involving the 4-Position or 4-Displacment:

R·R = (Δs)² = (ct)²-r·r = (ct)²-|r|²  
R·R = 0 for photonic signal

Spacetime position of an event wrt. an origin event

dR·dR = (ds)²
= (c dt)²-dr·dr
= (c dt)²-|dr|²
= (c dτ)²

Differential interval magnitude - the fundamental invariant differential form
This is supposed to be locally valid for all events, even in GR
If we let this occur in a "rest frame", then
dRo·dRo = (c dt_o)² = (c dτ)²

ΔR·ΔR = (Δs)² = (c Δt)²-Δr·Δr = (c Δt)²-|Δr|²

Spacetime displacement interval magnitude - used to derive SR
(Δs)² >0 TimeLike Interval (Δs)² = 0 LightLike Interval (Δs)² <0 SpaceLike Interval

∂·R = 4

The divergence of open spacetime is equal to the number of independent dimensions (t,x,y,z)

K·R = -φ_EM = (ωt-k·r)

Phase of a SR wave; Ψ = a E e ^-iK·R Photon Wave Equation (Solution to Maxwell Equation)

R·U = (ct,r)·γ(c,u ) = γ(c²t - r·u)

Part of expression used in Liénard-Wiechert potential

Relations involving the 4-Velocity:

U·U = c²

The magnitude of 4-Velocity is always c²

U₁·U₂ = γ[u₁]γ[u₂](c²-u₁·u₂) = γ[u_rel]c²

U₁·U₁ = U·U = c²

Relative Gamma Factor
(The scalar product of two uniformly moving event/particle 4-velocities is proportional to the γ factor of their relative velocities)

Also get the relativistic addition of velocities with this:
γ[u₁]γ[u₂](c²-u₁·u₂) = γ[u_rel]c²
γ₁γ₂(c²-u₁·u₂) = γ_relc²
γ₁γ₂(1-β₁·β₂) = γ_rel
γ₁^-2γ₂^-2(1-β₁·β₂)^-2 = γ_rel^-2
(1-β₁·β₁)(1-β₂·β₂)(1-β₁·β₂)^-2 = (1-β_rel·β_rel)
(β_rel·β_rel) = 1-(1-β₁·β₁)(1-β₂·β₂)(1-β₁·β₂)^-2
(β_rel)² = [(1-β₁·β₂)²-(1-β₁·β₁)(1-β₂·β₂)](1-β₁·β₂)^-2
(β_rel)² = [(β₁)²-2(β₁·β₂)+(β₂)²](1-β₁·β₂)^-2
(β_rel)² = [(β₁)-(β₂)]²(1-β₁·β₂)^-2
β_rel = [β₁-β₂]/(1-β₁·β₂)
u_rel = [u₁-u₂]/(1-β₁·β₂)
or
u_rel = [u₁+u₂]/(1+β₁·β₂) = [u₁+u₂]/(1+u₁·u₂/c²)

A₁·U₁ = 0, where A is dU/dτ

The 4-Acceleration of a given particle is always normal to its own worldline

P₁·U₂ = γ[u₂](E-p₁·u₂) = E_rel

P₁·U₁ = m_oU₁·U₁ = m_oc² = E_o

Relative Energy

The relative energy between events 1 and 2 is a Lorentz scalar - any observer (event 3) must measure the same relative energy between events 1 and 2

If measuring the 4-Momentum and 4-Velocity of the same event, then this simplifies to the rest energy of that event

K₁·U₂ = γ[u₂](ω-k₁·u₂) = ω_rel

K₁·U₁ = m_o/h_barU₁·U₁ = m_oc²/h_bar = E_o/h_bar = ω_o

Relative Ang. Frequency

The relative ang. frequency between events 1 and 2 is a Lorentz scalar - any observer (event 3) must measure the same relative ang. frequency between events 1 and 2

If measuring the 4-WaveVector and 4-Velocity of the same event, then this simplifies to the rest ang. frequency of that event

F·U =  (m_oA+(dm_o/dτ)U)·U = c²(dm_o/dτ) = γc²(dm_o/dt)

U·F = γ²(dE/dt-u·f_r) = γ dm_o/dt c²
(pure force if dm_o/dt = 0)

Power Law

U·∂ = γ(∂/∂t + u·del) = γ d/dt = d/dτ 

Relativistic Convective (Time) Derivative, Intrinsic Derivative
U·∂ = d/dτ = γ(∂/∂t + u·del) = γ d/dt

let u << c, then γ ~ 1
then U·∂ = d/dτ = γ(∂/∂t + u·del) ~ (∂/∂t + u·del) = d/dt
del_t(v) = (∂/∂t + u·del) v
is the gauge covariant derivative of a fluid where v is a velocity vector field of a fluid

∂·U = 0 (always??)

The General Continuity Equation, one might say the conservation of event flux.
I believe this is true generally but it needs checking...
If true, then it is the source of all the particle / charge / probability continuity equations

Relations involving the 4-Acceleration:

A·A = -a² =  -γ⁴[a² + (γ/c)²(u·a)²]

Magnitude squared of acceleration

U·A = 0, where A is dU/dτ

The 4-Acceleration of a given particle is always normal to its own worldline
U·U = c²
d/dτ(U·U) = d/dτ(c²) = 0
d/dτ(U·U) = 2*(U·dU/dτ) = 2*(U·A) = 0

Others:

P·P = (m_oc)² = (E_o/c)²
= m²(c²-u·u)
= (m²c²-p·p) = (E²/c²-p·p)
= 0 for light-like/photonic

Square Magnitude of the 4-Momentum

P₁·P₂ = γ[u₁]γ[u₂]m_o1 m_o2(c²-u₁·u₂)
= γ[u_r12]m_o1 m_o2c²

Relativistic Billiards...
P + Q = P' + Q': Momenta before and after collision generally - Conservation of 4-Momentum
P·Q = P'·Q': Momenta in an elastic (rest-mass preserving) collision - Relative velocities conserved

N·N = (n_oc)²

Square Magnitude of the 4-NumberCurrentDensity

J·J = (p_oc)² = (qn_oc)²

Square Magnitude of the 4-ElectricCurrentDensity

K·K = (m_oc / h_bar)² = (ω_o/c)²
= 0 for light-like/photonic

Square Magnitude of the 4-WaveVector

∂·∂ = (∂/c∂t,-del)·(∂/c∂t,-del) = ∂²/c²∂t²-del·del = -(m_oc / h_bar)²

Klein-Gordon Relativistic Wave Eqn.
** ∂·∂ is also known as the D'alembertian (Wave Operator) **

Note: Solutions to the wave equation are linear and allow superposition!
This is a general feature of the wave equation, not something that is only from QM

∂·J = ∂ρ/∂t +del·j = 0

Continuity Equation - Conservation of Electric Charge
No sources or sinks
Charge is neither created nor destroyed

E·K = 0

The Polarization of a photon is orthogonal to direction of wave motion (cancellation of "scalar" polarization)

E·E* = -1

The Polarization of a photon is always unit magnitude and space-like (cancellation of the longitudinal polarization)

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

c = √[U·U]

Speed of Light: c (in vacuum) E ~ cp

h = √[P·P/L·L] = P·L / L·L
h_bar = √[P·P/K·K] = P·K / K·K

Planck's const: h, E ~ hν
Dirac's const: h_bar, E ~ h_barω

k_B = √[P·P/P_T·P_T] = P·P_T / P_T·P_T

Boltzmann's const: k_B, E ~ k_BT

γ_rel = V·U / U·U

Relative Relativistic Gamma Factor, between 4-velocities U and V

Δs = √[ΔR·ΔR] = √[c²Δt² - Δx² - Δy² - Δz²]

Displacement
< 0 Outside LightCone, Acausal SpaceLike Separation
Δs² = 0 "On The LightCone", LightLike Signal Separation
> 0 Inside LightCone, Causally TimeLike Separation

Δσ = √[-ΔR·ΔR] = √[-(c²Δt² - Δx² - Δy² - Δz²)]