New X'^{μ}  =  4Tensor Antisymmetric Lorentz Transform M^{μν} Rotations j = M^{ab} Boosts k = M^{0b} = M^{b0} 3 + 3 = 6 
Original X^{μ}  +  4Vector SpaceTime Translation ΔX^{μ} ~ P^{μ} Time Translation H = P^{0} Space Translation p = P^{i} 1 + 3 = 4  
 = 

 + 
 
Total of 6 + 4 = 10 parameters Poincaré Transform = Lorentz Transform + SpaceTime Translation X'^{μ} = M^{μ}_{ν} X^{ν} + ΔX^{μ} where colors indicate ^{ }

Translational Operator 
∂^{μ}  K^{μ}  P^{μ} 
Equivalent  = ∂^{μ}  = i∂^{μ}  = iћ∂^{μ} 
Normal Commutator 
[∂^{μ}, X^{ν}] = η^{μν}  [K^{μ},X^{ν}] = iη^{μν}  [P^{μ}, X^{ν}] = iћη^{μν} 
Reversed Commutator 
[X^{ν}, ∂^{μ}] = η^{μν}  [X^{ν}, K^{μ}] = iη^{μν}  [X^{ν}, P^{μ}] = iћη^{μν} 
Rotational Momentum Operator M 
M^{μν} 
M^{μν} 
M^{μν}  Dimensionless Rotational Operator O 
O^{μν}  O^{μν}  O^{μν} 
Equivalent  = iћ(X^{μ}∂^{ν}  X^{ν}∂^{μ})  = ћ(X^{μ}K^{ν}  X^{ν}K^{μ})  def. = (X^{μ}P^{ν}  X^{ν}P^{μ}) 
Equivalent  def. = (X^{μ}∂^{ν}  X^{ν}∂^{μ}) 
= (1/i)(X^{μ}K^{ν}  X^{ν}K^{μ})  = (1/iћ)(X^{μ}P^{ν}  X^{ν}P^{μ}) 
Normal Commutator 
[M^{μν}, ∂^{ρ}] = iћ(η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) 
[M^{μν}, K^{ρ}] = iћ(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) 
[M^{μν}, P^{ρ}] = iћ(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) 
Normal Commutator 
[O^{μν}, ∂^{ρ}] = (η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) 
[O^{μν}, K^{ρ}] = (1/i)(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) 
[O^{μν}, P^{ρ}] = (1/iћ)(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) 
Reversed Commutator 
[∂^{ρ}, M^{μν}] = iћ(η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) = iћ(η^{ρμ}∂^{ν}  η^{ρν}∂^{μ}) 
[K^{ρ}, M^{μν}] = iћ(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) = iћ(η^{ρμ}K^{ν}  η^{ρν}K^{μ}) 
[P^{ρ}, M^{μν}] = iћ(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) = iћ(η^{ρμ}P^{ν}  η^{ρν}P^{μ})  Reversed Commutator 
[∂^{ρ}, O^{μν}] = (η^{ρν}∂^{μ}  η^{ρμ}∂^{ν}) = (η^{ρμ}∂^{ν}  η^{ρν}∂^{μ})  [K^{ρ}, O^{μν}] = (1/i)(η^{ρν}K^{μ}  η^{ρμ}K^{ν}) = (1/i)(η^{ρμ}K^{ν}  η^{ρν}K^{μ})  [P^{ρ}, O^{μν}] = (1/iћ)(η^{ρν}P^{μ}  η^{ρμ}P^{ν}) = (1/iћ)(η^{ρμ}P^{ν}  η^{ρν}P^{μ}) 
Standard Model Elementary Particle 
Relativistic Wave Equations (RWE)  Relativistic Wave Equations (RWE)  Newtonian Limit ( v << c )  
Particle Type  Spin  Statistics  Field  RQM Massless (m_{o} = 0)  RQM Massive (m_{o} > 0)  QM Massive (m_{o} > 0) 
Fundamental  0  Boson  Lorentz Scalar ψ 
Scalar Wave (∂·∂)ψ = 0 
KleinGordon Equation (∂·∂ + (m_{o}c/ћ)^{2})ψ = 0 
Schrödinger Equation (iħ∂_{t})ψ ~ [(m_{o}c^{2})  (ħ∇)^{2}/2m_{o}]ψ 
Fundamental  1/2  Fermion  Spinor Ψ 
Weyl Equation [(iγ^{μ}∂_{μ})]Ψ = 0 → [(σ^{μ}∂_{μ})]Ψ = 0 
Dirac Equation, Majorana Equation [(iγ^{μ}∂_{μ})  (m_{o}c/ћ)]Ψ = 0 (Γ^{μ}P_{μ})Ψ = (m_{o}c)Ψ iћ(Γ^{μ}∂_{μ})Ψ = (m_{o}c)Ψ 
Pauli Equation (iħ∂_{t})Ψ ~ [(m_{o}c^{2}) + (σ·p)^{2}/2m_{o}]Ψ 
Fundamental  1  Boson  4Vector A 
Maxwell Equation (∂·∂)A = 0 
Proca Equation (∂·∂ + (m_{o}c/ћ)^{2})A = 0 
? 
Composites  3/2  Fermion  SpinorVector  Majorana RaritaSchwinger  RaritaSchwinger Equation  
??  2  Boson  (2,0)Tensor  Graviton?? 
Standard Model Elementary Particle 
Relativistic Wave Equations (RWE)  Relativistic Wave Equations (RWE)  Newtonian Limit ( v << c )  
Particle Type  Spin  Statistics  Field  RQM Massless (m_{o} = 0)  RQM Massive (m_{o} > 0)  QM Massive (m_{o} > 0) 
Fundamental  0  Boson  Lorentz Scalar ψ 
Scalar Wave (D·D)ψ = 0 
KleinGordon Equation (D·D + (m_{o}c/ћ)^{2})ψ = 0 
Schrödinger Equation (iħ∂_{tT})ψ ~ [qφ + (m_{o}c^{2}) + (iħ∇_{T} qa)^{2}/2m_{o}]ψ (iħ∂_{tT})ψ ~ [V + (iħ∇_{T} qa)^{2}/2m_{o}]ψ : with [V = qφ + (m_{o}c^{2})] 
Fundamental  1/2  Fermion  Spinor Ψ 
Weyl Equation ? 
Dirac Equation, Majorana Equation Γ^{μ}(P_{μ}qA_{μ})Ψ = (m_{o}c)Ψ Γ^{μ}(iћ∂_{μ}qA_{μ})Ψ = (m_{o}c)Ψ  Pauli Equation (iħ∂_{tT})Ψ ~ [qφ + (m_{o}c^{2}) + [σ·(p_{T} qa)]^{2}/(2m_{o})]Ψ (iħ∂_{tT})Ψ ~ [qφ + (m_{o}c^{2}) + ([(p_{T} qa)]^{2}  ћq[σ·B])/(2m_{o})]Ψ 
Fundamental  1  Boson  4Vector A 
Maxwell Equation (∂·∂)A = 0 (∂·∂)A^{ν} = μ_{o}J^{ν}: Classical source (∂·∂)A^{ν} = q(ψ̅ γ^{ν} ψ): QED source 
Proca Equation  ? 
Composites  3/2  Fermion  SpinorVector  Majorana RaritaSchwinger  RaritaSchwinger Equation  
??  2  Boson  (2,0)Tensor  Graviton?? 
Written" on the papers Einstein is holding: =================================== R_{μν}  (1/2)g_{μν}R = κT_{μν} (the theory of GR) eV = hν  A (the PhotoElectric Effect) E = mc^{2} (the Equivalence of Energy and Matter) =================================== 
Correlates to: ========== GR QM SR ========== 
V^{0}  V^{1 }  V^{2 }  V^{3 } 
temporal part = V^{0} 
spatial part = V^{i} 
γ  β_{x}γ  0  0 
β_{x}γ  γ  0  0 
0  0  1  0 
0  0  0  1 
cosh[ζ]  sinh[ζ]  0  0 
sinh[ζ]  cosh[ζ]  0  0 
0  0  1  0 
0  0  0  1 
1  0  0  0 
0  cos[θ]  sin[θ]  0 
0  sin[θ]  cos[θ]  0 
0  0  0  1 
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} 
Basis  3position Representation 
_{Lower} Metric g_{ij}  ^{Upper} Metric g^{ij}  Line Element dx·dx = dx^{i}g_{ij}dx^{j} = (dl)^{2} 
Euclidean Space Independent  x  η_{ij}  η^{ij}  (dl)^{2} = dx·dx = dx^{i}η_{ij}dx^{j} 
Euclidean Cartesian/Rectangular  x→(x,y,z)  η_{ij} → Diag[+1,+1,+1] = δ_{ij} = I_{ }  η^{ij} → Diag[+1,+1,+1]  (dl)^{2} = dx^{2} + dy^{2} + dz^{2} 
Euclidean Cylindrical/Polar  x→(r,θ,z)  η_{ij} → Diag[+1,+r^{2},+1]  η^{ij} → Diag[+1,+1/r^{2},+1]  (dl)^{2} = dr^{2} + r^{2}dθ^{2} + dz^{2} 
Euclidean Spherical  x→(r,θ,φ) x→(r,{Ω}) 
η_{ij} → Diag[+1,+r^{2},+(r·sin[θ])^{2}]  η^{ij} → Diag[+1,+1/r^{2},+1/(r·sin[θ])^{2}  (dl)^{2} = dr^{2} + r^{2}dθ^{2} + (r·sin[θ])^{2}dφ^{2} (dl)^{2} = dr^{2} + r^{2}dΩ^{2} 
Basis  4Position Representation 
_{Lower} Metric g_{μν}  ^{Upper} Metric g^{μν}  WorldLine Element dX·dX = dX^{μ}g_{μν}dX^{ν} = (cdτ)^{2} 
Minkowski SpaceTime Independent  X  η_{μν}  η^{μν}  (cdτ)^{2} = dX·dX = dX^{μ}η_{μν}dX^{ν} 
Minkowski TimeSpace  X→(ct,x)  η_{μν} → Diag[+1,I]  η^{μν} → Diag[+1,I]  (cdτ)^{2} = (cdt)^{2}  dx·dx 
Minkowski Cartesian/Rectangular  X→(ct,x,y,z)  η_{μν} → Diag[+1,1,1,1]  η^{μν} → Diag[+1,1,1,1]  (cdτ)^{2} = (cdt)^{2}  dx^{2}  dy^{2}  dz^{2} 
Minkowski Cylindrical/Polar  X→(ct,r,θ,z)  η_{μν} → Diag[+1,1,r^{2},1]  η^{μν} → Diag[+1,1,1/r^{2},1]  (cdτ)^{2} = (cdt)^{2}  dr^{2}  r^{2}dθ^{2}  dz^{2} 
Minkowski Spherical  X→(ct,r,θ,φ) X→(ct,r,{Ω}) 
η_{μν} → Diag[+1,1,r^{2},(r·sin[θ])^{2}]  η^{μν} → Diag[+1,1,1/r^{2},1/(r·sin[θ])^{2}  (cdτ)^{2} = (cdt)^{2}  dr^{2}  r^{2}dθ^{2}  (r·sin[θ])^{2}dφ^{2} (cdτ)^{2} = (cdt)^{2}  dr^{2}  r^{2}dΩ^{2} 
others...  
Newtonian Gravity Cartesian/Rectangular {weak gravity limitingcase φ<<1} {becomes Minkowski for φ→0} 
X→(ct,x,y,z)  g_{μν} → Diag[+(1+2φ),1,1,1]  g^{μν} → Diag[+1/(1+2φ),1,1,1]  (cdτ)^{2} = (1+2φ)(cdt)^{2}  dx^{2}  dy^{2}  dz^{2} 
Schwartzschild Spherical {becomes Minkowski for R_{S}→0 or r→∞} 
X→(ct,r,θ,φ)  g_{μν} → Diag[+(1R_{S}/r),1/(1R_{S}/r),r^{2},r^{2}sin(θ)]  g^{μν} → Diag[+1/(1R_{S}/r),(1R_{S}/r),1/r^{2},1/r^{2}sin(θ)]  (cdτ)^{2} = (1R_{S}/r)(cdt)^{2}  1/(1R_{S}/r)dr^{2}  r^{2}dθ^{2}  (r·sin[θ])^{2}dφ^{2} 
FLRW (or FRW) Spherical {assumes homogeneity & isotropy} a[t] is "scale factor" k is uniform curvature constant typically k={,0,+} 
X→(ct,r,θ,φ)  g_{μν} → Diag[+1,1/(a[t])^{2}{1/(1kr^{2}),r^{2},(r·sin[θ])^{2}}]  g^{μν} → Diag[+1,(a[t])^{2}{(1kr^{2}),1/r^{2},1/(r·sin[θ])^{2}}]  (cdτ)^{2} = (cdt)^{2}  (a[t])^{2}{1/(1kr^{2})dr^{2} + r^{2}dθ^{2} + (r·sin[θ])^{2}dφ^{2}} (cdτ)^{2} = (cdt)^{2}  (a[t])^{2}{1/(1kr^{2})dr^{2} + r^{2}dΩ2} (cdτ)^{2} = (cdt)^{2}  (a[t])^{2}{dΣ^{2}} 
Tensor Type  Representation  Index Type  ^{Upper} Index Count 
_{Lower }Index Count 
Alt Name  Further Definitions  
(0,0)Tensor  S  N/A  0  0  (Lorentz) (4)Scalar  Invariant component  
(1,0)Tensor (0,1)Tensor 
V^{μ} V_{μ} 
Contravariant Covariant 
1 0 
0 1 
4Vector 4Covector 
1 temporal, 3 spatial components  
(2,0)Tensor (1,1)Tensor (0,2)Tensor 
T^{μν} T^{μ}_{ν} or T_{μ}^{ν} T_{μν} 
Contravariant Mixed Covariant 
2 1 0 
0 1 2 
4Tensor 
1 temporal, 9 spatial, 6 mixed timespace components Independent Components: Symmetric: 10 AntiSymmetric: 6 Generic: 16 possible 
S 
V^{0}  V^{1}  V^{2}  V^{3} 
T^{00}  T^{01}  T^{02}  T^{03} 
T^{10}  T^{11}  T^{12}  T^{13} 
T^{20}  T^{21}  T^{22}  T^{23} 
T^{30}  T^{31}  T^{32}  T^{33} 
4Acceleration A = (A^{0},A^{i}) = γ(cγ̇ , γ̇u + γu̇) = γ(cγ̇ , γ̇u + γa)  = ( γ^{4}(a·u)/c , γ^{4}(a·u)u/c^{2} + γ^{2}a )  = γ^{4}( (a·u)/c , a + u x (a x u)/c^{2} ) 
= ( γ^{4}(a·β) , γ^{4}(a·β)β + γ^{2}a )  = γ^{4}( (a·β) , a + β x (a x β) )  
= γ^{4}( (a·β) , (a·β)β + a/γ^{2} ) 
Particle Count  Mass_Energy  (d/dτ)[Mass_Energy]  Entropy  EM Charge  WaveAngFreq  EM Potential  
(Lorentz Scalar) <Potential> 
Ω = X·U (free worldline) 
S_{act }= X·P (free particle action) 
Φ = X·K (free wave phase) 

d/dτ[<Potential>] <Charge>*c^{2} 
U·U = c^{2} 
E_{o} = U·P = U·∂[S] = d/dτ[S] E_{o} = m_{o}c^{2} 
ω_{o} = U·K = U·∂[Φ] = d/dτ[Φ] ω_{o} = (ω_{o}/c^{2})c^{2} 

<Charge>  N (usually 1) 
m_{o} = (E_{o}/c^{2})  (d/dτ)[m_{o}]  S_{ent}  q  (ω_{o}/c^{2})  (φ_{o}/c^{2}) 
Particle 4Vector <Charge>U 
U  P = ∂[S] P = m_{o}U = (E_{o}/c^{2})U 
F = (d/dτ)[m_{o}]U + m_{o}A  J_{q} = qU 
K = ∂[Φ] K = (ω_{o}/c^{2})U 
A = (φ_{o}/c^{2})U  
Density 4Vector Flux 4Vector <Charge>N <ChargeDensity>U 
N = U_{den} = n_{o}U 
G = P_{den} = n_{o}P G = u_{mo}U = m_{o}n_{o}U = m_{o}N G = U·T^{μν}/c^{2} 
F_{d} = F_{den} = n_{o}F F_{d} = ∂·T^{μν} 
S = s_{o}U = S_{ent}n_{o}U = S_{ent}N 
J = J_{qden} = n_{o}J_{q} J = ρ_{o}U = qn_{o}U = qN 
? = (ω_{o}/c^{2})N  ? = (φ_{o}/c^{2})N 
<ChargeDensity>  n_{o}  u_{mo} = (u_{eo}/c^{2}) = n_{o}m_{o} 
(d/dτ)[u_{mo}] 
s_{o} = n_{o}S_{Ent }  ρ_{o} = n_{o}q_{ }  n_{o}(ω_{o}/c^{2})  n_{o}(φ_{o}/c^{2}) 
4Divergence = 0 Conservation Law 
∂·N = 0 Conservation of Particle Count N 
∂·G = 0 Conservation of Mass_Energy m_{o} 
∂·F_{d} = 0 Conservation of Power?? 
∂·S = 0 Conservation of Entropy S_{ent} 
∂·J = 0 Conservation of Charge q 
∂·K = 0 Conservation of Wave_Freq? 
∂·A = 0 Conservation of EM Potential (Lorenz Gauge) 
4WaveVector 4AngularWaveVector 
K = (ω/c,k) = (ω/c,n̂ω/v_{phase}) = (ω/c,ωu/c^{2}) = (ω/c)(1,β) = (1/c 
Atomic #  1  2 
Element  H  He 
Electron Config 
1s^{1}  1s^{2} 
Orbital Added 
1s_{t}↑ ~ +t  1s_{t}↓ ~ t 
Atomic #  3  4  5  6  7  8  9  10 
Element  Li  Be  B  C  N  O  F  Ne 
Electron Config 
[He]2s^{1}  [He]2s^{2}  [He]2s^{2}2p^{1}  [He]2s^{2}2p^{2}  [He]2s^{2}2p^{3}  [He]2s^{2}2p^{4}  [He]2s^{2}2p^{5}  [He]2s^{2}2p^{6} 
Orbital Added 
2s_{t}↑ ~ +t  2s_{t}↓ ~ t  2p_{x}↑ ~ +x  2p_{x}↓ ~ x  2p_{y}↑ ~ +y  2p_{y}↓ ~ y  2p_{z}↑ ~ +z  2p_{z}↓ ~ z 
Alkali Metals Group 1 SBlock 
Alkaline Earth Metals Group 2 SBlock 
Icosagens Group 13 PBlock 
Crystallogens Group 14 PBlock 
Pnictogens Group 15 PBlock 
Chaocogens Group 16 PBlock 
Halogens Group 17 PBlock 
Aerogens  Noble Gases Group 18 PBlock 

Atomic #  11  12  13  14  15  16  17  18 
Element  Na  Mg  Al  Si  P  S  Cl  Ar 
Electron Config 
[Ne]3s^{1}  [Ne]3s^{2}  [Ne]3s^{2}3p^{1}  [Ne]3s^{2}3p^{2}  [Ne]3s^{2}3p^{3}  [Ne]3s^{2}3p^{4}  [Ne]3s^{2}3p^{5}  [Ne]3s^{2}3p^{6} 
Orbital Added 
3s_{t}↑ ~ +t  3s_{t}↓ ~ t  3p_{x}↑ ~ +x  3p_{x}↓ ~ x  3p_{y}↑ ~ +y  3p_{y}↓ ~ y  3p_{z}↑ ~ +z  3p_{z}↓ ~ z 
Event R  Mass m_{o} = ρ_{mo}V_{o} Energy E_{o} = m_{o}c^{2} 
MassDensity ρ_{mo} = n_{o}m_{o} EnergyDensity u_{eo} = ρ_{mo}c^{2} 

Derivative of 4Position 
d^{n}R/dτ^{n}  Event 4Vector 
particle  density 
0th  R d^{0}R/dτ^{0} 
pos: R = (ct,r)  m_{o} at R  ρ_{mo} at R 
1st  dR/dτ d^{1}R/dτ^{1} 
vel:U = dR/dτ  P = m_{o}dR/dτ P = m_{o}U = (E_{o}/c^{2})U 
G = ρ_{mo}dR/dτ G = ρ_{mo}U = (u_{eo}/c^{2})U 
2nd  d^{2}R/dτ^{2}  accel: A = dU/dτ  F = dP/dτ  F_{d} = dG/dτ 
3rd  d^{3}R/dτ^{3}  jerk: J = dA/dτ jolt, surge, lurch: alt names 

4th  d^{4}R/dτ^{4}  snap: S = dJ/dτ jounce: alt name 

5th  d^{5}R/dτ^{5}  crackle: C = dS/dτ  
6th  d^{6}R/dτ^{6}  pop: P = dC/dτ 
U_{1}·U_{2} = γ_{12}(c^{2}) = γ_{rel}(c^{2})  U·U = (c)^{2} 
T_{1}·T_{2} = γ_{12} = γ_{rel}  T·T = 1 
Particle Count  Mass_Energy  (d/dτ)[Mass_Energy]  Entropy  EM Charge  WaveAngFreq  EM Potential  
(Lorentz Scalar) <Potential> 
Ω = X·U (free worldline) 
S_{act }= X·P (free particle action) 
Φ = X·K (free wave phase) 

d/dτ[<Potential>] <Charge>*c^{2} 
U·U = c^{2}  E_{o} = U·P = U·∂[S] = d/dτ[S] E_{o} = m_{o}c^{2} 
ω_{o} = U·K = U·∂[Φ] = d/dτ[Φ] ω_{o} = (ω_{o}/c^{2})c^{2} 

<Charge>  N (usually 1)  m_{o} = (E_{o}/c^{2})  (d/dτ)[m_{o}]  S_{ent}  q  (ω_{o}/c^{2})  (φ_{o}/c^{2}) 
Particle 4Vector <Charge>U 
U  P = ∂[S] P = m_{o}U = (E_{o}/c^{2})U 
F = (d/dτ)[m_{o}]U + m_{o}A  J_{q} = qU  K = ∂[Φ] K = (ω_{o}/c^{2})U 
A = (φ_{o}/c^{2})U  
Density 4Vector Flux 4Vector <Charge>N <ChargeDensity>U 
N = U_{den} = n_{o}U 
G = P_{den} = n_{o}P G = u_{mo}U = m_{o}n_{o}U = m_{o}N G = U·T^{μν}/c^{2} 
F_{d} = F_{den} = n_{o}F F_{d} = ∂·T^{μν} 
S = s_{o}U = S_{ent}n_{o}U = S_{ent}N  J = J_{qden} = n_{o}J_{q} J = ρ_{o}U = qn_{o}U = qN  ? = (ω_{o}/c^{2})N  ? = (φ_{o}/c^{2})N 
<ChargeDensity>  n_{o}  u_{mo} = (u_{eo}/c^{2}) = n_{o}m_{o}  (d/dτ)[u_{mo}]  s_{o} = n_{o}S_{Ent }  ρ_{o} = n_{o}q_{ }  n_{o}(ω_{o}/c^{2})  n_{o}(φ_{o}/c^{2}) 
4Divergence = 0 Conservation Law 
∂·N = 0 Conservation of Particle Count N  ∂·G = 0 Conservation of Mass_Energy m_{o}  ∂·F_{d} = 0 Conservation of Power??  ∂·S = 0 Conservation of Entropy S_{ent}  ∂·J = 0 Conservation of Charge q  ∂·K = 0 Conservation of Wave_Freq?  ∂·A = 0 Conservation of EM Potential (Lorenz Gauge) 
Traditional Style  Projection Tensor Style 
T_{perfectfluid}^{μν} = (ρ_{eo} + p_{o})U^{μ}U^{ν}/c^{2}  p_{o}η^{μν}  T_{perfectfluid}^{μν} = (ρ_{eo})V^{μν}  (p_{o})H^{μν} 
Contract with the 4Velocity  Contract with the 4Velocity 
T^{μν}U_{ν} = (ρ_{mo} + p_{o}/c^{2})U^{μ}U^{ν}U_{ν}  p_{o}η^{μν}U_{ν}  T^{μν}U_{ν} = (ρ_{eo})V^{μν}U_{ν}  (p_{o})H^{μν}U_{ν} 
T^{μν}U_{ν} = (ρ_{mo} + p_{o}/c^{2})U^{μ}c^{2}  p_{o}U^{μ}  T^{μν}U_{ν} = (ρ_{eo})U^{μ}  (p_{o})(0^{μ}) 
T^{μν}U_{ν} = (c^{2}ρ_{mo} + p_{o})U^{μ}  p_{o}U^{μ}  T^{μν}U_{ν} = (ρ_{eo})U^{μ} 
T^{μν}U_{ν} = c^{2}ρ_{mo}U^{μ}  T^{μν}U_{ν} = c^{2}ρ_{mo}U^{μ} 
T^{μν}U_{ν} = c^{2}G^{μ} = c^{2}G  T^{μν}U_{ν} = c^{2}G^{μ} = c^{2}G 
c^{2}ρ_{mo} = ρ_{eo} = (1/2)ε_{o}(e^{2} + c^{2}b^{2})  cg = cε_{o}(e x b) 
cg = cε_{o}(e x b)  σ^{ij} = ε_{o}[e^{i}e^{j} + c^{2}b^{i}b^{j}  (1/2)δ^{ij}(e^{2} + c^{2}b^{2})] 
c^{2}ρ_{mo} = ρ_{eo} = (1/2)ε_{o}(e^{2} + c^{2}b^{2})  cg = cε_{o}(e x b) 
cg = cε_{o}(e x b)  σ^{ij} = ε_{o}[e^{i}e^{j} + c^{2}b^{i}b^{j}  (1/2)δ^{ij}(e^{2} + c^{2}b^{2})] 
Invariants  P = Pressure = P_{o}  N = ParticleNum = N_{o}  S = Entropy = S_{o} 
Variables  V = Volume = (1/γ)V_{o}  μ = ChemPoten = (1/γ)μ_{o}  T = Temperature = (1/γ)T_{o} 
γ  [Ė/c]  = d/dτ  [E/c]  = qγ  [c]  ·  [0  e^{i}/c]  = qγ  [c*0 + u·e/c] 
[ f ]  [p]  [u]  [e^{i}/c  ε_{ijk}b^{k}]  [e + u⨯b] 
0  e^{x}/c  e^{y}/c  e^{z}/c 
e^{x}/c  0  b^{z}  b^{y} 
e^{y}/c  b^{z}  0  b^{x} 
e^{z}/c  b^{y}  b^{x}  0 
0  e^{i}/c 
+e^{j}/c  ε_{ijk}b^{k} 
0  +e^{i}/c 
e^{j}/c  ε_{ijk}b^{k} 
Faraday Electromagnetic Tensor  MagnetizationPolarization Tensor  Electromagnetic Displacement Tensor  
F^{αβ} = (∂^{α}A_{EM}^{β}  ∂^{β}A_{EM}^{α}) 

F^{αβ} =
 M^{αβ} =
 D^{αβ}
=
 
e = e^{i} = electric field b = b^{k} = magnetic field  p = p^{i} = electric polarization (polarization) m = m^{k} = magnetic polarization (magnetization) 
d = d^{i} = electric displacement field h = h^{k} = auxiliary magnetic field  
F^{αβ}:SI Units [T] = [kg/A·s^{2}] = [kg/C·s] e: SI Units [kg·m/A·s^{3}] = [kg·m/C·s^{2}] e/c: SI Units [kg/C·s] b: SI Units [T] = [kg/A·s^{2}] = [kg/C·s]  M^{αβ}: SI Units [C/m·s] p: SI Units [C/m^{2}] pc: SI Units [C/m·s] m: SI Units [C/m·s]  D^{αβ}: SI Units [C/m·s] d: SI Units [C/m^{2}] dc: SI Units [C/m·s] h: SI Units [C/m·s]  
∂_{β} F^{αβ} = μ_{o} J^{α}  ∂_{β} M^{αβ} = J_{bound}^{α}  ∂_{β} D^{αβ} = J_{free}^{α} 
TimeSpace Components  Purely Spatial Components 
(dc) = (1/μ_{o})(e/c)  (pc) (dc) = (c^{2}ε_{o})(e/c)  (pc) (dc) = (ε_{o})(ec)  (pc) (d) = (ε_{o})(e)  (p) (d) = (ε_{o})(e) + (p) d = ε_{o}e + p  (h) = (1/μ_{o})(b)  (m) h = (1/μ_{o})b  m b = μ_{o}(h + m) 
0  d^{x}/c  d^{y}/c  d^{z}/c 
d^{x}/c  0  μ^{z}/c  μ^{y}/c 
d^{y}/c  μ^{z}/c  0  μ^{x}/c 
d^{z}/c  μ^{y}/c  μ^{x}/c  0 
0  +d^{i}/c 
d^{j}/c  ε_{ijk}μ^{k} 
Maxwell Eqn  ∂_{α}(∂^{α}A_{EM}^{ν}  ∂^{ν}A_{EM}^{α}) = μ_{o}J^{ν}  ∂·F^{αν} = (μ_{o})J  Divergence of Faraday EM Tensor 
Lorentz Force Eqn  U_{α}(∂^{ν}A_{EM}^{α}  ∂^{α}A_{EM}^{ν}) = (1/q)F^{ν}  U·F^{αν} = (1/q)F  Invariant Temporal Component of Faraday EM Tensor 
Hamiltonian  Lagrangian  Combo 
H  L  H + L = p_{T}·u 
γ(P_{T}·U)  (P_{T}·U)/γ  γ(P_{T}·U)β^{2} 
γH_{o}  L_{o}/γ  γH_{o} + L_{o}/γ 
Rest Hamiltonian 
Rest Lagrangian 
Rest Combo = 0 
H_{o}  L_{o}  H_{o} + L_{o} = 0 = p_{T}·0 
(P_{T}·U)  (P_{T}·U)  (1)(P_{T}·U)(0)^{2} = 0^{ } 
H/γ  γL  0 
Projection Tensor  Alt Name/Mnemonic  Definition / Representation  4Divergence  Trace 
SpaceTime η^{μν}  "(n)ow,here" Worldline Event 
η^{μν} = ∂^{μ}[X^{ν}] = V^{μν} + H^{μν} → Diag[1,1,1,1]  ∂_{μ}η^{μν} = 0^{ν}  Tr[η^{μν}] = 4 
Temporal V^{μν}  "(V)ertical" Worldline Tangent 
V^{μν} = T^{μ}T^{ν} → Diag[1,0,0,0]  ∂_{ν}V^{μν} = = ∂_{ν}T^{μ}T^{ν} = T^{ν}∂_{ν}[T^{μ}] + T^{μ}∂_{ν}[T^{ν}] = (d/cdτ)[T^{μ}] + T^{μ}(∂·T) = A^{μ}/c^{2} + T^{μ}(∂·T) 
Tr[V^{μν}] = 1 
Spatial H^{μν}  "(H)orizontal" Worldline Normal Hyperplanes orthogonal to Worldline 
H^{μν} = η^{μν}  T^{μ}T^{ν} → Diag[0,1,1,1]  ∂_{μ}H^{μν} = ∂_{μ}V^{μν} 
Tr[H^{μν}] = 3 
Null N^{μν}  "(N)ull" LightPath 
N^{μν} = N^{μ}N^{ν} = V^{μν}  (1/3)H^{μν} → Diag[1,1/3,1/3,1/3]  ∂_{μ}N^{μν} = = ∂_{μ}(V^{μν}  (1/3)H^{μν}) = ∂_{μ}(V^{μν} + (1/3)V^{μν}) = (4/3)∂_{μ}V^{μν} 
Tr[N^{μν}] = 0 
Projection N̂^{μν} 
(P)rojection to Hyperplanes orthogonal to N̂ 
N̂^{μν} = η^{μν}  N̂^{μ}N̂^{ν} → ?? 
T^{00}  T^{01}  T^{02}  T^{03} 
T^{10}  T^{11}  T^{12}  T^{13} 
T^{20}  T^{21}  T^{22}  T^{23} 
T^{30}  T^{31}  T^{32}  T^{33} 
A^{μ} = A_{∥}^{μ} + A_{⟂}^{μ} η^{μ}_{ν} = V^{μ}_{ν} + H^{μ}_{ν} η^{μ}_{ν} = T^{μ}T_{ν} + S^{μ}S_{ν} 
SpaceTime 4Vector A^{μ} 
Temporal 4Vector A_{∥}^{μ} 
Spatial 4Vector A_{⟂}^{μ} 
SpaceTime Projection η^{μ}_{ν}  η^{μ}_{ν}A^{ν} = A^{μ}  η^{μ}_{ν}A_{∥}^{ν} = A_{∥}^{μ}  η^{μ}_{ν}A_{⟂}^{ν} = A_{⟂}^{μ} 
Temporal Projection V^{μ}_{ν}  V^{μ}_{ν}A^{ν} = A_{∥}^{μ} = T^{μ}(T·A)  V^{μ}_{ν}A_{∥}^{ν} = A_{∥}^{μ}  V^{μ}_{ν}A_{⟂}^{ν} = 0^{μ} 
Spatial Projection H^{μ}_{ν}  H^{μ}_{ν}A^{ν} = A_{⟂}^{μ} = S^{μ}(S·A)  H^{μ}_{ν}A_{∥}^{ν} = 0^{μ}  H^{μ}_{ν}A_{⟂}^{ν} = A_{⟂}^{μ} 
Null Projection N^{μ}_{ν}  N^{μ}_{ν}A^{ν} = A_{∠}^{μ} = N^{μ}(N·A) 
η^{μν} = V^{μν} + H^{μν}  SpaceTime Projection "(n)ow" η_{μν} 
Temporal Projection "(V)ertical" V_{μν} 
Spatial Projection "(H)orizontal" H_{μν} = η_{μν}  V_{μν} 
Null Projection "(N)ull" N_{μν} = V_{μν}  (1/3)H_{μν} 
SpaceTime Tensor η^{μν}  Tr[η^{μν}] = η_{μν}η^{μν} = 4  V_{μν}η^{μν} = 1  H_{μν}η^{μν} = 3  N_{μν}η^{μν} = 0 
Temporal Tensor V^{μν}  Tr[V^{μν}] = η_{μν}V^{μν} = 1  V_{μν}V^{μν} = 1  H_{μν}V^{μν} = 0  N_{μν}V^{μν} = 1 
Spatial Tensor H^{μν}  Tr[H^{μν}] = η_{μν}H^{μν} = 3  V_{μν}H^{μν} = 0  H_{μν}H^{μν} = 3  N_{μν}H^{μν} = 1 
Null Tensor N^{μν}  Tr[N^{μν}] = η_{μν}N^{μν} = 0  V_{μν}N^{μν} = 1  H_{μν}N^{μν} = 1  N_{μν}N^{μν} = 4/3? Related I believe to the 4/3 problem of Electromagnetic Mass 
Tensor Form  A^{μ}η_{μν}B^{ν}  A^{μ}V_{μν}B^{ν}  A^{μ}H_{μν}B^{ν} 
4Vector Form  A·B  (A·B)_{∥}  (A·B)_{⟂} 
Component Form  a^{0}b^{0}  a·b  a^{0}b^{0}  a·b 
SpaceTime Projection  Tr[η^{μ}_{α}η^{ν}_{β}] = η^{μ}_{α}η_{μν}η^{ν}_{β} = η_{αβ} 
Temporal Projection  Tr[V^{μ}_{α}V^{ν}_{β}] = V^{μ}_{α}η_{μν}V^{ν}_{β} = V_{αβ} 
Spatial Projection  Tr[H^{μ}_{α}H^{ν}_{β}] = H^{μ}_{α}η_{μν}H^{ν}_{β} = H_{αβ} 
Energy Density ρ_{m}c^{2} = ρ_{e} timetime T^{00 }

Energy Flux/c s/c = cg timespace T^{0j }


Momentum Density*c cg = s/c spacetime T^{i0 }

Momentum Flux = Spatial Stress σ^{ij} spacespace T^{ij}

ρ_{eo}  s_{x}/c  s_{y}/c  s_{z}/c 
s_{x}/c  σ_{xx}  σ_{xy}  σ_{xz} 
s_{y}/c  σ_{yx}  σ_{yy}  σ_{yz} 
s_{z}/c  σ_{zx}  σ_{zy}  σ_{zz} 
γ^{2}(ρ_{eo} + p_{o})  p_{o}  γ^{2}(ρ_{eo} + p_{o})u_{x}/c  γ^{2}(ρ_{eo} + p_{o})u_{y}/c  γ^{2}(ρ_{eo} + p_{o})u_{z}/c 
γ^{2}(ρ_{eo} + p_{o})u_{x}/c  γ^{2}(ρ_{eo} + p_{o})u_{x}u_{x}/c^{2} + p_{o}  γ^{2}(ρ_{eo} + p_{o})u_{x}u_{y}/c^{2}  γ^{2}(ρ_{eo} + p_{o})u_{x}u_{z}/c^{2} 
γ^{2}(ρ_{eo} + p_{o})u_{y}/c  γ^{2}(ρ_{eo} + p_{o})u_{x}u_{y}/c^{2}  γ^{2}(ρ_{eo} + p_{o})u_{y}u_{y}/c^{2} + p_{o}  γ^{2}(ρ_{eo} + p_{o})u_{y}u_{z}/c^{2} 
γ^{2}(ρ_{eo} + p_{o})u_{z}/c  γ^{2}(ρ_{eo} + p_{o})u_{x}u_{z}/c^{2}  γ^{2}(ρ_{eo} + p_{o})u_{y}u_{z}/c^{2}  γ^{2}(ρ_{eo} + p_{o})u_{z}u_{z}/c^{2} + p_{o} 
ρ_{eo}  
p_{o}  
p_{o}  
p_{o} 
T^{00} = ρ_{eo}  T^{0j} = 0 
T^{i0} = 0  T^{ij} = p_{o}δ^{ij} 
Perfect Fluid StressEnergy Tensor T^{μν} = (ρ_{eo})V^{μν}  (p_{o})H^{μν} 
Invariant Tr[T^{μν}] = (ρ_{eo}  3p_{o}) 
Pressure p_{o} = wρ_{eo} = wρ_{mo}c^{2}^{ } 
EoS Parm (w = p_{o}/ρ_{eo}) 
n = 3(w + 1) 
Energy Density ρ_{eo} falls off as a^{n} = a^{3(w + 1) } 
a[t] = a_{0}t^{2/n}  Cosmological Solution eg. Matter Dominated, Radiation Dominated, etc. 
(ρ_{eo})V^{μν}  (3ρ_{eo}/3)H^{μν}  2ρ_{eo}  ρ_{eo}  1  6  ~Stiff Equation of State (Neutron Stars) ??  
(ρ_{eo})V^{μν}  (2ρ_{eo}/3)H^{μν}  1ρ_{eo}  2ρ_{eo}/3  2/3  5  ???  
(ρ_{eo})V^{μν}  (1ρ_{eo}/3)H^{μν} = NullDust = (ρ_{eo})N^{μν} = (p_{o})(4V^{μν}  η^{μν}) 
0ρ_{eo}  ρ_{eo}/3  1/3  4  ρ_{R} = ρ_{eo} ∝ a^{4}  a[t] ∝ t^{1/2}  Radiation/~UltraRelativistic Matter/Soft Equation of State NullDust/Photon Gas/Hot Dust/Relativistic Neutrinos T^{μν} = (ρ_{eo})N^{μν} 
(ρ_{eo})V^{μν}  ((v/c)^{2}ρ_{eo}/3)H^{μν}  [1(v/c)^{2}]ρ_{eo} = (γ^{2})ρ_{eo} 
(v/c)^{2}ρ_{eo}/3 = v^{2}ρ_{mo}/3 = ρ_{mo}RT 
{0..1/3}  {3..4}  Perfect Gas (v<<c) = Warm Dust v = v_{th} = √[3RT] = √[3K_{B}T/m] = {0..c} = characteristic rms 3D thermal speed of molecules essentially this smoothly varies from MatterDust (v~0) to NullDust (v~c) 

(ρ_{eo})V^{μν}  (0ρ_{eo}/3)H^{μν} = MatterDust = (ρ_{eo})V^{μν} 
1ρ_{eo}  0  0  3  ρ_{M} = ρ_{eo} ∝ a^{3}  a[t] ∝ t^{2/3}  (Cold) Dust = (Incoherent) Matter/CDM/Baryons Einsteinde Sitter (EdS) solution T^{μν} = (ρ_{eo})V^{μν} 
(ρ_{eo})V^{μν}  (1ρ_{eo}/3)H^{μν}  2ρ_{eo}  ρ_{eo}/3  1/3  2  ρ_{eo} ∝ a^{2}  a[t] ∝ t  Curvature = Einstein Static Universe/Cosmic Strings 
> 2ρ_{eo}  < ρ_{eo}/3  < 1/3  < 2  Everything Below has Accelerating Expansion of Universe  
(ρ_{eo})V^{μν}  (2ρ_{eo}/3)H^{μν}  3ρ_{eo}  2ρ_{eo}/3  2/3  1  ρ_{eo} ∝ a^{1}  a[t] ∝ t^{2}  ??? Domain Walls? 
(ρ_{eo})V^{μν}  (3ρ_{eo}/3)H^{μν} = Vacuum Energy = (ρ_{eo})η^{μν } = (p_{o})η^{μν} 
4ρ_{eo}  ρ_{eo}  1  0  ρ_{Λ} = ρ_{eo} ∝ a^{0} = constant^{ } 
a[t] ∝ e^{Ht}  (Quantum) Vacuum Energy/Dark Energy /Cosmological Constant/de Sitter T^{μν} = (ρ_{eo})η^{μν} 
> 4ρ_{eo}  < ρ_{eo}  < 1  < 0  Big Rip = Phantom Energy 
ρ_{eo}  
p = 0  
p = 0  
p = 0 
ρ_{eo}  
p = ρ_{eo}  
p = ρ_{eo}  
p = ρ_{eo} 
ρ_{eo}  
p = ρ_{eo}/3  
p = ρ_{eo}/3  
p = ρ_{eo}/3 
c^{2}ρ_{mo} = ρ_{eo} = = (1/2)ε_{o}(e^{2} + c^{2}b^{2}) = (1/2)(ε_{o}e^{2} + b^{2}/μ_{o}) 
c g = s^{j}/c = cε_{o}(e x b) = (e x b)/(cμ_{o}) 
c g = s^{i}/c = cε_{o}(e x b) = (e x b)/(cμ_{o}) 
σ^{ij} = ε_{o}[e^{i}e^{j} + c^{2}b^{i}b^{j}  (1/2)δ^{ij}(e^{2} + c^{2}b^{2})] = [ε_{o}e^{i}e^{j} + b^{i}b^{j}/μ_{o}  (1/2)δ^{ij}(ε_{o}e^{2} + c^{2}b^{2}/μ_{o})] = The Maxwell Stress Tensor 
c^{2}ρ_{mo} = ρ_{eo}  c g = s^{j}/c 
c g = s^{i}/c  σ^{ij} = ε_{o}[e^{i}e^{j} + c^{2}b^{i}b^{j}  (1/2)δ^{ij}(e^{2} + c^{2}b^{2})] 
Temporal Projection  Spatial Projection 
V^{σ}_{μ}∂_{ν}T^{μν} = V^{σ}_{μ}( ∂_{ν}[ρ_{eo}]V^{μν} + (ρ_{eo} + p_{o})∂_{ν}[V^{μν}]  ∂_{ν}[p_{o}]H^{μν} ) ∂_{ν}[ρ_{eo}]V^{σ}_{μ}V^{μν} + (ρ_{eo} + p_{o})V^{σ}_{μ}∂_{ν}[V^{μν}]  ∂_{ν}[p_{o}]V^{σ}_{μ}H^{μν} ∂_{ν}[ρ_{eo}]V^{σ}^{ν} + (ρ_{eo} + p_{o})T_{∥}^{σ}(∂·T)  ∂_{ν}[p_{o}](0) T_{∥}^{σ}T^{ν}∂_{ν}[ρ_{eo}] + (ρ_{eo} + p_{o})T_{∥}^{σ}(∂·T) T_{∥}^{σ}[(d/cdτ)[ρ_{eo}] + (ρ_{eo} + p_{o})(∂·T)] T_{∥}^{σ}[∂·[ρ_{eo}T] + (p_{o})(∂·T)] T_{∥}^{σ}[∂·[ρ_{eo}U] + (p_{o})(∂·U)]/c γ[∂_{ν}[ρ_{eo}U^{ν}] + (p_{o})(∂_{ν}U^{ν})]/c 
H^{σ}_{μ}∂_{ν}T^{μν} = H^{σ}_{μ}( ∂_{ν}[ρ_{eo}]V^{μν} + (ρ_{eo} + p_{o})∂_{ν}[V^{μν}]  ∂_{ν}[p_{o}]H^{μν} ) ∂_{ν}[ρ_{eo}]H^{σ}_{μ}V^{μν} + (ρ_{eo} + p_{o})H^{σ}_{μ}∂_{ν}[V^{μν}]  ∂_{ν}[p_{o}]H^{σ}_{μ}H^{μν} ∂_{ν}[ρ_{eo}](0) + (ρ_{eo} + p_{o})(A_{⊥}^{σ}/c^{2})  ∂_{ν}[p_{o}]H^{σ}^{ν} (ρ_{eo} + p_{o})(A_{⊥}^{σ}/c^{2})  ∂_{⊥}^{σ}[p_{o}] ((ρ_{eo} + p_{o})/c^{2})γ(cγ̇,γ̇u + γu̇)_{⊥}  (∂_{t}/c, ∇)_{⊥}[p_{o}] ((ρ_{eo} + p_{o})/c^{2})γ(γ̇u + γu̇)  (∇)[p_{o}] γ((ρ_{eo} + p_{o})/c^{2})(γ̇u + γu̇) + ∇[p_{o}] 
CoolWarm Dust Condition (p_{o}) << (ρ_{eo}) U_{μ}∂_{ν}T^{μν} = Temporal Component ∂·(ρ_{eo}U) (∂_{t}[γρ_{eo}] + ∇·[γρ_{eo}u]) (∂_{t}[ρ_{e}] + ∇·[ρ_{e}u]) = 0 if conserved 
CoolWarm Dust Condition (p_{o}) << (ρ_{eo}) H^{σ}_{μ}∂_{ν}T^{μν} = Spatial Components γ((ρ_{eo})/c^{2})(γ̇u + γu̇) + ∇[p_{o}] (ρ_{m})(γ̇u + γu̇) + ∇[p_{o}] = 0 if conserved 
Newtonian Limit: u << c (∂_{t}[ρ_{e}]  ∇·[ρ_{e}u]) Same as Warm Dust = 0 if conserved 
Newtonian Limit: u << c, γ→1, γ̇→0 (ρ_{m})(u̇) + ∇[p_{o}] (ρ_{m})(a) + ∇[p_{o}] = Euler Equations for Fluid Dynamics = 0 if conserved 
Wave Type:  Scalar Waves  Photonic/EM Waves (4Vectors)  Gravitational Waves (2,0)Tensors  Lanczos Potential Tensor Gravitational Waves  
Special Background Conditions:  None  None  Linearized Gravity = Weak Field limit g^{μν} = η^{μν} + h^{μν} where h^{μν} << 1 Minkowski SpaceTime limit h^{μν} acts like Tensor Field propagating in "flat" Minkowski SR 
SR  
Field Type:  (0,0)Tensor = Scalar 
(1,0)Tensor = 4Vector 
(2,0)Tensor  (3,0)Tensor  
Field Identifier:  Φ  A = A^{μ}  h_{TT}^{μν}  H^{μνρ} or L^{μνρ}  
Special Tensor Conditions:  None, 4 possible independent components. Lorentz Invariant Conditions will reduce # of independent component. 
(h_{TT}^{μν}) = (h_{TT}^{νμ}) h_{TT}^{μν} = h_{TT}^{(μν)} Symmetric 2Tensor  Reduces independent components from 16 down to 10 Other Lorentz Invariant Conditions will reduce it further 
L^{μνρ} + L^{νμρ} = 0 L^{μνρ} = L^{[μν]ρ} AntiSymmetric on first 2 indices L^{μνρ} + L^{ρμν} + L^{νρμ} = 0 L^{[μνρ]} = 0 Jacobi/Bianchi Identity  Reduces independent components from 64 down to 20 Other Lorentz Invariant Conditions will reduce it further  
Conservative Field Condition: 4Divergence = 0 4Divergenceless = Lorenz Gauge Carroll also uses Lorenz Gauge for gravitational wave in Intro to GR: SpaceTime and Geometry, pg. 301 Other names include: Harmonic Gauge Einstein Gauge Hilbert Gauge de Donder Gauge Fock Gauge 
N/A  (∂·A) = (∂_{μ}A^{μ}) = 0 A is conserved 
(∂·h_{TT}^{μν}) = (∂_{ν}h_{TT}^{μν}) = (h_{TT}^{μν}_{,ν}) = 0^{μ} h_{TT}^{μν} is conserved 
Lanczos differential gauge SR (∂·L^{μνρ}) = (∂_{ρ}L^{μνρ}) = L^{μνρ}_{,ρ} = 0^{μν}  
Purely Spatial Wave Condition: Orthogonal to 4Velocity U 
(U·A) = (U_{ν}A^{ν}) = 0 for a photonic wave Generally A_{EM}·U = (φ/c,a)·γ(c,u) = γ(φ  a·u) = φ_{o} _{}As we will see, this is a photonic wave and the rest potential φ_{o} will be zero in the same way that the rest mass m_{o} of a photon is zero In other words: There is no "atrest" frame for lightlike (U·A) = 0 = γ(c,u)·(φ/c,a) = γ[φ  u·a] = 0 Therefore, φ = u·a Therefore A = (u·a/c,a) To an atrest observer (u=0), A appears spatial A → (0,a) To an ânull observer (u=câ), A appears null A → (a,a) A 3rd situation is that u·a=0 via 3D orthogonality, in which case A appears spatial A → (0,a) 
(U·h_{TT}^{μν}) = (U_{ν}h_{TT}^{μν}) = 0^{μ}  
Traceless Condition: Equivalent to Null = Photonic Condition: 
Generally A·A = (A^{μ}η_{μν}A^{ν}) = (φ/c,a)·(φ/c,a) = (φ/c)^{2}  a·a From above, A = (u·a/c,a) A·A = (u·a/c,a)·(u·a/c,a) = [(u·a/c)^{2}  a·a)] To an atrest observer (u=0), A·A = ( a·a) appears spatial To an ânull observer (u=câ), A·A = (0) appears null A 3rd situation is that u·a=0 via 3D orthogonality, in which case A appears spatial A → (0,a) and A·A = ( a·a) appears spatial 
Tr[ h_{TT}^{μν} ] = (η_{μν}h_{TT}^{μν}) = h_{TTν}^{ν} = 0  Lanczos algebraic gauge SR Tr[ L_{μν}^{ρ} ] = (η^{ν}_{ρ}L_{μν}^{ρ}) = ( L_{μρ}^{ρ}) = 0_{μ}  
Transverse Condition: Occurs due to the combination of: Solution is Free Plane Wave: gives K·K = 0, K is null The Conservative Field Condition: gives K·C = 0 The Purely Spatial Condition: gives U_{o}·C = 0 The combination leads to the spatial k·c = 0 The wave is transverse 
The TransverseTraceless Gauge (TT) aka. the Radiation Gauge 

Wave Equation with Source:  (∂·∂)A^{ν} = μ_{o} J^{ν}  (∂·∂)h_{TT}^{μν} = 2 G^{μν} (∂·∂)h_{TT}^{μν}^{} = 16πG T^{μν} I'm not sure about the signs Also, this is linerized approx to GR 
complicated  
Wave Equation without Source: ie. Freelypropagating 
(∂·∂)Φ = 0  (∂·∂)A^{ν} = 0^{ν}  (∂·∂)h_{TT}^{μν} = 0^{μν}  (∂·∂)L^{μνρ} = 0^{μνρ}  
Free Wave Solution: Plane Wave with C or C^{ν} or C^{μν} or C^{μνρ} as respective wave amplitudes 
Φ^{} = C e^{(iK·X)}  A^{μ} = C^{μ} e^{(iK·X)}  h_{TT}^{μν} = C^{μν} e^{(iK·X)}  L^{μνρ} = C^{μνρ} e^{(iK·X)}  
Solution Check: assumes that the wave amplitude C^{...} is a constant 
(∂·∂)Φ^{} = η_{ρσ} ∂^{ρ}∂^{σ} C e^{(iK·X)} = i η_{ρσ} ∂^{ρ}K^{σ} C e^{(iK·X)} = i^{2} η_{ρσ} K^{ρ}K^{σ} C e^{(iK·X)} =  K_{σ}K^{σ} Φ^{} = 0 
(∂·∂)A^{μ} = η_{ρσ} ∂^{ρ}∂^{σ} C^{μ} e^{(iK·X)} = i η_{ρσ} ∂^{ρ}K^{σ} C^{μ} e^{(iK·X)} = i^{2} η_{ρσ} K^{ρ}K^{σ} C^{μ} e^{(iK·X)} =  K_{σ}K^{σ} A^{μ} = 0^{ν} ^{} 
(∂·∂)h_{TT}^{μν}^{} = η_{ρσ} ∂^{ρ}∂^{σ} C^{μν} e^{(iK·X)} = i η_{ρσ} ∂^{ρ}K^{σ} C^{μν} e^{(iK·X)} = i^{2} η_{ρσ} K^{ρ}K^{σ} C^{μν} e^{(iK·X)} =  K_{σ}K^{σ} h_{TT}^{μν}^{} = 0^{μν} 
(∂·∂)L^{μνρ}^{} = η_{ρσ} ∂^{ρ}∂^{σ} C^{μνρ} e^{(iK·X)} = i η_{ρσ} ∂^{ρ}K^{σ} C^{μνρ} e^{(iK·X)} = i^{2} η_{ρσ} K^{ρ}K^{σ} C^{μνρ} e^{(iK·X)} =  K_{σ}K^{σ} L^{μνρ}^{} = 0^{μνρ}  
Trivial Solution = No Wave = (Field = 0)  Φ^{} = 0  A^{μ} = 0^{μ}  h_{TT}^{μν} = 0^{μν}  L^{μνρ} = 0^{μνρ}  
Interesting Solution = Wave 4WaveVector K is Null Massless = LightLike = Photonic 
K_{σ}K^{σ} = 0  K_{σ}K^{σ} = 0  K_{σ}K^{σ} = 0  K_{σ}K^{σ} = 0  
4Divergenceless Check 4WaveVector K orthogonal to 4WaveAmplitude C^{...} 4WaveVector K orthogonal to 4Polarization E 4WaveVector K orthogonal to Polarization Tensor 
(∂·A) = (∂_{μ}A^{μ}) = 0 ∂_{μ}C^{μ} e^{(iK·X)} = 0 iK_{μ}C^{μ} e^{(iK·X)} = 0 K_{μ}C^{μ} = 0 
(∂·h_{TT}^{μν}) = (∂_{ν}h_{TT}^{μν}) = 0^{μ} ∂_{ν}C^{μν} e^{(iK·X)} = 0^{μ} iK_{ν}C^{μν} e^{(iK·X)} = 0^{μ} K_{ν}C^{μν} = 0^{μ} 
(∂·L_{}^{μνρ}) = (∂_{ρ}L_{}^{μνρ}) = 0^{μν} ∂_{ρ}C^{μνρ} e^{(iK·X)} = 0^{μν} iK_{ρ}C^{μνρ} e^{(iK·X)} = 0^{μν} K_{ρ}C^{μνρ} = 0^{μν}  
Examine Solutions: general null K = (ω/c) (1,n̂) assume null K = (ω/c,0,0,ω/c) in spatial zdirection 
C  C = C^{ν} = → (0,c_{1},c_{2},0) 
C^{μν} = →
 C^{μνρ} = →...  
Find Polarizations: **Note** The 4Polarization E = C^{ν} The Polarization Tensor C^{μν} both can have complex components. These give circular and elliptical polarizations Circular/elliptical polarizations should also carry angular momentum 
Nonpolarized  → (0,c_{1},c_{2},0) → (0,c_{x},c_{y},0) → (0,1,0,0) = xpolarized → (0,0,1,0) = ypolarized (rotated 90°) for photon travelling in zdirection using the Jones Vector formalism n = z / z and to the observer at rest C = (0,1,0,0) : xpolarized linear C = (0,0,1,0) : ypolarized linear C = √[1/2] (0,1,1,0) : 45° from xpolarized linear C = √[1/2] (0,1,i,0) : rightpolarized circular C = √[1/2] (0,1,i,0) : leftpolarized circular Generalpolarized (elliptical) for zphoton C = (0,Cos[θ]Exp[iα_{x}],Sin[θ]Exp[iα_{y}]),0) C* = (0,Cos[θ]Exp[iα_{x}],Sin[θ]Exp[iα_{y}]),0) 
h_{+} = plus pattern h_{x} = cross pattern (rotated 45°) h_{R} = (1/√2)(h_{+} + ih_{x}) = Right Circular h_{L} = (1/√2)(h_{+}  ih_{x}) = Left Circular etc. Presumably there could be Elliptical polarizations for gravwaves also Test Particle Masses: Plus (+) Polarization Cross (x) Polarization  ... 
A = A^{μ}  h_{TT}^{μν} = kA^{μ}A^{ν} = kA^{μ}⊗A^{ν} 
(∂·A) = (∂_{μ}A^{μ}) = 0  (∂·h_{TT}^{μν}) = (∂_{ν}h_{TT}^{μν}) = (∂_{ν}kA_{}^{μ}A^{ν}) = k(∂_{ν}A^{ν})A^{μ} = k(0)A^{μ} = 0^{μ } 
(U·A) = (U_{ν}A^{ν}) = 0 for a photonic wave Generally A_{EM}·U = (φ/c,a)·γ(c,u) = γ(φ  a·u) = φ_{o} _{}As we will see, this is a photonic wave and the rest potential φ_{o} will be zero in the same way that the rest mass m_{o} of a photon is zero In other words: There is no "atrest" frame for lightlike (U·A) = 0 = γ(c,u)·(φ/c,a) = γ[φ  u·a] = 0 Therefore, φ = u·a Therefore A = (u·a/c,a) To an atrest observer (u=0), A appears spatial A → (0,a) To an ânull observer (u=câ), A appears null A → (a,a) A 3rd situation is that u·a=0 via 3D orthogonality, in which case A appears spatial A → (0,a)  (U·h_{TT}^{μν}) = (U_{ν}h_{TT}^{μν}) = (U_{ν}kA_{}^{μ}A^{ν}) = k(U_{ν}A^{ν})A^{μ} = k(0)A^{μ} = 0^{μ} 
Generally A·A = (A^{μ}η_{μν}A^{ν}) = (φ/c,a)·(φ/c,a) = (φ/c)^{2}  a·a From above, A = (u·a/c,a) A·A = (u·a/c,a)·(u·a/c,a) = [(u·a/c)^{2}  a·a] To an atrest observer (u=0), A·A = ( a·a) appears spatial To an ânull observer (u=câ), A·A = (0) appears null A 3rd situation is that u·a=0 via 3D orthogonality, in which case A appears spatial A → (0,a) and A·A = ( a·a) appears spatial  Tr[ h_{TT}^{μν} ] = (η_{μν}h_{TT}^{μν}) = k(η_{μν}A_{}^{μ}A^{ν}) = k(A^{μ}η_{μν}A^{ν}) = k[(u·a/c)^{2}  a·a] If indeed (h_{TT}^{μν} = kA^{μ}A^{ν} = kA^{μ}⊗A^{ν}) decomposes this way, we should get Tr[ h_{TT}^{μν} ] = 0^{μ} for the ânull observer Let me think on this a bit... 
*Lorentz Scalar* <Potential>  4Vector = Gradient[<Potential>]  Rest value Temporal Component  
SR Phase (Φ)  Φ = (K·R)  4WaveVector K = ∂[Φ]  U·K = U·∂[Φ] = d/dτ[Φ] = ω_{o} 
SR TotalPhase (Φ_{T})  Φ_{T} = (K_{T}·R)  4TotalWaveVector K_{T} = ∂[Φ_{T}]  U·K_{T} = U·∂[Φ_{T}] = d/dτ[Φ_{T}] = ω_{To} 
SR Action (S_{act})  S_{act} = (P_{T}·R)  4TotalMomentum P_{T} = ∂[S_{act}]  U·P_{T} = U·∂[S_{act}] = d/dτ[S_{act}] = H_{o} 
U·P_{Tden} = L_{deno} = n_{o}(P_{T}·U) = H_{deno}  
SR StressEnergy (T^{μν})  eg. T_{perfectfluid}^{μν} = (ρ_{eo})V^{μν}  (p_{o})H^{μν}  4ForceDensity F_{den}^{μ} = ∂_{ν}[T^{μν}]  U_{μ}F_{den}^{μ} = U_{μ}∂_{ν}[T^{μν}] = γ_{f}Ė_{o} = γ_{f}ṁ_{o}c^{2} 
The magic behind the EM curtain...  ∂^{ν}[P_{T}^{μ}] = q∂^{μ}[A_{EM}^{ν}] 
0  e^{x}/c  e^{y}/c  e^{z}/c 
e^{x}/c  0  b^{z}  b^{y} 
e^{y}/c  b^{z}  0  b^{x} 
e^{z}/c  b^{y}  b^{x}  0 
0  e^{i}/c 
+e^{j}/c  ε_{ijk}b^{k} 
∂_{X} = ±iK 
[∂_{X},X] = η^{μν} = [∂_{K},K] [∂_{X},X] = [∂_{K},K] [∂_{X},X] = [K,∂_{K}] [±iK,X] = [K,∂_{K}] ±i[K,X] = [K,∂_{K}] ±i^{2}[K,X] = i[K,∂_{K}] ±(1)[K,X] = i[K,∂_{K}] ±[K,X] = i[K,∂_{K}] [K,X] = ±i[K,∂_{K}] [K,X] = [K,±i∂_{K}] 
X = ±i∂_{K} ∂_{K} = ∓iX 
*Relativistic* P = (E/c,p) = ħK = iħ∂ = iћ(∂_{t}/c,∇) 
*Classical* = limitingcase using { √[1 + x] ~ (1 + x/2 + ...O[x^{2}] ) } 

4Momentum  Einstein Energy Relation P·P = (E/c)^{2}  p^{2} = (m_{o}c)^{2} 

solved for temporal component  E = √[(m_{o}c^{2})^{2} + c^{2}p^{2}]  Newtonian Energy Relation E ~ [(m_{o}c^{2}) + p^{2}/2m_{o}] 
4Gradient  Free Particle KleinGordon RQM Equation ∂·∂ = (∂_{t}/c)^{2} ∇^{2} = (im_{o}c/ћ)^{2 }iћ∂·iћ∂ = (iћ∂_{t}/c)^{2}  (iћ∇)^{2} = (m_{o}c)^{2} 

solved for temporal component  (iћ∂_{t}) = √[(m_{o}c^{2})^{2} + c^{2}(iћ∇)^{2}] (iћ∂_{t}) = √[(m_{o}c^{2})^{2}  c^{2}(ћ∇)^{2}] 
Free Particle Schrödinger QM Equation (iħ∂_{t}) ~ [(m_{o}c^{2}) + (iħ∇)^{2}/2m_{o}] (iħ∂_{t}) ~ [(m_{o}c^{2})  (ħ∇)^{2}/2m_{o}] 
*Relativistic* P = (E/c,p) = ħK = iħ∂ = iћ(∂_{t}/c,∇) P_{T} = P + qA 
*Classical* = limitingcase using { √[1 + x] ~ (1 + x/2 + ...O[x^{2}] ) } 

4Momentum  Einstein Energy Relation P·P = (E/c)^{2}  p^{2} = (m_{o}c)^{2 } = (E_{T}/c  qφ/c)^{2}  (p_{T} qa)^{2} = (m_{o}c)^{2} 

solved for temporal component  E = √[(m_{o}c^{2})^{2} + c^{2}p^{2}] (E_{T} qφ) = √[(m_{o}c^{2})^{2} + c^{2}(p_{T} qa)^{2}] 
Newtonian Energy Relation E ~ [(m_{o}c^{2}) + p^{2}/2m_{o}] (E_{T} qφ) ~ [(m_{o}c^{2}) + (p_{T} qa)^{2}/2m_{o}] 
4Gradient  Free Particle KleinGordon RQM Equation ∂·∂ = (∂_{t}/c)^{2} ∇^{2} = (im_{o}c/ћ)^{2 }iћ∂·iћ∂ = (iћ∂_{t}/c)^{2}  (iћ∇)^{2} = (m_{o}c)^{2 } KleinGordon RQM Equation w/Potential (iħ∂_{tT} qφ)^{2} = (m_{o}c^{2})^{2} + c^{2}(iħ∇_{T} qa)^{2}: 

solved for temporal component  (iћ∂_{t}) = √[(m_{o}c^{2})^{2} + c^{2}(iћ∇)^{2}] (iћ∂_{t}) = √[(m_{o}c^{2})^{2}  c^{2}(ћ∇)^{2}] (iħ∂_{tT} qφ) = √[(m_{o}c^{2})^{2} + c^{2}(iħ∇_{T} qa)^{2}] (iħ∂_{tT}) = qφ + √[(m_{o}c^{2})^{2} + c^{2}(iħ∇_{T} qa)^{2}] 
Free Particle Schrödinger QM Equation (iħ∂_{t}) ~ [(m_{o}c^{2}) + (iħ∇)^{2}/2m_{o}] (iħ∂_{t}) ~ [(m_{o}c^{2})  (ħ∇)^{2}/2m_{o}] Schrödinger QM Equation w/Potential (iħ∂_{tT} qφ) ~ [(m_{o}c^{2}) + (iħ∇_{T} qa)^{2}/2m_{o}] (iħ∂_{tT}) ~ [qφ + (m_{o}c^{2}) + (iħ∇_{T} qa)^{2}/2m_{o}] (iħ∂_{tT}) ~ [V + (iħ∇_{T} qa)^{2}/2m_{o}] : with [V = qφ + (m_{o}c^{2})] (iħ∂_{tT}) ~ [V  (ħ∇_{T})^{2}/2m_{o}]: with a = 0 the Standard way it is usually seen 
(∂·∂ + (m_{o}c/ћ)^{2} )Ψ = 0  Ψ is a scalar, KleinGordon eqn for massive spin0 field, ex. the Higgs Boson 
(∂·∂ + (m_{o}c/ћ)^{2} )A = 0  A is a 4Vector, Proca eqn for massive spin1 field, Lorenz Gauge 
(∂·∂)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, Lorenz Gauge 
(∂·∂)A = μ_{o}J = ρ_{o}μ_{o}U = qn_{o}μ_{o}U = qμ_{o}N  A is a 4Vector, Maxwell eqn for massless (m_{o} =
0) spin1 field, current source J, Lorenz Gauge Classical EM, does not include effects of particle spin in the current source J 
(∂·∂)A = μ_{o}J = μ_{o}(qΨ ̅ γΨ) (∂·∂)A^{μ} = μ_{o}J = μ_{o}(qΨ ̅ γ^{μ}Ψ) where Ψ ̅ γ^{μ}Ψ has units of flux (#/m^{2}·s) 
QED, A is a 4Vector, Maxwell eqn for massless (m_{o} =
0) spin1 field, current source J, Lorenz Gauge Quantum EM, does include effects of particle spin in the current source J = Ψ ̅ γ^{μ}Ψ 
Just a note: The classical Maxwell EM equations do not have Spin included (∂·∂)A_{EM} = μ_{o}J = μ_{o}ρ_{o}U = μ_{o}qn_{o}U = μ_{o}qN = μ_{o}(q/V_{o})U = μ_{o}q(c/V_{o})T Once spin is included, the equations for QED emerge: (∂·∂)A_{EM} = μ_{o}qψ Γψ not sure if the μ_{o} factor is included or not 
Full Equation (ungauged)  Lorenz Gauge (∂·A = 0)  Field Type 
(∂·∂ + (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  
∂_{ν}F^{νμ} = 0^{μ} ∂_{ν}(∂^{ν}A^{μ}  ∂^{μ}A^{ν}) = 0^{μ} ∂_{ν}∂^{ν}A^{μ}  ∂^{μ}∂_{ν}A^{ν} = 0^{μ} (∂·∂)A^{μ}  ∂^{μ}(∂·A) = 0^{μ} 
(∂·∂)A = Z (∂·∂)A^{μ} = 0^{μ} 
A is a 4Vector, Maxwell eqn for massless (m_{o} = 0) spin1 field, no current sources 
(∂·∂)A = μ_{o}J = ρ_{o}μ_{o}U = qn_{o}μ_{o}U = qμ_{o}N  A is a 4Vector, Maxwell eqn for massless (m_{o} = 0) spin1 field, current source J  
∂_{ν}F^{νμ} = qΨ ̅ γ^{μ}Ψ ∂_{ν}(∂^{ν}A^{μ}  ∂^{μ}A^{ν}) = qΨ ̅ γ^{μ}Ψ ∂_{ν}∂^{ν}A^{μ}  ∂^{μ}∂_{ν}A^{ν} = qΨ ̅ γ^{μ}Ψ (∂·∂)A^{μ}  ∂^{μ}(∂·A) = qΨ ̅ γ^{μ}Ψ 
(∂·∂)A = qΨ ̅ γΨ (∂·∂)A^{μ} = qΨ ̅ γ^{μ}Ψ 
QED, A is a 4Vector, spin1 field, current source Ψ ̅ γ^{μ}Ψ 
(∂·∂)h_{TT}^{μν} = 0^{μν}  Gravitational Waves, h_{TT}^{μν} is a (T)ranverse (T)raceless 2Tensor representing gravitational radiation in the weakfield limit far from the source 
∂_{X} = ±iK 
[∂_{X},X] = η^{μν} = [∂_{K},K] [∂_{X},X] = [∂_{K},K] [∂_{X},X] = [K,∂_{K}] [±i K,X] = [K,∂_{K}] ±i[K,X] = [K,∂_{K}] ±i^{2}[K,X] = i[K,∂_{K}] ±(1)[K,X] = i[K,∂_{K}] ±[K,X] = i[K,∂_{K}] [K,X] = ±i[K,∂_{K}] [K,X] = [K,±i∂_{K}] 
X = ±i∂_{K} ∂_{K} = ∓iX 
dX/dτ = (U·∂)[X] = U  dX/dθ = (K·∂)[X] = K 
U·U = c^{2} U_{1}·U_{2} = (γ_{12})c^{2} U·U_{o} = (γ_{rel})c^{2} 
K·K = (ω_{o}/c)^{2} 
∂[U·U] = 2*U·∂[U] = ∂[c^{2}] = Z ∂[U_{1}·U_{2}] = U_{1}·∂[U_{2}] +U_{2}·∂[U_{1}] = ∂[(γ_{12})c^{2}] = c^{2} ∂[γ_{12}] ∂[U·U_{o}] = U·∂[U_{o}] +U_{o}·∂[U] = (0) +U_{o}·∂[U] = ∂[(γ_{rel})c^{2}] = c^{2} ∂[γ_{rel}] 
∂[K·K] = 2* K·∂[K] = ∂[(ω_{o}/c)^{2}] = Z, if ω_{o} is constant 
d/dτ[U·U] = 2*U·d/dτ[U] = 2*U·A = d/dτ[c^{2}] = 0 d/dτ[U_{1}·U_{2}] = U_{1}·d/dτ[U_{2}] +U_{2}·d/dτ[U_{1}] = U_{1}·A_{2} +U_{2}·A_{1} = d/dτ[(γ_{12})c^{2}] = c^{2}d/dτ[γ_{12}] d/dτ[U·U_{o}] = U·d/dτ[U_{o}] +U_{o}·d/dτ[U] = (0) +U_{o}·A = d/dτ[(γ_{rel})c^{2}] = c^{2}d/dτ[γ_{rel}] 

d^{2} X/dτ^{2} = dU/dτ = ? = (U·∂)[U] = U·∂[U] = Z but should be A instead try d^{2} X/dτ^{2} = dU/dτ = (U_{o}·∂)[U] = U_{o}·∂[U] = A = ? = c^{2} ∂[γ_{rel}] 
d^{2} X/dθ^{2} = dK/dθ = (K·∂)[K] = K·∂[K] = 0 ? 
R = (ct,r)  particle/location 
U = dR/dτ  movement/velocity 
P = m_{o}U  mass/momentum 
K = (1/ћ)P  wave/particle duality 
∂ = iK  SpaceTime/wave structure 
∂_{t}^{2}/c^{2} = ∇·∇  (m_{o}c/ћ)^{2} 
(∂·∂ + (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, Lorenz Gauge 
(∂·∂)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, Lorenz Gauge 
(∂·∂)A = μ_{o}J = ρ_{o}μ_{o}U = qn_{o}μ_{o}U = qμ_{o}N  A is a 4Vector, Maxwell eqn for massless (m_{o} =
0) spin1 field, current source 4Vector J, Lorenz Gauge Classical EM, does not include effects of particle spin in the source J = ρ_{o}U 
(∂·∂)A = μ_{o}J = μ_{o}qΨ ̅ ΓΨ (∂·∂)A^{ν} = μ_{o}J^{ν} = μ_{o}qΨ ̅ γ^{ν}Ψ 
QED, A is a 4Vector, Maxwell eqn for massless (m_{o} =
0) spin1 field, current source Spinor J, Lorenz Gauge Quantum EM, does include effects of particle spin in the source J = qΨ ̅ γ^{μ}Ψ 
[a  b]  [X]  =  [c 0]  [X] 
[b  a]  [Y]  [0 c]  [Y] 
([1 0]  a +  [0 1]  b  )[X]  = c  [1 0]  [X] 
([0 1]  [1 0]  )[Y]  [0 1]  [Y] 
([1 0]  a +  [0 1]  b  )[X]  = c  [1 0]  [X] 
([0 1]  [1 0]  )[Y]  [0 1]  [Y] 
([1 0]  ps^{0} +  [0 1]  ps  )[X]  = (m_{o}c)I  [X] 
([0 1]  [1 0]  )[Y]  [Y] 
([1 0]  σ^{0}p^{0} +  [0 1]  σ·p  )[X]  = (m_{o}c)I 
[X] 
([0 1]  [1 0]  )[Y]  [Y] 
([σ^{0} 0]  p^{0} +  [0 σ]  ·p  )[X]  = (m_{o}c)I 
[X] 
([0 σ^{0}]  [σ 0]  )[Y]  [Y] 
let Spinor Ψ =  [X]  and note that σ^{0} = I_{2} 
[Y] 
([I_{2} 0]  p^{0} +  [0 σ]  ·p  )  Ψ= (m_{o}c)IΨ 
([0 I_{2}]  [σ 0]  ) 
Relativistic Hamiltonian H = γ(P_{T}·U) 
Relativistic Lagrangain L = (P_{T}·U)/γ 
p_{T}·u = ( γβ^{2})(P_{T}·U) = H + L = γ(P_{T}·U) + (P_{T}·U)/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 = ±m_{o}c^{2}√[1 + (p_{T} qa)^{2}/(m_{o}^{2}c^{2})] + qφ 
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 + 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 
Rest Hamiltonian H_{o} = (P_{T}·U) = H/γ 
Rest Lagrangian L_{o} = (P_{T}·U) = γL 
H_{o} + L_{o} = 0 
Probability   ↑ ⟩ State   ↓ ⟩ State  Ket Tensor Product Representaion 
1/4  1, 2   ↑ ⟩ ↑ ⟩  
1/4  1  2   ↑ ⟩ ↓ ⟩ 
1/4  2  1   ↓ ⟩ ↑ ⟩ 
1/4  1, 2   ↓ ⟩ ↓ ⟩ 
Probability   ↑ ⟩ State   ↓ ⟩ State  Ket Tensor Product Representation 
1/3  x,x   ↑ ⟩ ↑ ⟩ =  1,1 ⟩  
1/3  x  x  1/√[2]*(  ↑ ⟩ ↓ ⟩ +  ↓ ⟩ ↑ ⟩ ) =  1,0 ⟩ 
1/3  x,x   ↓ ⟩ ↓ ⟩ =  1,1 ⟩ 
Probability   ↑ ⟩ State   ↓ ⟩ State  Ket Tensor Product Representaion 
1  x  x  1/√[2]*( ↑ ⟩ ↓ ⟩   ↓ ⟩ ↑ ⟩ ) =  0,0 ⟩ 
Particles  Statistics  Energy Occupation  Principle  Canonical Commutation  Both (+)  Both ()  One each ( + ,) 

Bosons  BoseEinstein  <N_{i}> = g_{i}/(e^[(ε_{i} μ)/kT]  1)  Agglutination or Congregation 
[b_{α},b_{β}] = [b^{†}_{α},b^{†}_{β}] = 0 [b_{α},b^{†}_{β}] = b_{α}b^{†}_{β}  b^{†}_{β}b_{α} = δ_{αβ} 
1/3  1/3  1/3 
Distinguishable  MaxwellBoltzmann  <N_{i}> = g_{i}/(e^[(ε_{i} μ)/kT] + 0)  Simple Random  1/4  1/4  1/2  
Fermions  FermiDirac  <N_{i}> = g_{i}/(e^[(ε_{i} μ)/kT] + 1)  Pauli Exclusion  {f_{α},f_{β}} = {f^{†}_{α},f^{†}_{β}} = 0 {f_{α},f^{†}_{β}} = f_{α}f^{†}_{β} + f^{†}_{β}f_{α} = δ_{αβ} 
0  0  1 
Particles  Field Operator  Annihilation,Annihilation  Creation,Creation  Annihilation,Creation 

Bosons  Φ_{b}(r) = Σ_{j}[e^(k_{j}·r)b_{j}]  [Φ_{b}(r),Φ_{b}(r')] = 0 or [Φ_{b}(r),Φ_{b}(r')]_{ } = 0 
[Φ^{†}_{b}(r),Φ^{†}_{b}(r')] = 0 or [Φ^{†}_{b}(r),Φ^{†}_{b}(r')]_{ } = 0 
[Φ_{b}(r),Φ^{†}_{b}(r')] = ⟨ rr'⟩ = δ^{3}(rr') or [Φ_{b}(r),Φ^{†}_{b}(r')]_{ } = ⟨ rr'⟩ = δ^{3}(rr') 
Fermions  Φ_{f}(r) = Σ_{j}[e^(k_{j}·r)f_{j}]  {Φ_{f}(r),Φ_{f}(r')} = 0 or [Φ_{f}(r),Φ_{f}(r')]_{ + } = 0 
{Φ^{†}_{f}(r),Φ^{†}_{f}(r')} = 0 or [Φ^{†}_{f}(r),Φ^{†}_{f}(r')]_{ + } = 0 
{Φ_{f}(r),Φ^{†}_{f}(r')} = ⟨ rr'⟩ = δ^{3}(rr') or [Φ_{f}(r),Φ^{†}_{f}(r')]_{ + } = ⟨ rr'⟩ = δ^{3}(rr') 
timetime T^{00 }

timespace T^{0j }


spacetime T^{i0 }

spacespace T^{ij }


T^{00}  T^{01}  T^{02}  T^{03} 
T^{10}  T^{11}  T^{12}  T^{13} 
T^{20}  T^{21}  T^{22}  T^{23} 
T^{30}  T^{31}  T^{32}  T^{33} 
 = 

 = 

 = 

(Vacuum) Field Equations  (Sourced) Field Equations Minkowski Metric Lorentz Gauge 
Equations of Motion  Potential Φ  Independent Parameters 

Newton CM  g^{ij}Φ_{,ij} = 0  g^{ij}Φ_{,ij} = ∇·∇Φ = 4πGρ_{m}  d^{2}/dt^{2}[X^{i}] = g^{ij}Φ_{, j} = ∂Φ/∂X^{i} d^{2}/dt^{2}[x] = a = ∇Φ 
Scalar = (0Tensor)  1 
Maxwell SR  g^{μν}Φ_{ρ,μν} = 0  g^{μν}Φ_{ρ,μν} = (∂·∂)Φ_{ρ} = μ_{o}J_{ρ} (∂·∂)A = μ_{o}J (∂·∂)(φ/c,a) = μ_{o}(ρ_{e}c,j) (∂·∂)φ = μ_{o}ρ_{e}c^{2} = ρ_{e}/ε_{o} ∇·∇φ = ρ_{e}/ε_{o} {in timeindependent potential) 
(these assume constant restmass m_{o}) d^{2}/dτ^{2}[X^{μ}] = (q/cm_{o})g^{μα}(Φ_{α,β}  Φ_{β,α})(dX^{β}/dτ) A^{μ} = (q/cm_{o})g^{μα}(Φ_{α,β}  Φ_{β,α})U^{β} F^{μ} = qU_{ν}(∂^{μ}A_{EM}^{ν}  ∂^{ν}A_{EM}^{μ}) = qU_{ν}F^{μν} 
4Vector = (1Tensor)  4 
Einstein GR  g^{μν}Φ_{ρσ,μν} + ... = 0  d^{2}/dτ^{2}[X^{μ}] = (1/2)g^{μα}(g_{αβ,γ} + g_{αγ,β}  g_{βγ,α}) (dX^{β}/dτ)(dX^{γ}/dτ)  Tensor = (2Tensor)  10 
Approximation Level  Equation of Motion (Positions)  Equation of Motion (Velocities)  Limiting Case  
Einstein GR (base/fundamental) 
d^{2}X^{σ}/dτ^{2} + (Γ^{σ}_{μν})(dX^{μ}/dτ)(dX^{ν}/dτ) = 0  dU^{σ}/dτ + (Γ^{σ}_{μν})(U^{μ})(U^{ν}) = 0  Geodesic Motion  no Symmetry/Charge Forces  
Einstein SR  d^{2}X^{σ}/dτ^{2} = 0  dU^{σ}/dτ = d/dτ[U^{σ}] = 0 γdU^{σ}/dt = γd/dt[U^{σ}] = 0 
Geodesic Motion  no Symmetry/Charge Forces "Flat" Minkowski SpaceTime (Γ^{σ}_{μν}) → 0 

Newton CM  d^{2}X^{σ}/dt^{2} = 0  dU^{σ}/dt = 0  Geodesic Motion  no Symmetry/Charge Forces "Flat" Minkowski SpaceTime (Γ^{σ}_{μν}) → 0 Low Velocity (v << c; γ → 1, τ → t) 
Field Equations in Lorenz Gauge (Divergence of Basic Field = 0) 
Full Field Equations  CurrentDensity  Higher Field Construction  Basic Field  
SR EM 4Vector Style 
(∂·∂)A_{EM} = μ_{o}J with (∂·A_{EM}) = 0 
∂[F^{ρσ}] = (∂·∂)A_{EM}  ∂(∂·A_{EM}) = μ_{o}J  4CurrentDensity J  4VectorPotential A_{EM}  
SR EM Tensor Style 
(∂_{μ}∂^{μ})A_{EM}^{ν} = μ_{o}J^{ν} with (∂_{ν}A_{EM}^{ν}) = 0 
∂_{μ}F^{μν} = ∂_{μ}(∂^{μ}A_{EM}^{ν}  ∂^{ν}A_{EM}^{μ}) = μ_{o}J^{ν}  4CurrentDensity J^{ν} 
SR Faraday Tensor F^{μν} = (∂^{μ}A_{EM}^{ν}  ∂^{ν}A_{EM}^{μ}) 
4VectorPotential A_{EM}^{μ} 
SR EM (,) Style 
F^{μν}_{,μ} = μ_{o}J^{ν}  
GR (;) Style 
(∂·∂)H_{abc} = (∂_{μ}∂^{μ})H_{abc} = J_{abc} 2R_{c}^{d}H_{abd}+R_{a}^{d}H_{bcd}+R_{b}^{d}H_{acd} +(H_{dbe}g_{ac}H_{dae}g_{bc})R^{de}+RH_{abc}/2 
(Jordan Formulation) C_{abc}^{d}_{;d} = (some constant)J_{abc} 
Cotton Tensor (~Matter Current) J_{abc} = R_{ca;b}R_{cb;a} +(g_{cb}R_{;a}g_{ca}R_{;b})/6 
Weyl Tensor C_{abcd} = H_{abc;d}H_{abd;c}+H_{cda;b}H_{cdb;a}  (g_{ac}(H_{bd}+H_{db})g_{ad}(H_{bc}+H_{cb})+g_{bd}(H_{ac}+H_{ca})g_{bc}(H_{ad}+H_{da}))/2 +2H^{ef}_{e;f}(g_{ac}g_{bd}g_{ad}g_{bc})/3 where H_{bd}=H_{b}^{e}_{d;e}H_{b}^{e}_{c;d} 
Lanczos TensorPotential H_{abc} 
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} 
Schrödinger 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 
Schrödinger Equation (with standard scalar external potential) (i ћ)( ∂_{t} ) = V[x]  (ћ)^{2}(∇)^{2}/2m_{o} 
SL(2,ℂ) ⋉ ℝ^{1,3}  U(1)  SU(2)  SU(3) 
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GR  Standard Model 
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