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} ~ e^P^{μ} Time Translation H = P^{0} Space Translation p = P^{i} 1 + 3 = 4  
 = e^ 

 +e^ 
 
Total of 6 + 4 = 10 parameters Poincaré Transform = Lorentz Transform + SpaceTime Translation X^{μ'} = Λ^{μ'}_{ν} X^{ν} + ΔX^{μ'} with Λ^{μ'}ν = exp[1/2 ωαβM^{αβ}]^{μ’}ν ΔX^{μ'} = exp[X∙P]^{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}^{ } 
Eqn. of State (EoS )Parm w = (p_{o}/ρ_{eo}) = (n/3)1 
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. 
Gravitational Pressure  Speed of Sound c_{s} = (speed of sound) = c * Sqrt[w]? 
Dimensional Type 
< 2ρeo  > ρ_{eo}  >1  >6  Speed of Sound>Speed of Light = Unphysical  
(ρ_{eo})V^{μν}  (3ρ_{eo}/3)H^{μν}  2ρ_{eo}  ρ_{eo}  1  6  ρ_{eo} ∝ a^{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} ∝ a^{5}  ???  
< 0ρeo  > ρ_{eo}/3  > 1/3  >4  "Ultralight", meaning ultraphotonic...  Unknown  
(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} 
Relativistic Point 

(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 Normal Matter Einsteinde Sitter (EdS) solution T^{μν} = (ρ_{eo})V^{μν} 
Point  
(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?  Replusive?  Imaginary, Instabilities  Line 
> 2ρ_{eo}  < ρ_{eo}/3  < 1/3  < 2  Everything Below has Accelerating Expansion of Universe  Repulsive  Imaginary, Instabilities  
(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?  Replusive  Imaginary, Instabilities  Sheet 
(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} _{with H =Hubble Const} 
(Quantum) Vacuum Energy/Dark Energy /Cosmological Constant Λ/de Sitter/(Inflation~ 1) T^{μν} = (ρ_{eo})η^{μν} 
Replusive  Imaginary, Instabilities  Volume 
> 4ρ_{eo}  < ρ_{eo}  < 1  < 0  Big Rip = Phantom Energy  Replusive  Imaginary, Instabilities Speed of Sound>Speed of Light = Unphysical 
Unknown 
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} 
SpaceTime Symmetry  Internal Particle Symmetries  
SL(2,ℂ) ⋉ ℝ^{1,3}  U(1)  SU(2)  SU(3) 
Gravity  EM  Weak  Color 
GR  Standard Model 
The Science Realm: John's Virtual SciTech
Universe John's Science & Math Stuff:  About John  Send email to John 4Vectors  Ambigrams  Antipodes  Covert Ops Fragments  Cyrillic Projector  Forced Induction (Sums Of Powers Of Integers)  Fractals  Frontiers  JavaScript Graphics  Kid Science  Kryptos  Philosophy  Photography  Prime Sieve  QM from SR  QM from SRSimple RoadMap  SR 4Vector & Tensor Calculator  Quantum Phase  Quotes  RuneQuest Cipher Challenge  Scientific Calculator (complex capable)  Secret Codes & Ciphers  Science+Math  SciPan Pantheist Poems  Stereograms  SuperMagicSqr4x4  Turkish Grammar  
Quantum Mechanics is derivable from Special Relativity See SRQM  QM from SR  Simple RoadMap (.html) See SRQM  QM from SR  Simple RoadMap (.pdf) 