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γ 
β_{x}γ 
0 
0 
β_{x}γ 
γ 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
So (a^{t}, a^{x}, a^{y}, a^{z})' = (γa^{t}  γβ_{x}a^{x} ,γβ_{x}a^{t} + γa^{x}, a^{y}, a^{z}){for x̂boost Lorentz Transform}
An Inverse Lorentz Transformation is equivalent to just reversing the direction of the boost, i.e. change the signs on the β.
So (a^{t}, a^{x}, a^{y}, a^{z})' = (γa^{t} + γβ_{x}a^{x} , γβ_{x}a^{t} + γa^{x}, a^{y}, a^{z}){for x̂boost Inverse Lorentz Transform}
General Lorentz Transformation, for a frame shift in any direction n̂:
Λ^{μ'}_{ν} = {for n̂boost}
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} 
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 x B] 
Maxwell Eqn  ∂_{α}(∂^{α}A_{EM}^{ν}  ∂^{ν}A_{EM}^{α}) = μ_{o}J^{ν}  ∂·F^{αν} = (μ_{o})J 
Lorentz Force Eqn  U_{α}(∂^{ν}A_{EM}^{α}  ∂^{α}A_{EM}^{ν}) = (1/q)F^{ν}  U·F^{αν} = (1/q)F 
T^{00} ρ_{m}c^{2} = ρ_{e} Energy Density 
T^{0j} cg Energy Flux 
T^{i0} cg Momentum Density 
T^{ij} σ^{ij} Momentum Flux 
T^{00} Energy Density 
T^{01}  T^{02}  T^{03} Energy Flux 
T^{10}  T^{20}  T^{30} Momentum Density 
T^{11}  T^{12}  T^{13}    T^{21}  T^{22}  T^{23}    T^{31}  T^{32}  T^{33} Momentum Flux 
T^{11} Pressure xx 
T^{12} Shear xy 
T^{13} Shear xz 
T^{21} Shear yx 
T^{22} Pressure yy 
T^{23} Shear yz 
T^{31} Shear zx 
T^{32} Shear zy 
T^{33} Pressure zz 
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)  p  γ^{2}(ρ_{eo}+p)u_{x}/c  γ^{2}(ρ_{eo}+p)u_{y}/c  γ^{2}(ρ_{eo}+p)u_{z}/c 
γ^{2}(ρ_{eo}+p)u_{x}/c  γ^{2}(ρ_{eo}+p)u_{x}u_{x}/c^{2} + p  γ^{2}(ρ_{eo}+p)u_{x}u_{y}/c^{2}  γ^{2}(ρ_{eo}+p)u_{x}u_{z}/c^{2} 
γ^{2}(ρ_{eo}+p)u_{y}/c  γ^{2}(ρ_{eo}+p)u_{x}u_{y}/c^{2}  γ^{2}(ρ_{eo}+p)u_{y}u_{y}/c^{2} + p  γ^{2}(ρ_{eo}+p)u_{y}u_{z}/c^{2} 
γ^{2}(ρ_{eo}+p)u_{z}/c  γ^{2}(ρ_{eo}+p)u_{x}u_{z}/c^{2}  γ^{2}(ρ_{eo}+p)u_{y}u_{z}/c^{2}  γ^{2}(ρ_{eo}+p)u_{z}u_{z}/c^{2} + p 
ρ_{eo} 



 p 



 p 



 p 
T^{00} = ρ_{eo}  T^{0j} = 0 
T^{i0} = 0  T^{ij} = pδ^{ij} 
ρ_{eo} 





 


 



ρ_{eo} 



 p = ρ_{eo}/3 



 p = ρ_{eo}/3 



 p = ρ_{eo}/3 
ρ_{eo} 



 p = ρ_{eo} 



 p = ρ_{eo} 



 p = ρ_{eo} 
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})] 
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} 
L = (P_{T}·U)/γ  H = γ(P_{T}·U)  H + L = p_{T}·u = γ(P_{T}·U)  (P_{T}·U)/γ 
L = (P_{T}·U)/γ L = ((P + Q)·U)/γ L = (P·U + Q·U)/γ L = P·U/γ  Q·U/γ L = m_{o}U·U/γ  qA·U/γ L = m_{o}c^{2}/γ  qA·U/γ L = m_{o}c^{2}/γ  q(φ/c, a)·γ(c, u)/γ L = m_{o}c^{2}/γ  q(φ/c, a)·(c,u) L = m_{o}c^{2}/γ  q(φ  a·u) L = m_{o}c^{2}/γ  qφ + qa·u L = m_{o}c^{2}/γ  qφ_{o}/γ L = (m_{o}c^{2} + qφ_{o})/γ 
H = γ(P_{T}·U) H = γ((P + Q)·U) H = γ(P·U + Q·U) H = γP·U + γQ·U H = γm_{o}U·U + γqA·U H = γm_{o}c^{2} + qγφ_{o} H = γm_{o}c^{2} + qφ H = ( γβ^{2} + 1/γ )m_{o}c^{2} + qφ H = ( γm_{o}β^{2}c^{2} + m_{o}c^{2}/γ) + qφ H = ( γm_{o}u^{2} + m_{o}c^{2}/γ) + qφ H = p·u + m_{o}c^{2}/γ + qφ H = E + qφ H = ± c√[m_{o}^{2}c^{2}+p^{2}] + qφ H = ± c√[m_{o}^{2}c^{2}+(p_{T}qa)^{2}] + qφ 
H + L = γ(P_{T}·U)  (P_{T}·U)/γ H + L = (γ  1/γ)(P_{T}·U) H + L = ( γβ^{2} )(P_{T}·U) H + L = ( γβ^{2} )((P + Q)·U) H + L = ( γβ^{2} )(P·U + Q·U) H + L = ( γβ^{2} )(m_{o}c^{2} + qφ_{o}) H + L = (γm_{o}β^{2}c^{2} + qγφ_{o}β^{2}) H + L = (γm_{o}u·uc^{2}/c^{2} + qφ_{o}γu·u/c^{2}) H + L = (γm_{o}u·u + qa·u) H + L = (p·u + qa·u) H + L = p_{T}·u 
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Quantum Mechanics is derivable from Special Relativity See QM from SRSimple RoadMap 