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

The 4-vectors (four-vectors) of Special Relativistic (SR) theory are fundamental entities that accurately, precisely, and beautifully describe the physical properties of the world around us. While it is known that SR is not the deepest theory, it is valid for the majority of the universe. It is believed to apply to all forms of interaction, including that of fundamental particles, and with only the exception being that of large-scale gravitational phenomena, where spacetime itself is significantly curved, for which General Relativity (GR) is required. The SR 4-vector notation is one of the most powerful tools in understanding the physics of the universe, as it simplifies a great many of the physical relations.


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Introduction

4-vectors are tensorial entities which display Poincare' Invariance, meaning they leave invariant the differential squared interval (ds)²=(cdt)²-dx²-dy²-dz². A consequence of this invariant measurement is that any physical equation which is written in Poincare' Invariant form is automatically valid for any inertial reference frame, regardless of how coordinate systems are arranged. Transformations which leave these vectors unchanged include fixed translations through space and/or time, rotations through space, and boosts (coordinate systems moving with constant velocity) through spacetime. Since 4-vectors are tensors, and Poincare' Invariant, they can be used to describe and explain the physical properties that are observed in nature. Although the vector components may change from one reference frame to another, the 4-vector itself is an invariant, meaning that it gives valid physical information for all inertial observers. Likewise, the scalar products of Lorentz Invariant 4-vectors are themselves invariant quantities, known as Lorentz Scalars. Lorentz Invariance is a special subset of Poincare' Invariance.

The reason that I really like this notation is that it beautifully and elegantly displays the relations between lots of different physical properties. It also devolves very nicely into the limiting/approximate Newtonian cases of v<<c by letting γ =>1 and dγ/dt =>0. SR tells us that several different physical properties are actually dual aspects of the same thing, with the only real difference being one's point of view, or reference frame. Examples include: (Time , Space), (Energy , Momentum), (Power , Force), (Frequency , WaveNumber), (ChargeDensity , CurrentDensity), (EM-Potential , EM-VectorPotential), (Time Differential, Gradient), etc. Also, things are even more related than that. The 4-Momentum is just a constant times 4-Velocity. The 4-WaveVector is just a constant times 4-Momentum. In addition, the very important conservation/continuity equations seem to just fall out of the notation. The universe apparently has some simple laws which can be easy to write down by using a little math and a super notation.

Abbreviations

QM=Quantum Mechanics SR=Special Relativity
SM=Statistical Mechanics GR=General Relativity

Units of Measure - (SI variant, mksC)

length	[m] meter <*> time [s] second	Count of the quantity of separation; Location of events in spacetime
mass	[kg] kilogram	Count of the quantity of stuff/matter
EMcharge	[C] Coulomb	Count of the quantity of electric charge; the Coulomb is more fundamental than the Ampere
temp	[ºK] Kelvin	Count of the quantity of heat (statistical)

Useful SR Quantities

η_μν=g_μν=g^μν=Diag[+1,-1,-1,-1]: Minkowski Flat Spacetime Metric

β = (v/c): Beta factor, the fraction of the speed of light c [dimensionless]
γ = (1 / √[1-β²] )=(1 / √[1-(v/c)²] ): Lorentz Gamma Scaling Factor [dimensionless] (~1 for v<<c, +big for v~c)
φ = ln[γ(1+ β)] = BoostParameter/Rapidity (which remains additive in SR, unlike v)

τ = t / γ : Proper Time = Rest Time (time as measured in a frame at rest)
dτ = dt / γ : Differential of Proper Time
d/dτ = γ d/dt = U·∂ : Differential wrt Proper Time

V·V=Vo·Vo : Invariant interval is often easier to calculate in rest frame coordinates
√[1+x] ~ (1+x/2) for x<<1 : Math relation often used to simplify Relativistic eqns. to Newtonian eqns.
δ^uv = Delta function = (1 if u=v, 0 if u≠v)
γ v = c √[γ²-1] , βγ = √[γ²-1] , c² dγ = γ³ v dv , d(γ v) = γ³ dv
c=1/√[ε_oµ_o]

SR Notation Used

There are several different SR notations available that are, mathematically speaking, equivalent.
However, some are easier to employ than others; I have used that one which seems the most practical and least error-prone.
Always check notation conventions in SR & 4-Vector references, they are all relative ;-)

V=(c v_t,v_x,v_y,v_z)=(v⁰,v¹,v²,v³)
Intervals: Time/Temporal (+interval)=0 coordinate
Light/Null (0 interval)
Space/Spatial (-interval)=1,2,3 coordinates
Temporal Components: Future(+), Now(0), Past(-)
4-Vector Name: always references the "Spatial" 3-vector component, c-factor always applied to "Temporal" scalar component, as necessary to give consistent units for all vector components
4-Vector Magnitude: usually references the "Temporal" scalar component
4-Vector Symbols: V=(cv⁰,v)=(cv⁰,v¹+v²+v³): 4-vector={BOLD UPPERCASE}, time component={regular lowercase}, space 3-vector component={bold lowercase}
Relativistic Component: v --> v_o in a rest-frame, typically v = γ v_o (dilation) or v = (1/γ) v_o (contraction)
eg. t = γ t_o (time dilation), L = (1/γ) L_o (length contraction)
Minkowski SR Metric: η_μν=g_μν=g^μν=Diag[+1,-1,-1,-1]=Diag[+1,-1]
Imaginary unit: i used only for QM, not for SR frame transformations or metric

Minkowski SR Spacetime Metric

The main assumption of SR, or GR for that matter, is that the structure of spacetime is described by a metric g_μν. A metric tells how the spacetime is put together, or how distances are measured within the spacetime. These distances are known as intervals. In GR, the metric may take a number of different values, depending on various circumstances which determine its curvature. We are interested in the flat/pseudo-Euclidean spacetime of SR, also known as the Minkowski Metric, for which η_μν=g_μν=g^μν=Diag[+1,-1,-1,-1].

"Flat" SpaceTime η_μν=g_μν{SR}=
t x y z	+1	0	0	0
	0	-1	0	0
	0	0	-1	0
	0	0	0	-1

There are other ways of defining the metrics and 4-vectors available in SR which lead to the same results, but this particular notation has some nice qualities which place it above the others. First, it shows the difference between time and space in the metric. We perceive time differently than space, despite there being only spacetime. Also, this metric gives all of the SR relations (frame transformations) without using the imaginary unit "i" in the transforms. This is important, as "i" is absolutely essential for the complex wave functions once we get to QM. It is not needed, and would only complicate and confuse matters in SR. This metric will allow us to separate the "real" SR stuff from the "complex/imaginary" QM stuff easily. It also allows for the possibility of complex components in SR 4-vectors. The choice of +1 for the time component simplifies the derived equations later on.

SR 4-Vector Basics

η_μν=g_μν=g^μν=DiagnolMatrix(1,-1,-1,-1): Minkowski Spacetime Metric-the "flat" spacetime of SR
A=A^u=(a^t,a^x,a^y,a^z)=(a⁰,a¹,a²,a³)=(a⁰,a): Typical SR 4-vector
A_u=(a_t,-a_x,-a_y,-a_z)=(a₀,-a₁,-a₂,-a₃)=(a₀,-a): Typical SR 4-covector, we can always get the 4-vector form with A^u=g^μνA_u
Basically, this has the effect of putting a minus sign on the space component
B=B^u=(b^t,b^x,b^y,b^z)=(b⁰,b¹,b²,b³)=(b⁰,b): Another typical SR 4-vector

A·B=g_uvA^uB^v=A_vB^v=A^u B_u=+a⁰b⁰-a¹b¹-a²b²-a³b³=(+a⁰b⁰-a·b): The Scalar Product relation, used to make SR invariants

A'^u=L^u_vA^v: Lorentz Transform (Transformation tensor which gives relations between alternate boosted inertial reference frames)

L^u_v = (for x-boost)
γ	-γ(v_x/c)	0	0
-γ(v_x/c)	γ	0	0
0	0	1	0
0	0	0	1

γ -β_xγ 0 0

-β_xγ γ 0 0

0 0 1 0

0 0 0 1

General Lorentz Transformation
γ -β_xγ -β_yγ -β_zγ

-β_xγ 1+(γ-1)(β_x/β)² ( γ-1)(β_xβ_y/β)² ( γ-1)(β_xβ_z/β)²

-β_yγ ( γ-1)(β_yβ_x/β)² 1+( γ-1)(β_y/β)² ( γ-1)(β_yβ_z/β)²

-β_zγ ( γ-1)(β_zβ_x/β)² ( γ-1)(β_zβ_y/β)² 1+( γ-1)(β_z/β)²

General Lorentz Boost Transform using just vectors & components-Thank you Jackson, Master of Vectors! Chap. 11
β=v/c, β=|β|, γ=1/√[1-β²]
a⁰'=γ(a⁰-β·a)
a'=a+(γ-1)/β²(β·a)β-γ β a⁰

We are also able to use the Rapidity
φ=ln[γ(1+ β)]
e^φ=γ(1+β)=√[(1+β)/(1-β)]
β=Tanh[φ], γ=Cosh[φ], γβ=Sinh[φ], φ=Rapidity (which remains additive in SR, unlike v)
L^u_v = (for x-boost)
Cosh[φ] -Sinh[φ] 0 0

-Sinh[φ] Cosh[φ] 0 0

0 0 1 0

0 0 0 1

Formally, this is like a rotation in 3-space, but becomes a hyperbolic rotation through spacetime for a Lorentz boost
a⁰'=γ(a⁰-β·a) Temporal component
a^||'=γ(a^||-βa⁰) Spatial parallel component
a^⊥'=a^⊥ Spatial perpendicular components

Time t = γ t_o --> Time Dilation
Length L = L_o/γ --> Length Contraction

Complex SR 4-Vectors

A few 4-vectors are known to have complex components. The Polarization 4-vector is one of these. It will be assumed that all physical 4-vectors may potentially be complex.

i = √[-1] :Imaginary Unit
e₀: Unit vector in the temporal direction (typically not used since the temporal unit is always considered a scalar)
e₁, e₂, e₃ :Unit Vectors in the spatial x, y, z directions (used instead of i, j, k so that there is no confusion with the imaginary unit i)

Note that for the following 4-vectors, the superscript is the tensor index, not exponentiation.

A=(a⁰_c + a¹_c e₁+ a²_c e₂+ a³_c e₃): Complex 4-vector has complex components, 1 along time and 3 along space
Scalar[A] = a⁰_c: Just the time component
Vector[A] = a¹_c e₁ + a²_c e₂ + a³_c e₃: Just the spatial components
A = Scalar[A] + Vector[A]

A=( (a⁰_r + a⁰_i ) + (a¹_r + a¹_i ) e₁ + (a²_r + a²_i ) e₂ + (a³_r + a³_i ) e₃ ): Complex 4-vector has real + imaginary components, 1 each along time and 3 each along space
Re[A] = ( (a⁰_r ) + (a¹_r ) e₁ + (a²_r ) e₂ + (a³_r ) e₃ ): Only the real components
Im[A] = ( (a⁰_i ) + (a¹_i ) e₁ + (a²_i ) e₂ + (a³_i ) e₃ ): Only the imaginary components
A = Re[A] + i Im[A]

A=(a⁰_r + i a⁰_i,a_r + i a_i) : Standard 4-vector
A^*=(a⁰_r - i a⁰_i,a_r - i a_i): Complex conjugate 4-vector, just changes the sign of the imaginary component

A=(a⁰_r + i a⁰_i,a_r + i a_i) : A^*=(a⁰_r - i a⁰_i,a_r - i a_i)
B=(b⁰_r + i b⁰_i,b_r+ i b_i) : B^*=(b⁰_r - i b⁰_i,b_r- i b_i)

A·B=[( a⁰_r b⁰_r - a_r·b_r ) - ( a⁰_i b⁰_i - a_i·b_i )] + i [( a⁰_r b⁰_i - a_r·b_i ) + ( a⁰_i b⁰_r - a_i·b_r )] : General scalar product
A·A=[( a⁰_r² - a_r·a_r ) - ( a⁰_i² - a_i·a_i )] + 2i [( a⁰_r a⁰_i - a_r·a_i )] = |A|² : Scalar product of 4-vector with itself gives the magnitude squared
A·A^*=[( a⁰_r² + a⁰_i² ) - ( a_r·a_r + a_i·a_i )] = Re[A·A^*]: Scalar product of 4-vector with its complex conjugate is Real, thus Im[A·A^*]=0

∂·B=[( ∂/c∂t_r b⁰_r + del_r·b_r ) - ( ∂/c∂t_i b⁰_i + del_i·b_i )] + i [( ∂/c∂t_r b⁰_i + del_r·b_i ) + ( ∂/c∂t_i b⁰_r + del_i·b_r )]
=[( ∂/c∂t_r b⁰_r + del_r·b_r ) - ( ∂/c∂t_i b⁰_i + del_i·b_i )]
=Re[∂·B]
The 4-Divergence of a Complex 4-Vector is Real, assuming that:
The real gradient acts only on real spaces & the imaginary gradient acts only on imaginary spaces, thus Im[∂·B]=0
I believe this is due to the physical functions being complex analytic functions.

Fundamental/Universal Mathematical Constants

i = √[-1] :Imaginary Unit
Pi = 3.14159265358979... :Circular Const

Fundamental/Universal Physical Constants (Lorentz Scalars)

c = Speed of Light Const
h_bar = Planck's Reduced Const (h / 2Pi) - relates particle to wave - Action
k_B = Boltzmann's Const - relates temperature to energy
m_o= Rest Mass Const (varies with particle type)
q_o= Electric Charge Const (varies with particle type)
Note: I do not set various fundamental physical constants to dimensionless unity, (i.e. c=h=G=k_B=1). While doing so may make the mathematics a bit easier, it ultimately obscures the physics.
While 4-Vectors may be math, SR 4-Vectors is physics. I keep the dimensions and units.

Fundamental/Universal Physical SR 4-Vectors (Lorentz Vectors)

4-Vector	Vector=(temporal comp, spatial comp)	Units - description
		Calculus
4-Displacement = 4-Delta	ΔR=(cΔt, Δr)	[m] Δt=Temporal Displacement, Δr=Spatial Displacement, (Finite Differences)
4-Differential	dR=(cdt, dr)	[m] dt=Temporal Differential, dr=Spatial Differential, (Infintesimals)
4-Gradient = 4-Del or 4-Partial	∂=∂/∂x_u=(∂/c∂t, -del)	[m^-1] ∂ is the partial derivative, -del = -(∂/∂x i + ∂/∂y j + ∂/∂z k)
		Dynamics
4-Position	R=(ct, r)	[m] t=Time, r=Position(spatial), Location of an Event, the most basic 4-vector
4-Velocity	U=γ(c, u)=dR/dτ	[m s^-1] c=Speed_of_Light, u=Velocity, "U is historically used instead of V"
4-Acceleration	A=γ(c dγ/dt, dγ/dt u+γ a)=dU/dτ =γ(c dγ/dt, a_r)	[m s^-2] a_r=Relativistic Acceleration, u = 3-velocity, a = 3-Newtonian Acceleration a_r=dγ/dt u+γ a=(γ³/c²)(u·a) u+γ a
		Kinematics
4-Momentum	P=(E/c, p)=(mc, p)=m_o γ(c,u) =m_oU=h_barK =(E_o/c²)U	[kg m s^-1] E=Energy, p=Momentum m_o=RestMass( 0 for photons, + for massive )
4-Force	F=γ(dE/cdt, f_r)=dP/dT =m_oA	[kg m s^-2] dE/dt=Power, f_r=Relativistic Force
		Connection to Waves
4-WaveVector	K=(ω/c, k)=(ω/c,ω/v_phasen) = (1/h_bar)(E/c,p) = (ω_o/c²) γ(c,u) = (1/h_bar)P = (ω_o/c²)U = (1/h_bar)P = (m_o/h_bar)U	[rad m^-1] ω=AngularFrequency, k=WaveNumber, n=UnitWaveNormalVector, v_phase=phase_velocity ω_o=RestAngularFrequency( 0 for photons, + for massive )
	Flux 4-Vectors all in form of : V={rest_charge_density} U V={rest_charge}n_o U where n = γn_o	Flux 4-Vectors all have units of [{charge} m^-2 s^-1]=[{charge_density} m s^-1] Flux is the amount of {charge} that flows through a unit area in a unit time Flux can also be thought of as {charge_density_velocity}={current_density}
4-NumberFlux	N=(cn, n_f)=n_o γ(c, u) =n_oU	[# m^-2 s^-1] n_o=RestNumberDensity, n=γn_o=NumberDensity, sometimes called SR "Dust" n_f=NumberFlux
4-ElectricCurrentDensity = 4-ElectricChargeFlux	J=(cρ, j)=ρ_o γ(c, u) =ρ_oU=q_on_oU =J_elec	[C m^-2 s^-1] ρ_o=RestElecChargeDensity, ρ=γρ_o=ElecChargeDensity j=γ ρ_o u=ElecCurrentDensity=ElecChargeFlux j=α(E+uxB), α=Conductivity q_o =Electric Charge
4-MagneticCurrentDensity = 4-MagneticChargeFlux =Null (so far..)	J_mag=(cρ_mag, j_mag)=ρ_{o_}_mag γ(c, u) =ρ_{o_mag}U=q_{o_mag}n_oU =Null (so far...)	[MagCharge m^-2 s^-1] ρ_{o_mag}=RestMagChargeDensity, ρ_mag=γρ_{o_mag}=MagChargeDensity j_mag=MagCurrentDensity=MagChargeFlux q_{o_mag} =Magnetic Charge to date: ρ_{o_mag}=0 and j_mag=0 -- no magnetic (monopole) charges yet discovered
4-VolumetricFlux??		[(m³) m^-2 s^-1]
4-ChemicalFlux??		[(mol) m^-2 s^-1]
4-MomentumDensity =4-MassFlux	G=(u/c, g)=(cp_m, g)=p_o_{_m} γ(c, u)=p_o_{_m}U ????p_o_{_m}=m_on_o??=(1/c²)S	[(kg) m^-2 s^-1] u=EnergyDen=ne, p_m=MassDen=u/c² g=MomentumDen=(u/c²)u=(e_o)ExB, f=g·u=MomentumFlux u=3-velocity, n=ParticleDen, e=EnergyPerParticle
4-PoyntingVector =4-EnergyFlux =4-RadiativeFlux??	S=(cu, s)=u_o γ(c, u)=u_oU=c²G ????u_o=E_on_o??	[(J) m^-2 s^-1] u=EnergyDen=ne, s=EnergyFlux=PoyntingVector=uu=c²g=Ne u=(e_oE·E+B·B/µ_o)/2, s=ExB/µ_o u=3-velocity, n=ParticleDen, e=EnergyPerParticle, N=ParticleFlux=nu
4-EntropyFlux	S=(cs, s_f)=s_o γ(c, u)=s_oU	[(J ºK^-1) m^-2 s^-1] s_o=RestEntropyDensity=q_o/T, s_f=EntropyFlux Entropy S=∫s_odV = k_B ln Ω, where Ω = # of microstates for a given macrostate
4-HeatFlux	Q=(cq, q_f)=q_o γ(c, u)=q_oU	[J / m² s]
4-TempFlux ??	T=P/k_B=(t⁰/c, t_f) ?? t⁰=Temperature(in ºK)??	[ºK / m² s] ?? ºK=Kelvins, k=Boltzmann's constant
		Angular Momentum/Spin
4-Spin	S=(u·s/c,s)? units?	[spin] s=Spin=IntrinsicAngMomentum, u·s/c=component such that U·S=0 4-Spin is orthogonal to 4-Velocity, so time component is zero in rest frame This is actually an axial 4-vector only 3 independent components, not 4
4-Polarization	Ε=(ε⁰, ε)=(ε·u/c,ε) for a massive particle =(ε⁰, ε)=((c/v_phase) ε·n,ε) for a wave	[1] ε=PolarizationVector This 4-vector has complex components in QM Like the 4-Spin, Ε orthogonal to U, or K, so time component=0 in rest frame Ε·U=0, Ε·K=0, Additionally, Ε·E=-1 (normalized to unity along a spatial direction)
		Electromagnetic Field Potentials
4-VectorPotential_EM	A_EM=(Φ_EM/c, a_EM)	[kg m C^-1 s^-1] Φ_EM=ScalarPotenial_EM a_EM=VectorPotenial_EM Electric Field E=-del[Φ_EM]-∂a_EM/∂t, Magnetic Field B=del x a_EM
4-Momentum_EM	P_EM=(E/c+qΦ_EM/c, p+qa_EM)=γ m_o(c,u) =P + q A_EM	[kg m s^-1] Momentum including effects of EM potentials also known as Canonical Momentum
4-Gradient_EM	D_EM=(∂/c∂t+iq/h_bar Φ_EM/c, -del+iq/h_bar a_EM) =d+(iq/h_bar)A_EM	[m^-1] Gradient including effects of EM potentials

4-Null or 4-Zero	Null=(0,0)	[*] All components are 0 in all reference frames, the only vector with this property

Fundamental/Universal Relations

Event Tracking Relations
Event R	Mass m_o	ChargeDensity ρ_o : Charge Q_o=ρ_oV_o	NumberDensity n_o : ParticleNumber N_o=n_oV_o	WaveAngFreq w_o
R=(ct, r)	m_o	ρ_o	n_o	w_o
U=dR/dτ	P=m_oU=(E_o/c²)U	J=ρ_oU	N=n_oU	K=(w_o/c²)U= (1/h_bar)P
A=dU/dτ	F=m_oA=dP/dτ

Flux 4-Vectors, 4-Vector "Charges", and the Continuity/Conservation Equation

∂·R=(∂/c∂t,-del)·(ct,r)=(∂/c∂t[ct]+del·r)=(∂/∂t[t]+del·r)=(1+3)=4
∂·R=4 The divergence of open spacetime is equal to the number of independent dimensions (1 time + 3 space)

d/dτ (∂·R) = d/dτ (4) = 0
d/dτ (∂·R) = d/dτ (∂) · R + ∂·d/dτ (R) = d/dτ (∂) · R + ∂·U = γ d/dt (∂) · R + ∂·U = γ d/dt (∂)·R + ∂·U = γ (d/dt(∂/c∂t), -d/dt(del))·(ct,r) + ∂·U = γ (d/dt(∂/c∂t)(ct)+d/dt(del))·r + ∂·U
= γ (d/dt(∂/∂t)(t)+d/dt(del))·r + ∂·U = γ (d/dt(1)+d/dt(3))+ ∂·U = ∂·U
thus,
∂·U = 0, which is the general continuity equation, one might say the conservation of event flux. Due to this property, any Lorentz scalar constant times 4-Velocity U is a conserved quantity.
For example, let N=n_oU, so ∂·N = ∂·n_oU = n_o∂·U = n_o(0) = 0. The quantity n_o is conserved.

Any "charge" constant becomes a 4-vector when multiplied by the 4-Velocity, and obeys the Conservation of Charge/Continuity equation ∂·J=dp/∂t +del·j=0 where J=ρ_oU
let Charge Q_o=ρ_oV_o, where ρ_o is the "rest charge density", ρ = γρ_o is the relativistic "charge density", V_o is the rest volume, and j=γρ_ou=ρu is the "ChargeDensity-Flux or Current Density"
then ChargeFlux 4-Vector=CurentDensity 4-Vector J=ρ_oU=ρ_o γ(c, u)=ρ(c, u)=(cρ, j)
In the case of "electric" charge, ρ_o is the "rest electric-charge density", and j is the ElectricChargedensity-flux=electric current density
In the case of "number" charge, ρ_o is the "rest number-charge density"
In the case of "mass" charge, ρ_o is the "rest mass density", and j is the mass-flux=mass current density=momentum density

SR Path to QM
Event(SR)	EventMovement	MassEnergy	Particle-WaveDuality	QuantumMechanics(QM)	SpaceTimeVariations
R=(ct, r)	dR/dτ=U=γ(c,u)	U=P/m_o	P= h_bar K	* K= i ∂ *	∂=(∂/c∂t,-del)

d/dτ[R] = (i h_bar / m_o) ∂ Event motion ~ spacetime structure - depends on i h_bar / m_o

So, the following assumptions within SR-Special Relativity lead to QM-Quantum Mechanics:

R = (ct,r) Location of an event (i.e. a particle) within spacetime

U = dR/dτ Velocity of the event is the derivative of position wrt. Proper Time

P = m_oU Momentum is just the Rest Mass of the particle times its velocity

K = 1/h_bar P A particle's wave vector is just the momentum divided by Planck's constant, but uncertain by a phase factor

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

R·R=(Δ s)²=(ct)²-r·r = (ct)²-|r|² : dR·dR=(ds)²=(c dt)²-dr·dr = (c dt)²-|dr|² : Invariant Interval
U·U=c²
P·P=(m_oc)²
K·K=(m_oc / h_bar)²=(w_o/c)²
∂·∂=(∂/c∂t,-del)·(∂/c∂t,-del)=∂²/c²∂t²-del·del=-(m_oc / h_bar)² : Klein-Gordon Relativistic Wave Eqn.
Each relation may seem simple, but there is a lot of complexity generated by each level.
*see QM from SR (Quantum Mechanics derived from Special Relativity)*

Momentum/Gradient Relations(Correspondences)
P=i h_bar ∂= -∂(S_act)	∂=(∂/c∂t,-del)	A_EM=(0,0) special case
P_EM=P+qA_EM=i h_bar D_EM	D_EM=∂+iq/h_bar A_EM	A_EM=(V_EM/c,a_EM)

Derived Physical Constants (Scalar Products of Lorentz Vectors = Lorentz Scalars)

R·R=(Δs)²=(ct)²-r·r = (ct)²-|r|² : dR·dR=(ds)²=(c dt)²-dr·dr = (c dt)²-|dr|²
U·U=c² : A·A=-a²
P·P=(m_oc)² : N·N=(n_oc)²: J·J=(p_oc)²=(q_on_oc)²
K·K=(m_oc / h_bar)²=(w_o/c)²
∂·∂=(∂/c∂t,-del)·(∂/c∂t,-del)=∂²/c²∂t²-del·del=-(m_oc / h_bar)² : ** ∂·∂ is also known as the D'alembertian (Wave Operator) **
A_EM·A_EM=(V_EM/c,a_EM)·(V_EM/c,a_EM)=(V_EM/c)²-a_EM·a_EM=????

P·U=m_oc²=E_o Rest Energy
K·U=m_oc²/hbar=E_o/hbar=w_o Rest Ang. Frequency
U·F=γ²(dE/dt-u·f)=γ dm_o/dt c² (pure force if dm_o/dt=0) ??? Power Law
U₁·U₂=γ[u₁]γ[u₂](c²-u₁·u₂)=γ[u_r]c² (The scalar product of two uniformly moving particles is proportional to the γ factor of their relative velocities)
U·∂=d/dτ=γ(∂/∂t + u·del)=γ d/dt
∂·R=4 The divergence of open spacetime is equal to the number of independent dimensions
∂·U=0, which is the general continuity equation, one might say the conservation of event flux.

Invariants & Conservation Laws

There is an important distinction between an invariant quantity and a conserved quantity.
An invariant quantity has the same value wrt. all inertial systems, but may change upon physical interaction (e.x a fusion reaction "redistributes" the rest masses).
A conserved quantity maintains the same value both before and after an interaction, although the component values may appear different in different frames.
In 4-vector notation:
An invariant quantity is a Lorentz Scalar, the dot product of two 4-Vectors, A·B=invariant.
A conserved quantity is a component of a 4-Vector that has 4-Divergence=0, d·V=0.

**Invariant Quantities** (Lorentz Scalars~A·B)
c = √[U·U]	Speed of Light: c (in vacuum) E ~ cp
h_bar = √[P·P/K·K] = P·K/K·K	Planck's const: h_bar E ~ h_barw
k_B = √[P·P/T·T] ??	Boltzmann's const: k_B E ~ k_BT

ds = √[dR·dR]	Differential Length of World Line
dτ = √[dR·dR/U·U]	Differential Proper Time
d/dτ =U·d=γ(∂/∂t + u·del) =γ d/dt	Derivative wrt Proper Time d/dτ d/dt=total time derivative, ∂/∂t=partial time derivative

m_o = √[P·P/U·U] = P·U/U·U	RestMass of a Particle m_o ( 0 for photons, + for massive )
q_o = √[J·J/N·N] =	RestElectricCharge of a Particle q_o

E_o = P·U = m_oc²	RestEnergy of a Particle ( 0 for photons, + for massive )
w_o = K·U = m_oc²/hbar	RestAngFrequency of a Particle ( 0 for photons, + for massive )

ø_EM = -K·R	* Phase of EM wave *
Sact = -P·R = Integral[dt L;t_i,t_f]	Action Variable S of Action Integral
γ L =	Relativistic Lagragian

n_o = √[N·N/U·U] = N·U/U·U	Particle RestNumberDensity (for stat mech)
s_o = √[S·S/U·U] = S·U/U·U	RestEntropyDensity (for stat mech)

N_o = N	Particle Number: N=nV=(n/γ)(γ V)=n_oV_o=N_o
P_o = P	Pressure of system: P=P_o
S_o = S	Entropy: S=sV=(s/γ)(γ V)=s_oV_o=S_o
T_o = γ T	RestTemperature
Q_o = γ Q	RestHeat
V_o = γ V	RestVolume

Conserved Quantities (components of V which have 4-Divergence d·V=0 )

∂·J=dp/∂t +del·j=0 Conservation of 4-CurrentDensity (EM charge): p & j
change in ChargeDen wrt. time balanced by flow of CurrentDen

∂·N=∂/∂t(γ n_o)+del·(γ n_ou)
      =∂n/∂t+del·n_f=0 Conservation of 4-NumberFlux (Particle NumberDen, NumFlux): n_&n_fchange in NumberDen wrt. time balanced by flow of NumFlux

∂·P=(1/c²)∂E/∂t +del·p=0

Sum[P_f-P_i]=Null Conservation of 4-Momentum (Energy~Mass, Momentum): E & p
change in Energy wrt. time balanced by flow of Momentum
Alternately, the Sum[(Final 4-Momenta) - (Initial 4-Momenta)] = Null Vector
Note: this conservation equation, while rarely used, is perfectly acceptable for single particles. It is only when a group of particles is treated as a continuous fluid that the Energy-Momentum (2,0)Tensor is required. Then, the diagonal pressure terms and off-diagonal shear terms are necessary, basically allowing statistical particle interaction.

∂·K=∂/c∂t(w/c)+del·k
      =(1/c²)∂w/∂t +del·k=0. Conservation of 4-WaveVec (AngFreq, WaveNum): w & k
change in AngFreq wrt. time balanced by flow of WaveNum

∂·A_EM=(1/c²)∂V_EM/∂t +del·a_EM=0 Conservation of 4-VectPotential_EM (applies in the Lorentz Gauge): V_EM & a_EMchange in ScalarPotential wrt. time balanced by flow of VectorPotential

∂·U=∂/∂t(γ[u])+del·(γ[u] u)
     =γ³ (u/c² ∂u/∂t + del·u)
      =∂·Uo?
=0 if event is in a conservative field or space Conservation of 4-Velocity: (Flux-Gauss' Law)??: γ & γ u
change in (γ) wrt. time balanced by flow of (γ u)
If this quantity equals zero, then any physical quantity that is just a (constant* 4-velocity) is conserved.
For example ∂·P=∂·(m_oU)=m_o(∂·U)=0
Also from d/dτ (∂·R) = ∂·U=0
see also Noether's Theorem

Derived Equations

E²= p·p c² + m_o²c⁴: Energy of a particle has a Momentum component and a RestMass component
Total Energy: E = mc² = γ[u] m_oc² = h_bar w
Kinetic Energy: T = mc²-m_oc² = (γ[u]-1) m_oc²
Rest Energy: E_o = m_oc²

|
|___
|     |
|     |γ[u]
| m |
|     |
|___|____
m_o
Relativistic(apparent) mass m=AreaLike=γ[u] * m_o=h_bar w/c²=E/c²
Theoretically, this would scale like a δ-function for photons{m_o -->0,u -->c,γ-->Infinity}
Thus, the relativistic mass of a photon is proportional to w, the angular frequency
There is also a rest frequency w_o=m_oc²/h_bar, even when the massive particle is at rest. Mass is always "spinning" about the time dimension.

U·U=c² , d(U·U)=d(c²)=0 , d(U·U)=2*(U·dU)=2*(U·A)=0
U·A=0: The 4-Acceleration is orthogonal to its own 4-Velocity (Any acceleration is orthogonal to its own world-line, i.e. you don't accelerate in time).
U plays the part of the tangent vector of the world-line, and A plays the part of the normal vector of the world-line.
The curvature of a world-line is given by a/c².

U₁·U₂=γ[u₁]γ[u₂](c²-u₁·u₂)=γ[u_r]c² (The scalar product of two uniformly moving particles is proportional to the γ factor of their relative velocities)

∂·R=(∂/c∂t,-del)·(ct,r)=(∂/c∂t[ct]+del·r)=(∂/∂t[t]+del·r)=(1+3)=4
∂·R=4 The divergence of open space is equal to the number of independent dimensions

d/dτ (∂·R) = d/dτ (4) = 0
d/dτ (∂·R) = d/dτ (∂) · R + ∂·d/dτ (R) = d/dτ (∂) · R + ∂·U = γ d/dt (∂) · R + ∂·U = γ d/dt (∂)·R + ∂·U = γ (d/dt(∂/c∂t), -d/dt(del))·(ct,r) + ∂·U = γ (d/dt(∂/c∂t)(ct)+d/dt(del))·r + ∂·U
                 = γ (d/dt(∂/∂t)(t)+d/dt(del))·r + ∂·U = γ (d/dt(1)+d/dt(3))+ ∂·U = ∂·U
thus,
∂·U = 0, which is the general continuity equation, one might say the conservation of event flux. Due to this property, any Lorentz scalar constant times 4-Velocity U is a conserved quantity.
For example, let N=n_oU, so ∂·N = ∂·n_oU = n_o∂·U = n_o(0) = 0. The quantity n_o is conserved.

Alternately, ∂·U = (∂/c∂t, -del)·γ(c, u) = ∂·Uo = (∂/c∂t, -del)·(c, 0) = ∂/c∂t (c) = ∂/∂t (1) = 0

P·P=(m_oc)²   ==>0 for photons
P_phot1·P_phot2=h_bar²K₁·K₂=(h_bar²w₁w₂/c²)(1-n₁·n₂)=(h_bar²w₁w₂/c²)(1-cos[ø])
P_phot·P_mass=h_barK·P=(h_barw/c)(1,n)·(E/c,p)=(h_barw/c)(E/c-n·p)=(h_barwE_o/c²)=(h_barwm_o)
P_phot + P_mass = P'_phot + P'_mass   Conservation of 4-Momentum in Photon-Massive Interaction
P_phot + P_mass - P'_phot = P'_mass   rearrange
(P_phot + P_mass - P'_phot)² = (P'_mass)² square to get scalars
(P_phot·P_phot + 2 P_phot·P_mass - 2 P_phot·P'_phot + P_mass·P_mass - 2 P_mass·P'_phot + P'_phot·P'_phot) = (P'_mass)²
(0 + 2 P_phot·P_mass - 2 P_phot·P'_phot + (m_oc)² - 2 P_mass·P'_phot + 0) = ((m_oc)²)²
P_phot·P_mass - P_mass·P'_phot = P_phot·P'_phot
(h_barwm_o)-(h_barw'm_o)=(h_bar²ww'/c²)(1-cos[ø])
(w-w')/(ww')=(h_bar/m_oc²)(1-cos[ø])
(1/w'-1/w)=(h_bar/m_oc²)(1-cos[ø])
(1/(2pi v')-1/(2pi v))=(h_bar/m_oc²)(1-cos[ø])
(1/(v')-1/(v))=(h/m_oc²)(1-cos[ø])
(λ'/c-λ/c)=(h/m_oc²)(1-cos[ø])
(λ'-λ)=(h/m_oc)(1-cos[ø])
(λ'-λ)=(h/m_oc)(2sin²[ø/2]) Compton scattering with Compton wavelength (h/m_oc)

Relativistic Stat-Mech (SM)/Thermodynamic stuff

U=γ(c, u), P = (E/c,p), d(P)=(dE/c,dp)
U·d(P)=γ(c dE/c-u·dp)=γ(dE-u·dp)=γ(T dS - P dV + µ dN)=(T_o dS_o - P_o dV_o + µ_o dN_o) = 0 ??

U·d(P)=γ(dE-u·dp)=(T_odS_o - P_odV_o + µ_odN_o) = const = ? 0 ?

U·d(P)=γ(dE-u·dp)=γ(T dS - P dV + Sum[µ_i dN_i] + w·dL + E·dP + B·dM) ???
E=Energy, [Total energy of system]
u=Velocity, p=Momentum, [Translational/Kinetic energy]
T=Temperature, S=Entropy [Heat energy]
P=Pressure, V=Volume [Mechanical compression energy?]
µ=ChemicalPotenital, N=ParticleNumber, ["Chemical" energy=energy per particle] (Sum over different particle types)
w=Angular Velocity, L=Angular Momentum, [Rotational energy]
E=Electric Field, P=Polarization, [Electrical energy]
B=Magnetic Field, M=Magnetisation, [Magnetic energy]
Always have (intensive var * differential extensive var), intensive=sys size independent, extensive=sys size proportional

U=γ(c, u), P = (E/c,p), U·U=c², P·P=(m_oc)²
U·P=γ(c E/c-u·p)=γ(E-u·p)=γ(T S - P V + µ N)=(T_o S_o - P_o V_o + µ_o N_o) ?

U·P=γ(E-u·p)=(T_o S_o - P_o V_o + µ_o N_o) = m_oc²? for a spatially homogeneous system: relativistic Gibbs-Duhem eqn.

Invariants	P=Pressure=P_o	N=ParticleNum=N_o	S=Entropy=S_o
Variables	V=Volume=(1/γ)Vol_o	µ=ChemPoten=(1/γ)µ_o	T=Tempurature=(1/γ)Temp_o

V*P (particle superstructure = Vol*Press)
µ*N (particle structure = ChemPoten*ParticleNum)
T*S (particle substructure = Temp*Entropy)

Time t=γ t_o
Length L=L_o/γ

Heat Q=Q_o/γ
dQ_o=T_odS_o
InertialMassDen(of radiation field) q=P/vV=γ q_o

Total Particle Number N=N_o is an invariant, because the NumberDensity n varies as n=γ n_o, but this is balanced by Volume V=V_o/γ
NumberDenstiy n = γ n_o where NumberFlux 4-Vector N=(cn,n_f)=n_o γ(c, u)=n_oU, n_o=N_o/(Δ_x_o*Δ_y_o*Δ_z_o)
N = n * V = (γ n_o)*(V_o/γ) = n_o* V_o = N_o
N·N=(n_oc)²

Total Entropy S=S_o is an invariant, because the EntropyDensity s varies as s=γ s_o, but this is balanced by Volume V=V_o/γ
EntropyDensity s = γ s_o where EntropyFlux 4-Vector S=(cs,s_f)=s_o γ(c, u)=s_oU, s_o=S_o/(Δ_x_o*Δ_y_o*Δ_z_o)
S = s * V = (γ s_o)*(V_o/γ) = s_o* V_o = S_o
S·S=(s_oc)²

Relativistic Analytical-Mech

Action S=S(ct,x,y,z)
dS/dτ=0
dS/dτ =U·d(S)=γ(∂S/∂t + u·del(S))=0
see Menzel pg.172

Relativistic Quantum Mechanics

∂·∂=(∂/c∂t,-del)·(∂/c∂t,-del)=∂²/c²∂t²-del·del=-(m_oc / h_bar)²: Klein-Gordon Relativistic Wave eqn.
D_EM=(∂/c∂t+iq/h_bar V_EM/c, -del+iq/h_bar a_EM)
=d+(iq/h_bar)A_EM
D_EM·D_EM=-(m_oc / h_bar)²: Klein-Gordon Relativistic Wave eqn. in electromagnetic potentials
(d+(iq/h_bar)A_EM)·(d+(iq/h_bar)A_EM)=-(m_oc / h_bar)²: Klein-Gordon Relativistic Wave eqn. w/ electromagnetic potentials
(∂·∂)+(iq/h_bar)(∂·A_EM+A_EM·∂)+(iq/h_bar)²(A_EM·A_EM)=-(m_oc / h_bar)²: Klein-Gordon Relativistic Wave eqn. w/ electromagnetic potentials
if (∂·A_EM+A_EM·∂)=0
then (∂/c∂t,-del)·(∂/c∂t,-del)+(iq/h_bar)²((V_EM/c, a_EM)·(V_EM/c, a_EM))=-(m_oc / h_bar)²
(∂²/c²∂t²-del·del)+(iq/h_bar)²( (V_EM/c)²-(a_EM·a_EM) )=-(m_oc / h_bar)²
(∂²/c²∂t²+(iq/ch_bar)²(V_EM²)-(del·del+(iq/h_bar)²(a_EM·a_EM) )=-(m_oc / h_bar)²
if ∂·A_EM & A_EM·∂ = 0
then (∂²/c²∂t²+(iq/ch_bar)²(V_EM²)-( del·del+(iq/h_bar)²(a_EM·a_EM) )=-(m_oc / h_bar)²
...not finished...

Newtonian Approximations

E²= p·p c² + m_o²c⁴: Relativistic Energy of a particle
E²= m_o²c⁴ + p·p c²
E²= m_o²c⁴ * [1 + p·p / m_o²c²]
E = ± m_oc² * √[ 1 + p·p / m_o²c² ]
E ~ ± m_oc² * [ 1 + p·p / 2m_o²c² + ...] for p²<<m_o²c²
E ~ ± [m_oc² + p·p / 2m_o + ...]
choosing the positive root and discarding higher order terms...
E ~ E_o +|p|² / 2m_oTotal Energy = Rest Energy + Newtonian Momentum term

∂·∂=(∂/c∂t,-del)·(∂/c∂t,-del)=∂²/c²∂t²-del·del=-(m_oc / h_bar)²: Klein-Gordon Relativistic Wave eqn.
∂²/c²∂t²= del·del-(m_oc / h_bar)²
∂²/c²∂t²= (im_oc / h_bar)²+del·del
(i h_bar)²∂²/c²∂t²= (i h_bar)²(im_oc / h_bar)²+(i h_bar)²del·del
(i h_bar)²∂²/c²∂t²= (m_oc)²+(i h_bar)²del·del
(i h_bar)²∂²/∂t²= (m_oc²)²*[1 + (i h_bar/m_oc)²del·del]
(i h_bar)∂/∂t = ± (m_oc²)*√[1 + (i h_bar/m_oc)²del·del]
(i h_bar)∂/∂t ~ ± (m_oc²)*[1 + (1/2)*(i h_bar/m_oc)²del·del + ...] for (h_bar)²*del·del<<(m_oc)² ,generally a very good approx. for non-relativistic systems
(i h_bar)∂/∂t ~ ± [(m_oc²) + (i² h_bar²/2m_o)del·del + ...]
choosing the positive root and discarding higher order terms...
(i h_bar)∂/∂t ~ (m_oc²) - (h_bar²/2m_o)|del|²
(i h_bar)∂/∂t ~ V(t,r) - ( h_bar²/2m_o)|del|² becomes the time dependent Schrödinger eqn. by equating rest energy with the potential energy of the particle

Interesting Relations

(K =m_o/h_barU=w_o/c²U) gives (c²/v_phase n = u) Both the wave vector and particle velocity point in the same direction; along the worldline. The product of the phase velocity and the particle velocity always equals c² (v_phase * u = c²). In the case of photons, the phase velocity=particle velocity=c. In the case of matter particles, the phase velocity v_phase = c²/u > c and particle velocity u<c. What does this mean? Suppose that you have a collection of particles traveling at identical velocities that all flash at the same time. The v_phase is the speed at which the flash moves in other reference frames, and can be considered the speed of propagation of simultaneity. For particles which are at rest, the v_phase is infinite, which makes sense since they all appear to flash simultaneously.

(∂·∂)A_EM=µ₀ J+∂(∂·A_EM) Inhomogeneous Maxwell Equation
(∂·∂)A_EM=µ₀ J Homogeneous Maxwell/Lorentz Equation (if ∂·A_EM=0 Lorentz Gauge)
∂·J=dp/∂t +del·j=0 Conservation of EMcurrent
Psi=a E e^-iK·R Photon Wave Equation (Solution to Maxwell Equation)
E·K=0 The Polarization of a photon is orthogonal to the WaveVector of that photon

P·U_obs=E/c γ[u_obs]c-p·γ[u_obs] u_obs=γ[u_obs](E-p·u_obs)
P·U_obs[u_obs=0]=E (RestFrame Invariant expression for energy)
K·U_obs=w/c γ[u_obs]c-k·γ[u_obs] u_obs=γ[u_obs](w-k·u_obs)
K·U_obs[u_obs=0]=w (RestFrame Invariant expression for angular frequency)
R·U_obs=ct γ[u_obs]c-r·γ[u_obs] u_obs=γ[u_obs](c²t-p·u_obs)
R·U_obs[u_obs=0]/c²=t (RestFrame Invariant expression for time)
J·U_obs=cp γ[u_obs]c-j·γ[u_obs] u_obs=γ[u_obs](pc²-j·u_obs)
J·U_obs[u_obs=0]/c²=p (RestFrame Invariant expression for ElecChargeDensity)

F^uv=∂^uA^v-∂^vA^u Electromagnetic Field Tensor (F⁰ⁱ=-Eⁱ,F^ij=e^ijkB^k)
L = -1/4 F_uvF^uv - J_u A^u : Lagrangian Density for EM field
L= -m_oc²/γ -V: Relativistic Lagrangian function of a Particle in a Conservative Potential
V_EM=q U·A_EM/γ: Potential of EM field
L_EM = -m_oc²/γ - q U·A_EM/γ = - (P·P/m₀ + q₀U·A_EM)/γ = - (m₀U·U + q₀U·A_EM)/γ

d/dτ = U·∂ = γ d/dt
U·∂/γ=∂/∂t + u·del=d/dt : Convective Derivative

Quantum Commutation & SR Uncertainty Relations

[ ∂^u , R^v ] = ∂^uR^v - R^v∂^u = g^μν   quantum commutator with pure 4-gradient      [since ( R^v∂^u = 0) generally???]
but K = i d
[ K^u , R^v ] = i g^μν   quantum commutator with 4-wave vector
but P = h_bar K
[ P^u , R^v ] = i h_bar g^μν   quantum commutator with 4-momentum

[ R^u , P^v ] = R^uP^v - P^vR^u = (- i h_bar g^μν)   SR quantum commutator with 4-momentum

this gives
[ x , p_x ] = [ y , p_y ] = [ z , p_z ] = (i h_bar)
[ ct , E/c ] = [ t , E ] = (- i h_bar) :assuming that one can treat the time as an operator...

both of these yield the familiar uncertainty relations:
Generalized Uncertainty relation: (Δ A) * (Δ B) >= (1/2) |< i[A,B] >| see Sudbury pg. 59 for a great derivation

(Δ x * Δ p_x >= h_bar / 2) and (Δ t * Δ E >= h_bar / 2)
or more generally
(Δ R^u * Δ P^v >= h_bar d^uv / 2)
or
(Δ R^u * Δ K^v >= d^uv / 2)
(Δ x * Δ k_x >= 1/2) and (Δ t * Δ w >= 1/2)

[ R^u , R^v ] = R^uR^v - R^vR^u = 0 : All position coordinates commute
[ P^u , P^v ] = P^uP^v - P^vP^u = 0 : All momentum coordinates commute

While I'm at it, a small comment about the quantum uncertainty relation. A great many books state that the quantum uncertainty relations mean that a "particle" cannot simultaneously have precise properties of position and momentum. I disagree with that interpretation. The uncertainty relations, the mathematical structure of the argument, say nothing about "simultaneous" measurements. They do say something about "sequential" measurements. A measurement of one variable places the system in a state such that if the next measurement is that of a non-commuting variable of the first, then the uncertainty must be of a minimum>0 amount. Also, note that the uncertainty relations are not necessarily about the size of h. Nor are they about the factor of i in the commutation relation. It would appear that they are about the metric g^μν itself, which has a non-zero result for sequential, non-commuting measurements.

Also, a comment on the EPR results. Based on SR, one cannot say that the measurement of one particle immediately "collapses" the physical state of the other. Since the two entangled particles can be setup such that they are space-like separated at the "events" of their respective measurement, there exist coordinate frames in which the measurement of the 1st particle occurs before that of the 2nd, exactly at the same time as the 2nd, and after that of the 2nd. Thus, how is the first particle to "know" that it must collapse the wavefunction of the 2nd, or that it must itself be collapsed by the 2nd?
--------
need to derive:
(Δ phi_x * Δ L_x >= h_bar / 2)
where phi_x is angle about x, and L_x is angular momentum about x

Light Cone

| time-like interval(+)

/ light-like interval(0)
worldline

      |
      |       c
\   future /
  \   |   /
   \ | /         -- space-like interval(-)
      \|/now
      /|\
    / |  \         elsewhere
/   |   \
/   past   \
      |       -c

(0,0) Zero-Null Vector

(+a,0) Future Pointing Pure TimeLike
(-a,0) Past Pointing Pure TimeLike
(0,b) Pure SpaceLike

(a,b) |a|>|b| TimeLike
(a,b) |a|=|b| Photonic-LightCone
(a,b) |a|<|b| SpaceLike

References (on 4-Vectors in SR & QM in SR)

Classical Dynamics of Particles & Systems, 3rd Ed., Jerry B. Marion & Stephen T. Thornton (Chap14)
Classical Electrodynamics, 2nd Ed., J.D. Jackson (Chap11,12)
Classical Mechanics, 2nd Ed., Herbert Goldstein (Chap7,12)
Electromagnetic Field, The, Albert Shadowitz (Chap13-15)
First Course in General Relativity, A, Bernard F. Schutz (Chap1-4)
Fundamental Formulas of Physics, Donald H. Menzel (Chap6)
Introduction to Electrodynamics, 2nd Ed., David J. Griffiths (Chap10)
Introduction to Modern Optics, 2nd Ed., Grant R. Fowles (var)
Introduction to Special Relativity, 2nd Ed., Wolfgang Rindler (All) (**pg60-65,82-86**)
Lectures on Quantum Mechanics, Gordon Baym (Chap22,23)
Modern Elementary Particle Physics: The Fundamental Particles and Forces?, Gordon Kane (Chap2+)
Path Integrals and Quantum Processes, Mark Swanson (var)
Quantum Electrodynamics, Richard P. Feynman (Lec7-rest)
Quantum Mechanics, Albert Messiah (Chap20)
Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians, Anthony Sudbery (Chap7)
Relativistic Quantum Fields, Mark Hindmarsh, Sussex, UK
Spacetime and Geometry: An Introduction to General Relativity, Sean M. Carroll (var)
Statistical Mechanics, R.K. Pathria (Chap6.5)
Theory of Spinors, The, E'lie Cartan (var)
Topics in Advanced Quantum Mechanics, Barry R. Holstein (Chap3,6,7)

This remains a work in progress.
Please, send comments/corrections to John