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four-gradient
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In differential geometry, the four-gradient (or 4-gradient) mathbf{partial} is the four-vector analogue of the gradient vec{mathbf{nabla}} from vector calculus.In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors.

Notation

This article uses the {{nowrap|(+ − − −)}} metric signature.SR and GR are abbreviations for special relativity and general relativity respectively.(c) indicates the speed of light in vacuum.eta_{munu} = operatorname{diag}[1,-1,-1,-1] is the flat spacetime metric of SR.There are alternate ways of writing four-vector expressions in physics:
mathbf{A} cdot mathbf{B} is a four-vector style, which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. vec{mathbf{a}} cdot vec{mathbf{b}}. Most of the 3-space vector rules have analogues in four-vector mathematics.
A^mu eta_{munu} B^nu is a Ricci calculus style, which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as F^{munu} = partial^mu A^nu - partial^nu A^mu.
The Latin tensor index ranges in {{nowrap|{1, 2, 3},}} and represents a 3-space vector, e.g. A^i = (a^1,a^2,a^3) = vec{mathbf{a}}.The Greek tensor index ranges in {{nowrap|{0, 1, 2, 3},}} and represents a 4-vector, e.g. A^mu = (a^0,a^1,a^2,a^3) = mathbf{A}.In SR physics, one typically uses a concise blend, e.g. mathbf{A} = (a^0, vec{mathbf{a}}), where a^0 represents the temporal component and vec{mathbf{a}} represents the spatial 3-component.The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 56,151–152,158–161
,weblink


mathbf{A} cdot mathbf{B} = A^mu eta_{munu} B^nu = A_nu B^nu = A^mu B_mu = sum_{mu=0}^{3} a^mu b_mu = a^0 b^0 - sum_{i=1}^{3} a^i b^i = a^0 b^0 - vec{mathbf{a}} cdot vec{mathbf{b}}

Definition

The 4-gradient covariant components compactly written in Ricci calculus notation are:The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, {{ISBN|978-0-521-57507-2}}BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5
, 16,


dfrac{partial}{partial X^mu} = left(frac{1}{c}frac{partial}{partial t}, vec{nabla}right) = left(frac{partial_t}{c}, vec{nabla}right) = partial_mu = {}_{,mu}
The comma in the last part above {}_{,mu} implies the partial differentiation with respect to 4-position X^mu.The contravariant components are:The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, {{ISBN|978-0-521-57507-2}}BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5
, 16,


mathbf{partial} = partial^alpha = eta^{alpha beta} partial_beta = left(frac{1}{c} frac{partial}{partial t}, -vec{nabla} right)= left(frac{partial_t}{c}, -vec{nabla}right) = left(frac{partial_t}{c}, -partial_x,-partial_y,-partial_zright)
Alternative symbols to partial_alpha are Box and D (although Box can also signify partial^mu partial_mu, the d'Alembert operator).In GR, one must use the more general metric tensor g^{alpha beta}, and the tensor covariant derivative nabla_{mu} = {}_{;mu}, (not to be confused with the vector 3-gradient vec{nabla}).The covariant derivative nabla_{nu} incorporates the 4-gradient partial_nu plus spacetime curvature effects via the Christoffel symbols Gamma^{mu}{}_{sigma nu} The strong equivalence principle can be stated as:BOOK
, A first course in general relativity
, 1st
, Bernard F.
, Shultz
, Cambridge University Press
, 1985
, 0-521-27703-5
, 184
, "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as the "comma to semi-colon rule".So, for example, if T^{munu}{}_{,mu} = 0 in SR, then T^{munu}{}_{;mu} = 0 in GR.On a (1,0)-tensor or 4-vector this would be:BOOK
, A first course in general relativity
, 1st
, Bernard F.
, Shultz
, Cambridge University Press
, 1985
, 0-521-27703-5
, 136–139
,
nabla_beta V^alpha = partial_beta V^alpha + V^mu Gamma^{alpha}{}_{mubeta} V^{alpha}{}_{ ;beta} = V^{alpha}{}_{ ,beta} + V^mu Gamma^{alpha}{}_{mubeta}
On a (2,0)-tensor this would be:
nabla_{nu} T^{mu nu} = partial_nu T^{mu nu} + Gamma^{mu}{}_{sigma nu}T^{sigma nu} + Gamma^{nu}{}_{sigma nu} T^{mu sigma} T^{mu nu}{}_{;nu} = T^{mu nu}{}_{,nu} + Gamma^{mu}{}_{sigma nu}T^{sigma nu} + Gamma^{nu}{}_{sigma nu} T^{mu sigma}

Usage

The 4-gradient is used in a number of different ways in special relativity (SR):Throughout this article the formulas are all correct for the flat spacetime Minkowski coordinates of SR,but have to be modified for the more general curved space coordinates of general relativity (GR).

As a 4-divergence and source of conservation laws

Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point.The 4-divergence of the 4-position X^mu = (ct,vec{mathbf{x}}) gives the dimension of spacetime:
mathbf{partial} cdot mathbf{X} = partial^mu eta_{munu} X^nu = partial_nu X^nu = left(frac{partial_t}{c},-vec{nabla}right)cdot (ct,vec{x}) = frac{partial_t}{c}(ct) + vec{nabla}cdot vec{x} = (partial_t t) + (partial_x x+partial_y y+partial_z z) = (1) + (3) = 4
The 4-divergence of the 4-current density J^mu = (rho c,vec{mathbf{j}}) = rho_o U^mu = rho_o gamma(c,vec{mathbf{u}}) = (rho c,rho vec{mathbf{u}}) gives a conservation law – the conservation of charge:BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 103–107
,weblink


mathbf{partial} cdot mathbf{J} = partial^mu eta_{munu} J^nu = partial_nu J^nu = left(frac{partial_t}{c},-vec{nabla}right)cdot (rho c,vec{j}) = frac{partial_t}{c}(rho c) + vec{nabla}cdot vec{j} =partial_t rho + vec{nabla}cdot vec{j} = 0
This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density partial_t rho = -vec{nabla}cdot vec{j}.In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.The 4-divergence of the 4-number flux (4-dust) N^mu = (nc,vec{mathbf{n}}) = n_o U^mu = n_o gamma(c,vec{mathbf{u}}) = (nc,nvec{mathbf{u}}) is used in particle conservation:BOOK
, A first course in general relativity
, 1st
, Bernard F.
, Shultz
, Cambridge University Press
, 1985
, 0-521-27703-5
, 90–110,


mathbf{partial} cdot mathbf{N} = partial^mu eta_{munu} N^nu =partial_nu N^nu = left(frac{partial_t}{c},-vec{nabla}right)cdot left(nc,nvec{mathbf{u}}right) = frac{partial_t}{c}left(ncright) + vec{nabla}cdot nvec{mathbf{u}} =partial_t n + vec{nabla}cdot nvec{mathbf{u}} = 0
This is a conservation law for the particle number density, typically something like baryon number density.The 4-divergence of the electromagnetic 4-potential A^mu = left(frac{phi}{c},vec{mathbf{a}}right) is used in the Lorenz gauge condition:BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 105–107
,weblink


mathbf{partial} cdot mathbf{A} = partial^mu eta_{munu} A^nu =partial_nu A^nu = left(frac{partial_t}{c},-vec{nabla}right)cdot left(frac{phi}{c},vec{a}right) = frac{partial_t}{c}left(frac{phi}{c}right) + vec{nabla}cdot vec{a} =frac{partial_t phi}{c^2} + vec{nabla}cdot vec{a} = 0
This is the equivalent of a conservation law for the EM 4-potential.The 4-divergence of the transverse traceless 2-tensor h^{munu}_{TT} representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).
mathbf{partial} cdot h^{munu}_{TT} = partial_mu h^{munu}_{TT} = 0 :Transverse condition
is the equivalent of a conservation equation for freely propagating gravitational waves.The 4-divergence of the stress–energy tensor T^{mu nu}, the conserved Noether current associated with spacetime translations, gives four conservation laws in SR:BOOK
, A first course in general relativity
, 1st
, Bernard F.
, Shultz
, Cambridge University Press
, 1985
, 0-521-27703-5
, 101–106
, The conservation of energy (temporal direction) and the conservation of linear momentum (3 separate spatial directions).
mathbf{partial} cdot T^{mu nu} = partial_{nu} T^{mu nu} = T^{mu nu}{}_{,nu} = 0^mu = (0,0,0,0)
It is often written as:
partial_{nu} T^{mu nu} = T^{mu nu}{}_{,nu} = 0
where it is understood that the single zero is actually a 4-vector zero 0^mu = (0,0,0,0).When the conservation of the stress–energy tensor (partial_{nu} T^{mu nu} = 0^mu ) for a perfect fluid is combined with the conservation of particle number density (mathbf{partial} cdot mathbf{N} = 0), both utilizing the 4-gradient, one can derive the relativistic Euler equations, which in fluid mechanics and astrophysics are a generalization of the Euler equations that account for the effects of special relativity.These equations reduce to the classical Euler equations if the fluid 3-space velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density.In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum (relativistic angular momentum) is also conserved:
partial_nu(x^{alpha} T^{mu nu} - x^{mu} T^{alpha nu}) =(x^{alpha} T^{mu nu} - x^{mu} T^{alpha nu})_{,nu} = 0^{alpha mu}
where this zero is actually a (2,0)-tensor zero.

As a Jacobian matrix for the SR Minkowski metric tensor

The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.The 4-gradient partial^mu acting on the 4-position X^nu gives the SR Minkowski space metric eta^{munu}:BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5
, 16,


mathbf{partial} [mathbf{X}] = partial^mu[X^nu] = X^{nu_,mu} = left(frac{partial_t}{c},-vec{nabla}right)[(ct,vec{x})] = left(frac{partial_t}{c},-partial_x,-partial_y,-partial_zright)[(ct,x,y,z)],
= begin{bmatrix}frac{partial_t}{c} ct & frac{partial_t}{c} x & frac{partial_t}{c} y & frac{partial_t}{c} z -partial_x ct & -partial_x x & -partial_x y & -partial_x z -partial_y ct & -partial_y x & -partial_y y & -partial_y z -partial_z ct & -partial_z x & -partial_z y & -partial_z zend{bmatrix} = begin{bmatrix}1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1end{bmatrix} = operatorname{diag}[1,-1,-1,-1]
mathbf{partial} [mathbf{X}] = eta^{munu}.
For the Minkowski metric, the components [eta^{mumu}] = 1/[eta_{mumu}] {mu not summed} , with non-diagonal components all zero.For the Cartesian Minkowski Metric, this gives eta^{munu} = eta_{munu} = operatorname{diag}[1,-1,-1,-1].Generally, eta_mu^nu = delta_mu^nu = operatorname{diag}[1,1,1,1], where delta_mu^nu is the 4D Kronecker delta.

As a way to define the Lorentz transformations

The Lorentz transformation is written in tensor form asBOOK
, A first course in general relativity
, 1st
, Bernard F.
, Shultz
, Cambridge University Press
, 1985
, 0-521-27703-5
, 69,


X^{mu'} = Lambda^{mu'}_nu X^nu
and since Lambda^{mu'}_nu are just constants, then
partial X^{mu'} / partial X^nu = Lambda^{mu'}_nu
Thus, by definition of the 4-gradient
partial_nu [X^{mu'}] = (partial / partial X^nu)[X^{mu'}] = partial X^{mu'} / partial X^nu = Lambda^{mu'}_nu
This identity is fundamental. Components of the 4-gradient transform according to the inverse of the components of 4-vectors. So the 4-gradient is the "archetypal" one-form.

As part of the total proper time derivative

The scalar product of 4-velocity U^mu with the 4-gradient gives the total derivative with respect to proper time frac{d}{dtau}:BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 58–59
,weblink


mathbf{U} cdot mathbf{partial} =U^mu eta_{munu} partial^nu = gamma (c,vec{u}) cdot left(frac{partial_t}{c},-vec{nabla}right) = gamma left(c frac{partial_t}{c} + vec{u} cdot vec{nabla} right)= gamma left(partial_t + frac{dx}{dt} partial_x + frac{dy}{dt} partial_y + frac{dz}{dt} partial_z right) = gamma frac{d}{dt} = frac{d}{dtau}
frac{d}{dtau} = frac{dX^mu}{dX^mu} frac{d}{dtau} = frac{dX^mu}{dtau} frac{d}{dX^mu} = U^mu partial_mu = mathbf{U} cdot mathbf{partial}
The fact that mathbf{U} cdot mathbf{partial} is a Lorentz scalar invariant shows that the total derivative with respect to proper time frac{d}{dtau} is likewise a Lorentz scalar invariant.So, for example, the 4-velocity U^mu is the derivative of the 4-position X^mu with respect to proper time:
frac{d}{dtau} mathbf{X} = (mathbf{U} cdot mathbf{partial})mathbf{X} = mathbf{U} cdot mathbf{partial}[mathbf{X}] = U^alpha cdot eta^{munu} = U^alpha eta_{alpha nu} eta^{munu} = U^alpha delta_alpha^mu = U^mu = mathbf{U}
or
frac{d}{dtau} mathbf{X} = gammafrac{d}{dt} mathbf{X} = gammafrac{d}{dt} (ct,vec{x}) = gamma left(frac{d}{dt}ct,frac{d}{dt}vec{x} right) = gamma (c,vec{u}) = mathbf{U}
Another example, the 4-acceleration A^mu is the proper-time derivative of the 4-velocity U^mu:
frac{d}{dtau} mathbf{U} = (mathbf{U} cdot mathbf{partial})mathbf{U} = mathbf{U} cdot mathbf{partial}[mathbf{U}] = U^alpha eta_{alphamu}partial^mu[U^nu]
= U^alpha eta_{alphamu}begin{bmatrix} frac{partial_t}{c} gamma c & frac{partial_t}{c} gamma vec{u} -vec{nabla}gamma c & -vec{nabla}gamma vec{u} end{bmatrix} = U^alpha begin{bmatrix} frac{partial_t}{c} gamma c & 0 0 & vec{nabla}gamma vec{u} end{bmatrix} = gamma left(c frac{partial_t}{c} gamma c , vec{u} cdot nablagamma vec{u} right)= gamma left(c partial_t gamma, frac{d}{dt}[gamma vec{u}] right) = gamma (c dot{gamma}, dot{gamma} vec{u} + gamma dot{vec{u}} )= mathbf{A}
or
frac{d}{dtau} mathbf{U} =gamma frac{d}{dt} (gamma c,gamma vec{u}) =gamma left(frac{d}{dt}[gamma c],frac{d}{dt}[gamma vec{u}] right) = gamma (c dot{gamma}, dot{gamma} vec{u} + gamma dot{vec{u}} ) = mathbf{A}

As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations

The Faraday electromagnetic tensor F^{munu} is a mathematical object that describes the electromagnetic field in spacetime of a physical system.BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 101–128
,weblink
BOOK
, Quantum mechanics and the particles of nature: An outline for mathematicians
, 1st
, Anthony
, Sudbury
, Cambridge University Press
, 1986
, 0-521-27765-5
, 314,
BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5
, 17–18,
BOOK
, An Introduction to General Relativity: Spacetime and Geometry
, 1st
, Sean M.
, Carroll
, Addison-Wesley Publishing Co.
, 2004
, 0-8053-8732-3
, 29–30,
BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 4,
Applying the 4-gradient to make an antisymmetric tensor, one gets:
F^{munu} = partial^mu A^nu - partial^nu A^mu =
begin{bmatrix}E_x/c & 0 & -B_z & B_y E_y/c & B_z & 0 & -B_x E_z/c & -B_y & B_x & 0end{bmatrix}where:
Electromagnetic 4-potential A^mu = mathbf{A} = left(frac{phi}{c}, vec{mathbf{a}}right), not to be confused with the 4-acceleration mathbf{A} = gamma (c dot{gamma}, dot{gamma} vec{u} + gamma dot{vec{u}} )
phi is the electric scalar potential, and vec{mathbf{a}} is the magnetic 3-space vector potential.By applying the 4-gradient again, and defining the 4-current density as J^{beta} = mathbf{J} = ( crho, vec{mathbf{j}} ) one can derive the tensor form of the Maxwell equations:
partial_{alpha} F^{alphabeta} = mu_o J^{beta}
partial_gamma F_{ alpha beta } + partial_alpha F_{ beta gamma } + partial_beta F_{ gamma alpha } = 0_{alpha beta gamma}
where the second line is a version of the Bianchi identity (Jacobi identity).

As a way to define the 4-wavevector

A wavevector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagationThe 4-wavevector K^mu is the 4-gradient of the negative phase Phi (or the negative 4-gradient of the phase) of a wave in Minkowski Space:BOOK
, An Introduction to General Relativity: Spacetime and Geometry
, 1st
, Sean M.
, Carroll
, Addison-Wesley Publishing Co.
, 2004
, 0-8053-8732-3
, 387,


K^mu = mathbf{K} = left(frac{omega}{c}, vec{mathbf{k}}right) = mathbf{partial} [-Phi]= -mathbf{partial} [Phi]
This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave):
mathbf{K} cdot mathbf{X} = omega t - vec{mathbf{k}} cdot vec{mathbf{x}} = -Phi
where 4-position mathbf{X} = (ct, vec{mathbf{x}}), omega is the temporal angular frequency, vec{mathbf{k}} is the spatial 3-space wavevector, and Phi is the Lorentz scalar invariant phase.
partial [mathbf{K} cdot mathbf{X}] = partial [omega t - vec{mathbf{k}} cdot vec{mathbf{x}}] = left(frac{partial_t}{c},-nablaright)[omega t - vec{mathbf{k}} cdot vec{mathbf{x}}] = left(frac{partial_t}{c}[omega t - vec{mathbf{k}} cdot vec{mathbf{x}}],-nabla[omega t - vec{mathbf{k}} cdot vec{mathbf{x}}]right) = left(frac{partial_t}{c}[omega t],-nabla[- vec{mathbf{k}} cdot vec{mathbf{x}}]right) = left(frac{omega}{c},vec{mathbf{k}}right) = mathbf{K}
with the assumption that the plane wave omega and vec{mathbf{k}} are not explicit functions of t or vec{mathbf{x}}The explicit form of an SR plane wave Psi_n(mathbf{X}) can be written as:BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 9,


Psi_n(mathbf{X})= A_ne^{-i(mathbf{K_n}cdotmathbf{X})} = A_ne^{i(Phi_n)} where A_n is a (possibly complex) amplitude.
A general wave Psi(mathbf{X}) would be the superposition of multiple plane waves:
Psi(mathbf{X})= sum_{ n }[Psi_n(mathbf{X})] = sum_{ n }[ A_{n}e^{-i(mathbf{K_n}cdotmathbf{X})}] = sum_{ n }[ A_{n}e^{i(Phi_n)}]
Again using the 4-gradient,
partial [Psi(mathbf{X})] = partial[Ae^{-i(mathbf{K}cdotmathbf{X})}] = -imathbf{K} [Ae^{-i(mathbf{K}cdotmathbf{X})}] = -imathbf{K} [Psi(mathbf{X})]
or
mathbf{partial} = -i mathbf{K}, which is the 4-gradient version of complex-valued plane waves

As the d'Alembertian operator

In special relativity, electromagnetism and wave theory, the d'Alembert operator, also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert.The square of mathbf{partial} is the 4-Laplacian, which is called the d'Alembert operator:BOOK
, Quantum mechanics and the particles of nature: An outline for mathematicians
, 1st
, Anthony
, Sudbury
, Cambridge University Press
, 1986
, 0-521-27765-5
, 300,
BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5
, 17–18,
BOOK
, An Introduction to General Relativity: Spacetime and Geometry
, 1st
, Sean M.
, Carroll
, Addison-Wesley Publishing Co.
, 2004
, 0-8053-8732-3
, 41,
BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 4,


mathbf{partial} cdot mathbf{partial} = partial^mu cdot partial^nu = partial^mu eta_{munu} partial^nu = partial_nu partial^nu = frac{1}{c^2}frac{partial^2}{partial t^2} - vec{nabla}^2 = left(frac{partial_t}{c}right)^2 - vec{nabla}^2.
As it is the dot product of two 4-vectors, the d'Alembertian is a Lorentz invariant scalar.Occasionally, in analogy with the 3-dimensional notation, the symbols Box and Box^2 are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol Box is reserved for the d'Alembertian.Some examples of the 4-gradient as used in the d'Alembertian follow:In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson):
[(mathbf{partial} cdot mathbf{partial}) + left(frac {m_0 c}{hbar}right)^2]psi = [left(frac{partial_t^2}{c^2} - vec{nabla}^2right) + left(frac {m_0 c}{hbar}right)^2] psi = 0
In the wave equation for the electromagnetic field { using Lorenz gauge (mathbf{partial} cdot mathbf{A}) = (partial_mu A^mu) = 0 }:
(mathbf{partial} cdot mathbf{partial}) mathbf{A} = mathbf{0} {in vacuum} (mathbf{partial} cdot mathbf{partial}) mathbf{A} = mu_0 mathbf{J} {with a 4-current source, not including the effects of spin} (mathbf{partial} cdot mathbf{partial}) A^{mu}=ebar{psi} gamma^{mu} psi {with quantum electrodynamics source, including effects of spin}
where:
Electromagnetic 4-potential mathbf{A} = A^{alpha} = left(frac{phi}{c},mathbf{vec{a}}right) is an electromagnetic vector potential 4-current density mathbf{J} = J^{alpha} = (rho c,mathbf{vec{j}}) is an electromagnetic current density Dirac Gamma matrices gamma^alpha = ( gamma^0, gamma^1, gamma^2, gamma^3 ) provide the effects of spin
In the wave equation of a gravitational wave { using a similar Lorenz gauge (partial_mu h^{munu}_{TT}) = 0 }BOOK
, An Introduction to General Relativity: Spacetime and Geometry
, 1st
, Sean M.
, Carroll
, Addison-Wesley Publishing Co.
, 2004
, 0-8053-8732-3
, 274–322,


(mathbf{partial} cdot mathbf{partial}) h^{munu}_{TT} = 0
where h^{munu}_{TT} is the transverse traceless 2-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).Further conditions on h^{munu}_{TT} are:
mathbf{U} cdot h^{munu}_{TT} = h^{0nu}_{TT} = 0 :Purely spatial eta_{munu} h^{munu}_{TT} = h^{nu}_{TTnu} = 0 :Traceless mathbf{partial} cdot h^{munu}_{TT} = partial_mu h^{munu}_{TT} = 0 :Transverse
In the 4-dimensional version of Green's function:
(mathbf{partial} cdot mathbf{partial}) G[mathbf{X}-mathbf{X'}] = delta^{(4)}[mathbf{X}-mathbf{X'}]
where the 4D Delta function is:
delta^{(4)}[mathbf{X}] = frac{1}{(2 pi)^4} int d^4 mathbf{K} e^{-i(mathbf{K} cdot mathbf{X})}

As a component of the 4D Gauss' Theorem / Stokes' Theorem / Divergence Theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
int_Omega d^4X (partial_mu V^mu ) = oint_{partial Omega} dS (V^mu N_mu)
or
int_Omega d^4X (mathbf{partial} cdot mathbf{V}) = oint_{partial Omega} dS (mathbf{V} cdot mathbf{N})
where
mathbf{V} = V^mu is a 4-vector field defined in Omega
mathbf{partial}cdotmathbf{V} = partial_mu V^mu is the 4-divergence of V
mathbf{V}cdotmathbf{N} = V^mu N_mu is the component of V along direction N
Omega is a 4D simply connected region of Minkowski spacetime
partial Omega = S is its 3D boundary with its own 3D volume element dS
mathbf{N} = N^mu is the outward-pointing normal
d^4X = (c,dt) (d^3x) = (c,dt) (dx,dy,dz) is the 4D differential volume element

As a component of the SR Hamilton–Jacobi equation in relativistic analytic mechanics

The Hamilton–Jacobi equation (HJE) is a formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particleThe generalized relativistic momentum mathbf{P_T} of a particle can be written asBOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 93–96
,weblink


mathbf{P_T} = mathbf{P} + qmathbf{A}
where mathbf{P} = left(frac{E}{c}, vec{mathbf{p}}right) and mathbf{A} = left(frac{phi}{c}, vec{mathbf{a}}right)This is essentially the 4-total momentum mathbf{P_T} = left(frac{E_T}{c}, vec{mathbf{p_T}}right) of the system; a test particle in a field using the minimal coupling rule. There is the inherent momentum of the particle mathbf{P}, plus momentum due to interaction with the EM 4-vector potential mathbf{A} via the particle charge q.The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action S.
mathbf{P_T} = -mathbf{partial} [S]
mathbf{P_T} = left(frac{E_T}{c}, vec{mathbf{p_T}}right) = left(frac{H}{c}, vec{mathbf{p_T}}right) = -mathbf{partial} [S] = -left(frac{partial_t}{c},-vec{mathbf{nabla}}right)[S]
The temporal component gives: E_T = H =-partial_t[S]The spatial components give: vec{mathbf{p_T}} = vec{mathbf{nabla}}[S]where H is the Hamiltonian.This is actually related to the 4-wavevector being equal the negative 4-gradient of the phase from above.K^mu = mathbf{K} = left(frac{omega}{c}, vec{mathbf{k}}right) = -mathbf{partial} [Phi]To get the HJE, one first uses the Lorentz scalar invariant rule on the 4-momentum:
mathbf{P} cdot mathbf{P} = (m_0 c)^2
But from the minimal coupling rule:
mathbf{P} = mathbf{P_T} - qmathbf{A}
So:
(mathbf{P_T} - qmathbf{A}) cdot (mathbf{P_T} - qmathbf{A}) = (m_0 c)^2
(mathbf{P_T} - qmathbf{A})^2 = (m_0 c)^2
(-mathbf{partial}[S] - qmathbf{A})^2 = (m_0 c)^2
Breaking into the temporal and spatial components:
(-partial_t[S]/c - q phi /c)^2 - (mathbf{nabla}[S] - q mathbf{a})^2 = (m_0 c)^2
(mathbf{nabla}[S] - q mathbf{a})^2 - (1/c)^2(-partial_t[S] - q phi)^2 + (m_0 c)^2 = 0
(mathbf{nabla}[S] - q mathbf{a})^2 - (1/c)^2(partial_t[S] + q phi)^2 + (m_0 c)^2 = 0
where the final is the relativistic Hamilton–Jacobi equation.

As a component of the Schrödinger relations in quantum mechanics

The 4-gradient is connected with quantum mechanics.The relation between the 4-momentum mathbf{P} and the 4-gradient mathbf{partial} gives the Schrödinger QM relations.BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 3–5,
mathbf{P} = left(frac{E}{c},vec{p}right) = ihbar mathbf{partial} = ihbar left(frac{partial_t}{c},-vec{nabla}right)The temporal component gives: E = ihbar partial_tThe spatial components give: vec{p} = -ihbar vec{nabla}This can actually be composed of two separate steps.First:BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
, 82–84
,weblink
mathbf{P} = left(frac{E}{c},vec{p}right) = hbar mathbf{K} = hbar left(frac{omega}{c},vec{k}right) which is the full 4-vector version of:The (temporal component) Planck–Einstein relation E = hbar omega
The (spatial components) de Broglie matter wave relation vec{p} = hbar vec{k}Second:BOOK
, Quantum mechanics and the particles of nature: An outline for mathematicians
, 1st
, Anthony
, Sudbury
, Cambridge University Press
, 1986
, 0-521-27765-5
, 300,
mathbf{K} = left(frac{omega}{c},vec{k}right) = i mathbf{partial} = i left(frac{partial_t}{c},-vec{nabla}right) which is just the 4-gradient version of the wave equation for complex-valued plane wavesThe temporal component gives: omega = i partial_tThe spatial components give: vec{k} = - i vec{nabla}

As a component of the covariant form of the quantum commutation relation

In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).
[P^mu,X^nu] = i hbar [partial^mu,X^nu] = i hbar partial^mu[X^nu] = i hbar eta^{mu nu}BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 4,


[p^j,x^k] = i hbar eta^{j k}: Taking the spatial components:
[p^j,x^k] = - i hbar delta^{j k}: because eta^{mu nu} = operatorname{diag}[1,-1,-1,-1]
[x^k,p^j] = i hbar delta^{k j}: because [a,b] = -[b,a]
[x^j,p^k] = i hbar delta^{j k}: relabeling indices gives the usual quantum commutation rules

As a component of the wave equations and probability currents in relativistic quantum mechanics

The 4-gradient is a component in several of the relativistic wave equations:BOOK
, Quantum mechanics and the particles of nature: An outline for mathematicians
, 1st
, Anthony
, Sudbury
, Cambridge University Press
, 1986
, 0-521-27765-5
, 300–309,
BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5
, 25,30–31,55–69,
In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson):BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 5,


[(partial^mu partial_mu) + left(frac {m_0 c}{hbar}right)^2]psi = 0
In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons):BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 130,


[i gamma^mu partial_mu - frac{m_0 c}{hbar}] psi = 0
where gamma^mu are the Dirac gamma matrices and psi is a relativistic wave function.psi is Lorentz scalar for the Klein–Gordon equation, and a spinor for the Dirac equation.It is nice that the gamma matrices themselves refer back to the fundamental aspect of SR, the Minkowski metric:BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 129,


{gamma^mu,gamma^nu} = gamma^mu gamma^nu + gamma^nu gamma^mu = 2 eta^{munu}I_4 ,
Conservation of 4-probability current density follows from the continuity equation:BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 6,


mathbf{partial} cdot mathbf{J} = partial_t rho + vec{mathbf{nabla}} cdot vec{mathbf{j}} = 0
The 4-probability current density has the relativistically covariant expression:BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 6,


J_{prob}^mu = frac{ihbar}{2m_0}(psi^*partial^mupsi - psipartial^mupsi^*)
The 4-charge current density is just the charge (q) times the 4-probability current density:BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 8,


J_{charge}^mu = frac{ihbar q}{2m_0}(psi^*partial^mupsi - psipartial^mupsi^*)

As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity

Relativistic wave equations use 4-vectors in order to be covariant.BOOK
, Modern Elementary Particle Physics: The Fundamental Particles and Forces
, Updated
, Gordon
, Kane
, Addison-Wesley Publishing Co.
, 1994
, 0-201-62460-5,
BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8,
Start with the standard SR 4-vectors:BOOK
, Introduction to Special Relativity
, 2nd
, Wolfgang
, Rindler
, Oxford Science Publications
, 1991
, 0-19-853952-5
,weblink


4-position mathbf{X} = (ct,vec{mathbf{x}}) 4-velocity mathbf{U} = gamma(c,vec{mathbf{u}}) 4-momentum mathbf{P} = left(frac{E}{c},vec{mathbf{p}}right) 4-wavevector mathbf{K} = left(frac{omega}{c},vec{mathbf{k}}right) 4-gradient mathbf{partial} = left(frac{partial_t}{c},-vec{mathbf{nabla}}right)
Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar:
mathbf{U} = frac{d}{dtau} mathbf{X}, where tau is the proper time mathbf{P} = m_0 mathbf{U}, where m_0 is the rest mass mathbf{K} = (1/hbar) mathbf{P}, which is the 4-vector version of the Planck–Einstein relation & the de Broglie matter wave relation mathbf{partial} = -i mathbf{K}, which is the 4-gradient version of complex-valued plane waves
Now, just apply the standard Lorentz scalar product rule to each one:
mathbf{U} cdot mathbf{U} = (c)^2 mathbf{P} cdot mathbf{P} = (m_0 c)^2 mathbf{K} cdot mathbf{K} = left(frac{m_0 c}{hbar}right)^2 mathbf{partial} cdot mathbf{partial} = left(frac{-i m_0 c}{hbar}right)^2 = -left(frac{m_0 c}{hbar}right)^2
The last equation (with the 4-gradient scalar product) is a fundamental quantum relation.When applied to a Lorentz scalar field psi, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations:BOOK
, Relativistic Quantum Mechanics: Wave Equations
, 3rd
, Walter
, Greiner
, Springer
, 2000
, 3-540-67457-8
, 5–8,


[mathbf{partial} cdot mathbf{partial} + left(frac{m_0 c}{hbar}right)^2]psi = 0
The Schrödinger equation is the low-velocity limiting case {|v|

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