Four-gradient explained

In differential geometry, the four-gradient (or 4-gradient)

\boldsymbol{\partial} is the four-vector analogue of the gradient \vec{\boldsymbol{\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 metric signature.

SR and GR are abbreviations for special relativity and general relativity respectively.

c indicates the speed of light in vacuum.η _{\mu\nu} = \operatorname{diag}[ 1 , -1 , -1 , -1 ] is the flat spacetime metric of SR.

There are alternate ways of writing four-vector expressions in physics:

• The four-vector style can be used:

A ⋅ B , 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{ a } \cdot \vec. Most of the 3-space vector rules have analogues in four-vector mathematics.

• The Ricci calculus style can be used:

A ^{\mu} η _{\mu\nu} B ^{\nu} , which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as F ^{\mu\nu} = \partial ^{\mu} A ^{\nu} - \partial ^{\nu} A ^{\mu} .

The Latin tensor index ranges in and represents a 3-space vector, e.g.

A ^{i} = \left(a ^{1,} a ^{2,} a ^{3\right)} = \vec{ a }.

The Greek tensor index ranges in and represents a 4-vector, e.g.

A ^{\mu} = \left(a ^{0,} a ^{1,} a ^{2,} a ^{3\right)} = A .

In SR physics, one typically uses a concise blend, e.g.

A = \left(a ^{0,} \vec{ a }\right), where a ^{0} represents the temporal component and \vec{ a } represents the spatial 3-component.

Tensors in SR are typically 4D

(m,n) -tensors, with m upper indices and n lower indices, with the 4D indicating 4 dimensions = the number of values each index can take.

The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):^[1] $$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; A^\backslash mu\; \backslash eta\_\; B^\backslash nu\; =\; A\_\backslash nu\; B^\backslash nu\; =\; A^\backslash mu\; B\_\backslash mu\; =\; \backslash sum\_^\; a^\backslash mu\; b\_\backslash mu\; =\; a^0\; b^0\; -\; \backslash sum\_^\; a^i\; b^i\; =\; a^0\; b^0\; -\; \backslash vec\; \backslash cdot\; \backslash vec$$

Definition

The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:^{[2] [3]} $$\backslash dfrac\; =\; \backslash left(\backslash partial\_0,\backslash partial\_1,\backslash partial\_2,\backslash partial\_3\backslash right)\; =\; \backslash left(\backslash partial\_0,\backslash partial\_i\backslash right)\; =\; \backslash left(\backslash frac\backslash frac,\; \backslash vec\backslash right)\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; \backslash left(\backslash frac,\; \backslash partial\_x,\backslash partial\_y,\backslash partial\_z\backslash right)\; =\; \backslash partial\_\backslash mu\; =\; \_$$

The comma in the last part above

{} _{,\mu} implies the partial differentiation with respect to 4-position X ^{\mu} .

The contravariant components are:^{[2] [3]} $$\backslash boldsymbol\; =\; \backslash partial^\backslash alpha\; =\; \backslash eta^\; \backslash partial\_\backslash beta\; =\; \backslash left(\backslash partial^0,\backslash partial^1,\backslash partial^2,\backslash partial^3\backslash right)\; =\; \backslash left(\backslash partial^0,\backslash partial^i\backslash right)\; =\; \backslash left(\backslash frac\; \backslash frac,\; -\backslash vec\; \backslash right)\; =\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)\; =\; \backslash left(\backslash frac,\; -\backslash partial\_x,-\backslash partial\_y,-\backslash partial\_z\backslash right)$$

Alternative symbols to

\partial _{\alpha} are \Box and D (although \Box can also signify \partial ^{\mu} \partial _{\mu} as the d'Alembert operator).g ^{\alpha} 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}

The strong equivalence principle can be stated as:^[4]

"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 ^{\mu\nu} {} _{,\mu} = 0 in SR, then T ^{\mu\nu} {} _{;\mu} = 0 in GR.

On a (1,0)-tensor or 4-vector this would be:^[4]

On a (2,0)-tensor this would be:

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. Note that in this metric signature [+,−,−,−] the 4-Gradient has a negative spatial component. It gets canceled when taking the 4D dot product since the Minkowski Metric is Diagonal[+1,−1,−1,−1].

X ^{\mu} = \left(ct, \vec{ x }\right) gives the dimension of spacetime:$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash partial^\backslash mu\; \backslash eta\_\; X^\backslash nu\; =\; \backslash partial\_\backslash nu\; X^\backslash nu\; =\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)\; \backslash cdot\; (ct,\backslash vec)\; =\; \backslash frac(ct)\; +\; \backslash vec\backslash cdot\; \backslash vec\; =\; (\backslash partial\_t\; t)\; +\; (\backslash partial\_x\; x\; +\; \backslash partial\_y\; y\; +\; \backslash partial\_z\; z)\; =\; (1)\; +\; (3)\; =\; 4$$

The 4-divergence of the 4-current density $$J^\backslash mu\; =\; \backslash left(\backslash rho\; c,\; \backslash vec\backslash right)\; =\; \backslash rho\_o\; U^\backslash mu\; =\; \backslash rho\_o\; \backslash gamma\backslash left(c,\; \backslash vec\backslash right)\; =\; \backslash left(\backslash rho\; c,\; \backslash rho\; \backslash vec\backslash right)$$ gives a conservation law – the conservation of charge:^[1] $$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash partial^\backslash mu\; \backslash eta\_\; J^\backslash nu\; =\; \backslash partial\_\backslash nu\; J^\backslash nu\; =\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)\; \backslash cdot\; (\backslash rho\; c,\backslash vec)\; =\; \backslash frac\; (\backslash rho\; c)\; +\; \backslash vec\; \backslash cdot\; \backslash vec\; =\; \backslash partial\_t\; \backslash rho\; +\; \backslash vec\; \backslash cdot\; \backslash vec\; =\; 0$$

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} ⋅ \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} = \left(nc, \vec{ n }\right) = n_o U^\mu = n_o \gamma\left(c, \vec\right) = \left(nc, n\vec\right) is used in particle conservation:^[4] $$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash partial^\backslash mu\; \backslash eta\_\; N^\backslash nu\; =\; \backslash partial\_\backslash nu\; N^\backslash nu\; =\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)\; \backslash cdot\; \backslash left(nc,\; n\backslash vec\backslash right)\; =\; \backslash frac\; \backslash left(nc\backslash right)\; +\; \backslash vec\; \backslash cdot\; n\; \backslash vec\; =\; \backslash partial\_t\; n\; +\; \backslash vec\backslash cdot\; n\backslash vec\; =\; 0$$

This is a conservation law for the particle number density, typically something like baryon number density.

The 4-divergence of the electromagnetic 4-potential $A^\backslash mu\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)$ is used in the Lorenz gauge condition:^[1] $$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash partial^\backslash mu\; \backslash eta\_\; A^\backslash nu\; =\; \backslash partial\_\backslash nu\; A^\backslash nu\; =\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)\; \backslash cdot\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; \backslash frac\; \backslash left(\backslash frac\backslash right)\; +\; \backslash vec\; \backslash cdot\; \backslash vec\; =\; \backslash frac\; +\; \backslash vec\; \backslash cdot\; \backslash vec\; =\; 0$$

This is the equivalent of a conservation law for the EM 4-potential.

The 4-divergence of the transverse traceless 4D (2,0)-tensor

h ^{\mu\nu} _{TT} representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source).

The transverse condition $$\backslash boldsymbol\; \backslash cdot\; h^\_\; =\; \backslash partial\_\backslash mu\; h^\_\; =\; 0$$is the equivalent of a conservation equation for freely propagating gravitational waves.

T ^{\mu} as the conserved Noether current associated with spacetime translations, gives four conservation laws in SR:^[4]

The conservation of energy (temporal direction) and the conservation of linear momentum (3 separate spatial directions).$$\backslash boldsymbol\; \backslash cdot\; T^\; =\; \backslash partial\_\; T^\; =\; T^\_\; =\; 0^\backslash mu\; =\; (0,0,0,0)$$

It is often written as:$$\backslash partial\_\; T^\; =\; T^\_\; =\; 0$$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 for a perfect fluid is combined with the conservation of particle number density (

\boldsymbol{\partial} ⋅ 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:$$\backslash partial\_\backslash nu\; \backslash left(x^\; T^\; -\; x^\; T^\backslash right)\; =\; \backslash left(x^\; T^\; -\; x^\; T^\backslash right)\_\; =\; 0^$$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 η ^{\mu\nu} :^[3]

For the Minkowski metric, the components

\left[ η ^{\mu\mu} \right] = 1 /\left[ η _{\mu\mu} \right] (\mu not summed), with non-diagonal components all zero.

For the Cartesian Minkowski Metric, this gives

η ^{\mu\nu} = η _{\mu\nu} = \operatorname{diag}[ 1 , -1 , -1 , -1 ] .

Generally,

η ^{\nu} _{\mu} = \delta ^{\nu} _{\mu} = \operatorname{diag}[ 1 , 1 , 1 , 1 ] , where \delta ^{\nu} _{\mu} is the 4D Kronecker delta.

As a way to define the Lorentz transformations

The Lorentz transformation is written in tensor form as^[4] $$X^\; =\; \backslash Lambda^\_\; X^\backslash nu$$and since

Λ ^{\mu'} _{\nu} are just constants, then$$\backslash dfrac\; =\; \backslash Lambda^\_\backslash nu$$

Thus, by definition of the 4-gradient$$\backslash partial\_\backslash nu\; \backslash left[X^\{\backslash mu\text{'}\}\backslash right]\; =\; \backslash left(\backslash dfrac\backslash right)\backslash left[X^\{\backslash mu\text{'}\}\backslash right]\; =\; \backslash dfrac\; =\; \backslash Lambda^\_\backslash nu$$

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

U ^{\mu} with the 4-gradient gives the total derivative with respect to proper time (d)/(d\tau) :^[1]

The fact that

U ⋅ \boldsymbol{\partial} is a Lorentz scalar invariant shows that the total derivative with respect to proper time (d)/(d\tau) is likewise a Lorentz scalar invariant.U ^{\mu} is the derivative of the 4-position X ^{\mu} with respect to proper time:$$\backslash frac\; \backslash mathbf\; =\; (\backslash mathbf\; \backslash cdot\; \backslash boldsymbol)\backslash mathbf\; =\; \backslash mathbf\; \backslash cdot\; \backslash boldsymbol[\backslash mathbf\{X\}]\; =\; U^\backslash alpha\; \backslash cdot\; \backslash eta^\; =\; U^\backslash alpha\; \backslash eta\_\; \backslash eta^\; =\; U^\backslash alpha\; \backslash delta\_\backslash alpha^\backslash mu\; =\; U^\backslash mu\; =\; \backslash mathbf$$or$$\backslash frac\; \backslash mathbf\; =\; \backslash gamma\backslash frac\; \backslash mathbf\; =\; \backslash gamma\backslash frac\; \backslash left(ct,\; \backslash vec\backslash right)\; =\; \backslash gamma\; \backslash left(\backslash fracct,\backslash frac\backslash vec\backslash right)\; =\; \backslash gamma\; \backslash left(c,\; \backslash vec\backslash right)\; =\; \backslash mathbf$$A ^{\mu} is the proper-time derivative of the 4-velocity U ^{\mu} :

or$$\backslash frac\; \backslash mathbf\; =\; \backslash gamma\; \backslash frac\; (\backslash gamma\; c,\backslash gamma\; \backslash vec)\; =\backslash gamma\; \backslash left(\backslash frac\; [\backslash gamma\; c],\backslash frac\; [\backslash gamma\; \backslash vec\{u\}]\; \backslash right)\; =\; \backslash gamma\; (c\; \backslash dot,\; \backslash dot\; \backslash vec\; +\; \backslash gamma\; \backslash dot)\; =\; \backslash mathbf$$

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

F ^{\mu\nu} is a mathematical object that describes the electromagnetic field in spacetime of a physical system.^{[1] [5] [3] [6] [7]}

Applying the 4-gradient to make an antisymmetric tensor, one gets:where:

A ^{\mu} = A = \left( (\phi)/(c) , \vec{ a }\right), not to be confused with the 4-acceleration A = \gamma \left(c • \gamma , • \gamma \vec{u} + \gamma • \vec{u }\right)

• The electric scalar potential is

\phi

• The magnetic 3-space vector potential is

\vec{ a }

By applying the 4-gradient again, and defining the 4-current density as

J ^{\beta} = J = \left(c\rho, \vec{ j }\right) one can derive the tensor form of the Maxwell equations:$$\backslash partial\_\; F^\; =\; \backslash mu\_o\; J^$$$$\backslash partial\_\backslash gamma\; F\_\; +\; \backslash partial\_\backslash alpha\; F\_\; +\; \backslash partial\_\backslash beta\; F\_\; =\; 0\_$$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 propagation

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:^[6] $$K^\backslash mu\; =\; \backslash mathbf\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; \backslash boldsymbol\; [-\backslash Phi]\; =\; -\backslash boldsymbol\; [\backslash Phi]$$

This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave):$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash omega\; t\; -\; \backslash vec\; \backslash cdot\; \backslash vec\; =\; -\backslash Phi$$

where 4-position

X = \left(ct, \vec{ x }\right), \omega is the temporal angular frequency, \vec{ k } is the spatial 3-space wavevector, and \Phi is the Lorentz scalar invariant phase.

$$\backslash partial\; [\backslash mathbf\{K\}\; \backslash cdot\; \backslash mathbf\{X\}]\; =\; \backslash partial\; \backslash left[\backslash omega\; t\; -\; \backslash vec\{\backslash mathbf\{k\}\}\; \backslash cdot\; \backslash vec\{\backslash mathbf\{x\}\}\backslash right]\; =\; \backslash left(\backslash frac,\; -\backslash nabla\backslash right)\backslash left[\backslash omega\; t\; -\; \backslash vec\{\backslash mathbf\{k\}\}\; \backslash cdot\; \backslash vec\{\backslash mathbf\{x\}\}\backslash right]\; =\; \backslash left(\backslash frac\backslash left[\backslash omega\; t\; -\; \backslash vec\{\backslash mathbf\{k\}\}\; \backslash cdot\; \backslash vec\{\backslash mathbf\{x\}\}\backslash right],\; -\backslash nabla\backslash left[\backslash omega\; t\; -\; \backslash vec\{\backslash mathbf\{k\}\}\; \backslash cdot\; \backslash vec\{\backslash mathbf\{x\}\}\backslash right]\backslash right)\; =\; \backslash left(\backslash frac[\backslash omega\; t],\; -\backslash nabla\backslash left[-\; \backslash vec\{\backslash mathbf\{k\}\}\; \backslash cdot\; \backslash vec\{\backslash mathbf\{x\}\}\backslash right]\backslash right)\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; \backslash mathbf$$with the assumption that the plane wave

\omega and \vec{ k } are not explicit functions of t or \vec{ x }.

The explicit form of an SR plane wave

\Psi _{n(X)} can be written as:^[7]

$$\backslash Psi\_n(\backslash mathbf)\; =\; A\_ne^\; =\; A\_ne^$$ where

A _{n} is a (possibly complex) amplitude.

A general wave

\Psi( X ) would be the superposition of multiple plane waves:$$\backslash Psi(\backslash mathbf)\; =\; \backslash sum\_[\backslash Psi\_n(\backslash mathbf\{X\})]\; =\; \backslash sum\_\backslash left[A\_\{n\}\; e^\{-i(\backslash mathbf\{K\_n\}\backslash cdot\; \backslash mathbf\{X\})\}\backslash right]\; =\; \backslash sum\_\backslash left[A\_\{n\}\; e^\{i(\backslash Phi\_n)\}\backslash right]$$

Again using the 4-gradient,$$\backslash partial\; [\backslash Psi(\backslash mathbf\{X\})]\; =\; \backslash partial\backslash left[Ae^\{-i(\backslash mathbf\{K\}\backslash cdot\backslash mathbf\{X\})\}\backslash right]\; =\; -i\backslash mathbf\; \backslash left[Ae^\{-i(\backslash mathbf\{K\}\backslash cdot\backslash mathbf\{X\})\}\backslash right]\; =\; -i\backslash mathbf\; [\backslash Psi(\backslash mathbf\{X\})]$$or$$\backslash boldsymbol\; =\; -i\; \backslash mathbf$$ 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

\boldsymbol{\partial} is the 4-Laplacian, which is called the d'Alembert operator:^{[5] [3] [6] [7]}

$$\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol\; =\; \backslash partial^\backslash mu\; \backslash cdot\; \backslash partial^\backslash nu\; =\; \backslash partial^\backslash mu\; \backslash eta\_\; \backslash partial^\backslash nu\; =\; \backslash partial\_\backslash nu\; \backslash partial^\backslash nu\; =\; \backslash frac\backslash frac\; -\; \backslash vec^2\; =\; \backslash left(\backslash frac\backslash right)^2\; -\; \backslash vec^2.$$

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):$$\backslash left[(\backslash boldsymbol\{\backslash partial\}\; \backslash cdot\; \backslash boldsymbol\{\backslash partial\})\; +\; \backslash left(\backslash frac\{m\_0\; c\}\{\backslash hbar\}\backslash right)^2\backslash right]\backslash psi\; =\; \backslash left[\backslash left(\backslash frac\{\backslash partial\_t^2\}\{c^2\}\; -\; \backslash vec\{\backslash nabla\}^2\backslash right)\; +\; \backslash left(\backslash frac\{m\_0\; c\}\{\backslash hbar\}\backslash right)^2\backslash right]\backslash psi\; =\; 0$$

(\boldsymbol{\partial} ⋅ A ) = \left(\partial _{\mu} A ^{\mu\right)} = 0 ):

• In vacuum: $$(\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; \backslash mathbf\; =\; (\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; A^\; =\; \backslash mathbf\; =\; 0^$$
• With a 4-current source, not including the effects of spin: $$(\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; \backslash mathbf\; =\; (\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; A^\; =\; \backslash mu\_0\; \backslash mathbf\; =\; \backslash mu\_0\; J^$$
• With quantum electrodynamics source, including effects of spin: $$(\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; \backslash mathbf\; =\; (\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; A^\; =\; e\backslash bar\; \backslash gamma^\; \backslash psi$$

where:

A = A ^{\alpha} = \left( (\phi)/(c) , \vec{a }\right) is an electromagnetic vector potentialJ = J ^{\alpha} = \left(\rho c, \vec{j }\right) is an electromagnetic current density\gamma ^{\alpha} = \left(\gamma ^{0,} \gamma ^{1,} \gamma ^{2,} \gamma ^{3\right)} provide the effects of spin\left(\partial _{\mu} h ^{\mu\nu} _{TT} \right) = 0 )^[6] $$(\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; h^\_\; =\; 0$$where h ^{\mu\nu} _{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 ^{\mu\nu} _{TT} are:

• Purely spatial: $$\backslash mathbf\; \backslash cdot\; h^\_\; =\; h^\_\; =\; 0$$
• Traceless: $$\backslash eta\_\; h^\_\; =\; h^\_\; =\; 0$$
• Transverse: $$\backslash boldsymbol\; \backslash cdot\; h^\_\; =\; \backslash partial\_\backslash mu\; h^\_\; =\; 0$$

In the 4-dimensional version of Green's function:$$(\backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol)\; G\backslash left[\backslash mathbf\{X\}\; -\; \backslash mathbf\{X\text{'}\}\backslash right]\; =\; \backslash delta^\backslash left[\backslash mathbf\{X\}\; -\; \backslash mathbf\{X\text{'}\}\backslash right]$$where the 4D Delta function is:$$\backslash delta^[\backslash mathbf\{X\}]\; =\; \backslash frac\; \backslash int\; d^4\; \backslash mathbf\; e^$$

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.

$$\backslash int\_\backslash Omega\; d^4X\; \backslash left(\backslash partial\_\backslash mu\; V^\backslash mu\backslash right)\; =\; \backslash oint\_\; dS\; \backslash left(V^\backslash mu\; N\_\backslash mu\backslash right)$$or$$\backslash int\_\backslash Omega\; d^4X\; \backslash left(\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\backslash right)\; =\; \backslash oint\_\; dS\; \backslash left(\backslash mathbf\; \backslash cdot\; \backslash mathbf\backslash right)$$where

V = V ^{\mu} is a 4-vector field defined in \Omega \boldsymbol{\partial} ⋅ V = \partial _{\mu} V ^{\mu} is the 4-divergence of V V ⋅ 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 N = N ^{\mu} is the outward-pointing normald ^{4X} = (cdt) \left(d ^{3x\right)} = (cdt) (dxdydz) 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 particle

The generalized relativistic momentum

P _{T} of a particle can be written as^[1] $$\backslash mathbf\; =\; \backslash mathbf\; +\; q\backslash mathbf$$where P = \left( (E)/(c) , \vec{ p }\right) and A = \left( (\phi)/(c) , \vec{ a }\right)

This is essentially the 4-total momentum

P _{T} = \left( (E _{T})/(c) , \vec{ 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 P , plus momentum due to interaction with the EM 4-vector potential A via the particle charge q .S .$$\backslash mathbf\; =\; -\backslash boldsymbol\; [S]\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; -\backslash boldsymbol\; [S]\; =\; -\backslash left(\backslash frac,\; -\backslash vec\backslash right)[S]$$

The temporal component gives:

E _{T} = H = - \partial _{t[S]}

The spatial components give:

\vec{ p _{T} } = \vec[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} = K = \left( (\omega)/(c) , \vec{ k }\right) = -\boldsymbol [\Phi]

To get the HJE, one first uses the Lorentz scalar invariant rule on the 4-momentum:$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; (m\_0\; c)^2$$

But from the minimal coupling rule:$$\backslash mathbf\; =\; \backslash mathbf\; -\; q\backslash mathbf$$

So:

Breaking into the temporal and spatial components:

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.

P and the 4-gradient \boldsymbol{\partial} gives the Schrödinger QM relations.^[7] $$\backslash mathbf\; =\; \backslash left(\backslash frac,\; \backslash vec\backslash right)\; =\; i\backslash hbar\; \backslash boldsymbol\; =\; i\backslash hbar\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)$$

The temporal component gives:

E = i\hbar \partial _{t}

The spatial components give:

\vec{p} = - i\hbar \vec{\nabla}

This can actually be composed of two separate steps.

First:^[1]

$$\backslash mathbf\; =\; \backslash left(\backslash frac,\backslash vec\backslash right)\; =\; \backslash hbar\; \backslash mathbf\; =\; \backslash hbar\; \backslash left(\backslash frac,\backslash vec\backslash right)$$ which is the full 4-vector version of:

E = \hbar \omega

The (spatial components) de Broglie matter wave relation

\vec{p} = \hbar \vec{k}

Second:^[5]

$$\backslash mathbf\; =\; \backslash left(\backslash frac,\backslash vec\backslash right)\; =\; i\; \backslash boldsymbol\; =\; i\; \backslash left(\backslash frac,\; -\backslash vec\backslash right)$$ which is just the 4-gradient version of the wave equation for complex-valued plane waves

The temporal component gives:

\omega = i \partial _{t}

The 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).

• According to:^[7] $$\backslash left[P^\backslash mu,\; X^\backslash nu\backslash right]\; =\; i\; \backslash hbar\; \backslash left[\backslash partial^\backslash mu,\; X^\backslash nu\backslash right]\; =\; i\; \backslash hbar\; \backslash partial^\backslash mu\backslash left[X^\backslash nu\backslash right]\; =\; i\; \backslash hbar\; \backslash eta^$$
• Taking the spatial components, $$\backslash left[p^j,\; x^k\backslash right]\; =\; i\; \backslash hbar\; \backslash eta^$$
• Since

η ^{\mu} = \operatorname{diag}[ 1 , -1 , -1 , -1 ] , $$\backslash left[p^j,\; x^k\backslash right]\; =\; -\; i\; \backslash hbar\; \backslash delta^$$

• Since

[a, b] = - [b, a] , $$\backslash left[x^k,\; p^j\backslash right]\; =\; i\; \backslash hbar\; \backslash delta^$$

• And, relabeling indices gives the usual quantum commutation rules: $$\backslash left[x^j,\; p^k\backslash right]\; =\; i\; \backslash hbar\; \backslash delta^$$

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:^{[5] [3]}

In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson):^[7] $$\backslash left[\backslash left(\backslash partial^\backslash mu\; \backslash partial\_\backslash mu\backslash right)\; +\; \backslash left(\backslash frac\{m\_0\; c\}\{\backslash hbar\}\backslash right)^2\backslash right]\backslash psi\; =\; 0$$

In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons):^[7] $$\backslash left[i\; \backslash gamma^\backslash mu\; \backslash partial\_\backslash mu\; -\; \backslash frac\{m\_0\; c\}\{\backslash hbar\}\backslash right]\; \backslash psi\; =\; 0$$

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:^[7] $$\backslash left\backslash \; =\; \backslash gamma^\backslash mu\; \backslash gamma^\backslash nu\; +\; \backslash gamma^\backslash nu\; \backslash gamma^\backslash mu\; =\; 2\; \backslash eta^I\_4$$

Conservation of 4-probability current density follows from the continuity equation:^[7] $$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash partial\_t\; \backslash rho\; +\; \backslash vec\; \backslash cdot\; \backslash vec\; =\; 0$$

The 4-probability current density has the relativistically covariant expression:^[7] $$J\_\backslash text^\backslash mu\; =\; \backslash frac\backslash left(\backslash psi^*\; \backslash partial^\backslash mu\backslash psi\; -\; \backslash psi\backslash partial^\backslash mu\; \backslash psi^*\backslash right)$$

The 4-charge current density is just the charge times the 4-probability current density:^[7] $$J\_\backslash text^\backslash mu\; =\; \backslash frac\backslash left(\backslash psi^*\; \backslash partial^\backslash mu\backslash psi\; -\; \backslash psi\backslash partial^\backslash mu\backslash psi^*\backslash right)$$

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.^{[3] [7]}

Start with the standard SR 4-vectors:^[1]

X = \left(ct, \vec{ x }\right)U = \gamma\left(c, \vec{ u }\right)P = \left( (E)/(c) , \vec{ p }\right)K = \left( (\omega)/(c) , \vec{ k }\right)

• 4-gradient

\boldsymbol{\partial} = \left( (\partial _{t})/(c) , - \vec{\boldsymbol{\nabla}}\right)

Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar:

• 4-velocity

U = (d)/(d\tau) X , where \tau is the proper time

• 4-momentum

P = m _{0} U , where m _{0} is the rest mass

• 4-wavevector

K = (1)/(\hbar) P , which is the 4-vector version of the Planck–Einstein relation & the de Broglie matter wave relation

• 4-gradient

\boldsymbol{\partial} = - i K , which is the 4-gradient version of complex-valued plane waves

Now, just apply the standard Lorentz scalar product rule to each one:

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:^[7] $$\backslash left[\backslash boldsymbol\{\backslash partial\}\; \backslash cdot\; \backslash boldsymbol\{\backslash partial\}\; +\; \backslash left(\backslash frac\{m\_0\; c\}\{\backslash hbar\}\backslash right)^2\backslash right]\backslash psi\; =\; 0$$

The Schrödinger equation is the low-velocity limiting case of the Klein–Gordon equation.^[7]

If the quantum relation is applied to a 4-vector field

A ^{\mu} instead of a Lorentz scalar field \psi , then one gets the Proca equation:^[7] $$\backslash left[\backslash boldsymbol\{\backslash partial\}\; \backslash cdot\; \backslash boldsymbol\{\backslash partial\}\; +\; \backslash left(\backslash frac\{m\_0\; c\}\{\backslash hbar\}\backslash right)^2\backslash right]A^\backslash mu\; =\; 0^\backslash mu$$

If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation:$$[\backslash boldsymbol\{\backslash partial\}\; \backslash cdot\; \backslash boldsymbol\{\backslash partial\}]A^\backslash mu\; =\; 0^\backslash mu$$

More complicated forms and interactions can be derived by using the minimal coupling rule:

As a component of the RQM covariant derivative (internal particle spaces)

In modern elementary particle physics, one can define a gauge covariant derivative which utilizes the extra RQM fields (internal particle spaces) now known to exist.

The version known from classical EM (in natural units) is:^[3] $$D^\backslash mu\; =\; \backslash partial^\backslash mu\; -\; i\; g\; A^\backslash mu$$

The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is:^[3]

$$D^\backslash mu\; =\; \backslash partial^\backslash mu\; -\; i\; g\_1\; \backslash frac\; Y\; B^\backslash mu\; -\; i\; g\_2\; \backslash frac\backslash tau\_i\; \backslash cdot\; W\_i^\backslash mu\; -\; i\; g\_3\; \backslash frac\; \backslash lambda\_a\; \backslash cdot\; G\_a^\backslash mu$$or$$\backslash mathbf\; =\; \backslash boldsymbol\; -\; i\; g\_1\; \backslash frac\; Y\; \backslash mathbf\; -\; i\; g\_2\; \backslash frac\; \backslash boldsymbol\_i\; \backslash cdot\; \backslash mathbf\_i\; -\; i\; g\_3\; \backslash frac\; \backslash boldsymbol\_a\; \backslash cdot\; \backslash mathbf\_a$$

where the scalar product summations (

⋅ ) here refer to the internal spaces, not the tensor indices:B ^{\mu} corresponds to U(1) invariance = (1) EM force gauge bosonW ^{\mu} _{i} corresponds to SU(2) invariance = (3) weak force gauge bosons (i = 1, …, 3)G ^{\mu} _{a} corresponds to SU(3) invariance = (8) color force gauge bosons (a = 1, …, 8)(g _{1,} g _{2,} g _{3)} are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the g _{i} are fixed for one representation, they are known for all representations.

These internal particle spaces have been discovered empirically.^[3]

Derivation

In three dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to 4 dimensions should be:$$\backslash partial^\backslash alpha\; \backslash overset\; \backslash left(\backslash frac,\; \backslash vec\; \backslash right),$$ which is incorrect.

However, a line integral involves the application of the vector dot product, and when this is extended to 4-dimensional spacetime, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of spacetime. In this article, we place a negative sign on the spatial coordinates (the time-positive metric convention

η ^{\mu\nu} = \operatorname{diag}[ 1 , -1 , -1 , -1 ] ). The factor of (1/c) is to keep the correct unit dimensionality, [length], for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of 4-gradient:^{[1] [3]} $$\backslash partial^\backslash alpha\; =\; \backslash left(\backslash frac\; \backslash frac,\; -\backslash vec\; \backslash right)$$

See also

• Four-vector
• Four-position
• Four-velocity
• Four-acceleration
• Four-momentum
• Four-force
• Four-current
• Four-potential
• Four-frequency
• Four-wavevector
• Four-spin
• Ricci calculus
• Index notation
• Tensor
• Antisymmetric tensor
• Einstein notation
• Raising and lowering indices
• Abstract index notation
• Covariance and contravariance of vectors

References

Note about References

Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations. For instance, some use

m for invariant rest mass, others use m _{0} for invariant rest mass and use m for relativistic mass. Many authors set factors of c and \hbar and G to dimensionless unity. Others show some or all the constants. Some authors use v for velocity, others use u . Some use K as a 4-wavevector (to pick an arbitrary example). Others use k or K or k ^{\mu} or k _{\mu} or K ^{\nu} or N , etc. Some write the 4-wavevector as \left( (\omega)/(c) , k \right) , some as \left( k , (\omega)/(c) \right) or \left(k ^{0,} k \right) or \left(k ^{0,} k ^{1,} k ^{2,} k ^{3\right)} or \left(k ^{1,} k ^{2,} k ^{3,} k ^{4\right)} or \left(k _{t,} k _{x,} k _{y,} k _{z\right)} or \left(k ^{1,} k ^{2,} k ^{3,} i k ^{4\right)} . Some will make sure that the dimensional units match across the 4-vector, others do not. Some refer to the temporal component in the 4-vector name, others refer to the spatial component in the 4-vector name. Some mix it throughout the book, sometimes using one then later on the other. Some use the metric, others use the metric . Some don't use 4-vectors, but do everything as the old style E and 3-space vector p. The thing is, all of these are just notational styles, with some more clear and concise than the others. The physics is the same as long as one uses a consistent style throughout the whole derivation.^[7]

Further reading

• S. Hildebrandt, "Analysis II" (Calculus II),, 2003
• L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988
• J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley

Notes and References

1. Book: Introduction to Special Relativity . 2nd . Wolfgang . Rindler . Oxford Science Publications . 1991 . 0-19-853952-5 .
2. The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010,
3. Book: Modern Elementary Particle Physics: The Fundamental Particles and Forces . Updated . Gordon . Kane . Addison-Wesley Publishing Co. . 1994 . 0-201-62460-5.
4. Book: A first course in general relativity . 1st . Bernard F. . Shultz . Cambridge University Press . 1985 . 0-521-27703-5.
5. Book: Quantum mechanics and the particles of nature: An outline for mathematicians . 1st . Anthony . Sudbury . Cambridge University Press . 1986 . 0-521-27765-5 .
6. Book: An Introduction to General Relativity: Spacetime and Geometry . 1st . Sean M. . Carroll . Addison-Wesley Publishing Co. . 2004 . 0-8053-8732-3 .
7. Book: Relativistic Quantum Mechanics: Wave Equations . 3rd . Walter . Greiner . Springer . 2000 . 3-540-67457-8.