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Physics[gamma_] - The gamma_ 3D metric tensor

 Calling Sequence gamma_[$i,j$] gamma_[$i,j$, matrix] gamma_[keyword]

Parameters

 i, j - the indices, as names representing integer numbers between 1 and the space dimension (i.e. spacetime dimension - 1), they can also be the numbers themselves matrix - optional, returns the matrix form for the given (covariant or contravariant) indices; if passed without indices it returns the covariant metric keyword - optional, it can be definition, determinant, line_element, matrix, nonzero

Description

 • The gamma_[i, j], displayed as ${\mathrm{\gamma }}_{i,j}$ (without _ in between $\mathrm{\gamma }$ and its indices), is a computational representation for the space 3D metric tensor. Note: to work with gamma_[i, j], a kind of letter to represent a space index must be set first using Setup, for instance entering Setup(spaceindices = lowercaselatin_is).
 • When Physics is loaded, the dimension of spacetime is automatically set to 4 and the metric is automatically set to be galilean, representing a Minkowski spacetime with signature (-, -, -, +), so the value 0, referring to time, points to the fourth place in the list of coordinates (indexing with 0 or 4 is the same). The relationship between the 3D metric gamma_[i, j] and the space components of the 4D metric g_ is as in equations (84.9) and (84.7) of the Landau and Lifshitz book [1]

${\mathrm{\gamma }}_{}^{i,j}=-{g}_{}^{i,j}$

${\mathrm{\gamma }}_{i,j}=-{g}_{i,j}+\frac{{g}_{0,i}{g}_{0,j}}{{g}_{0,0}}$

 where the signature is (- - - +). These definitions depend on the signature set and automatically follow any change of it that you can perform using the Setup command, as in Setup(signature = ...) where the right-hand side is any of (- - - +), (+ - - -), (+ + + -) and (- + + +).
 • An important distinction is made between the 3D spatial metric represented by Physics:-gamma_, related to the infinitesimal distance measured by one observer at a fixed point in space, expressed in terms of the spatial components of the metric g_ as shown above, and the 3D spatial metric of a hypersurface defined by a fixed constant value of t, the time, represented by Physics:-ThreePlusOne:-gamma3_, where the 3D line element is the infinitesimal spatial distance on this hypersurface, the distance measured by an Eulerian observer (whose velocity is perpendicular to the hypersurface), and so can be obtained from the 4D one by restricting it to constant time surfaces, namely ${\mathrm{\gamma }}_{i,j}={g}_{i,j}$ where the signature is (- + + +), as used in the context of the ThreePlusOne package.
 • By replacing g_ by gamma_ and all spacetime indices by space indices in the definitions of the Christoffel symbols and the Ricci and Riemann tensors you obtain 3D tensors, whose indices are lowered and raised using the 3D metric gamma_[i, j], the 3D versions of the corresponding 4D tensors (see after sec.88 in [1]).
 • When the indices of gamma_ assume integer values, it is expected they are between 0 and the space dimension (the spacetime dimension - 1), and the corresponding value of gamma_ is returned. The value 0 always points to the position of the time-like component, so with the signatures (- - - +) and (+ + + -) indexing with 0 is the same as indexing with 4, while with the signatures (+ - - -) and (- + + +) it is the same as indexing with 1. When the indices have symbolic values, say as in $i,j$, if $i=j$ gamma_ returns the dimension of space. If the system can prove that $i\ne j$ (e.g. via assumptions), gamma_ returns zero whenever the metric is diagonal. Otherwise gamma_ returns unevaluated, after normalizing its indices taking into account that this 3D space metric is symmetric.
 • You can change the value of the 3D metric gamma_ in different ways by changing the value of the 4D metric g_, using Setup or g_ itself as explained in its help page. For example, among others, the simplest way to set the Schwarzschild metric and related coordinates is to enter g_[sc]
 • Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices - see . and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~i; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.
 • Besides being indexed with two indices, gamma_ accepts the following keywords as an index:
 – definition: when passed alone, gamma_ returns its definition, that is an equation with the metric on the left-hand side and the matrix form of the metric on the right-hand side, the same as when you enter the metric without indices (gamma_[]).
 – determinant: when passed alone, gamma_ returns the determinant of the all-covariant metric ${\mathrm{\gamma }}_{i,j}$ . Recall that the components of the covariant ${\mathrm{\gamma }}_{i,j}$ are not equal to the space components of ${g}_{\mathrm{\mu },\mathrm{\nu }}$ unless ${g}_{0,0}=1$ and ${g}_{0,j}=0$. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting determinant takes into account the character of the indices.
 – line_element: (synonym: lineelement) when passed alone, gamma_ returns the 3D line element for the 4D current metric expressing the differentials of the coordinates using d_.
 – matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in gamma_[]) returns a Matrix that when indexed with numerical values from 1 to the dimension of space it returns the value of each of the components of gamma_. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting Matrix takes into account the character of the indices.
 – nonzero: returns a set of equations, with the left-hand side as a sequence of two positive numbers identifying the element of $\mathrm{γ__i,j}$ and the corresponding value on the right-hand side. Note that this set is actually the output of the ArrayElems command when passing to it the Matrix obtained with the keyword matrix.
 • In turn, some predefined values for the 4D spacetime metric can be set by indexing the metric with a name or a portion of it - see g_ - and in that way you can indirectly set the values of the 3D gamma_ metric.
 • The %gamma_ command is the inert form of gamma_, so it represents the same tensor but entering it does not result in performing any computation. To perform the related computations as if %gamma_ were gamma_, use value.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true},\mathrm{spaceindices}=\mathrm{lowercase_is}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}{,}{\mathrm{spaceindices}}{=}{\mathrm{lowercaselatin_is}}\right]$ (1)

When Physics is initialized, the default spacetime metric is of Minkowski type. You can see the metric querying Setup, as in Setup(metric), or directly entering the metric as g_[], with no indices

 > $\mathrm{g_}\left[\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\begin{array}{cccc}{-1}& {0}& {0}& {0}\\ {0}& {-1}& {0}& {0}\\ {0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]$ (2)

The default spacetime of the Physics package is a Minkowski spacetime with dimension 4 and the time-like component in position 4, so when indexing 4D tensors the values 0 and 4 represent the same object. For example, when indexing the metric g_,

 > $\mathrm{g_}\left[0,0\right]=\mathrm{g_}\left[4,4\right]$
 ${1}{=}{1}$ (3)
 > $\mathrm{g_}\left[0,3\right]$
 ${0}$ (4)
 > $\mathrm{g_}\left[\mathrm{line_element}\right]$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(\mathrm{x1}{,}\mathrm{x2}{,}\mathrm{x3}{,}\mathrm{x4}\right)\right\}$
 ${-}{{\mathbf{ⅆ}}{}\left({\mathrm{x1}}\right)}^{{2}}{-}{{\mathbf{ⅆ}}{}\left({\mathrm{x2}}\right)}^{{2}}{-}{{\mathbf{ⅆ}}{}\left({\mathrm{x3}}\right)}^{{2}}{+}{{\mathbf{ⅆ}}{}\left({\mathrm{x4}}\right)}^{{2}}$ (5)
 > $\mathrm{gamma_}\left[\mathrm{line_element}\right]$
 ${{\mathbf{ⅆ}}{}\left({\mathrm{x1}}\right)}^{{2}}{+}{{\mathbf{ⅆ}}{}\left({\mathrm{x2}}\right)}^{{2}}{+}{{\mathbf{ⅆ}}{}\left({\mathrm{x3}}\right)}^{{2}}$ (6)

Note the character positive definite of this quadratic form. The symmetry property of gamma_ is automatically taken into account when the indices have symbolic values

 > $\mathrm{gamma_}\left[i,j\right]-\mathrm{gamma_}\left[j,i\right]$
 ${0}$ (7)

You can always query about the letters used to represent spacetime and space indices as well as the dimension of spacetime (and so of space) via

 > $\mathrm{Setup}\left(\mathrm{spacetimeindices},\mathrm{spaceindices},\mathrm{dimension}\right)$
 $\left[{\mathrm{dimension}}{=}{4}{,}{\mathrm{spaceindices}}{=}{\mathrm{lowercaselatin_is}}{,}{\mathrm{spacetimeindices}}{=}{\mathrm{greek}}\right]$ (8)

The trace of the spacetime metric g_ is equal to the dimension and the trace of the space gamma_ metric is equal to the dimension - 1. Note the use of the sum rule for repeated indices, both for spacetime and space only indices

 > $\mathrm{g_}\left[\mathrm{\mu },\mathrm{\nu }\right]$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$ (9)
 > $\mathrm{eval}\left(,\mathrm{\mu }=\mathrm{\nu }\right)$
 ${4}$ (10)
 > $\mathrm{gamma_}\left[i,j\right]$
 ${{\mathrm{\gamma }}}_{{i}{,}{j}}$ (11)
 > $\mathrm{eval}\left(,i=j\right)$
 ${3}$ (12)

To set the scenario as a curved spacetime set the metric using Setup, for instance indicating the square of the spacetime interval. This can be done directly from g_. Choose for example the keyword Tolman and this also, automatically, implies on setting spherical coordinates as the differentiation variables for d_

 > $\mathrm{g_}\left[\mathrm{Tolman}\right]$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}\mathrm{The Tolman metric in coordinates}{}\left[r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right]$
 $\mathrm{Parameters:}\left[R{}\left(t{,}r\right){,}E{}\left(r\right)\right]$
 $\mathrm{Signature:}\left(\mathrm{- - - +}\right)$
 ${}\mathrm{_______________________________________________________}$
 ${{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\left[\begin{array}{cccc}{-}\frac{{\left(\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{R}{}\left({t}{,}{r}\right)\right)}^{{2}}}{{1}{+}{2}{}{E}{}\left({r}\right)}& {0}& {0}& {0}\\ {0}& {-}{{R}{}\left({t}{,}{r}\right)}^{{2}}& {0}& {0}\\ {0}& {0}& {-}{{R}{}\left({t}{,}{r}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]$ (13)

Use a compact display for these functions to avoid redundant repeated display of their functionality plus having derivatives displayed indexed by the differentiation variables

 > $\mathrm{CompactDisplay}\left(\right)$
 ${E}{}\left({r}\right){}{\mathrm{will now be displayed as}}{}{E}$
 ${R}{}\left({t}{,}{r}\right){}{\mathrm{will now be displayed as}}{}{R}$ (14)

The corresponding 3D line element

 > $\mathrm{gamma_}\left[\mathrm{line_element}\right]$
 $\frac{{{{R}}_{{r}}}^{{2}}{}{{\mathbf{ⅆ}}{}\left({r}\right)}^{{2}}}{{1}{+}{2}{}{E}}{+}{{R}}^{{2}}{}{{\mathbf{ⅆ}}{}\left({\mathrm{\theta }}\right)}^{{2}}{+}{{R}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{\mathbf{ⅆ}}{}\left({\mathrm{\phi }}\right)}^{{2}}$ (15)

The matrix form: all of gamma_[], gamma_[matrix] and gamma_[i, j, matrix] return the all-covariant matrix

 > $\mathrm{gamma_}\left[i,j,\mathrm{matrix}\right]$
 ${{\mathrm{\gamma }}}_{{i}{,}{j}}{=}\left[\begin{array}{ccc}\frac{{{{R}}_{{r}}}^{{2}}}{{1}{+}{2}{}{E}}& {0}& {0}\\ {0}& {{R}}^{{2}}& {0}\\ {0}& {0}& {{R}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}\end{array}\right]$ (16)

Or, for the matrix components of the all contravariant ${\mathrm{\gamma }}_{}^{\mathrm{&i;},\mathrm{&j;}}$, which as a matrix is equal to the inverse of the all-covariant ${\mathrm{\gamma }}_{i,j}$, you can use gamma_[~i, ~j, matrix], with contravariant indices (prefixed by ~), or directly

 > $\mathrm{gamma_}\left[\mathrm{~}\right]$
 ${{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{i}{,}{j}}}^{\phantom{{}}{i}{,}{j}}{=}\left[\begin{array}{ccc}\frac{{1}{+}{2}{}{E}}{{{{R}}_{{r}}}^{{2}}}& {0}& {0}\\ {0}& \frac{{1}}{{{R}}^{{2}}}& {0}\\ {0}& {0}& \frac{{{\mathrm{csc}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{{{R}}^{{2}}}\end{array}\right]$ (17)
 > $\mathrm{gamma_}\left[\mathrm{determinant}\right]$
 $\frac{{{{R}}_{{r}}}^{{2}}{}{{R}}^{{4}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{{1}{+}{2}{}{E}}$ (18)

The nonzero components

 > $\mathrm{gamma_}\left[\mathrm{nonzero}\right]$
 ${{\mathrm{\gamma }}}_{{i}{,}{j}}{=}\left\{\left({1}{,}{1}\right){=}\frac{{{{R}}_{{r}}}^{{2}}}{{1}{+}{2}{}{E}}{,}\left({2}{,}{2}\right){=}{{R}}^{{2}}{,}\left({3}{,}{3}\right){=}{{R}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}\right\}$ (19)

The nonzero components of the inverse of the metric, that is, the all-contravariant ${g}_{}^{\mathrm{&i;},\mathrm{&j;}}$

 > $\mathrm{gamma_}\left[\mathrm{~},\mathrm{nonzero}\right]$
 ${{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{i}{,}{j}}}^{\phantom{{}}{i}{,}{j}}{=}\left\{\left({1}{,}{1}\right){=}\frac{{1}{+}{2}{}{E}}{{{{R}}_{{r}}}^{{2}}}{,}\left({2}{,}{2}\right){=}\frac{{1}}{{{R}}^{{2}}}{,}\left({3}{,}{3}\right){=}\frac{{{\mathrm{csc}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{{{R}}^{{2}}}\right\}$ (20)

By replacing g_ by gamma_ and all spacetime indices by space indices in the definitions of Christoffel you obtain the corresponding 3D tensor, i.e., the 3D Christoffel symbols, that allow you to express covariant derivatives in a 3D curved space (not spacetime)

 > $\mathrm{Christoffel}\left[\mathrm{definition}\right]$
 ${{\mathrm{\Gamma }}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\frac{{{\partial }}_{{\mathrm{\nu }}}{}\left({{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}}\right)}{{2}}{+}\frac{{{\partial }}_{{\mathrm{\mu }}}{}\left({{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}\right)}{{2}}{-}\frac{{{\partial }}_{{\mathrm{\alpha }}}{}\left({{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}\right)}{{2}}$ (21)
 > $\mathrm{subs}\left(\mathrm{\mu }=i,\mathrm{\nu }=j,\mathrm{\alpha }=k,\mathrm{g_}=\mathrm{gamma_},\right)$
 ${{\mathrm{\Gamma }}}_{{k}{,}{i}{,}{j}}{=}\frac{{{\partial }}_{{j}}{}\left({{\mathrm{\gamma }}}_{{k}{,}{i}}\right)}{{2}}{+}\frac{{{\partial }}_{{i}}{}\left({{\mathrm{\gamma }}}_{{k}{,}{j}}\right)}{{2}}{-}\frac{{{\partial }}_{{k}}{}\left({{\mathrm{\gamma }}}_{{i}{,}{j}}\right)}{{2}}$ (22)

You can use this to construct 3D versions, for instance for Christoffel and the Ricci and Riemann tensors, next define them using Define, and operate with them in 3D as with any other tensor of the Physics package, with the geometrical meaning of the Ricci and Riemann 3D tensors in curved space (see problems after sections 88, 89, 90, 92 and 95 of [1])

 > $\mathrm{C3}\left[i,j,k\right]=\mathrm{rhs}\left(\right)$
 ${{\mathrm{C3}}}_{{i}{,}{j}{,}{k}}{=}\frac{{{\partial }}_{{j}}{}\left({{\mathrm{\gamma }}}_{{i}{,}{k}}\right)}{{2}}{+}\frac{{{\partial }}_{{i}}{}\left({{\mathrm{\gamma }}}_{{j}{,}{k}}\right)}{{2}}{-}\frac{{{\partial }}_{{k}}{}\left({{\mathrm{\gamma }}}_{{i}{,}{j}}\right)}{{2}}$ (23)
 > $\mathrm{Define}\left(\right)$
 $\mathrm{Defined objects with tensor properties}$
 $\left\{{{\mathrm{C3}}}_{{i}{,}{j}{,}{k}}{,}{{▿}}_{{\mathrm{\mu }}}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{,}{{\mathrm{\sigma }}}_{{\mathrm{\mu }}}{,}{{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{{C}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{{\partial }}_{{\mathrm{\mu }}}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\gamma }}}_{{i}{,}{j}}{,}{{\mathrm{\Gamma }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{{G}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\epsilon }}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{X}}_{{\mathrm{\mu }}}\right\}$ (24)
 > $\mathrm{C3}\left[\mathrm{definition}\right]$
 ${{\mathrm{C3}}}_{{i}{,}{j}{,}{k}}{=}\frac{{{\partial }}_{{j}}{}\left({{\mathrm{\gamma }}}_{{i}{,}{k}}\right)}{{2}}{+}\frac{{{\partial }}_{{i}}{}\left({{\mathrm{\gamma }}}_{{j}{,}{k}}\right)}{{2}}{-}\frac{{{\partial }}_{{k}}{}\left({{\mathrm{\gamma }}}_{{i}{,}{j}}\right)}{{2}}$ (25)
 > $\mathrm{C3}\left[\mathrm{nonzero}\right]$
 ${{\mathrm{C3}}}_{{i}{,}{j}{,}{k}}{=}\left\{\left({1}{,}{1}{,}{1}\right){=}\frac{{{R}}_{{r}}{}\left({1}{+}{2}{}{E}\right){}{{R}}_{{r}{,}{r}}{-}{{{R}}_{{r}}}^{{2}}{}{{E}}_{{r}}}{{\left({1}{+}{2}{}{E}\right)}^{{2}}}{,}\left({1}{,}{2}{,}{2}\right){=}{R}{}{{R}}_{{r}}{,}\left({1}{,}{3}{,}{3}\right){=}{R}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{R}}_{{r}}{,}\left({2}{,}{1}{,}{2}\right){=}{R}{}{{R}}_{{r}}{,}\left({2}{,}{2}{,}{1}\right){=}{-}{R}{}{{R}}_{{r}}{,}\left({2}{,}{3}{,}{3}\right){=}{{R}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){,}\left({3}{,}{1}{,}{3}\right){=}{R}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{R}}_{{r}}{,}\left({3}{,}{2}{,}{3}\right){=}{{R}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){,}\left({3}{,}{3}{,}{1}\right){=}{-}{R}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{{R}}_{{r}}{,}\left({3}{,}{3}{,}{2}\right){=}{-}{{R}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\right\}$ (26)
 > $\mathrm{C3}\left[\mathrm{~1},j,k,\mathrm{matrix}\right]$
 ${{\mathrm{C3}}}_{\phantom{{}}\phantom{{1}}{j}{,}{k}}^{\phantom{{}}{1}\phantom{{j}{,}{k}}}{=}\left[\begin{array}{ccc}\frac{\left({1}{+}{2}{}{E}\right){}{{R}}_{{r}{,}{r}}{-}{{R}}_{{r}}{}{{E}}_{{r}}}{{{R}}_{{r}}{}\left({1}{+}{2}{}{E}\right)}& {0}& {0}\\ {0}& \frac{\left({1}{+}{2}{}{E}\right){}{R}}{{{R}}_{{r}}}& {0}\\ {0}& {0}& \frac{\left({1}{+}{2}{}{E}\right){}{R}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}{{{R}}_{{r}}}\end{array}\right]$ (27)

References

 [1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.