The g_[mu, nu], displayed as $g_{\mathrm{\mu },\mathrm{\nu }}$ (without _ in between $g$ and its indices), is a computational representation for the spacetime metric tensor. When Physics is loaded, the dimension of spacetime is set to 4 and the metric is automatically set to be galilean, representing a Minkowski spacetime with signature (-, -, -, +), so time in the fourth place.

When working in curvilinear coordinates so that the components of the Christoffel symbols are not zero it is sometimes convenient to rewrite tensorial expressions in terms of the metric g_ and its derivatives. For example, this is the case of the Christoffel symbols and any of the general relativity tensors that are defined in terms of them. For this purpose you can use convert(expression, g_) - see the Examples section.

When the indices of g_ assume integer values, it is expected they are between 0 and the spacetime dimension and the corresponding value of g_ is returned. The values 0 and 4, or for the case any dimension set for the spacetime, represent the same object. When the indices have symbolic values, say as in $\mathrm{\mu },\mathrm{\nu }$, if $\mathrm{\mu }=\mathrm{\nu }$ g_ returns the dimension of spacetime when $\mathrm{\mu }$ is a spacetime index, or the dimension of space when $\mathrm{\mu }$ is a space index. If the system can prove that $\mathrm{\mu }\ne \mathrm{\nu }$ (e.g. via assumptions), g_ returns zero whenever the metric is diagonal. Otherwise g_ returns unevaluated, after normalizing its indices taking into account that the spacetime metric is symmetric.

You can change the value of the metric g_ in different ways, using Setup or, when choosing a predefined set of values, you can also change the metric using g_ itself, indexing it with the related metric name; for example as in g_[Minkowski]. The predefined sets are: arbitrary, Euclidean, Minkowski, Schwarzschild, Tolman or related to any of the metrics of the book "Exact Solutions of Einstein's Field Equations" - see references at the end. Note that you can pass the keyword incomplete or misspelled, in which case a searchtext is performed among the keywords understood by g_ and if a match is found the metric is set accordingly, or if many matches are found then corresponding information is displayed on the screen. See the Examples section. Using predefined sets, either through Setup or g_, automatically sets the necessary systems of coordinates and differentiation variables for the spacetime differentiation operator d_ accordingly.

Any of the 991 metrics of the book "Exact Solutions of Einstein's Field Equations" can also be load by indexing g_ with a list of three numbers, for example as in g_[[12, 16, 1]], respectively representing the book's chapter where the metric is found, the equation number and the case.

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 ~mu; 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.

Note: since Maple 2019, g_[mu, ~nu], where one of its indices is contravariant as in ~nu while the other remains covariant, is displayed using the $\mathrm{\delta }$ greek letter, as ${\mathrm{\delta }}_{\mathrm{\mu }}^{\mathrm{\nu }}$, as is standard in physics textbooks. Also, KroneckerDelta[mu, nu] is not considered a tensor (of type Physics:-Library:-PhysicsType:-Tensor), even if $\mathrm{\mu }$ and $\mathrm{\nu }$ are letters representing tensor indices, unless one of such indices is contravariant as in ~nu, while the other remains covariant, in which case KroneckerDelta[mu, ~nu] is automatically transformed into the spacetime metric g_[mu, nu], which is a tensor.

During a Maple session, the value of any component of the general relativity tensors of Physics, that is: Christoffel, the covariant derivative D_, Einstein, Ricci, Riemann and Weyl, automatically follow the value or any changes introduced in the components of g_, the spacetime metric, provided these changes are made using Setup or g_ itself.

Besides being indexed with two indices, g_ accepts the following keywords as an index:

definition: when passed alone, g_ 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 (g_[]).

determinant: when passed alone, g_ returns the determinant of the all-covariant metric g[mu, nu] . 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, g_ returns the line element for the current metric expressing the differentials of the coordinates using d_.

matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in g_[]) returns a Matrix that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of g_. 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 $\mathrm{g\_\_\μ,\ν}$ 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.

<metric name>: some predefined sets of values for the spacetime metric can be used by giving the metric name or a portion of it; currently these are Euclidean, Minkowski, Tolman, Schwarzschild and the metrics of Chapter 12 of "Exact Solutions of Einstein's Field Equations" (second edition).

Some automatic checking and normalization are carried out each time you enter g_[...]. The checking is concerned with possible unexpected values of the indices. The automatic normalization takes into account the symmetry of g_[mu,nu] with respect to interchanging the positions of the indices mu and nu.

The %g_ command is the inert form of g_, so it represents the same tensor but entering it does not result in performing any computation. To perform the related computations as if %g_ were g_, use value.

$\mathrm{with}\left(\mathrm{Physics}\right)\&colon;$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{g\_}\left[\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{g\_}\left[\mathrm{line\_element}\right]$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\&comma;\mathrm{x4}\right)\right\}$

$-{\mathbf{\&DifferentialD;}\left(\mathrm{x1}\right)}^2-{\mathbf{\&DifferentialD;}\left(\mathrm{x2}\right)}^2-{\mathbf{\&DifferentialD;}\left(\mathrm{x3}\right)}^2+{\mathbf{\&DifferentialD;}\left(\mathrm{x4}\right)}^2$

$\mathrm{g\_}\left[0,0\right]=\mathrm{g\_}\left[4,4\right]$

$1=1$

$\mathrm{g\_}\left[0,3\right]$

$0$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]-\mathrm{g\_}\left[\mathrm{\nu },\mathrm{\mu }\right]$

$0$

$\mathrm{Setup}\left(\mathrm{spacetimeindices}\&comma;\mathrm{dimension}\right)$

$\left[\mathrm{dimension}=4\&comma;\mathrm{spacetimeindices}=\mathrm{greek}\right]$

$\mathrm{Setup}\left(\mathrm{spaceindices}=\mathrm{lowercaselatin\_ah}\right)$

$\left[\mathrm{spaceindices}=\mathrm{lowercaselatin\_ah}\right]$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{eval}\left(\&comma;\mathrm{\mu }=\mathrm{\nu }\right)$

$4$

$\mathrm{g\_}\left[a,b\right]$

$g_{a,b}$

$\mathrm{eval}\left(\&comma;a=b\right)$

$3$

$\mathrm{Christoffel}\left[\mathrm{nonzero}\right]$

${\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\varnothing$

$\mathrm{Christoffel}\left[\mathrm{\mu },\mathrm{\alpha },\mathrm{\beta }\right]$

$0$

$\mathrm{g\_}\left[\mathrm{Tolman}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\mathrm{The\; Tolman\; metric\; in\; coordinates}\left[r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right]$

$\mathrm{Parameters:}\left[R\left(t\&comma;r\right)\&comma;E\left(r\right)\right]$

$\mathrm{Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{g\_}\left[\mathrm{line\_element}\right]$

$\frac{-{\left(\frac{\partial }{\partial r}R\left(t\&comma;r\right)\right)}^2{\mathbf{\&DifferentialD;}\left(r\right)}^2-\left(1+2E\left(r\right)\right){R\left(t\&comma;r\right)}^2\left({\sin \left(\mathrm{\theta }\right)}^2{\mathbf{\&DifferentialD;}\left(\mathrm{\phi }\right)}^2+{\mathbf{\&DifferentialD;}\left(\mathrm{\theta }\right)}^2\right)}{1+2E\left(r\right)}+{\mathbf{\&DifferentialD;}\left(t\right)}^2$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{g\_}\left[1,1\right]$

$-\frac{{\left(\frac{\partial }{\partial r}R\left(t\&comma;r\right)\right)}^2}{1+2E\left(r\right)}$

$\mathrm{g\_}\left[\mathrm{~1},\mathrm{~1}\right]$

$-\frac{1+2E\left(r\right)}{{\left(\frac{\partial }{\partial r}R\left(t\&comma;r\right)\right)}^2}$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu },\mathrm{matrix}\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{g\_}\left[\mathrm{~mu},\mathrm{~nu},\mathrm{matrix}\right]$

$g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Setup}\left(\mathrm{coordinates}\&comma;\mathrm{differentiation}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{differentiation}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{differentiationvariables}\text{'}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\&comma;\mathrm{differentiationvariables}=\left[X\right]\right]$

$\mathrm{Coordinates}\left(\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\left\{X\right\}$

$\mathrm{ds2}≔r^2{\mathrm{dtheta}}^2+r^2{\sin \left(\mathrm{\theta }\right)}^2{\mathrm{dphi}}^2-2\mathrm{dt}\mathrm{dr}-2{k\left(r\&comma;t\right)}^2{\mathrm{dt}}^2$

$\mathrm{ds2}≔r^2{\mathrm{d\&theta;}}^2+r^2{\sin \left(\mathrm{\theta }\right)}^2{\mathrm{d\&phi;}}^2-2\mathrm{dt}\mathrm{dr}-2{k\left(r\&comma;t\right)}^2{\mathrm{dt}}^2$

$\mathrm{Setup}\left(\mathrm{metric}=\mathrm{ds2}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Coordinates:}\left[r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right]\mathrm{.\; Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{metric}=\left\{\left(1\&comma;4\right)=\mathrm{-1}\&comma;\left(2\&comma;2\right)=r^2\&comma;\left(3\&comma;3\right)=r^2{\sin \left(\mathrm{\theta }\right)}^2\&comma;\left(4\&comma;4\right)=-2{k\left(r\&comma;t\right)}^2\right\}\right]$

$\mathrm{g\_}\left[1,1\right]$

$0$

$\mathrm{g\_}\left[\mathrm{~1},\mathrm{~1}\right]$

$2{k\left(r\&comma;t\right)}^2$

$\mathrm{g\_}\left[\mathrm{determinant}\right]$

$-r^4{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{g\_}\left[\mathrm{nonzero}\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\&comma;4\right)=\mathrm{-1}\&comma;\left(2\&comma;2\right)=r^2\&comma;\left(3\&comma;3\right)=r^2{\sin \left(\mathrm{\theta }\right)}^2\&comma;\left(4\&comma;4\right)=-2{k\left(r\&comma;t\right)}^2\right\}$

$\mathrm{g\_}\left[\mathrm{~mu},\mathrm{~nu},\mathrm{nonzero}\right]$

$g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\&comma;1\right)=2{k\left(r\&comma;t\right)}^2\&comma;\left(1\&comma;4\right)=\mathrm{-1}\&comma;\left(2\&comma;2\right)=\frac{1}{r^2}\&comma;\left(3\&comma;3\right)=\frac{{\csc \left(\mathrm{\theta }\right)}^2}{r^2}\right\}$

$\mathrm{Christoffel}\left[\mathrm{\mu },\mathrm{\alpha },\mathrm{\beta }\right]$

${\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\alpha },\mathrm{\beta }}$

$\mathrm{convert}\left(\&comma;\mathrm{g\_}\right)$

$\frac{{\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\alpha },\mathrm{\mu }}\right)}{2}+\frac{{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\beta },\mathrm{\mu }}\right)}{2}-\frac{{\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)}{2}$

$\mathrm{Define}\left(A\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A\&comma;{▿}_{\mathrm{\mu }}\&comma;{\mathrm{\gamma }}_{\mathrm{\mu }}\&comma;{\mathrm{\sigma }}_{\mathrm{\mu }}\&comma;R_{\mathrm{\mu },\mathrm{\nu }}\&comma;R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\&comma;C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\&comma;{\partial }_{\mathrm{\mu }}\&comma;g_{\mathrm{\mu },\mathrm{\nu }}\&comma;{\mathrm{\gamma }}_{a,b}\&comma;{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\&comma;G_{\mathrm{\mu },\mathrm{\nu }}\&comma;{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\&comma;X_{\mathrm{\mu }}\right\}$

$\mathrm{D\_}\left[\mathrm{\mu }\right]\left(A\left[\mathrm{~nu}\right]\left(X\right)\right)$

${▿}_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)$

$\mathrm{expand}\left(\right)$

${\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)$

$\mathrm{convert}\left(\&comma;\mathrm{d\_}\right)$

${\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)$

$\mathrm{convert}\left(\&comma;\mathrm{g\_}\right)$

${\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)+\frac{g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}\left({\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)+{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\beta },\mathrm{\mu }}\right)-{\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\alpha },\mathrm{\mu }}\right)\right)A_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)}{2}$

$\mathrm{convert}\left(\&comma;\mathrm{Christoffel}\right)$

${\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)+g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}{\mathrm{\Gamma }}_{\mathrm{\beta },\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)$

$\mathrm{Simplify}\left(\right)$

${\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\mu }\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\mu }}\mathrm{\alpha }}{A}_{\mathrm{\alpha }}\left(X\right)+{\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)$

$\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\beta }\right]\right)$

${\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)$

$\mathrm{convert}\left(\&comma;\mathrm{Christoffel}\right)$

${\mathrm{\Gamma }}_{\mathrm{\beta },\mathrm{\alpha },\mathrm{\mu }}+{\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu }}$

$\mathrm{Ricci}\left[\mathrm{\alpha },\mathrm{\beta }\right]$

$R_{\mathrm{\alpha },\mathrm{\beta }}$

$\mathrm{convert}\left(\&comma;\mathrm{Christoffel}\right)$

${\partial }_{\mathrm{\mu }}\left({\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\alpha },\mathrm{\beta }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}\right)-{\partial }_{\mathrm{\beta }}\left({\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\mu },\mathrm{\nu }}}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\beta },\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\beta },\mathrm{\nu }}}$

$\mathrm{convert}\left(\&comma;\mathrm{g\_}\right)$

$\frac{{\partial }_{\mathrm{\mu }}\left(g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\right)\left({\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)+{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\beta },\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)\right)}{2}+\frac{g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\left({\partial }_{\mathrm{\beta }}\left({\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)\right)+{\partial }_{\mathrm{\alpha }}\left({\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\beta },\mathrm{\nu }}\right)\right)-{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)\right)\right)}{2}-\frac{{\partial }_{\mathrm{\beta }}\left(g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\right){\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\mu },\mathrm{\nu }}\right)}{2}-\frac{g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\partial }_{\mathrm{\alpha }}\left({\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\mu },\mathrm{\nu }}\right)\right)}{2}+\frac{g_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\nu }}}^{\phantom{}\mathrm{\lambda },\mathrm{\nu }}\left({\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\alpha },\mathrm{\lambda }}\right)+{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\beta },\mathrm{\lambda }}\right)-{\partial }_{\mathrm{\lambda }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)\right)g_{\phantom{}\phantom{\mathrm{\kappa },\mathrm{\mu }}}^{\phantom{}\mathrm{\kappa },\mathrm{\mu }}{\partial }_{\mathrm{\nu }}\left(g_{\mathrm{\kappa },\mathrm{\mu }}\right)}{4}-\frac{g_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\nu }}}^{\phantom{}\mathrm{\lambda },\mathrm{\nu }}\left({\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\lambda }}\right)+{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\lambda },\mathrm{\mu }}\right)-{\partial }_{\mathrm{\lambda }}\left(g_{\mathrm{\alpha },\mathrm{\mu }}\right)\right)g_{\phantom{}\phantom{\mathrm{\kappa },\mathrm{\mu }}}^{\phantom{}\mathrm{\kappa },\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(g_{\mathrm{\beta },\mathrm{\kappa }}\right)+{\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\kappa },\mathrm{\nu }}\right)-{\partial }_{\mathrm{\kappa }}\left(g_{\mathrm{\beta },\mathrm{\nu }}\right)\right)}{4}$

$\mathrm{\%g\_}\left[\mathrm{~1},1\right]$

${\mathrm{\delta }}_{\phantom{}\phantom{1}1}^{\phantom{}1\phantom{1}}$

$\mathrm{value}\left(\right)$

$1$

$\mathrm{g\_}\left[\mathrm{sik}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 34,\; 1]\; =\; [Authors\; =\; [Kaigorodov\; (1962),\; Cahen\; (1964),\; Siklos\; (1981),\; Ozsvath\; (1987)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [Homogeneous],\; Comments\; =\; [All\; metrics\; with\; \_epsilon\; <>\; 0\; are\; equivalent\; to\; the\; cases\; \_epsilon\; =\; +1,\; -1,\; \_epsilon\; =\; 0\; is\; anti-deSitter\; space]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 35,\; 1]\; =\; [Authors\; =\; [Kaigorodov\; (1962),\; Cahen\; (1964),\; Siklos\; (1981),\; Ozsvath\; (1987)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [Homogeneous,\; SimpleTransitive]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 38,\; 1]\; =\; [Authors\; =\; [Siklos\; (1985)],\; PrimaryDescription\; =\; PureRadiation,\; SecondaryDescription\; =\; [Homogeneous,\; PureRadiation],\; Comments\; =\; [Generic\; k\; value,\; k\; <>\; -1\; (see\; [12,\; 6,\; 1]),\; 1/2\; (see\; [12,38,2]),\; 3/2\; (see\; [12,38,\; 5]),\; 2\; (see\; [12,38,\; 3]\; and\; [12,\; 38,\; 4]).\; Note\; the\; k\; =\; 1/2\; is\; Petrov\; type\; "O"\; .]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 38,\; 2]\; =\; [Authors\; =\; [Siklos\; (1985)],\; PrimaryDescription\; =\; PureRadiation,\; SecondaryDescription\; =\; [PureRadiation,\; Homogeneous],\; Comments\; =\; [This\; is\; k\; =\; 1/2\; in\; [12,\; 38,\; 1].\; This\; is\; of\; Petrov\; type\; "O"]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 38,\; 3]\; =\; [Authors\; =\; [Siklos\; (1985)],\; PrimaryDescription\; =\; EinsteinMaxwell,\; SecondaryDescription\; =\; [PureRadiation,\; Homogeneous],\; Comments\; =\; [This\; is\; \_k\; =\; 2\; and\; \_epsilon\; =\; 1\; in\; [12,\; 38,\; 1].\; Only\; \_epsilon\; =\; 1\; gives\; a\; Maxwell\; field]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 38,\; 4]\; =\; [Authors\; =\; [Siklos\; (1985)],\; PrimaryDescription\; =\; PureRadiation,\; SecondaryDescription\; =\; [PureRadiation,\; Homogeneous],\; Comments\; =\; [This\; is\; \_k\; =\; 2\; and\; \_epsilon\; =\; -1\; in\; [12,\; 38,\; 1].\; Only\; \_epsilon\; =\; 1\; gives\; a\; Maxwell\; field]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[12,\; 38,\; 5]\; =\; [Authors\; =\; [Siklos\; (1985),\; Nariai(1950)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [Homogeneous],\; Comments\; =\; [This\; is\; k\; =\; 3/2\; in\; [12,\; 38,\; 1].\; It\; is\; equivalent\; to\; [12,\; 34,\; 1]]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[13,\; 64,\; 1]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; []]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[13,\; 64,\; 3]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [],\; Comments\; =\; [See\; also\; (13,\; 64,\; 4)\; for\; Riemannian\; orbits]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[13,\; 64,\; 4]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [],\; Comments\; =\; [See\; also\; (13,\; 64,\; 3)\; for\; PseudoRiemannian\; orbits]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[13,\; 65,\; 1]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [],\; Comments\; =\; [We\; can\; use\; the\; transformation\; x\; ->\; -\; x\; to\; assume\; the\; parameter\; \_b\; >\; 0,\; A\; boost\; and\; a\; spatial\; rotation\; are\; required\; to\; take\; the\; null\; tetrad\; to\; an\; adapted\; null\; tetrad]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[13,\; 67,\; 1]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [TwistingSolution]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[13,\; 67,\; 2]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Einstein,\; SecondaryDescription\; =\; [TwistingSolution]]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{[38,\; 1,\; 1]\; =\; [Authors\; =\; [Siklos\; (1981)],\; PrimaryDescription\; =\; Hypersurface-Homogeneous,\; SecondaryDescription\; =\; [AlgebraicallySpecial]]}$

$\mathit{Found\; more\; than\; one\; match\; for\; the\; keyword\; \text{'}}\mathit{sik}\mathit{\text{'},\; as\; seen\; above.\; Please\; refine\; your\; \text{'}keyword\text{'}\; or\; re-enter\; the\; metric\; g\_}\left[\mathit{...}\right]\mathit{with\; the\; list\; of\; three\; numbers\; identifying\; the\; metric,\; for\; example\; as\; in\; g\_}\left[\left[\mathbf{12}\&comma;\mathbf{34}\&comma;\mathbf{1}\right]\right]\mathit{or}\mathit{Setup}\left(\mathit{metric}\mathbf{=}\left[\mathbf{12}\&comma;\mathbf{34}\&comma;\mathbf{1}\right]\right)$

$\mathrm{g\_}\left[\left[12\&comma;38\&comma;1\right]\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(v\&comma;y\&comma;z\&comma;u\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(v\&comma;y\&comma;z\&comma;u\right)\right\}$

$\mathrm{The}\mathrm{Siklos\; (1985)}\mathrm{metric\; in\; coordinates}\left[v\&comma;y\&comma;z\&comma;u\right]$

$\mathrm{Parameters:}\left[k\&comma;\mathrm{\epsilon }\&comma;\mathrm{\Lambda }\&comma;\mathrm{\&kappa;0}\right]$

$\mathrm{Comments:}\mathrm{G\&ExponentialE;n\&ExponentialE;ric\; k\; valu\&ExponentialE;,\; k\; <>\; -1\; (s\&ExponentialE;\&ExponentialE;\; [12,\; 6,\; 1]),\; 1/2\; (s\&ExponentialE;\&ExponentialE;\; [12,38,2]),\; 3/2\; (s\&ExponentialE;\&ExponentialE;\; [12,38,\; 5]),\; 2\; (s\&ExponentialE;\&ExponentialE;\; [12,38,\; 3]\; an\&DifferentialD;\; [12,\; 38,\; 4]).\; Not\&ExponentialE;\; th\&ExponentialE;\; k\; =\; 1/2\; is\; P\&ExponentialE;trov\; typ\&ExponentialE;\; "O"\; .}$

$\mathrm{Domain:}\left[0<y\right]$

$\mathrm{Assumptions:}\left[\mathrm{\epsilon }=\mathrm{-1}\&comma;\mathrm{\epsilon }=1\&comma;0<\mathrm{\&kappa;0}\&comma;\mathrm{\Lambda }<0\right]$

$\mathrm{Resetting\; the\; signature\; of\; spacetime\; from}\left(\mathrm{-\; -\; -\; +}\right)\mathrm{to}\left(\mathrm{+\; +\; +\; -}\right)\mathrm{in\; order\; to\; match\; the\; signature\; in\; the\; database\; of\; metrics}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Define}\left(A\&comma;B\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{B\&comma;A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\&comma;{▿}_{\mathrm{\mu }}\&comma;{\mathrm{\gamma }}_{\mathrm{\mu }}\&comma;{\mathrm{\sigma }}_{\mathrm{\mu }}\&comma;R_{\mathrm{\mu },\mathrm{\nu }}\&comma;R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\&comma;C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\&comma;{\partial }_{\mathrm{\mu }}\&comma;g_{\mathrm{\mu },\mathrm{\nu }}\&comma;{\mathrm{\gamma }}_{a,b}\&comma;{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\&comma;G_{\mathrm{\mu },\mathrm{\nu }}\&comma;{\mathrm{{\rm T}}}_{\mathrm{\mu },\mathrm{\nu }}\&comma;{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\&comma;X_{\mathrm{\mu }}\right\}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]$

$A_{\mathrm{\mu }}{\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\alpha }}\mathrm{\mu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]·A\left[\mathrm{\mu }\right]$

$A_{\mathrm{\alpha }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]B\left[\mathrm{\nu }\right]$

$A_{\mathrm{\mu }}{B}_{\mathrm{\nu }}{\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\alpha }}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}$

$\mathrm{Simplify}\left(\&comma;\mathrm{indices}\right)$

$A_{\mathrm{\nu }}{B}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]·A\left[\mathrm{\mu }\right]\right)\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]B\left[\mathrm{\nu }\right]$

$A_{\mathrm{\alpha }}{B}_{\mathrm{\nu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]·B\left[\mathrm{\nu }\right]\right)$

$B_{\mathrm{\alpha }}{A}_{\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\mu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\mu }}$

$\left(\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]\right)·\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]\right)B\left[\mathrm{\nu }\right]$

$A_{\mathrm{\nu }}{B}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]\left(A\left[\mathrm{\mu }\right]+B\left[\mathrm{\mu }\right]\right)$

$\left(A_{\mathrm{\mu }}+B_{\mathrm{\mu }}\right){\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\alpha }}\mathrm{\mu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]·\left(A\left[\mathrm{\mu }\right]+B\left[\mathrm{\mu }\right]\right)$

$A_{\mathrm{\alpha }}+B_{\mathrm{\alpha }}$

$\mathrm{g\_}\left[\mathrm{sc}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right)\right\}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\&comma;t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{+\; +\; +\; -}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]$

$R_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}{R}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]·\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]$

$\frac{48m^2}{r^6}$

$\mathrm{KroneckerDelta}\left[\mathrm{\mu },\mathrm{\mu }\right]$

$1$

$\mathrm{\%KroneckerDelta}\left[\mathrm{\mu },\mathrm{\mu }\right]$

${\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\mu }}$

$\mathrm{value}\left(\right)$

$1$

$\mathrm{KroneckerDelta}\left[\mathrm{\mu },\mathrm{\nu }\right]$

${\mathrm{\delta }}_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{type}\left(\&comma;\mathrm{Physics}:-\mathrm{Library}:-\mathrm{PhysicsType}:-\mathrm{Tensor}\right)$

$\mathrm{false}$

$\mathrm{KroneckerDelta}\left[\mathrm{\mu },\mathrm{~nu}\right]=\mathrm{g\_}\left[\mathrm{\mu },\mathrm{~nu}\right]$

${\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}={\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

$\mathrm{lprint}\left(\right)$

Physics:-g_[mu,~nu] = Physics:-g_[mu,~nu]

$\mathrm{eval}\left(\&comma;\mathrm{\mu }=\mathrm{\nu }\right)$

$4=4$

$\mathbf{assuming}\mathrm{\mu }=\mathrm{\nu }$

$4=4$

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

Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C. Herlt, E.  Exact Solutions of Einstein's Field Equations, Cambridge Monographs on Mathematical Physics, second edition. Cambridge University Press, 2003.

The Physics[g_] command was updated in Maple 2019.