The TransformCoordinates, performs a transformation of coordinates in an algebraic expression that is also tensorial, i.e. one that has free and/or repeated spacetime indices that are either covariant or contravariant (see Check to determine the indices).

The transformation performed with TransformCoordinates is the standard one for tensors. Consider for instance a transformation from one coordinate system, $x_{}^0,x_{}^1,x_{}^2,x_{}^3$ to another $y_{}^0,y_{}^1,y_{}^2,y_{}^3$

where the $f_{}^{\mathrm{\mu }}$ are certain functions. When you transform the coordinates, every contravariant tensor of one index, say $A_{}^{\mathrm{\mu }}\left(x\right)$, a vector, transforms into a contravariant $A_{}^{\mathrm{\mu }}\left(y\right)$ in the new system of coordinates according to

and every covariant vector transforms according to

and in the case of more than one index, for each contravariant or covariant index there is a factor as in the corresponding right-hand sides of these formulas.

CAVEAT: in curvilinear coordinates, none of the Christoffel symbols, the noncovariant d_ operator and the SpaceTimeVector $X^{\mathrm{\mu }}$ are tensors, even when the algebraic manipulation of their indices is done as if they were. This version of TransformCoordinates does not transform coordinates on these objects.

When the given tr is from the old variables - say $x_{}^{\mathrm{\mu }}$ - on the left-hand sides of the transformation equations to the new variables - say $y_{}^{\mathrm{\mu }}$ - on the right-hand sides, as in the previous item, the output is entirely expressed in terms of the new coordinates $y_{}^{\mathrm{\mu }}$. This is accomplished in two steps:

In the given expression, change variables from the old ones $x$ to the new ones $y$ to get everything expressed in terms of $y$. (If options known = ... and unknown = ... are received by TransformCoordinates, they are forwarded to dchange.  See dchange for info on the use of those options.)

Transform the resulting tensorial expression expressed in terms of $y$ regarding each of its free indices (i.e.: multiply by the appropriate factors mentioned, $\frac{\partial y_{}^{\mathrm{\mu }}}{\partial x_{}^{\mathrm{\nu }}}$ or $\frac{\partial x_{}^{\mathrm{\nu }}}{\partial y_{}^{\mathrm{\mu }}}$ and sum over the corresponding repeated indices).

The result represents the tensorial expression in the new coordinate system, expressed in terms of the new coordinates.

It frequently happens, however, that the transformation can only be expressed in simple form if it is from the new variables on the left-hand sides to the old variables on the right-hand sides. You can pass the transformation tr in this way too, in which case you must also pass the two lists X_new and X_old respectively as third and fourth arguments. The result will then be the tensor in the new coordinate system, entirely expressed in terms of the coordinates $x_{}^{\mathrm{\mu }}$ of the old system, .

The output of TransformCoordinates is an Array with dimension equal to the number of free indices of expression. In the special cases where the number of indices is 2 or 1, instead of an Array, the result is respectively expressed as a Matrix or a column Vector. In all cases, giving values to these indices you retrieve each of the components of the tensorial expression in the new coordinate system, entirely expressed in terms of the new coordinates X_new whenever the given transformation is from the old to the new variables (see the two previous paragraphs).

The transformation can be given as an equation or a set or list of them, and does not need to include equations for all the coordinates - for instance: some may not be changing. Also, because the transformation of coordinates is equivalent to a change of the system of reference, it is possible to continue using the same geometrical coordinates (same names representing the coordinates in both systems) so that the right-hand sides of the transformation equations can involve the list of X_new variables, even if they are the same as the old ones, for instance as in transforming [t = -t], performed by TransformCoordinates in two steps: transform $t=-\mathrm{\tau }$ then evaluate at $\mathrm{\tau }=t$.

You can retrieve all the nonzero components of the transformed expression at once by passing the output of TransformCoordinates to the ArrayElems command.

When expression is the spacetime metric g_[mu,nu], you can optionally request the output to be the corresponding line element by passing output = line_element instead of a Matrix.

By default, when changing variables in the metric (i.e. expression is the metric itself), the transformed metric is not automatically set as the new metric. To change this behavior, pass the optional argument setmetric, in which case a call to Setup with metric = ... transformed metric ..., coordinates = ... new coordinates ... will be automatically performed before returning the result.

By default, the new variables are not set as a coordinate system (when they are not already) nor they are set as the differentiation variables. To change this behavior, pass the optional argument setcoordinates. If not given, the default value of this option is false, unless you pass the option setmetric in which case the new variables are set as a coordinate system and as new differentiation variables together with the metric g_. When the new variables are not already set as a coordinate system, the label used for the new variables will always be the label originally used for the old variables.

By default, in the returned result, summation is explicitly performed over all the repeated indices found in expression, taking into account the covariant/contravariant character of each index. To avoid performing this summation and keep repeated indices not summed pass the optional argument performsumoverrepeatedindices = false.

By default, all the components of the returned result are simplified using simplify. This frequently results in a desired form. You can change the simplifier to be any other one by indicating it on the right-hand side of the optional argument simplifier = ....

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

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

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

$\mathrm{Coordinates}\left(X\,Y\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

$\left\{X\,Y\right\}$

$\mathrm{tr}≔\left[\mathrm{x1}=\mathrm{X1}\left(Y\right)\,\mathrm{x2}=\mathrm{X2}\left(Y\right)\,\mathrm{x3}=\mathrm{X3}\left(Y\right)\,\mathrm{x4}=\mathrm{X4}\left(Y\right)\right]$

$\mathrm{tr}≔\left[\mathrm{x1}=\mathrm{X1}\left(Y\right)\,\mathrm{x2}=\mathrm{X2}\left(Y\right)\,\mathrm{x3}=\mathrm{X3}\left(Y\right)\,\mathrm{x4}=\mathrm{X4}\left(Y\right)\right]$

$\mathrm{itr}≔\left[\mathrm{y1}=\mathrm{Y1}\left(X\right)\,\mathrm{y2}=\mathrm{Y2}\left(X\right)\,\mathrm{y3}=\mathrm{Y3}\left(X\right)\,\mathrm{y4}=\mathrm{Y4}\left(X\right)\right]$

$\mathrm{itr}≔\left[\mathrm{y1}=\mathrm{Y1}\left(X\right)\,\mathrm{y2}=\mathrm{Y2}\left(X\right)\,\mathrm{y3}=\mathrm{Y3}\left(X\right)\,\mathrm{y4}=\mathrm{Y4}\left(X\right)\right]$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\mathrm{map}\left(\mathrm{rhs}\,\mathrm{tr}\right)\right)$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\mathrm{map}\left(\mathrm{rhs}\,\mathrm{itr}\right)\right)$

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

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

$\left\{A\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\,Y_{\mathrm{\mu }}\right\}$

$A\left[\mathrm{\mu }\right]$

$A_{\mathrm{\mu }}$

$\mathrm{TransformCoordinates}\left(\mathrm{tr}\,\right)$

$\left[\begin{array}{c}\left(\frac{\partial \mathrm{X1}}{\partial \mathrm{y1}}\right)A_1+\left(\frac{\partial \mathrm{X2}}{\partial \mathrm{y1}}\right)A_2+\left(\frac{\partial \mathrm{X3}}{\partial \mathrm{y1}}\right)A_3+\left(\frac{\partial \mathrm{X4}}{\partial \mathrm{y1}}\right)A_4\\ \left(\frac{\partial \mathrm{X1}}{\partial \mathrm{y2}}\right)A_1+\left(\frac{\partial \mathrm{X2}}{\partial \mathrm{y2}}\right)A_2+\left(\frac{\partial \mathrm{X3}}{\partial \mathrm{y2}}\right)A_3+\left(\frac{\partial \mathrm{X4}}{\partial \mathrm{y2}}\right)A_4\\ \left(\frac{\partial \mathrm{X1}}{\partial \mathrm{y3}}\right)A_1+\left(\frac{\partial \mathrm{X2}}{\partial \mathrm{y3}}\right)A_2+\left(\frac{\partial \mathrm{X3}}{\partial \mathrm{y3}}\right)A_3+\left(\frac{\partial \mathrm{X4}}{\partial \mathrm{y3}}\right)A_4\\ \left(\frac{\partial \mathrm{X1}}{\partial \mathrm{y4}}\right)A_1+\left(\frac{\partial \mathrm{X2}}{\partial \mathrm{y4}}\right)A_2+\left(\frac{\partial \mathrm{X3}}{\partial \mathrm{y4}}\right)A_3+\left(\frac{\partial \mathrm{X4}}{\partial \mathrm{y4}}\right)A_4\end{array}\right]$

$A\left[\mathrm{~mu}\right]$

$A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{TransformCoordinates}\left(\mathrm{itr}\,\right)$

$\left[\begin{array}{c}\left(\frac{\partial \mathrm{Y1}}{\partial \mathrm{x1}}\right)A_{\phantom{}\phantom{1}}^{\phantom{}1}+\left(\frac{\partial \mathrm{Y1}}{\partial \mathrm{x2}}\right)A_{\phantom{}\phantom{2}}^{\phantom{}2}+\left(\frac{\partial \mathrm{Y1}}{\partial \mathrm{x3}}\right)A_{\phantom{}\phantom{3}}^{\phantom{}3}+\left(\frac{\partial \mathrm{Y1}}{\partial \mathrm{x4}}\right)A_{\phantom{}\phantom{4}}^{\phantom{}4}\\ \left(\frac{\partial \mathrm{Y2}}{\partial \mathrm{x1}}\right)A_{\phantom{}\phantom{1}}^{\phantom{}1}+\left(\frac{\partial \mathrm{Y2}}{\partial \mathrm{x2}}\right)A_{\phantom{}\phantom{2}}^{\phantom{}2}+\left(\frac{\partial \mathrm{Y2}}{\partial \mathrm{x3}}\right)A_{\phantom{}\phantom{3}}^{\phantom{}3}+\left(\frac{\partial \mathrm{Y2}}{\partial \mathrm{x4}}\right)A_{\phantom{}\phantom{4}}^{\phantom{}4}\\ \left(\frac{\partial \mathrm{Y3}}{\partial \mathrm{x1}}\right)A_{\phantom{}\phantom{1}}^{\phantom{}1}+\left(\frac{\partial \mathrm{Y3}}{\partial \mathrm{x2}}\right)A_{\phantom{}\phantom{2}}^{\phantom{}2}+\left(\frac{\partial \mathrm{Y3}}{\partial \mathrm{x3}}\right)A_{\phantom{}\phantom{3}}^{\phantom{}3}+\left(\frac{\partial \mathrm{Y3}}{\partial \mathrm{x4}}\right)A_{\phantom{}\phantom{4}}^{\phantom{}4}\\ \left(\frac{\partial \mathrm{Y4}}{\partial \mathrm{x1}}\right)A_{\phantom{}\phantom{1}}^{\phantom{}1}+\left(\frac{\partial \mathrm{Y4}}{\partial \mathrm{x2}}\right)A_{\phantom{}\phantom{2}}^{\phantom{}2}+\left(\frac{\partial \mathrm{Y4}}{\partial \mathrm{x3}}\right)A_{\phantom{}\phantom{3}}^{\phantom{}3}+\left(\frac{\partial \mathrm{Y4}}{\partial \mathrm{x4}}\right)A_{\phantom{}\phantom{4}}^{\phantom{}4}\end{array}\right]$

$\mathrm{TR}≔r={\left(1+\frac{m}{2\mathrm{\rho }}\right)}^2\mathrm{\rho }$

$\mathrm{TR}≔r={\left(1+\frac{m}{2\mathrm{\rho }}\right)}^2\mathrm{\rho }$

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

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

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

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

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

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

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{r}{2m-r}& 0& 0& 0\\ 0& -r^2& 0& 0\\ 0& 0& -r^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& \frac{r-2m}{r}\end{array}\right]$

$\mathrm{TransformCoordinates}\left(\mathrm{TR}\,\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]\,\left[\mathrm{\rho }\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\,\mathrm{output}=\mathrm{lineelement}\right)$

$-\frac{{\left(2\mathrm{\rho }+m\right)}^4\left({\left(\mathbf{\ⅆ}\left(\mathrm{\rho }\right)\right)}^2+\left({\sin \left(\mathrm{\theta }\right)}^2{\left(\mathbf{\ⅆ}\left(\mathrm{\phi }\right)\right)}^2+{\left(\mathbf{\ⅆ}\left(\mathrm{\theta }\right)\right)}^2\right){\mathrm{\rho }}^2\right)}{16{\mathrm{\rho }}^4}+\frac{{\left(m-2\mathrm{\rho }\right)}^2{\left(\mathbf{\ⅆ}\left(t\right)\right)}^2}{{\left(2\mathrm{\rho }+m\right)}^2}$

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

$\frac{r{\mathbf{\ⅆ}\left(r\right)}^2}{2m-r}-r^2{\mathbf{\ⅆ}\left(\mathrm{\theta }\right)}^2-r^2{\sin \left(\mathrm{\theta }\right)}^2{\mathbf{\ⅆ}\left(\mathrm{\phi }\right)}^2+\frac{\left(r-2m\right){\mathbf{\ⅆ}\left(t\right)}^2}{r}$

$\mathrm{TransformCoordinates}\left(\mathrm{TR}\,\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]\,\left[\mathrm{\rho }\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\,\mathrm{output}=\mathrm{lineelement}\,\mathrm{setmetric}\right)$

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

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}-\frac{{\left(2\mathrm{\rho }+m\right)}^4}{16{\mathrm{\rho }}^4}& 0& 0& 0\\ 0& -\frac{{\left(2\mathrm{\rho }+m\right)}^4}{16{\mathrm{\rho }}^2}& 0& 0\\ 0& 0& -\frac{{\left(2\mathrm{\rho }+m\right)}^4{\sin \left(\mathrm{\theta }\right)}^2}{16{\mathrm{\rho }}^2}& 0\\ 0& 0& 0& \frac{{\left(m-2\mathrm{\rho }\right)}^2}{{\left(2\mathrm{\rho }+m\right)}^2}\end{array}\right]$

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

$-\frac{{\left(2\mathrm{\rho }+m\right)}^4\left({\left(\mathbf{\ⅆ}\left(\mathrm{\rho }\right)\right)}^2+\left({\sin \left(\mathrm{\theta }\right)}^2{\left(\mathbf{\ⅆ}\left(\mathrm{\phi }\right)\right)}^2+{\left(\mathbf{\ⅆ}\left(\mathrm{\theta }\right)\right)}^2\right){\mathrm{\rho }}^2\right)}{16{\mathrm{\rho }}^4}+\frac{{\left(m-2\mathrm{\rho }\right)}^2{\left(\mathbf{\ⅆ}\left(t\right)\right)}^2}{{\left(2\mathrm{\rho }+m\right)}^2}$

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{\rho }\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

$\left\{X\,Y\right\}$

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

$-\frac{{\left(2\mathrm{\rho }+m\right)}^4\left({\mathbf{\ⅆ}\left(\mathrm{\rho }\right)}^2+\left({\sin \left(\mathrm{\theta }\right)}^2{\mathbf{\ⅆ}\left(\mathrm{\phi }\right)}^2+{\mathbf{\ⅆ}\left(\mathrm{\theta }\right)}^2\right){\mathrm{\rho }}^2\right)}{16{\mathrm{\rho }}^4}+\frac{{\left(m-2\mathrm{\rho }\right)}^2{\mathbf{\ⅆ}\left(t\right)}^2}{{\left(2\mathrm{\rho }+m\right)}^2}$

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

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

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

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

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

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{r}{2m-r}& 0& 0& 0\\ 0& -r^2& 0& 0\\ 0& 0& -r^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& \frac{r-2m}{r}\end{array}\right]$

$\mathrm{Coordinates}\left(K=\left[U\,\mathrm{\theta }\,\mathrm{\phi }\,V\right]\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{K=\left(U\,\mathrm{\theta }\,\mathrm{\phi }\,V\right)\,X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

$\left\{K\,X\,Y\right\}$

$\mathrm{tr}≔\left[V=\exp \left(\frac{v}{4m}\right)\,U=-\exp \left(-\frac{u}{4m}\right)\right]$

$\mathrm{tr}≔\left[V={\ⅇ}^{\frac{v}{4m}}\,U=-{\ⅇ}^{-\frac{u}{4m}}\right]$

$v≔t+r+2m\ln \left(r-2m\right)$

$v≔t+r+2m\ln \left(r-2m\right)$

$u≔t-r-2m\ln \left(r-2m\right)$

$u≔t-r-2m\ln \left(r-2m\right)$

$\mathrm{TransformCoordinates}\left(\mathrm{tr}\,\mathrm{g\_}\left[i,j\right]\,\mathrm{output}=\mathrm{lineelement}\right)$

$-r^2{\mathbf{\ⅆ}\left(\mathrm{\theta }\right)}^2-r^2{\sin \left(\mathrm{\theta }\right)}^2{\mathbf{\ⅆ}\left(\mathrm{\phi }\right)}^2+\frac{16{\ⅇ}^{-\frac{r}{2m}}{m}^2\mathbf{\ⅆ}\left(U\right)\mathbf{\ⅆ}\left(V\right)}{r}$

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

$g_{\mathrm{\mu },\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_{\mathrm{\nu }}$

$M≔\mathrm{TransformCoordinates}\left(r=r\,\right)$

$M≔\left[\begin{array}{cccc}\frac{A_1{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}& \frac{A_2{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}& \frac{A_3{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}& \frac{A_4{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}\\ -A_1{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}& -A_2{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}& -A_3{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}& -A_4{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}\\ -A_1{A}_{\phantom{}\phantom{3}}^{\phantom{}3}{r}^2{\sin \left(\mathrm{\theta }\right)}^2& -A_2{A}_{\phantom{}\phantom{3}}^{\phantom{}3}{r}^2{\sin \left(\mathrm{\theta }\right)}^2& -A_3{A}_{\phantom{}\phantom{3}}^{\phantom{}3}{r}^2{\sin \left(\mathrm{\theta }\right)}^2& -A_4{A}_{\phantom{}\phantom{3}}^{\phantom{}3}{r}^2{\sin \left(\mathrm{\theta }\right)}^2\\ \frac{A_1{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}& \frac{A_2{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}& \frac{A_3{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}& \frac{A_4{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}\end{array}\right]$

$\mathrm{TransformCoordinates}\left(r=r\,\,\mathrm{performsumoverrepeatedindices}=\mathrm{false}\right)$

$\left[\begin{array}{cccc}{g}_{1,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_1& g_{1,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_2& g_{1,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_3& g_{1,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_4\\ g_{2,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_1& g_{2,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_2& g_{2,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_3& g_{2,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_4\\ g_{3,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_1& g_{3,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_2& g_{3,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_3& g_{3,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_4\\ g_{4,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_1& g_{4,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_2& g_{4,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_3& g_{4,\mathrm{\rho }}{A}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{A}_4\end{array}\right]$

$M\left[1,1\right]$

$\frac{A_1{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}$

$M\left[4,1\right]$

$\frac{A_1{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}$

$\mathrm{Define}\left(T\left[\mathrm{\mu }\right]\right)$

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

$\left\{A_{\mathrm{\mu }}\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,T_{\mathrm{\mu }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,K_{\mathrm{\mu }}\,X_{\mathrm{\mu }}\,Y_{\mathrm{\mu }}\right\}$

$\mathrm{Define}\left(R\left[\mathrm{\mu }\right]\,\mathrm{relativeweight}=1\right)$

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

$\left\{A_{\mathrm{\mu }}\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,T_{\mathrm{\mu }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,K_{\mathrm{\mu }}\,X_{\mathrm{\mu }}\,Y_{\mathrm{\mu }}\right\}$

$\mathrm{TR}$

$r={\left(1+\frac{m}{2\mathrm{\rho }}\right)}^2\mathrm{\rho }$

$\mathrm{TransformCoordinates}\left(\mathrm{TR}\,T\left[\mathrm{\mu }\right]\,\left[\mathrm{\rho }\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\right)$

$\left[\begin{array}{c}-\frac{\left(m^2-4{\mathrm{\rho }}^2\right)T_1}{4{\mathrm{\rho }}^2}\\ T_2\\ T_3\\ T_4\end{array}\right]$

$\mathrm{TransformCoordinates}\left(\mathrm{TR}\,R\left[\mathrm{\mu }\right]\,\left[\mathrm{\rho }\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\right)$

$\left[\begin{array}{c}\frac{{\left(m^2-4{\mathrm{\rho }}^2\right)}^2{R}_1}{16{\mathrm{\rho }}^4}\\ -\frac{\left(m^2-4{\mathrm{\rho }}^2\right)R_2}{4{\mathrm{\rho }}^2}\\ -\frac{\left(m^2-4{\mathrm{\rho }}^2\right)R_3}{4{\mathrm{\rho }}^2}\\ -\frac{\left(m^2-4{\mathrm{\rho }}^2\right)R_4}{4{\mathrm{\rho }}^2}\end{array}\right]$

$\mathrm{J\_\_matrix}≔\mathrm{simplify}\left(\mathrm{VectorCalculus}:-\mathrm{Jacobian}\left(\left[\mathrm{rhs}\left(\mathrm{TR}\right)\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\,\left[\mathrm{\rho }\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\right)\right)$

$\mathrm{J\_\_matrix}≔\left[\begin{array}{cccc}\frac{-m^2+4{\mathrm{\rho }}^2}{4{\mathrm{\rho }}^2}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$J=\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(\mathrm{J\_\_matrix}\right)$

$J=\frac{-m^2+4{\mathrm{\rho }}^2}{4{\mathrm{\rho }}^2}$

$\mathrm{TR}$

$r={\left(1+\frac{m}{2\mathrm{\rho }}\right)}^2\mathrm{\rho }$

[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.

[2] Lovelock, D., and Rund, H. Tensors, Differential Forms and Variational Principles, Dover, 1989.