The ADMEquations command returns the ADM equations in inert form, that is, the 3+1 decomposition of Einstein's equations expressed in terms of the Lapse, Shift, the 3D metric gamma3_, the ExtrinsicCurvature and the EnergyMomentum tensor.

You can set the values of the Lapse and Shift using Setup and its lapseandshift keyword. There are three possible values for lapseandshift: standard, arbitrary, or a list of algebraic expressions representing the Lapse and Shift. The value chosen determines the values of the components of all the ThreePlusOne tensors and so of tensorial expressions involving them (e.g. the ExtrinsicCurvature and ADMEquations). Those components are always computed first in terms of the Lapse and Shift and the space part $g_{i,j}$ of the 4D metric, then in a second step, if lapseandshift = standard, the Lapse and the Shift are replaced by their expressions in terms of the $g_{0,\mathrm{\mu }}$ part of the 4D metric, according to ${\mathbf{\alpha }}^2={\left(-g_{}^{0,0}\right)}^{-1}$ and ${\mathbf{\beta }}_j=g_{0,j}$, or if lapseandshift was set passing a list to Setup then in terms of the values indicated in that list (these values are not used to set the $g_{0,\mathrm{\mu }}$ part of the 4D metric). When lapseandshift = arbitrary, the second step is not performed and the Lapse and Shift evaluate to themselves, representing an arbitrary value for them. In the three cases, standard, arbitrary, or a list of algebraic expressions, the components of the ThreePlusOne tensors are computed in terms of a metric with line element ${\mathrm{ds}}^2=-{\mathit{\alpha }}^2{\mathrm{dt}}^2+{\mathit{\gamma }}_{i,j}\left({\mathrm{dx}}^i+{\mathit{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}\mathrm{dt}\right)\left({\mathrm{dx}}^j+{\mathit{\beta }}_{\phantom{}\phantom{j}}^{\phantom{}j}\mathrm{dt}\right)$, where $\mathbf{\alpha }$ and ${\mathbf{\beta }}_{}^i$ have the Lapse and Shift values mentioned, and ${\mathbf{\gamma }}_{i,j}=g_{i,j}$ is the ThreePlusOne:-gamma3_ metric of the 3D hypersurface. This design permits working with any 4D metric set and, without changing its value, experimenting with different values of the Lapse and Shift (different values of $g_{0,\mathrm{\mu }}$) for the 3+1 decomposition of Einstein's equations. See also LapseAndShiftConditions.

The output of ADMEquations consists of a column vector with four equations: the Hamiltonian and momentum constraints, and the evolution equations for the ExtrinsicCurvature and for the 3D gamma3_ metric. These equations are expressed using the inert form of the tensors involved in order to allow for different kinds of manipulations. Once the system has the desired form, the inert representations can be transformed into active ones and the computations represented be performed using the value command, possibly followed by convert to g_, with or without the option only = {Lapse, Shift}, to express the Lapse and Shift in terms of the spacetime metric. Before or after that you can also use SumOverRepeatedIndices, TensorArray or Decompose to do the corresponding manipulations on these ADM equations. These different steps can be performed in any preferred order.

Alternatively, by passing the option inert = false, first all occurrences of the Lapse and Shift get rewritten in terms of the spacetime metric g_, and then the inert representations are transformed into active using value.

The free indices in the equations returned are 4D spacetime indices. To request these equations with 3D space indices use the option output = 3D (on the worksheet, when you input $\mathrm{3D}$ it will get converted to the product 3*D, which for the purpose of requesting 3D indices works fine, the same way as the symbol `3D`)

To define the EnergyMomentum tensor that enters the equations returned by ADMEquations, use the Define command.

with(Physics): with(ThreePlusOne):

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

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

$\mathrm{Defined\; as\; 4D\; spacetime\; tensors}\left(\mathrm{see\; ?Physics,ThreePlusOne}\right),{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }},{\mathit{▿}}_{\mathrm{\mu }},{\mathbf{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }},{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }},{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }},{\mathbf{\beta }}_{\mathrm{\mu }},{\mathit{n}}_{\mathrm{\mu }},{\mathit{t}}_{\mathrm{\mu }},{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{Changing\; the\; signature\; of\; spacetime\; from}\left(\mathrm{-\; -\; -\; +}\right)\mathrm{to}\left(\mathrm{+\; +\; +\; -}\right)\mathrm{in\; order\; to\; match\; the\; signature\; customarily\; used\; in\; the\; ADM\; formalism}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

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

Setup(mathematicalnotation = true);

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

g_[sc];

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

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

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

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

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

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

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

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

EnergyMomentum[];

${\mathrm{{\rm T}}}_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

EnergyMomentum[definition];

${\mathrm{{\rm T}}}_{\mathrm{\mu },\mathrm{\nu }}=\frac{G_{\mathrm{\mu },\mathrm{\nu }}}{8\mathrm{\pi }}$

TensorArray((5));

$\left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]$

ADMEquations();

$\left[\begin{array}{c}-{\mathbf{{\rm K}}}_{\mathrm{\alpha },\mathrm{\beta }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}+{\mathbf{{\rm K}}}_{\mathrm{\upsilon }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\upsilon }}\mathrm{\upsilon }}{\mathbf{{\rm K}}}_{\mathrm{\chi }\phantom{\mathrm{\chi }}}^{\phantom{\mathrm{\chi }}\mathrm{\chi }}+{\mathit{R}}_{\mathrm{\upsilon }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\upsilon }}\mathrm{\upsilon }}=16\mathrm{\pi }{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}\\ {\mathit{▿}}_{\mathrm{\beta }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}\right)-{\mathit{▿}}_{\mathrm{\mu }}\left({\mathbf{{\rm K}}}_{\mathrm{\upsilon }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\upsilon }}\mathrm{\upsilon }}\right)=-8\mathrm{\pi }{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}{\mathit{n}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{\mathrm{{\rm T}}}_{\mathrm{\beta },\mathrm{\tau }}\\ {ℒ}_{\mathit{t}}\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\right)=-{\mathit{▿}}_{\mathrm{\mu }}\left({\mathit{▿}}_{\mathrm{\nu }}\left(\mathbf{\alpha }\right)\right)+\mathbf{\alpha }\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{{\rm K}}}_{\mathrm{\upsilon }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\upsilon }}\mathrm{\upsilon }}-2{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu }}}+{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }}\right)-8\mathrm{\pi }\mathbf{\alpha }\left({\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}{\mathrm{{\rm T}}}_{\mathrm{\kappa },\mathrm{\lambda }}-\frac{{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\left({\mathbf{\gamma }}_{\mathrm{\kappa }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\kappa }}\mathrm{\lambda }}{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\kappa },\mathrm{\sigma }}}^{\phantom{}\mathrm{\kappa },\mathrm{\sigma }}{\mathrm{{\rm T}}}_{\mathrm{\lambda },\mathrm{\sigma }}-{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}\right)}{2}\right)+{ℒ}_{\mathbf{\beta }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\right)\\ {ℒ}_{\mathit{t}}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)=-2\mathbf{\alpha }{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}+{ℒ}_{\mathbf{\beta }}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)\end{array}\right]$

ADMEquations(inert=false);

$\left[\begin{array}{c}-{\mathbf{{\rm K}}}_{\mathrm{\alpha },\mathrm{\beta }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}=16\mathrm{\pi }{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}\\ {\mathit{▿}}_{\mathrm{\beta }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}\right)=-8\mathrm{\pi }{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}{\mathit{n}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{\mathrm{{\rm T}}}_{\mathrm{\beta },\mathrm{\tau }}\\ {\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{{\rm K}}}_{\mathrm{\tau },\mathrm{\nu }}{▿}_{\mathrm{\mu }}\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)+{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{▿}_{\mathrm{\nu }}\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)=-\frac{3{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\nu }}\mathrm{\tau }}\left(-\frac{{\partial }_{\mathrm{\tau }}\left(r\right)}{2m-r}-\frac{r{\partial }_{\mathrm{\tau }}\left(r\right)}{{\left(2m-r\right)}^2}\right)\left(-\frac{{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\mu }}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{2m-r}-\frac{r{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\mu }}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{{\left(2m-r\right)}^2}\right)}{4{\left(-\frac{r}{2m-r}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}+\frac{{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\nu }}\mathrm{\tau }}\left(-\frac{{\mathit{▿}}_{\mathrm{\mu }}\left({\partial }_{\mathrm{\tau }}\left(r\right)\right)}{2m-r}-\frac{2{\partial }_{\mathrm{\tau }}\left(r\right){\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\mu }}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{{\left(2m-r\right)}^2}-\frac{2r{\partial }_{\mathrm{\tau }}\left(r\right){\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\mu }}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{{\left(2m-r\right)}^3}-\frac{r\left({\mathit{▿}}_{\mathrm{\mu }}\left({\partial }_{\mathrm{\tau }}\left(r\right)\right)\right)}{{\left(2m-r\right)}^2}\right)}{2{\left(-\frac{r}{2m-r}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}+\frac{-2{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu }}}+{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }}}{\sqrt{-\frac{r}{2m-r}}}-\frac{8\mathrm{\pi }\left({\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}{\mathrm{{\rm T}}}_{\mathrm{\kappa },\mathrm{\lambda }}-\frac{{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\left({\mathbf{\gamma }}_{\mathrm{\kappa }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\kappa }}\mathrm{\lambda }}{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\kappa },\mathrm{\sigma }}}^{\phantom{}\mathrm{\kappa },\mathrm{\sigma }}{\mathrm{{\rm T}}}_{\mathrm{\lambda },\mathrm{\sigma }}-{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}\right)}{2}\right)}{\sqrt{-\frac{r}{2m-r}}}+{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{{\rm K}}}_{\mathrm{\tau },\mathrm{\nu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\mu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\mu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}\\ {\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{\gamma }}_{\mathrm{\tau },\mathrm{\nu }}{▿}_{\mathrm{\mu }}\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)+{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\tau }}{▿}_{\mathrm{\nu }}\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)=-\frac{2{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}}{\sqrt{-\frac{r}{2m-r}}}+{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{\gamma }}_{\mathrm{\tau },\mathrm{\nu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\mu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\mu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}\end{array}\right]$

TensorArray((8), simplifier=simplify);

$\left[\begin{array}{c}0=0\\ \left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]\end{array}\right]$

ADMEquations(inert = false, output = 3*D);

$\left[\begin{array}{c}-{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=16\mathrm{\pi }{\mathit{n}}_{\mathrm{\mu }}{\mathit{n}}_{\mathrm{\nu }}{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\\ {\mathit{▿}}_{\mathrm{\nu }}\left({\mathbf{{\rm K}}}_{i\phantom{\mathrm{\nu }}}^{\phantom{i}\mathrm{\nu }}\right)=-8\mathrm{\pi }{\mathbf{\gamma }}_{i\phantom{\mathrm{\nu }}}^{\phantom{i}\mathrm{\nu }}{\mathit{n}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{\mathrm{{\rm T}}}_{\mathrm{\alpha },\mathrm{\nu }}\\ {\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{{\rm K}}}_{i,j}\right)+{\mathbf{{\rm K}}}_{\mathrm{\tau },j}{▿}_i\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)+{\mathbf{{\rm K}}}_{i,\mathrm{\tau }}{▿}_j\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)=-\frac{3{\mathbf{\gamma }}_{j\phantom{\mathrm{\tau }}}^{\phantom{j}\mathrm{\tau }}\left(-\frac{{\partial }_{\mathrm{\tau }}\left(r\right)}{2m-r}-\frac{r{\partial }_{\mathrm{\tau }}\left(r\right)}{{\left(2m-r\right)}^2}\right)\left(-\frac{{\mathbf{\gamma }}_{i\phantom{\mathrm{\upsilon }}}^{\phantom{i}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{2m-r}-\frac{r{\mathbf{\gamma }}_{i\phantom{\mathrm{\upsilon }}}^{\phantom{i}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{{\left(2m-r\right)}^2}\right)}{4{\left(-\frac{r}{2m-r}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}+\frac{{\mathbf{\gamma }}_{j\phantom{\mathrm{\tau }}}^{\phantom{j}\mathrm{\tau }}\left(-\frac{{\mathit{▿}}_i\left({\partial }_{\mathrm{\tau }}\left(r\right)\right)}{2m-r}-\frac{2{\partial }_{\mathrm{\tau }}\left(r\right){\mathbf{\gamma }}_{i\phantom{\mathrm{\upsilon }}}^{\phantom{i}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{{\left(2m-r\right)}^2}-\frac{2r{\partial }_{\mathrm{\tau }}\left(r\right){\mathbf{\gamma }}_{i\phantom{\mathrm{\upsilon }}}^{\phantom{i}\mathrm{\upsilon }}{\partial }_{\mathrm{\upsilon }}\left(r\right)}{{\left(2m-r\right)}^3}-\frac{r\left({\mathit{▿}}_i\left({\partial }_{\mathrm{\tau }}\left(r\right)\right)\right)}{{\left(2m-r\right)}^2}\right)}{2{\left(-\frac{r}{2m-r}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}+\frac{-2{\mathbf{{\rm K}}}_{\mathrm{\alpha },i}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\alpha }}j}^{\phantom{}\mathrm{\alpha }\phantom{j}}+{\mathit{R}}_{i,j}}{\sqrt{-\frac{r}{2m-r}}}-\frac{8\mathrm{\pi }\left({\mathbf{\gamma }}_{i\phantom{\mathrm{\kappa }}}^{\phantom{i}\mathrm{\kappa }}{\mathbf{\gamma }}_{j\phantom{\mathrm{\lambda }}}^{\phantom{j}\mathrm{\lambda }}{\mathrm{{\rm T}}}_{\mathrm{\kappa },\mathrm{\lambda }}-\frac{{\mathbf{\gamma }}_{i,j}\left({\mathbf{\gamma }}_{\mathrm{\kappa }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\kappa }}\mathrm{\lambda }}{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\kappa },\mathrm{\sigma }}}^{\phantom{}\mathrm{\kappa },\mathrm{\sigma }}{\mathrm{{\rm T}}}_{\mathrm{\lambda },\mathrm{\sigma }}-{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\mathit{n}}_{\mathrm{\mu }}{\mathit{n}}_{\mathrm{\nu }}\right)}{2}\right)}{\sqrt{-\frac{r}{2m-r}}}+{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{{\rm K}}}_{i,j}\right)+{\mathbf{{\rm K}}}_{\mathrm{\tau },j}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}i,\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{i,\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{{\rm K}}}_{i,\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}j,\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{j,\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}\\ {\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{i,j}\right)+{\mathbf{\gamma }}_{\mathrm{\tau },j}{▿}_i\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)+{\mathbf{\gamma }}_{i,\mathrm{\tau }}{▿}_j\left({\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}\right)=-\frac{2{\mathbf{{\rm K}}}_{i,j}}{\sqrt{-\frac{r}{2m-r}}}+{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{i,j}\right)+{\mathbf{\gamma }}_{\mathrm{\tau },j}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}i,\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{i,\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{\gamma }}_{i,\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}j,\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{j,\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}\end{array}\right]$

map(Check, (10), all);

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\mathrm{The\; products\; in\; the\; given\; expression\; check\; ok.}$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\mathrm{The\; products\; in\; the\; given\; expression\; check\; ok.}$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\begin{array}{c}\left(\left[\left\{\mathrm{\mu }\,\mathrm{\nu }\right\}\right]\,\varnothing \right)=\left(\left[\left\{\mathrm{\mu }\,\mathrm{\nu }\right\}\right]\,\varnothing \right)\\ \left(\left[\left\{\mathrm{\nu }\right\}\right]\,\left\{i\right\}\right)=\left(\left[\left\{\mathrm{\alpha }\,\mathrm{\nu }\right\}\right]\,\left\{i\right\}\right)\\ \left(\left[\left\{\mathrm{\tau }\right\}\,\left\{\mathrm{\tau }\right\}\,\left\{\mathrm{\tau }\right\}\right]\,\left\{i\,j\right\}\right)=\left(\left[\left\{\mathrm{\tau }\,\mathrm{\upsilon }\right\}\,\left\{\mathrm{\tau }\,\mathrm{\upsilon }\right\}\,\left\{\mathrm{\alpha }\right\}\,\left\{\mathrm{\kappa }\,\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\nu }\,\mathrm{\sigma }\right\}\,\left\{\mathrm{\tau }\right\}\,\left\{\mathrm{\tau }\,\mathrm{\upsilon }\right\}\,\left\{\mathrm{\tau }\,\mathrm{\upsilon }\right\}\right]\,\left\{i\,j\right\}\right)\\ \left(\left[\left\{\mathrm{\tau }\right\}\,\left\{\mathrm{\tau }\right\}\,\left\{\mathrm{\tau }\right\}\right]\,\left\{i\,j\right\}\right)=\left(\left[\varnothing \,\left\{\mathrm{\tau }\right\}\,\left\{\mathrm{\tau }\,\mathrm{\upsilon }\right\}\,\left\{\mathrm{\tau }\,\mathrm{\upsilon }\right\}\right]\,\left\{i\,j\right\}\right)\end{array}\right]$

TensorArray((10), simplifier=simplify);

$\left[\begin{array}{c}0=0\\ \left[\begin{array}{ccc}0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{ccc}0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{ccc}0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\end{array}\right]\end{array}\right]$

g_[tol];

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

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

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

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)}^2}{1+2E\left(r\right)}& 0& 0& 0\\ 0& {R\left(t\,r\right)}^2& 0& 0\\ 0& 0& {R\left(t\,r\right)}^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]$

CompactDisplay((13));

$E\left(r\right)\mathrm{will\; now\; be\; displayed\; as}E$

$R\left(t\,r\right)\mathrm{will\; now\; be\; displayed\; as}R$

EnergyMomentum[~mu, nu] = -rho__M(t, r)*g_[`~mu`, 0]*g_[nu, `~0`]-rho__Lambda*g_[`~mu`, nu];

${\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}}=-\mathrm{\ρ\_\_M}\left(t\,r\right){\mathrm{\delta }}_{4\phantom{\mathrm{\mu }}}^{\phantom{4}\mathrm{\mu }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{4}}^{\phantom{\mathrm{\nu }}4}-\mathrm{\ρ\_\_\Λ}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\nu }}\mathrm{\mu }}$

Define((15));

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

$\left\{{\mathit{▿}}_{\mathrm{\mu }}\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\mathbf{\beta }}_{\mathrm{\mu }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,{\mathbf{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{{\rm T}}}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\mathit{t}}_{\mathrm{\mu }}\,{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\,{\mathit{n}}_{\mathrm{\mu }}\,X_{\mathrm{\mu }}\right\}$

CompactDisplay((15));

$\mathrm{\ρ\_\_M}\left(t\,r\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\ρ\_\_M}$

isolate((5), Einstein[mu, nu]);

$G_{\mathrm{\mu },\mathrm{\nu }}=8{\mathrm{{\rm T}}}_{\mathrm{\mu },\mathrm{\nu }}\mathrm{\pi }$

EQ4 := TensorArray((5));

$\mathrm{EQ4}≔\left[\begin{array}{cccc}-\frac{\mathrm{\ρ\_\_\Λ}{R_r}^2}{1+2E}=-\frac{{R_r}^2\left(2R\stackrel{\mathbf{..}}{R}+{{\stackrel{\mathbf{.}}{R}}^{}}^2-2E\right)}{8R^2\left(1+2E\right)\mathrm{\pi }}& 0=0& 0=0& 0=0\\ 0=0& -\mathrm{\ρ\_\_\Λ}{R}^2=-\frac{R\left(\left(\frac{{\partial }^{3}R}{\partial t^2\partial r}\right)R+\stackrel{\mathbf{..}}{R}{R}_r+\stackrel{\mathbf{.}}{R}\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)-E_r\right)}{8R_r\mathrm{\pi }}& 0=0& 0=0\\ 0=0& 0=0& -\mathrm{\ρ\_\_\Λ}{R}^2{\sin \left(\mathrm{\theta }\right)}^2=-\frac{R{\sin \left(\mathrm{\theta }\right)}^2\left(\left(\frac{{\partial }^{3}R}{\partial t^2\partial r}\right)R+\stackrel{\mathbf{..}}{R}{R}_r+\stackrel{\mathbf{.}}{R}\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)-E_r\right)}{8R_r\mathrm{\pi }}& 0=0\\ 0=0& 0=0& 0=0& \mathrm{\ρ\_\_M}+\mathrm{\ρ\_\_\Λ}=-\frac{-{{\stackrel{\mathbf{.}}{R}}^{}}^2{R}_r-2\stackrel{\mathbf{.}}{R}\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R+2E_{r}R+2R_{r}E}{8R^2{R}_r\mathrm{\pi }}\end{array}\right]$

M(r) = -1/2*(-(diff(R(t, r), t))^2+2*E(r))*R(t, r);

$M\left(r\right)=-\frac{\left(-{{\stackrel{\mathbf{.}}{R}}^{}}^2+2E\right)R}{2}$

simplify(EQ4[4,4], {(20)}, {E(r)});

$\mathrm{\ρ\_\_M}+\mathrm{\ρ\_\_\Λ}=\frac{M_r}{4R^2{R}_r\mathrm{\pi }}$

eq := ADMEquations(inert = false);

$\mathrm{eq}≔\left[\begin{array}{c}\frac{{\left(2\stackrel{\mathbf{.}}{R}{R}_r+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R\right)}^2}{{R_r}^2{R}^2}-{\mathbf{{\rm K}}}_{\mathrm{\alpha },\mathrm{\beta }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}-\frac{4\left(R_{r}E+E_{r}R\right)}{R_r{R}^2}=16\mathrm{\pi }{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}\\ {\mathit{▿}}_{\mathrm{\beta }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}\right)-{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\mu }}\mathrm{\tau }}\left(-\frac{2\left(\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right){\partial }_{\mathrm{\tau }}\left(r\right)+\stackrel{\mathbf{..}}{R}{\partial }_{\mathrm{\tau }}\left(t\right)\right)R_r+2\stackrel{\mathbf{.}}{R}\left(R_{r,r}{\partial }_{\mathrm{\tau }}\left(r\right)+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right){\partial }_{\mathrm{\tau }}\left(t\right)\right)+\left(\left(\frac{{\partial }^{3}R}{\partial t\partial r^2}\right){\partial }_{\mathrm{\tau }}\left(r\right)+\left(\frac{{\partial }^{3}R}{\partial t^2\partial r}\right){\partial }_{\mathrm{\tau }}\left(t\right)\right)R+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right){\partial }_{\mathrm{\tau }}\left(R\right)}{R_{r}R}+\frac{\left(2\stackrel{\mathbf{.}}{R}{R}_r+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R\right)\left(R_{r,r}{\partial }_{\mathrm{\tau }}\left(r\right)+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right){\partial }_{\mathrm{\tau }}\left(t\right)\right)}{{R_r}^{2}R}+\frac{\left(2\stackrel{\mathbf{.}}{R}{R}_r+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R\right){\partial }_{\mathrm{\tau }}\left(R\right)}{R_r{R}^2}\right)=-8\mathrm{\pi }{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}{\mathit{n}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{\mathrm{{\rm T}}}_{\mathrm{\beta },\mathrm{\tau }}\\ {\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{{\rm K}}}_{\mathrm{\tau },\mathrm{\nu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\mu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\mu },\mathrm{\upsilon }}}{\mathit{t}}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu },\mathrm{\upsilon }}}{\mathit{t}}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}=-\frac{\left(2\stackrel{\mathbf{.}}{R}{R}_r+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R\right){\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}}{R_{r}R}-2{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu }}}+{\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }}-8\mathrm{\pi }\left({\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}{\mathrm{{\rm T}}}_{\mathrm{\kappa },\mathrm{\lambda }}-\frac{{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\left({\mathbf{\gamma }}_{\mathrm{\kappa }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\kappa }}\mathrm{\lambda }}{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\kappa },\mathrm{\sigma }}}^{\phantom{}\mathrm{\kappa },\mathrm{\sigma }}{\mathrm{{\rm T}}}_{\mathrm{\lambda },\mathrm{\sigma }}-{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}\right)}{2}\right)+{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{{\rm K}}}_{\mathrm{\tau },\mathrm{\nu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\mu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\mu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}\\ {\mathit{t}}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{\gamma }}_{\mathrm{\tau },\mathrm{\nu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\mu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\mu },\mathrm{\upsilon }}}{\mathit{t}}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu },\mathrm{\upsilon }}}{\mathit{t}}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}=-2{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}+{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{▿}_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)+{\mathbf{\gamma }}_{\mathrm{\tau },\mathrm{\nu }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\mu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\mu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}+{\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\tau }}{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\tau }}\mathrm{\nu },\mathrm{\upsilon }}^{\phantom{}\mathrm{\tau }\phantom{\mathrm{\nu },\mathrm{\upsilon }}}{\mathbf{\beta }}_{\phantom{}\phantom{\mathrm{\upsilon }}}^{\phantom{}\mathrm{\upsilon }}\end{array}\right]$

eq[1];

$\frac{{\left(2\stackrel{\mathbf{.}}{R}{R}_r+\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R\right)}^2}{{R_r}^2{R}^2}-{\mathbf{{\rm K}}}_{\mathrm{\alpha },\mathrm{\beta }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}-\frac{4\left(R_{r}E+E_{r}R\right)}{R_r{R}^2}=16\mathrm{\pi }{\mathit{n}}_{\mathrm{\alpha }}{\mathit{n}}_{\mathrm{\beta }}{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}$

SumOverRepeatedIndices(eq[1]);

$-\frac{{{\stackrel{\mathbf{.}}{R}}^{}}^2{R}_r{\sin \left(\mathrm{\theta }\right)}^2{\csc \left(\mathrm{\theta }\right)}^2-3{{\stackrel{\mathbf{.}}{R}}^{}}^2{R}_r-4\stackrel{\mathbf{.}}{R}\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R+4R_{r}E+4E_{r}R}{R_r{R}^2}=16\mathrm{\pi }\left(\mathrm{\ρ\_\_M}+\mathrm{\ρ\_\_\Λ}\right)$

simplify(isolate((24), rho__M(t, r) + rho__Lambda));

$\mathrm{\ρ\_\_M}+\mathrm{\ρ\_\_\Λ}=\frac{2\stackrel{\mathbf{.}}{R}\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)R+\left({{\stackrel{\mathbf{.}}{R}}^{}}^2-2E\right)R_r-2E_{r}R}{8R^2{R}_r\mathrm{\pi }}$