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.

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

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

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

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

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

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

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

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

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

${\mathrm{g\_}}_{\mathrm{sc}}$

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

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

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

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

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

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

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}-\frac{r}{2m-r}& 0& 0& 0\\ 0& r^2& 0& 0\\ 0& 0& r^2{\sin \left(\mathrm{\θ}\right)}^2& 0\\ 0& 0& 0& \frac{2m-r}{r}\end{array}\right]\right)$

${\mathrm{EnergyMomentum}}_{\[\]}$

${\mathrm{EnergyMomentum}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{rrrr}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\right)$

${\mathrm{EnergyMomentum}}_{\mathrm{definition}}$

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

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

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

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

$\left[\begin{array}{c}-{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\α}\,\mathrm{\β}}{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{~alpha}\,\mathrm{~beta}}\+{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\χ}\,\mathrm{~chi}}{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\υ}\,\mathrm{~upsilon}}\+{\mathrm{\%Ricci3}}_{\mathrm{\υ}\,\mathrm{~upsilon}}\=16\mathrm{\π}{\mathrm{\%UnitNormalVector}}_{\mathrm{\α}}{\mathrm{\%UnitNormalVector}}_{\mathrm{\β}}{\mathrm{\%EnergyMomentum}}_{\mathrm{~alpha}\,\mathrm{~beta}}\\ {\mathrm{\%D3\_}}_{\mathrm{\β}}\left({\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{~beta}}\right)-{\mathrm{\%D3\_}}_{\mathrm{\μ}}\left({\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\υ}\,\mathrm{~upsilon}}\right)\=-8\mathrm{\π}{\mathrm{\%gamma3\_}}_{\mathrm{\μ}\,\mathrm{~beta}}{\mathrm{\%UnitNormalVector}}_{\mathrm{~tau}}{\mathrm{\%EnergyMomentum}}_{\mathrm{\β}\,\mathrm{\τ}}\\ {\mathrm{\%LieDerivative}}_{\mathrm{TimeVector}}\left({\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\right)\=-{\mathrm{\%D3\_}}_{\mathrm{\μ}}\left({\mathrm{\%D3\_}}_{\mathrm{\ν}}\left(\mathrm{Lapse}\right)\right)\+\mathrm{Lapse}\left({\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\υ}\,\mathrm{~upsilon}}-2{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{~tau}\,\mathrm{\ν}}\+{\mathrm{\%Ricci3}}_{\mathrm{\μ}\,\mathrm{\ν}}\right)-8\mathrm{\π}\mathrm{Lapse}\left({\mathrm{\%gamma3\_}}_{\mathrm{\μ}\,\mathrm{~kappa}}{\mathrm{\%gamma3\_}}_{\mathrm{\ν}\,\mathrm{~lambda}}{\mathrm{\%EnergyMomentum}}_{\mathrm{\κ}\,\mathrm{\λ}}-\frac{1}{2}{\mathrm{\%gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\left({\mathrm{\%EnergyMomentum}}_{\mathrm{\λ}\,\mathrm{\σ}}{\mathrm{\%gamma3\_}}_{\mathrm{\κ}\,\mathrm{~lambda}}{\mathrm{\%gamma3\_}}_{\mathrm{~kappa}\,\mathrm{~sigma}}-{\mathrm{\%EnergyMomentum}}_{\mathrm{~alpha}\,\mathrm{~beta}}{\mathrm{\%UnitNormalVector}}_{\mathrm{\α}}{\mathrm{\%UnitNormalVector}}_{\mathrm{\β}}\right)\right)\+{\mathrm{\%LieDerivative}}_{\mathrm{Shift}}\left({\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\right)\\ {\mathrm{\%LieDerivative}}_{\mathrm{TimeVector}}\left({\mathrm{\%gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\right)\=-2\mathrm{Lapse}{\mathrm{\%ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\+{\mathrm{\%LieDerivative}}_{\mathrm{Shift}}\left({\mathrm{\%gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\right)\end{array}\right]$

$\mathrm{ADMEquations}\left(\mathrm{inert}\=\mathrm{false}\right)$

$\left[\begin{array}{c}-{\mathrm{ExtrinsicCurvature}}_{\mathrm{\α}\,\mathrm{\β}}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~alpha}\,\mathrm{~beta}}\=16\mathrm{\π}{\mathrm{UnitNormalVector}}_{\mathrm{\α}}{\mathrm{UnitNormalVector}}_{\mathrm{\β}}{\mathrm{EnergyMomentum}}_{\mathrm{~alpha}\,\mathrm{~beta}}\\ {\mathrm{D3\_}}_{\mathrm{\β}}\left({\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{~beta}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\=-8\mathrm{\π}{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~beta}}{\mathrm{UnitNormalVector}}_{\mathrm{~tau}}{\mathrm{EnergyMomentum}}_{\mathrm{\β}\,\mathrm{\τ}}\\ {\mathrm{TimeVector}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{D\_}}_{\mathrm{\μ}}\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{D\_}}_{\mathrm{\ν}}\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\=-\frac{3}{4}\frac{{\mathrm{gamma3\_}}_{\mathrm{\ν}\,\mathrm{~tau}}\left(-\frac{{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{2m-r}-\frac{r{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}\right)\left(-\frac{{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{2m-r}-\frac{r{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}\right)}{{\left(-\frac{r}{2m-r}\right)}^{5\/2}}\+\frac{1}{2}\frac{{\mathrm{gamma3\_}}_{\mathrm{\ν}\,\mathrm{~tau}}\left(-\frac{{\mathrm{D3\_}}_{\mathrm{\μ}}\left({\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{2m-r}-\frac{2{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right){\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}-\frac{2r{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right){\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^3}-\frac{r{\mathrm{D3\_}}_{\mathrm{\μ}}\left({\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}\right)}{{\left(-\frac{r}{2m-r}\right)}^{3\/2}}\+\frac{-2{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~tau}\,\mathrm{\ν}}\+{\mathrm{Ricci3}}_{\mathrm{\μ}\,\mathrm{\ν}}}{\sqrt{-\frac{r}{2m-r}}}-\frac{8\mathrm{\π}\left({\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~kappa}}{\mathrm{gamma3\_}}_{\mathrm{\ν}\,\mathrm{~lambda}}{\mathrm{EnergyMomentum}}_{\mathrm{\κ}\,\mathrm{\λ}}-\frac{1}{2}{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\left({\mathrm{gamma3\_}}_{\mathrm{\κ}\,\mathrm{~lambda}}{\mathrm{gamma3\_}}_{\mathrm{~kappa}\,\mathrm{~sigma}}{\mathrm{EnergyMomentum}}_{\mathrm{\λ}\,\mathrm{\σ}}-{\mathrm{EnergyMomentum}}_{\mathrm{~alpha}\,\mathrm{~beta}}{\mathrm{UnitNormalVector}}_{\mathrm{\α}}{\mathrm{UnitNormalVector}}_{\mathrm{\β}}\right)\right)}{\sqrt{-\frac{r}{2m-r}}}\+{\mathrm{Shift}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\μ}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\ν}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\\ {\mathrm{TimeVector}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{D\_}}_{\mathrm{\μ}}\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{D\_}}_{\mathrm{\ν}}\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\=-\frac{2{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}}{\sqrt{-\frac{r}{2m-r}}}\+{\mathrm{Shift}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\μ}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\+{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\ν}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\end{array}\right]$

$\mathrm{TensorArray}\left(\,\mathrm{simplifier}\=\mathrm{simplify}\right)$

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

$\mathrm{ADMEquations}\left(\mathrm{inert}\=\mathrm{false}\,\mathrm{output}\=3\mathrm{D}\right)$

$\left[\begin{array}{c}-{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~mu}\,\mathrm{~nu}}\=16\mathrm{\π}{\mathrm{UnitNormalVector}}_{\mathrm{\μ}}{\mathrm{UnitNormalVector}}_{\mathrm{\ν}}{\mathrm{EnergyMomentum}}_{\mathrm{~mu}\,\mathrm{~nu}}\\ {\mathrm{D3\_}}_{\mathrm{\ν}}\left({\mathrm{ExtrinsicCurvature}}_{i\,\mathrm{~nu}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\=-8\mathrm{\π}{\mathrm{gamma3\_}}_{i\,\mathrm{~nu}}{\mathrm{UnitNormalVector}}_{\mathrm{~alpha}}{\mathrm{EnergyMomentum}}_{\mathrm{\α}\,\mathrm{\ν}}\\ {\mathrm{TimeVector}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{ExtrinsicCurvature}}_{i\,j}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\τ}\,j}{\mathrm{D\_}}_i\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{i\,\mathrm{\τ}}{\mathrm{D\_}}_j\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\=-\frac{3}{4}\frac{{\mathrm{gamma3\_}}_{j\,\mathrm{~tau}}\left(-\frac{{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{2m-r}-\frac{r{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}\right)\left(-\frac{{\mathrm{gamma3\_}}_{i\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{2m-r}-\frac{r{\mathrm{gamma3\_}}_{i\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}\right)}{{\left(-\frac{r}{2m-r}\right)}^{5\/2}}\+\frac{1}{2}\frac{{\mathrm{gamma3\_}}_{j\,\mathrm{~tau}}\left(-\frac{{\mathrm{D3\_}}_i\left({\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{2m-r}-\frac{2{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right){\mathrm{gamma3\_}}_{i\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}-\frac{2r{\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right){\mathrm{gamma3\_}}_{i\,\mathrm{~upsilon}}{\mathrm{d\_}}_{\mathrm{\υ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^3}-\frac{r{\mathrm{D3\_}}_i\left({\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{{\left(2m-r\right)}^2}\right)}{{\left(-\frac{r}{2m-r}\right)}^{3\/2}}\+\frac{-2{\mathrm{ExtrinsicCurvature}}_{\mathrm{\α}\,i}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~alpha}\,j}\+{\mathrm{Ricci3}}_{i\,j}}{\sqrt{-\frac{r}{2m-r}}}-\frac{8\mathrm{\π}\left({\mathrm{gamma3\_}}_{i\,\mathrm{~kappa}}{\mathrm{gamma3\_}}_{j\,\mathrm{~lambda}}{\mathrm{EnergyMomentum}}_{\mathrm{\κ}\,\mathrm{\λ}}-\frac{1}{2}{\mathrm{gamma3\_}}_{i\,j}\left({\mathrm{gamma3\_}}_{\mathrm{\κ}\,\mathrm{~lambda}}{\mathrm{gamma3\_}}_{\mathrm{~kappa}\,\mathrm{~sigma}}{\mathrm{EnergyMomentum}}_{\mathrm{\λ}\,\mathrm{\σ}}-{\mathrm{EnergyMomentum}}_{\mathrm{~mu}\,\mathrm{~nu}}{\mathrm{UnitNormalVector}}_{\mathrm{\μ}}{\mathrm{UnitNormalVector}}_{\mathrm{\ν}}\right)\right)}{\sqrt{-\frac{r}{2m-r}}}\+{\mathrm{Shift}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{ExtrinsicCurvature}}_{i\,j}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\τ}\,j}{\mathrm{Christoffel}}_{\mathrm{~tau}\,i\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\+{\mathrm{ExtrinsicCurvature}}_{i\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,j\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\\ {\mathrm{TimeVector}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{gamma3\_}}_{i\,j}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\τ}\,j}{\mathrm{D\_}}_i\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{i\,\mathrm{\τ}}{\mathrm{D\_}}_j\left({\mathrm{TimeVector}}_{\mathrm{~tau}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\=-\frac{2{\mathrm{ExtrinsicCurvature}}_{i\,j}}{\sqrt{-\frac{r}{2m-r}}}\+{\mathrm{Shift}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{gamma3\_}}_{i\,j}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\τ}\,j}{\mathrm{Christoffel}}_{\mathrm{~tau}\,i\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\+{\mathrm{gamma3\_}}_{i\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,j\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\end{array}\right]$

$\mathrm{map}\left(\mathrm{Check}\,\,\mathrm{all}\right)$

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

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

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

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

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

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

$\left[\begin{array}{c}\left(\left[\left\{\mathrm{\μ}\,\mathrm{\ν}\right\}\right]\,\left\{\right\}\right)\=\left(\left[\left\{\mathrm{\μ}\,\mathrm{\ν}\right\}\right]\,\left\{\right\}\right)\\ \left(\left[\left\{\mathrm{\ν}\right\}\right]\,\left\{i\right\}\right)\=\left(\left[\left\{\mathrm{\α}\,\mathrm{\ν}\right\}\right]\,\left\{i\right\}\right)\\ \left(\left[\left\{\mathrm{\τ}\right\}\,\left\{\mathrm{\τ}\right\}\,\left\{\mathrm{\τ}\right\}\right]\,\left\{i\,j\right\}\right)\=\left(\left[\left\{\mathrm{\τ}\,\mathrm{\υ}\right\}\,\left\{\mathrm{\τ}\,\mathrm{\υ}\right\}\,\left\{\mathrm{\α}\right\}\,\left\{\mathrm{\κ}\,\mathrm{\λ}\,\mathrm{\μ}\,\mathrm{\ν}\,\mathrm{\σ}\right\}\,\left\{\mathrm{\τ}\right\}\,\left\{\mathrm{\τ}\,\mathrm{\υ}\right\}\,\left\{\mathrm{\τ}\,\mathrm{\υ}\right\}\right]\,\left\{i\,j\right\}\right)\\ \left(\left[\left\{\mathrm{\τ}\right\}\,\left\{\mathrm{\τ}\right\}\,\left\{\mathrm{\τ}\right\}\right]\,\left\{i\,j\right\}\right)\=\left(\left[\left\{\right\}\,\left\{\mathrm{\τ}\right\}\,\left\{\mathrm{\τ}\,\mathrm{\υ}\right\}\,\left\{\mathrm{\τ}\,\mathrm{\υ}\right\}\right]\,\left\{i\,j\right\}\right)\end{array}\right]$

$\mathrm{TensorArray}\left(\,\mathrm{simplifier}\=\mathrm{simplify}\right)$

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

${\mathrm{g\_}}_{\mathrm{tol}}$

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

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

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

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

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}\frac{{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)}^2}{1\+2E\left(r\right)}& 0& 0& 0\\ 0& {R\left(t\,r\right)}^2& 0& 0\\ 0& 0& {R\left(t\,r\right)}^2{\sin \left(\mathrm{\θ}\right)}^2& 0\\ 0& 0& 0& -1\end{array}\right]\right)$

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

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

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

${\mathrm{EnergyMomentum}}_{\mathrm{`~mu`}\,\mathrm{\ν}}\=-\mathit{rho\_\_M}\left(t\,r\right){\mathrm{g\_}}_{\mathrm{`~mu`}\,0}{\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{`~0`}}-\mathit{rho\_\_Lambda}{\mathrm{g\_}}_{\mathrm{`~mu`}\,\mathrm{\ν}}$

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

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

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

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

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

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

$\mathrm{isolate}\left(\,{\mathrm{Einstein}}_{\mathrm{\μ}\,\mathrm{\ν}}\right)$

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

$\mathrm{EQ4}≔\mathrm{TensorArray}\left(\right)$

$\left[\begin{array}{cccc}-\frac{\mathrm{\ρ\_\_\Λ}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)}^2}{1\+2E\left(r\right)}\=-\frac{1}{8}\frac{{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)}^2\left(2R\left(t\,r\right)\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\right)\+{\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)}^2-2E\left(r\right)\right)}{{R\left(t\,r\right)}^2\left(1\+2E\left(r\right)\right)\mathrm{\π}}& 0\=0& 0\=0& 0\=0\\ 0\=0& -\mathrm{\ρ\_\_\Λ}{R\left(t\,r\right)}^2\=-\frac{1}{8}\frac{R\left(t\,r\right)\left(\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)\right)R\left(t\,r\right)\+\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)-\left(\frac{\ⅆ}{\ⅆr}E\left(r\right)\right)\right)}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\mathrm{\π}}& 0\=0& 0\=0\\ 0\=0& 0\=0& -\mathrm{\ρ\_\_\Λ}{R\left(t\,r\right)}^2{\sin \left(\mathrm{\θ}\right)}^2\=-\frac{1}{8}\frac{R\left(t\,r\right){\sin \left(\mathrm{\θ}\right)}^2\left(\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)\right)R\left(t\,r\right)\+\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)-\left(\frac{\ⅆ}{\ⅆr}E\left(r\right)\right)\right)}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\mathrm{\π}}& 0\=0\\ 0\=0& 0\=0& 0\=0& \mathrm{\ρ\_\_M}\left(t\,r\right)\+\mathrm{\ρ\_\_\Λ}\=-\frac{1}{8}\frac{-\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right){\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)}^2-2R\left(t\,r\right)\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)\+2\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)E\left(r\right)\+2\left(\frac{\ⅆ}{\ⅆr}E\left(r\right)\right)R\left(t\,r\right)}{{R\left(t\,r\right)}^2\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\mathrm{\π}}\end{array}\right]$

$M\left(r\right)\=-\frac{1\left(-{\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)}^2\+2E\left(r\right)\right)R\left(t\,r\right)}{2}$

$M\left(r\right)\=-\frac{1}{2}\left(-{\left(\mathrm{diff}\left(R\left(t\,r\right)\,t\right)\right)}^2\+2E\left(r\right)\right)R\left(t\,r\right)$

$\mathrm{simplify}\left({\mathrm{EQ4}}_{4\,4}\,\left\{\right\}\,\left\{E\left(r\right)\right\}\right)$

$\mathrm{\ρ\_\_M}\left(t\,r\right)\+\mathrm{\ρ\_\_\Λ}\=\frac{1}{4}\frac{\mathrm{diff}\left(M\left(r\right)\,r\right)}{{R\left(t\,r\right)}^2\left(\mathrm{diff}\left(R\left(t\,r\right)\,r\right)\right)\mathrm{\π}}$

$\mathrm{eq}≔\mathrm{ADMEquations}\left(\mathrm{inert}\=\mathrm{false}\right)$

$\left[\begin{array}{c}\frac{{\left(2\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)R\left(t\,r\right)\right)}^2}{{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)}^2{R\left(t\,r\right)}^2}-{\mathrm{ExtrinsicCurvature}}_{\mathrm{\α}\,\mathrm{\β}}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~alpha}\,\mathrm{~beta}}-\frac{4\left(\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)E\left(r\right)\+\left(\frac{\ⅆ}{\ⅆr}E\left(r\right)\right)R\left(t\,r\right)\right)}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right){R\left(t\,r\right)}^2}\=16\mathrm{\π}{\mathrm{UnitNormalVector}}_{\mathrm{\α}}{\mathrm{UnitNormalVector}}_{\mathrm{\β}}{\mathrm{EnergyMomentum}}_{\mathrm{~alpha}\,\mathrm{~beta}}\\ {\mathrm{D3\_}}_{\mathrm{\β}}\left({\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{~beta}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)-{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~tau}}\left(-\frac{2\left(\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(t\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+2\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\left(\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(t\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\right)\+\left(\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(t\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\right)R\left(t\,r\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(R\left(t\,r\right)\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)R\left(t\,r\right)}\+\frac{\left(2\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)R\left(t\,r\right)\right)\left(\left(\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(r\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(t\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\right)}{{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)}^{2}R\left(t\,r\right)}\+\frac{\left(2\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)R\left(t\,r\right)\right){\mathrm{d\_}}_{\mathrm{\τ}}\left(R\left(t\,r\right)\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right){R\left(t\,r\right)}^2}\right)\=-8\mathrm{\π}{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~beta}}{\mathrm{UnitNormalVector}}_{\mathrm{~tau}}{\mathrm{EnergyMomentum}}_{\mathrm{\β}\,\mathrm{\τ}}\\ {\mathrm{TimeVector}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\μ}\,\mathrm{\υ}}{\mathrm{TimeVector}}_{\mathrm{~upsilon}}\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\ν}\,\mathrm{\υ}}{\mathrm{TimeVector}}_{\mathrm{~upsilon}}\=-\frac{\left(2\left(\frac{\partial }{\partial t}R\left(t\,r\right)\right)\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\+\left(\frac{\partial }{\partial t}\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)\right)R\left(t\,r\right)\right){\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}}{\left(\frac{\partial }{\partial r}R\left(t\,r\right)\right)R\left(t\,r\right)}-2{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~tau}\,\mathrm{\ν}}\+{\mathrm{Ricci3}}_{\mathrm{\μ}\,\mathrm{\ν}}-8\mathrm{\π}\left({\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{~kappa}}{\mathrm{gamma3\_}}_{\mathrm{\ν}\,\mathrm{~lambda}}{\mathrm{EnergyMomentum}}_{\mathrm{\κ}\,\mathrm{\λ}}-\frac{1}{2}{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\left({\mathrm{gamma3\_}}_{\mathrm{\κ}\,\mathrm{~lambda}}{\mathrm{gamma3\_}}_{\mathrm{~kappa}\,\mathrm{~sigma}}{\mathrm{EnergyMomentum}}_{\mathrm{\λ}\,\mathrm{\σ}}-{\mathrm{EnergyMomentum}}_{\mathrm{~alpha}\,\mathrm{~beta}}{\mathrm{UnitNormalVector}}_{\mathrm{\α}}{\mathrm{UnitNormalVector}}_{\mathrm{\β}}\right)\right)\+{\mathrm{Shift}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\μ}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\+{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\ν}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\\ {\mathrm{TimeVector}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\μ}\,\mathrm{\υ}}{\mathrm{TimeVector}}_{\mathrm{~upsilon}}\+{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\ν}\,\mathrm{\υ}}{\mathrm{TimeVector}}_{\mathrm{~upsilon}}\=-2{\mathrm{ExtrinsicCurvature}}_{\mathrm{\μ}\,\mathrm{\ν}}\+{\mathrm{Shift}}_{\mathrm{~tau}}{\mathrm{D\_}}_{\mathrm{\τ}}\left({\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\,\left[r\,\mathrm{\θ}\,\mathrm{\φ}\,t\right]\right)\+{\mathrm{gamma3\_}}_{\mathrm{\τ}\,\mathrm{\ν}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\μ}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\+{\mathrm{gamma3\_}}_{\mathrm{\μ}\,\mathrm{\τ}}{\mathrm{Christoffel}}_{\mathrm{~tau}\,\mathrm{\ν}\,\mathrm{\υ}}{\mathrm{Shift}}_{\mathrm{~upsilon}}\end{array}\right]$