 ADMEquations - Maple Help

Physics[ThreePlusOne][ADMEquations] - The ADM Equations as the 3+1 decomposition of the Einstein equations Calling Sequence ADMEquations() ADMEquations(keyword = ...) Parameters

 keyword = ... - optional, the left-hand side can be inert and the right-hand side can be true (default) or false; or the left-hand side can be output and the right hand side can be 4D (default) or 3D, to return with the free indices of spacetime or space kind. Description

 • 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 ${\mathrm{ds}}^{2}=-{\mathbit{\alpha }}^{2}{\mathrm{dt}}^{2}+{\mathbit{\gamma }}_{i,j}\left({\mathrm{dx}}^{i}+{\mathbit{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}\mathrm{dt}\right)\left({\mathrm{dx}}^{j}+{\mathbit{\beta }}_{\phantom{}\phantom{j}}^{\phantom{}j}\mathrm{dt}\right)$, 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. Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$$\mathrm{with}\left(\mathrm{ThreePlusOne}\right):$
 ${}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}\mathrm{lowercaselatin_is}\mathrm{letters to represent}\mathrm{space}\mathrm{indices}$
 $\mathrm{Defined as 4D spacetime tensors}\left(\mathrm{see ?Physics,ThreePlusOne}\right){,}{\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{▿}}_{{\mathrm{\mu }}}{,}{\mathbf{\Gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{\mathbf{\beta }}_{{\mathrm{\mu }}}{,}{\mathbit{n}}_{{\mathrm{\mu }}}{,}{\mathbit{t}}_{{\mathrm{\mu }}}{,}{\mathbf{Κ}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$
 $\mathrm{Changing the signature of spacetime from}\left(\mathrm{- - - +}\right)\mathrm{to}\left(\mathrm{+ + + -}\right)\mathrm{in order to match the signature customarily used in the ADM formalism}$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 ${}\mathrm{_______________________________________________________}$ (1)
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (2)

Set the Schwarzschild metric in spherical coordinates (see g_)

 > ${\mathrm{g_}}_{\mathrm{sc}}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{Systems of spacetime coordinates are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}\mathrm{The Schwarzschild metric in coordinates}{}\left[r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right]$
 $\mathrm{Parameters:}\left[m\right]$
 $\mathrm{Signature:}\left(\mathrm{+ + + -}\right)$
 ${}\mathrm{_______________________________________________________}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}-\frac{r}{2{}m-r}& 0& 0& 0\\ 0& {r}^{2}& 0& 0\\ 0& 0& {r}^{2}{}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}& 0\\ 0& 0& 0& \frac{2{}m-r}{r}\end{array}\right]\right)$ (3)

Check the current (default) value of the EnergyMomentum tensor

 > ${\mathrm{EnergyMomentum}}_{[]}$
 ${{\mathrm{EnergyMomentum}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{rrrr}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\right)$ (4)

This value coincides with the value of this tensor for a Schwarzschild solution in vacuum. Check the definition of the EnergyMomentum tensor, in a curved spacetime, it is the source of the gravitational field

 > ${\mathrm{EnergyMomentum}}_{\mathrm{definition}}$
 ${{\mathrm{Τ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}\frac{{{G}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}}{{8}{}{\mathrm{\pi }}}$ (5)

The above are actually Einstein's equations with the Einstein tensor on the right-hand side. Compute a tensor array with the equation components of this tensorial equation

 > $\mathrm{TensorArray}\left(\right)$
 $\left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]$ (6)

The ADM equations are the 3+1 split of Einstein's equations, presented by ADMEquations as a column vector of four equations: first the two constraints then the two evolution equations

 > $\mathrm{ADMEquations}\left(\right)$
 $\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]$ (7)

Where $ℒ$ represents the LieDerivative and $\mathbit{t}$ and $\mathbf{\beta }$ indexing $ℒ$ respectively represent the TimeVector and the Shift. Note the display of tensors in gray; they are all inert representations - to activate them use value.

Having set the Schwarzschild metric, if we now compute the components of each of these equations we should obtain $0=0$ for all of them. For that purpose, first compute these equations in active form (not inert as in the above), then compute a tensor array for them

 > $\mathrm{ADMEquations}\left(\mathrm{inert}=\mathrm{false}\right)$
 $\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)}{2{}m-r}-\frac{r{}{\mathrm{d_}}_{\mathrm{τ}}{}\left(r,\left[r,\mathrm{θ},\mathrm{φ},t\right]\right)}{{\left(2{}m-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)}{2{}m-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(2{}m-r\right)}^{2}}\right)}{{\left(-\frac{r}{2{}m-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)}{2{}m-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(2{}m-r\right)}^{2}}-\frac{2{}r{}{\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(2{}m-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(2{}m-r\right)}^{2}}\right)}{{\left(-\frac{r}{2{}m-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}{2{}m-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}{2{}m-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}{2{}m-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]$ (8)
 > $\mathrm{TensorArray}\left(,\mathrm{simplifier}=\mathrm{simplify}\right)$
 $\left[\begin{array}{c}0=0\\ \left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]\end{array}\right]$ (9)

To obtain the 3D version of these equations, pass the optional argument 3D that you can input as the product 3*D or as the symbol 3D

 > $\mathrm{ADMEquations}\left(\mathrm{inert}=\mathrm{false},\mathrm{output}=3\mathrm{D}\right)$
 $\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)}{2{}m-r}-\frac{r{}{\mathrm{d_}}_{\mathrm{τ}}{}\left(r,\left[r,\mathrm{θ},\mathrm{φ},t\right]\right)}{{\left(2{}m-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)}{2{}m-r}-\frac{r{}{\mathrm{gamma3_}}_{i,\mathrm{~upsilon}}{}{\mathrm{d_}}_{\mathrm{υ}}{}\left(r,\left[r,\mathrm{θ},\mathrm{φ},t\right]\right)}{{\left(2{}m-r\right)}^{2}}\right)}{{\left(-\frac{r}{2{}m-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)}{2{}m-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(2{}m-r\right)}^{2}}-\frac{2{}r{}{\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(2{}m-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(2{}m-r\right)}^{2}}\right)}{{\left(-\frac{r}{2{}m-r}\right)}^{3/2}}+\frac{-2{}{\mathrm{ExtrinsicCurvature}}_{\mathrm{α},i}{}{\mathrm{ExtrinsicCurvature}}_{\mathrm{~alpha},j}+{\mathrm{Ricci3}}_{i,j}}{\sqrt{-\frac{r}{2{}m-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}{2{}m-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}{2{}m-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]$ (10)

Check the repeated and free indices of each of these four equations

 > $\mathrm{map}\left(\mathrm{Check},,\mathrm{all}\right)$
 $\mathrm{The repeated indices per term are:}\left[\left\{\mathrm{...}\right\}{,}\left\{\mathrm{...}\right\}{,}\mathrm{...}\right]\mathrm{, the free indices are:}\left\{\mathrm{...}\right\}$
 $\mathrm{The products in the given expression check ok.}$
 $\mathrm{The repeated indices per term are:}\left[\left\{\mathrm{...}\right\}{,}\left\{\mathrm{...}\right\}{,}\mathrm{...}\right]\mathrm{, the free indices are:}\left\{\mathrm{...}\right\}$
 $\mathrm{The products in the given expression check ok.}$
 $\mathrm{The repeated indices per term are:}\left[\left\{\mathrm{...}\right\}{,}\left\{\mathrm{...}\right\}{,}\mathrm{...}\right]\mathrm{, the free indices are:}\left\{\mathrm{...}\right\}$
 $\mathrm{The repeated indices per term are:}\left[\left\{\mathrm{...}\right\}{,}\left\{\mathrm{...}\right\}{,}\mathrm{...}\right]\mathrm{, the free indices are:}\left\{\mathrm{...}\right\}$
 $\left[\begin{array}{c}\left(\left[\left\{\mathrm{μ},\mathrm{ν}\right\}\right],\left\{{}\right\}\right)=\left(\left[\left\{\mathrm{μ},\mathrm{ν}\right\}\right],\left\{{}\right\}\right)\\ \left(\left[\left\{\mathrm{ν}\right\}\right],\left\{i\right\}\right)=\left(\left[\left\{\mathrm{α},\mathrm{ν}\right\}\right],\left\{i\right\}\right)\\ \left(\left[\left\{\mathrm{τ}\right\},\left\{\mathrm{τ}\right\},\left\{\mathrm{τ}\right\}\right],\left\{i,j\right\}\right)=\left(\left[\left\{\mathrm{τ},\mathrm{υ}\right\},\left\{\mathrm{τ},\mathrm{υ}\right\},\left\{\mathrm{α}\right\},\left\{\mathrm{κ},\mathrm{λ},\mathrm{μ},\mathrm{ν},\mathrm{σ}\right\},\left\{\mathrm{τ}\right\},\left\{\mathrm{τ},\mathrm{υ}\right\},\left\{\mathrm{τ},\mathrm{υ}\right\}\right],\left\{i,j\right\}\right)\\ \left(\left[\left\{\mathrm{τ}\right\},\left\{\mathrm{τ}\right\},\left\{\mathrm{τ}\right\}\right],\left\{i,j\right\}\right)=\left(\left[\left\{{}\right\},\left\{\mathrm{τ}\right\},\left\{\mathrm{τ},\mathrm{υ}\right\},\left\{\mathrm{τ},\mathrm{υ}\right\}\right],\left\{i,j\right\}\right)\end{array}\right]$ (11)

In this result we see the free indices $i,j$ are 3D space indices, and all the repeated ones are 4D spacetime indices. For this 3D version of the ADM equations we also expect all of them identically satisfied and equal to 0

 > $\mathrm{TensorArray}\left(,\mathrm{simplifier}=\mathrm{simplify}\right)$
 $\left[\begin{array}{c}0=0\\ \left[\begin{array}{ccc}0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{ccc}0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\end{array}\right]\\ \left[\begin{array}{ccc}0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\\ 0=0& 0=0& 0=0\end{array}\right]\end{array}\right]$ (12)

Consider now the Lemaitre-Tolman-Bondi metric, that in the Maple database of solutions to Einstein's equations can be retrieved directly using a portion of the word Tolman as an index to the metric g_

 > ${\mathrm{g_}}_{\mathrm{tol}}$
 ${}\mathrm{_______________________________________________________}$
 ${}\mathrm{The Tolman metric in coordinates}{}\left[r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right]$
 $\mathrm{Parameters:}\left[R{}\left(t{,}r\right){,}E{}\left(r\right)\right]$
 $\mathrm{Signature:}\left(\mathrm{+ + + -}\right)$
 ${}\mathrm{_______________________________________________________}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}\frac{{\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)}^{2}}{1+2{}E{}\left(r\right)}& 0& 0& 0\\ 0& {R{}\left(t,r\right)}^{2}& 0& 0\\ 0& 0& {R{}\left(t,r\right)}^{2}{}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}& 0\\ 0& 0& 0& -1\end{array}\right]\right)$ (13)
 > $\mathrm{CompactDisplay}\left(\right)$
 ${E}{}\left({r}\right){}{\mathrm{will now be displayed as}}{}{E}$
 ${R}{}\left({t}{,}{r}\right){}{\mathrm{will now be displayed as}}{}{R}$ (14)

The EnergyMomentum tensor for this metric is given by

 > ${\mathrm{EnergyMomentum}}_{\mathrm{~mu},\mathrm{ν}}=-\mathit{rho__M}\left(t,r\right){\mathrm{g_}}_{\mathrm{~mu},0}{\mathrm{g_}}_{\mathrm{ν},\mathrm{~0}}-\mathit{rho__Lambda}{\mathrm{g_}}_{\mathrm{~mu},\mathrm{ν}}$
 ${{\mathrm{Τ}}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}{\mathrm{\nu }}}^{\phantom{{}}{\mathrm{\mu }}\phantom{{\mathrm{\nu }}}}{=}{-}\mathrm{ρ__M}{}\left({t}{,}{r}\right){}{{\mathrm{g_}}}_{{4}{,}{\mathrm{~mu}}}{}{{\mathrm{g_}}}_{{\mathrm{ν}}{,}{\mathrm{~4}}}{-}\mathrm{ρ__Λ}{}{{\mathrm{g_}}}_{{\mathrm{ν}}{,}{\mathrm{~mu}}}$ (15)

where ${\mathrm{\rho }}_{M}\left(t,r\right)$ is the matter density, ${u}_{}^{\mathrm{\mu }}={\mathrm{\delta }}_{0}^{\mathrm{\mu }}$ is the 4-velocity of the matter that is comoving and we keep the vacuum energy ${\mathrm{\rho }}_{\mathrm{\Lambda }}=\mathrm{constant}$ for illustration purposes only. Define this tensor and use a CompactDisplay for ${\mathrm{\rho }}_{M}\left(t,r\right)$

 > $\mathrm{Define}\left(\right)$
 $\mathrm{Defined objects with tensor properties}$
 $\left\{{\mathbit{▿}}_{{\mathrm{\mu }}}{,}{{▿}}_{{\mathrm{\mu }}}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{,}{{\mathrm{\sigma }}}_{{\mathrm{\mu }}}{,}{{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{\mathbf{\beta }}_{{\mathrm{\mu }}}{,}{{C}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{{\partial }}_{{\mathrm{\mu }}}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\gamma }}}_{{i}{,}{j}}{,}{{\mathrm{\Gamma }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{\mathbf{\Gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{{G}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{Τ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\epsilon }}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{\mathbit{t}}_{{\mathrm{\mu }}}{,}{\mathbf{Κ}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{n}}_{{\mathrm{\mu }}}{,}{{X}}_{{\mathrm{\mu }}}\right\}$ (16)
 > $\mathrm{CompactDisplay}\left(\right)$
 $\mathrm{ρ__M}{}\left({t}{,}{r}\right){}{\mathrm{will now be displayed as}}{}\mathrm{ρ__M}$ (17)

Take now the 4D form of Einstein's equations and derive an expression for ${\mathrm{\rho }}_{M}\left(t,r\right)$ as a function of $R,E$

 > $\mathrm{isolate}\left(,{\mathrm{Einstein}}_{\mathrm{μ},\mathrm{ν}}\right)$
 ${{G}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}{8}{}{{\mathrm{Τ}}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{\mathrm{\pi }}$ (18)
 > $\mathrm{EQ4}≔\mathrm{TensorArray}\left(\right)$
 $\left[\begin{array}{cccc}-\frac{\mathrm{ρ__Λ}{}{\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)}^{2}}{1+2{}E{}\left(r\right)}=-\frac{1}{8}{}\frac{{\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)}^{2}{}\left(2{}R{}\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}-2{}E{}\left(r\right)\right)}{{R{}\left(t,r\right)}^{2}{}\left(1+2{}E{}\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}{}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}=-\frac{1}{8}{}\frac{R{}\left(t,r\right){}{\mathrm{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}-2{}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)+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]$ (19)

The relationship we are looking for is in ${\mathrm{EQ4}}_{4,4}$ and can be simplified further if we introduce $M\left(r\right)$, the gravitational mass within the comoving sphere at radius $r$

 > $M\left(r\right)=-\frac{1\left(-{\left(\frac{\partial }{\partial t}R\left(t,r\right)\right)}^{2}+2E\left(r\right)\right)R\left(t,r\right)}{2}$
 ${M}{}\left({r}\right){=}{-}\frac{{1}}{{2}}{}\left({-}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{+}{2}{}{E}{}\left({r}\right)\right){}{R}{}\left({t}{,}{r}\right)$ (20)

So simplify the expression obtained for ${\mathrm{\rho }}_{M}+{\mathrm{\rho }}_{\mathrm{\Lambda }}$ introducing $M\left(r\right)$ and eliminating $E$ (see simplify, siderelations)

 > $\mathrm{simplify}\left({\mathrm{EQ4}}_{4,4},\left\{\right\},\left\{E\left(r\right)\right\}\right)$
 $\mathrm{ρ__M}{}\left({t}{,}{r}\right){+}\mathrm{ρ__Λ}{=}\frac{{1}}{{4}}{}\frac{{\mathrm{diff}}{}\left({M}{}\left({r}\right){,}{r}\right)}{{{R}{}\left({t}{,}{r}\right)}^{{2}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}$ (21)

Consider now the 3 + 1 ADM equations equivalent to the 4D Einstein equations $\mathrm{EQ4}$, and show that it results in the same expression for ${\mathrm{\rho }}_{M}+{\mathrm{\rho }}_{\mathrm{\Lambda }}$ and that both systems of equations are one and the same. Start from the active form of the ADM equations

 > $\mathrm{eq}≔\mathrm{ADMEquations}\left(\mathrm{inert}=\mathrm{false}\right)$
 $\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]$ (22)

The expression for ${\mathrm{\rho }}_{M}+{\mathrm{\rho }}_{\mathrm{\Lambda }}$ in terms of $M\left(r\right)$ is obtained now from ${\mathrm{eq}}_{1}$

 > ${\mathrm{eq}}_{1}$
 $\frac{{\left({2}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{R}{}\left({t}{,}{r}\right)\right)}^{{2}}}{{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right)}^{{2}}{}{{R}{}\left({t}{,}{r}\right)}^{{2}}}{-}{{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{α}}{,}{\mathrm{β}}}{}{{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{~alpha}}{,}{\mathrm{~beta}}}{-}\frac{{4}{}\left(\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{E}{}\left({r}\right){+}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)\right)}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{{R}{}\left({t}{,}{r}\right)}^{{2}}}{=}{16}{}{\mathrm{π}}{}{{\mathrm{UnitNormalVector}}}_{{\mathrm{α}}}{}{{\mathrm{UnitNormalVector}}}_{{\mathrm{β}}}{}{{\mathrm{EnergyMomentum}}}_{{\mathrm{~alpha}}{,}{\mathrm{~beta}}}$ (23)
 > $\mathrm{SumOverRepeatedIndices}\left({\mathrm{eq}}_{1}\right)$
 ${-}\frac{{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}{{\mathrm{csc}}{}\left({\mathrm{θ}}\right)}^{{2}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{3}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{4}{}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){+}{4}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{E}{}\left({r}\right){+}{4}{}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{{R}{}\left({t}{,}{r}\right)}^{{2}}}{=}{16}{}{\mathrm{π}}{}\left(\mathrm{ρ__M}{}\left({t}{,}{r}\right){+}\mathrm{ρ__Λ}\right)$ (24)
 > $\mathrm{simplify}\left(\mathrm{isolate}\left(,\mathit{rho__M}\left(t,r\right)+\mathit{rho__Lambda}\right)\right)$
 $\mathrm{ρ__M}{}\left({t}{,}{r}\right){+}\mathrm{ρ__Λ}{=}\frac{{1}}{{8}}{}\frac{{2}{}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){+}\left({\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{2}{}{E}{}\left({r}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){-}{2}{}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)}{{{R}{}\left({t}{,}{r}\right)}^{{2}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}$ (25)
 > $\mathrm{simplify}\left(,\left\{\right\},\left\{E\left(r\right)\right\}\right)$
 $\mathrm{ρ__M}{}\left({t}{,}{r}\right){+}\mathrm{ρ__Λ}{=}\frac{{1}}{{4}}{}\frac{{\mathrm{diff}}{}\left({M}{}\left({r}\right){,}{r}\right)}{{{R}{}\left({t}{,}{r}\right)}^{{2}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}$ (26)

The second equation, ${\mathrm{eq}}_{2}$, is identically satisfied

 > ${\mathrm{eq}}_{2}$
 ${{\mathrm{D3_}}}_{{\mathrm{β}}}{}\left({{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{μ}}{,}{\mathrm{~beta}}}{,}\left[{X}\right]\right){-}{{\mathrm{gamma3_}}}_{{\mathrm{μ}}{,}{\mathrm{~tau}}}{}\left({-}\frac{{2}{}\left(\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({r}{,}\left[{X}\right]\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({t}{,}\left[{X}\right]\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}{2}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left(\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{r}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({r}{,}\left[{X}\right]\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({t}{,}\left[{X}\right]\right)\right){+}\left(\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{r}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({r}{,}\left[{X}\right]\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({t}{,}\left[{X}\right]\right)\right){}{R}{}\left({t}{,}{r}\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({R}{}\left({t}{,}{r}\right){,}\left[{X}\right]\right)}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)}{+}\frac{\left({2}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{R}{}\left({t}{,}{r}\right)\right){}\left(\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{r}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({r}{,}\left[{X}\right]\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({t}{,}\left[{X}\right]\right)\right)}{{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right)}^{{2}}{}{R}{}\left({t}{,}{r}\right)}{+}\frac{\left({2}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{R}{}\left({t}{,}{r}\right)\right){}{{\mathrm{d_}}}_{{\mathrm{τ}}}{}\left({R}{}\left({t}{,}{r}\right){,}\left[{X}\right]\right)}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{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}}}$ (27)
 > $\mathrm{TensorArray}\left({\mathrm{eq}}_{2},\mathrm{simplifier}=\mathrm{simplify}\right)$
 $\left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\end{array}\right]$ (28)

The fourth equation, ${\mathrm{eq}}_{4}$, is also identically satisfied (basically, this is the definition of the ExtrinsicCurvature)

 > ${\mathrm{eq}}_{4}$
 ${\mathbit{t}}_{\phantom{{}}\phantom{{\mathrm{\tau }}}}^{\phantom{{}}{\mathrm{\tau }}}{}{{▿}}_{{\mathrm{\tau }}}{}\left({\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}\right){+}{\mathbf{\gamma }}_{{\mathrm{\nu }}{,}{\mathrm{\tau }}}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\tau }}}{\mathrm{\mu }}{,}{\mathrm{\upsilon }}}^{\phantom{{}}{\mathrm{\tau }}\phantom{{\mathrm{\mu }}{,}{\mathrm{\upsilon }}}}{}{\mathbit{t}}_{\phantom{{}}\phantom{{\mathrm{\upsilon }}}}^{\phantom{{}}{\mathrm{\upsilon }}}{+}{\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\tau }}}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\tau }}}{\mathrm{\nu }}{,}{\mathrm{\upsilon }}}^{\phantom{{}}{\mathrm{\tau }}\phantom{{\mathrm{\nu }}{,}{\mathrm{\upsilon }}}}{}{\mathbit{t}}_{\phantom{{}}\phantom{{\mathrm{\upsilon }}}}^{\phantom{{}}{\mathrm{\upsilon }}}{=}{-}{2}{}{\mathbf{Κ}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{+}{\mathbf{\beta }}_{\phantom{{}}\phantom{{\mathrm{\tau }}}}^{\phantom{{}}{\mathrm{\tau }}}{}{{▿}}_{{\mathrm{\tau }}}{}\left({\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}\right){+}{\mathbf{\gamma }}_{{\mathrm{\nu }}{,}{\mathrm{\tau }}}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\tau }}}{\mathrm{\mu }}{,}{\mathrm{\upsilon }}}^{\phantom{{}}{\mathrm{\tau }}\phantom{{\mathrm{\mu }}{,}{\mathrm{\upsilon }}}}{}{\mathbf{\beta }}_{\phantom{{}}\phantom{{\mathrm{\upsilon }}}}^{\phantom{{}}{\mathrm{\upsilon }}}{+}{\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\tau }}}{}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\tau }}}{\mathrm{\nu }}{,}{\mathrm{\upsilon }}}^{\phantom{{}}{\mathrm{\tau }}\phantom{{\mathrm{\nu }}{,}{\mathrm{\upsilon }}}}{}{\mathbf{\beta }}_{\phantom{{}}\phantom{{\mathrm{\upsilon }}}}^{\phantom{{}}{\mathrm{\upsilon }}}$ (29)
 > $\mathrm{TensorArray}\left({\mathrm{eq}}_{2},\mathrm{simplifier}=\mathrm{simplify}\right)$
 $\left[\begin{array}{cccc}0=0& 0=0& 0=0& 0=0\end{array}\right]$ (30)

So it is in ${\mathrm{eq}}_{3}$ where the evolution of the system is encoded, in terms of the functions $\left\{{\mathrm{\rho }}_{M},E,R\right\}$ and their derivatives

 > ${\mathrm{eq}}_{3}$
 ${{\mathrm{TimeVector}}}_{{\mathrm{~tau}}}{}{{\mathrm{D_}}}_{{\mathrm{τ}}}{}\left({{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{,}\left[{X}\right]\right){+}{{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{ν}}{,}{\mathrm{τ}}}{}{{\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({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}{R}{}\left({t}{,}{r}\right)\right){}{{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{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[{X}\right]\right){+}{{\mathrm{ExtrinsicCurvature}}}_{{\mathrm{ν}}{,}{\mathrm{τ}}}{}{{\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}}}$ (31)
 > $\mathrm{EQ3}≔\mathrm{TensorArray}\left({\mathrm{eq}}_{3},\mathrm{simplifier}=\mathrm{simplify}\right)$
 $\left[\begin{array}{cccc}\frac{-{\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){}\left(\frac{\partial }{\partial t}{}\left(\frac{\partial }{\partial t}{}\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)\right)\right)}{1+2{}E{}\left(r\right)}=\frac{-{\left(\frac{\partial }{\partial t}{}\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)\right)}^{2}{}R{}\left(t,r\right)+2{}\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){}\left(\frac{\partial }{\partial t}{}R{}\left(t,r\right)\right)-8{}\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right){}\left(\mathrm{π}{}R{}\left(t,r\right){}\left(\mathrm{ρ__Λ}+\frac{1}{2}{}\mathrm{ρ__M}{}\left(t,r\right)\right){}\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)+\frac{1}{4}{}\frac{ⅆ}{ⅆr}{}E{}\left(r\right)\right)}{R{}\left(t,r\right){}\left(1+2{}E{}\left(r\right)\right)}& 0=0& 0=0& 0=0\\ 0=0& -{\left(\frac{\partial }{\partial t}{}R{}\left(t,r\right)\right)}^{2}-R{}\left(t,r\right){}\left(\frac{\partial }{\partial t}{}\left(\frac{\partial }{\partial t}{}R{}\left(t,r\right)\right)\right)=\frac{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(-8{}\left(\mathrm{ρ__Λ}+\frac{1}{2}{}\mathrm{ρ__M}{}\left(t,r\right)\right){}\mathrm{π}{}{R{}\left(t,r\right)}^{2}-2{}E{}\left(r\right)\right){}\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)-\left(\frac{ⅆ}{ⅆr}{}E{}\left(r\right)\right){}R{}\left(t,r\right)}{\frac{\partial }{\partial r}{}R{}\left(t,r\right)}& 0=0& 0=0\\ 0=0& 0=0& -{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}{}\left({\left(\frac{\partial }{\partial t}{}R{}\left(t,r\right)\right)}^{2}+R{}\left(t,r\right){}\left(\frac{\partial }{\partial t}{}\left(\frac{\partial }{\partial t}{}R{}\left(t,r\right)\right)\right)\right)=-\frac{{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}{}\left(-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(8{}\left(\mathrm{ρ__Λ}+\frac{1}{2}{}\mathrm{ρ__M}{}\left(t,r\right)\right){}\mathrm{π}{}{R{}\left(t,r\right)}^{2}+2{}E{}\left(r\right)\right){}\left(\frac{\partial }{\partial r}{}R{}\left(t,r\right)\right)+\left(\frac{ⅆ}{ⅆr}{}E{}\left(r\right)\right){}R{}\left(t,r\right)\right)}{\frac{\partial }{\partial r}{}R{}\left(t,r\right)}& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]$ (32)

To demonstrate that this system of equations $\mathrm{EQ3}$ together with the constraint ${\mathrm{eq}}_{1}$ (23), that is the ADMEquations decomposition, is equivalent to the 4D system of equations $\mathrm{EQ4}$ obtained directly from Einstein's equations in (19), it suffices to show that each of these two systems entirely reduces the other one. For this purpose, convert these arrays of equations to sets of equations

 > $\mathrm{EQ4}≔\mathrm{convert}\left(\mathrm{EQ4},\mathrm{setofequations}\right)$
 $\left\{{0}{=}{0}{,}{-}\mathrm{ρ__Λ}{}{{R}{}\left({t}{,}{r}\right)}^{{2}}{=}{-}\frac{{1}}{{8}}{}\frac{{R}{}\left({t}{,}{r}\right){}\left(\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right){,}{t}\right)\right){}{R}{}\left({t}{,}{r}\right){+}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){-}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right)\right)}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}{,}{-}\mathrm{ρ__Λ}{}{{R}{}\left({t}{,}{r}\right)}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{=}{-}\frac{{1}}{{8}}{}\frac{{R}{}\left({t}{,}{r}\right){}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}\left(\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right){,}{t}\right)\right){}{R}{}\left({t}{,}{r}\right){+}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){-}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right)\right)}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}{,}{-}\frac{\mathrm{ρ__Λ}{}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right)}^{{2}}}{{1}{+}{2}{}{E}{}\left({r}\right)}{=}{-}\frac{{1}}{{8}}{}\frac{{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right)}^{{2}}{}\left({2}{}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right){,}{t}\right)\right){+}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{2}{}{E}{}\left({r}\right)\right)}{{{R}{}\left({t}{,}{r}\right)}^{{2}}{}\left({1}{+}{2}{}{E}{}\left({r}\right)\right){}{\mathrm{π}}}{,}\mathrm{ρ__M}{}\left({t}{,}{r}\right){+}\mathrm{ρ__Λ}{=}{-}\frac{{1}}{{8}}{}\frac{{-}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{2}{}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){+}{2}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{E}{}\left({r}\right){+}{2}{}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)}{{{R}{}\left({t}{,}{r}\right)}^{{2}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}\right\}$ (33)
 > $\mathrm{EQ3}≔\mathrm{convert}\left(\mathrm{EQ3},\mathrm{setofequations}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∪\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left\{\right\}$
 $\left\{{0}{=}{0}{,}\frac{{-}{\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right)}^{{2}}{-}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right){,}{t}\right)\right)}{{1}{+}{2}{}{E}{}\left({r}\right)}{=}\frac{{-}{\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right)}^{{2}}{}{R}{}\left({t}{,}{r}\right){+}{2}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){-}{8}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}\left({\mathrm{π}}{}{R}{}\left({t}{,}{r}\right){}\left(\mathrm{ρ__Λ}{+}\frac{{1}}{{2}}{}\mathrm{ρ__M}{}\left({t}{,}{r}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\frac{{1}}{{4}}{}{\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right)}{{R}{}\left({t}{,}{r}\right){}\left({1}{+}{2}{}{E}{}\left({r}\right)\right)}{,}{-}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}\left({\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{+}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right){,}{t}\right)\right)\right){=}{-}\frac{{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}{}\left({-}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){+}\left({8}{}\left(\mathrm{ρ__Λ}{+}\frac{{1}}{{2}}{}\mathrm{ρ__M}{}\left({t}{,}{r}\right)\right){}{\mathrm{π}}{}{{R}{}\left({t}{,}{r}\right)}^{{2}}{+}{2}{}{E}{}\left({r}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){+}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)\right)}{{\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)}{,}{-}{\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right){,}{t}\right)\right){=}\frac{{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){+}\left({-}{8}{}\left(\mathrm{ρ__Λ}{+}\frac{{1}}{{2}}{}\mathrm{ρ__M}{}\left({t}{,}{r}\right)\right){}{\mathrm{π}}{}{{R}{}\left({t}{,}{r}\right)}^{{2}}{-}{2}{}{E}{}\left({r}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){-}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)}{{\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)}{,}\mathrm{ρ__M}{}\left({t}{,}{r}\right){+}\mathrm{ρ__Λ}{=}\frac{{1}}{{8}}{}\frac{{2}{}{R}{}\left({t}{,}{r}\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right){}\left({\mathrm{diff}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right){,}{t}\right)\right){+}\left({\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{t}\right)\right)}^{{2}}{-}{2}{}{E}{}\left({r}\right)\right){}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){-}{2}{}\left({\mathrm{diff}}{}\left({E}{}\left({r}\right){,}{r}\right)\right){}{R}{}\left({t}{,}{r}\right)}{{{R}{}\left({t}{,}{r}\right)}^{{2}}{}\left({\mathrm{diff}}{}\left({R}{}\left({t}{,}{r}\right){,}{r}\right)\right){}{\mathrm{π}}}\right\}$ (34)

The differential reductions can now be performed using PDEtools:-ReducedForm

 > $\mathrm{simplify}\left(\mathrm{PDEtools}:-\mathrm{ReducedForm}\left(\mathrm{EQ4},\mathrm{EQ3}\right)\right)$
 ${\mathrm{casesplit/ans}}{}\left(\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{}\right]\right)$ (35)

The reduction the other way around

 > $\mathrm{simplify}\left(\mathrm{PDEtools}:-\mathrm{ReducedForm}\left(\mathrm{EQ3},\mathrm{EQ4}\right)\right)$
 ${\mathrm{casesplit/ans}}{}\left(\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{}\right]\right)$ (36) See Also References

  Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
  Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.
  Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.
  Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf. Compatibility

 • The Physics[ThreePlusOne][ADMEquations] command was introduced in Maple 2017.