The LapseAndShiftConditions() command returns a list of keywords associated to the most common slicing and spatial gauge conditions for the Lapse and Shift that fix the freedom of choice of coordinates of the 4D formulation of Einstein's equations describing gravity.

The LapseAndShiftConditions(keyword) returns a Vector column with the equations for the Lapse and Shift associated to that keyword, expressed in terms of the inert forms of the tensors involved; these are mainly the ExtrinsicCurvature, the gamma3_ metric of the spacelike hypersurface of the 3+1 splitting, the NormalUnitVector, and the EnergyMomentum tensor.

When the option evaluate is passed, all the inert forms of tensors are first transformed into active by calling value, then the Vector column of equations is transformed into an array of equations calling TensorArray. If in addition the option simplifier = ... is given, the right-hand side of this option is applied to each of the elements of the resulting array.

The evaluated equations returned for the Lapse and Shift can also then be expressed in terms of the components of the 4D metric $g_{\mathrm{\mu },\mathrm{\nu }}$; for that purpose use $\mathrm{convert}\left(\mathrm{equations}\,\mathrm{g\_}\right)$. This rewriting in terms of the metric can also be done for the non-evaluated equations returned by default (when no options are passed), and can be applied selectively to one or several of the tensors appearing in the equations, e.g. via $\mathrm{convert}\left(\mathrm{equations}\,\mathrm{g\_}\,\mathrm{only}=\left\{\mathrm{ExtrinsicCurvature}\right\}\right)$ where the set can contain one or several tensors entering equations.

Depending on the keyword and the 4D metric set, in some cases it is possible to solve these slicing and gauge conditions for the Lapse and Shift exactly. To attempt that solution you can either convert the array of equations returned into a set of equations, then use PDEtools:-Solve to tackle them, or just pass the option output = solution.

When the equations for the Lapse and Shift can be solved, by passing the optional argument setconditions, LapseAndShiftConditions will automatically set these values using Setup and its lapseandshift keyword.

You can also set the value of the lapseandshift directly using Setup; there are three possible values: 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 (e.g. ExtrinsicCurvature) and so of tensorial expressions involving them (e.g. the 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.

Depending on the manipulations to be done with the equations for the Lapse and Shift it is frequently convenient to set the value of lapseandshift via $\mathrm{Setup}\left(\mathrm{lapseandshift}=\mathrm{arbitrary}\right)$ in order to avoid their spontaneous evaluation in terms of the components of the metric $g_{0,0}$ and $g_{0,j}$ during intermediate steps.

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

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

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

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

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

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

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

$\left[\mathrm{ADMEquations}\,\mathrm{Christoffel3}\,\mathrm{D3\_}\,\mathrm{ExtrinsicCurvature}\,\mathrm{Lapse}\,\mathrm{LapseAndShiftConditions}\,\mathrm{Ricci3}\,\mathrm{Riemann3}\,\mathrm{Shift}\,\mathrm{TimeVector}\,\mathrm{UnitNormalVector}\,\mathrm{gamma3\_}\right]$

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

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

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

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]$

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

$\left[\mathrm{advectiveonepluslog}\,\mathrm{generalizedharmonicslicing}\,\mathrm{geodesicslicing}\,\mathrm{harmoniccoordinates}\,\mathrm{harmonicslicing}\,\mathrm{kdriver}\,\mathrm{maximalslicing}\,\mathrm{onepluslog}\right]$

$$\begin{gathered} \mathbf{for}\mathrm{key}\mathbf{in}\mathrm{LapseAndShiftConditions}\left(\right)\mathbf{do} \\ \mathrm{key}=\mathrm{LapseAndShiftConditions}\left(\mathrm{key}\right) \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\mathrm{advectiveonepluslog}=\left[\begin{array}{c}{ℒ}_{\mathit{n}}\left(\mathbf{\alpha }\right)=-2{\mathbf{{\rm K}}}_{\mathrm{trace}}\\ {\partial }_0\left(\mathbf{\alpha }\right)-{\mathbf{\beta }}_{\phantom{}\phantom{j}}^{\phantom{}j}\left({\partial }_j\left(\mathbf{\alpha }\right)\right)=-2\mathbf{\alpha }{\mathbf{{\rm K}}}_{\mathrm{trace}}\end{array}\right]$

$\mathrm{generalizedharmonicslicing}=\left[\begin{array}{c}{\partial }_0\left(\mathbf{\alpha }\right)=-{\mathbf{\alpha }}^{2}f\left(\mathbf{\alpha }\right){\mathbf{{\rm K}}}_{\mathrm{trace}}\\ {\mathbf{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}=0\end{array}\right]$

$\mathrm{geodesicslicing}=\left[\begin{array}{c}\mathbf{\alpha }=1\\ {\mathbf{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}=0\end{array}\right]$

$\mathrm{harmoniccoordinates}=\left[\begin{array}{c}{\partial }_0\left(\mathbf{\alpha }\right)-{\mathbf{\beta }}_{\phantom{}\phantom{j}}^{\phantom{}j}\left({\partial }_j\left(\mathbf{\alpha }\right)\right)=-{\mathbf{\alpha }}^2{\mathbf{{\rm K}}}_{\mathrm{trace}}\\ {\partial }_0\left({\mathbf{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}\right)-{\mathbf{\beta }}_{\phantom{}\phantom{j}}^{\phantom{}j}\left({\partial }_j\left({\mathbf{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}\right)\right)=-{\mathbf{\alpha }}^2\left({\mathbf{\gamma }}_{\phantom{}\phantom{i,j}}^{\phantom{}i,j}\left({\partial }_j\left(\ln \left(\mathbf{\alpha }\right)\right)\right)+{\mathbf{\gamma }}_{\phantom{}\phantom{j,k}}^{\phantom{}j,k}{\mathbf{\Gamma }}_{\phantom{}\phantom{i}j,k}^{\phantom{}i\phantom{j,k}}\right)\end{array}\right]$

$\mathrm{harmonicslicing}=\left[\begin{array}{c}{\partial }_0\left(\mathbf{\alpha }\right)=-{\mathbf{\alpha }}^2{\mathbf{{\rm K}}}_{\mathrm{trace}}\\ \mathbf{\alpha }=C\left(X\right)\sqrt{\left|\mathbf{\gamma }\right|}\\ {\mathbf{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}=0\end{array}\right]$

$\mathrm{kdriver}=\left[\begin{array}{c}{\partial }_0\left({\mathbf{{\rm K}}}_{\mathrm{trace}}\right)=c{\mathbf{{\rm K}}}_{\mathrm{trace}}\\ {\partial }_0\left(\mathbf{\alpha }\right)=-\mathrm{\epsilon }\left({\partial }_0\left({\mathbf{{\rm K}}}_{\mathrm{trace}}\right)+c{\mathbf{{\rm K}}}_{\mathrm{trace}}\right)\\ 0<c,0<\mathrm{\epsilon }\end{array}\right]$

$\mathrm{maximalslicing}=\left[\begin{array}{c}{\mathbf{{\rm K}}}_{\mathrm{trace}}=0,{\partial }_0\left({\mathbf{{\rm K}}}_{\mathrm{trace}}\right)=0\\ {▿}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left({▿}_{\mathrm{\mu }}\left(\mathbf{\alpha }\right)\right)=\mathbf{\alpha }\left({\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}+4\mathrm{\pi }\left({\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\alpha }}}^{\phantom{}\mathrm{\mu },\mathrm{\alpha }}{\mathrm{{\rm T}}}_{\mathrm{\nu },\mathrm{\alpha }}+{\mathit{n}}_{\mathrm{\mu }}{\mathit{n}}_{\mathrm{\nu }}{\mathrm{{\rm T}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\right)\right)\end{array}\right]$

$\mathrm{onepluslog}=\left[\begin{array}{c}{\partial }_0\left(\mathbf{\alpha }\right)=-2\mathbf{\alpha }{\mathbf{{\rm K}}}_{\mathrm{trace}}\\ \mathbf{\alpha }=1+\ln \left(\left|\mathbf{\gamma }\right|\right)\\ {\mathbf{\beta }}_{\phantom{}\phantom{i}}^{\phantom{}i}=0\end{array}\right]$

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

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

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

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

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

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

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

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

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

$\mathrm{LapseAndShiftConditions}\left(\mathrm{kdriver}\right)$

$\left[\begin{array}{c}{\partial }_0\left({\mathbf{{\rm K}}}_{\mathrm{trace}}\right)=c{\mathbf{{\rm K}}}_{\mathrm{trace}}\\ {\partial }_0\left(\mathbf{\alpha }\right)=-\mathrm{\epsilon }\left({\partial }_0\left({\mathbf{{\rm K}}}_{\mathrm{trace}}\right)+c{\mathbf{{\rm K}}}_{\mathrm{trace}}\right)\\ 0<c,0<\mathrm{\epsilon }\end{array}\right]$

$\mathrm{LapseAndShiftConditions}\left(\mathrm{kdriver}\&comma;\mathrm{evaluate}\right)$

$\left[\begin{array}{c}\frac{2\left(-{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\mathbf{\alpha }r\left(\frac{{\partial }^2}{\partial t\partial \mathrm{\theta }}\mathbf{\alpha }\right)-\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\mathbf{\alpha }{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}r\left(\frac{{\partial }^2}{\partial \mathrm{\phi }\partial t}\mathbf{\alpha }\right)-\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\mathbf{\alpha }r{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\left(\frac{{\partial }^2}{\partial r\partial t}\mathbf{\alpha }\right)-\left(-\stackrel{\mathbf{..}}{\mathbf{\alpha }}r\mathbf{\alpha }+\stackrel{\mathbf{.}}{\mathbf{\alpha }}\left(2\stackrel{\mathbf{.}}{\mathbf{\alpha }}r-2\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\theta }}\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}-2\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\phi }}\right)r{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}-2\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;r}\right)r{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}+\mathbf{\alpha }\left(\cot \left(\mathrm{\theta }\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}+2{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\right)\right)\right)\left(m-\frac{r}{2}\right)\right)}{r{\mathbf{\alpha }}^3\left(2m-r\right)}=-\frac{2c\left({\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\theta }}\right)+{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\phi }}\right)+{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;r}\right)-\left(m-\frac{r}{2}\right)\left(\stackrel{\mathbf{.}}{\mathbf{\alpha }}r+\mathbf{\alpha }\left(\cot \left(\mathrm{\theta }\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}+2{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\right)\right)\right)}{r{\mathbf{\alpha }}^2\left(2m-r\right)}\\ \stackrel{\mathbf{.}}{\mathbf{\alpha }}=-\frac{2\left(-{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\mathbf{\alpha }r\left(\frac{{\partial }^2}{\partial t\partial \mathrm{\theta }}\mathbf{\alpha }\right)-\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\mathbf{\alpha }{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}r\left(\frac{{\partial }^2}{\partial \mathrm{\phi }\partial t}\mathbf{\alpha }\right)-\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\mathbf{\alpha }r{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\left(\frac{{\partial }^2}{\partial r\partial t}\mathbf{\alpha }\right)+r\mathbf{\alpha }\left(m-\frac{r}{2}\right)\stackrel{\mathbf{..}}{\mathbf{\alpha }}+\left(-2mr+r^2\right){{\stackrel{\mathbf{.}}{\mathbf{\alpha }}}^{}}^2+\left(2\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\theta }}\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}+2\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\phi }}\right)r{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}+2\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;r}\right)r{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}+\mathbf{\alpha }\left(-\cot \left(\mathrm{\theta }\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}+cr-2{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\right)\right)\left(m-\frac{r}{2}\right)\stackrel{\mathbf{.}}{\mathbf{\alpha }}+\left(-{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\theta }}\right)-{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\phi }}\right)-{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;r}\right)+\left(m-\frac{r}{2}\right)\left(\cot \left(\mathrm{\theta }\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}+2{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\right)\mathbf{\alpha }\right)c\mathbf{\alpha }\right)\mathrm{\epsilon }}{r{\mathbf{\alpha }}^3\left(2m-r\right)}\\ 0<c,0<\mathrm{\epsilon }\end{array}\right]$

$\mathrm{LapseAndShiftConditions}\left(\mathrm{harmonicslicing}\&comma;\mathrm{evaluate}\right)$

$\left[\begin{array}{c}0=0\\ \stackrel{\mathbf{.}}{\mathbf{\alpha }}=\frac{2\left({\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\theta }}\right)+{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;\mathrm{\phi }}\right)+{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}r\left(\frac{\left(-{\mathbf{\alpha }}^2-1\right)r}{2}+m\right)\left(\frac{\&DifferentialD;\mathbf{\alpha }}{\&DifferentialD;r}\right)-\left(m-\frac{r}{2}\right)\left(\stackrel{\mathbf{.}}{\mathbf{\alpha }}r+\mathbf{\alpha }\left(\cot \left(\mathrm{\theta }\right)r{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}+2{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}\right)\right)\right)}{r\left(2m-r\right)}\\ \mathbf{\alpha }=\frac{C\left(X\right)r^2\sin \left(\mathrm{\theta }\right)}{\mathbf{\alpha }}\\ \left[\begin{array}{ccc}{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}=0& {\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}=0& {\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}=0\end{array}\right]\end{array}\right]$

$\mathrm{LapseAndShiftConditions}\left(\mathrm{harmonicslicing}\&comma;\mathrm{output}=\mathrm{solution}\right)$

$\left\{\mathbf{\alpha }=-\sqrt{\sin \left(\mathrm{\theta }\right)}\sqrt{\mathrm{f\_\_1}\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)}r\&comma;m=m\&comma;{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}=0\&comma;{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}=0\&comma;{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}=0\&comma;C\left(X\right)=\mathrm{f\_\_1}\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)\right\},\left\{\mathbf{\alpha }=\sqrt{\sin \left(\mathrm{\theta }\right)}\sqrt{\mathrm{f\_\_1}\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)}r\&comma;m=m\&comma;{\mathbf{\beta }}_{\phantom{}\phantom{1}}^{\phantom{}1}=0\&comma;{\mathbf{\beta }}_{\phantom{}\phantom{2}}^{\phantom{}2}=0\&comma;{\mathbf{\beta }}_{\phantom{}\phantom{3}}^{\phantom{}3}=0\&comma;C\left(X\right)=\mathrm{f\_\_1}\left(r\&comma;\mathrm{\theta }\&comma;\mathrm{\phi }\right)\right\}$

[1] Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.

[2] Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.

[3] Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf.