The ExtrinsicCurvature[mu, nu], displayed as $\mathrm{\Κ\_\_\μ,\ν}$, in black as all the tensors of ThreePlusOne, is a computational representation for a 4D symmetric tensor whose space components are the extrinsic curvature of a 3D hypersurface, defined by some global function $\mathrm{{\rm T}}\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,t\right)=\mathrm{constant}$. Note: to work with ExtrinsicCurvature[mu, nu], a kind of letter to represent a space index must be set first using Setup, for instance entering Setup(spaceindices = lowercaselatin_is). In turn this kind of letter is automatically set when you load ThreePlusOne.

The value of the components of the ExtrinsicCurvature are related to the values of the Lapse and Shift, that you can set 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 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.

When the spacetime indices of ExtrinsicCurvature assume integer values, it is expected they are between 0 and 3, and the corresponding value of ExtrinsicCurvature is returned. The presentation in this page also assumes the signature (+ + + -), automatically set when ThreePlusOne is loaded. This signature is physically equivalent to (- + + +) but (+ + + -) has the computational advantage of having space indices running from 1 to 3 (instead of from 2 to 4) and the value 0 of a spacetime index pointing to position 4 (instead of position 1).

The components of ExtrinsicCurvature[mu, nu] are related to the space components of the 4D spacetime metric $\mathrm{g\_\_\μ,\ν}$ (represented by g_) through the 3D metric $\mathrm{\γ\_\_\μ,\ν}$ (represented by gamma3_) entering the definition

where $ℒ$ is the LieDerivative operator, $\mathit{n}$ is the UnitNormalVector vector, and $\mathrm{\γ\_\_\μ,\ν}$ is the gamma3_ 3D metric. The ExtrinsicCurvature is thus a measure of change in the UnitNormalVector $\mathrm{n\_\_\μ}$ as we parallel transport it to a nearby point: in general, the new vector will not be normal to the hypersurface. In this sense, an alternative equivalent definition is

i.e., the projection onto the 3D hypersurface of the covariant derivative (4D gradient) of the UnitNormalVector. From this definition it is clear that $\mathrm{\Κ\_\_\μ,\ν}$ is a purely spatial tensor, ${\mathit{n}}_{}^{\mathrm{\mu }}{\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}=0$, so that, in an adapted system of coordinates (see Lapse), ${\mathbf{{\rm K}}}_{}^{0\mathrm{\mu }}=0$. Consequently, ${\mathbf{{\rm K}}}_{\mathrm{\mu }}^{\mathrm{\mu }}={\mathbf{{\rm K}}}_i^i$ and ${\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}{\mathbf{{\rm K}}}_{}^{\mathrm{\mu },\mathrm{\nu }}={\mathbf{{\rm K}}}_{i,j}{\mathbf{{\rm K}}}_{}^{i,j}$, where $i,j$ are space tensor indices.

In the context of a Hamiltonian description of gravity (see [5]), the ExtrinsicCurvature is also related to the "conjugate momentum" $\mathrm{\Π\_\_i,j}$ of the 3D metric $\mathrm{\γ\_\_\μ,\ν}$ (represented by gamma3_) by

Although $\mathrm{\Π\_\_i,j}$ is not a predefined tensor of the ThreePlusOne package, you can make it work as if it were by simply defining it passing the equation above to the Define command. After that, $\mathrm{\Π\_\_i,j}$ will be understood by the system as any of the tensors of the Physics package.

Regarding the geometry in this 3D hypersurface described by $\mathrm{\γ\_\_i,j}$, a 3D covariant derivative, related Christoffel symbols, and the Ricci and Riemann tensors are represented respectively by the D3_, Christoffel3, Ricci3 and Riemann3 commands. The definition of these commands is obtained by replacing g_ by the 3D metric gamma3_ in the definitions of the 4D spacetime D_, Christoffel, Ricci and Riemann tensors. All of D3_, Christoffel3, Ricci3 and Riemann3 are also tensors in 3D when replacing spacetime indices by space indices.

You can change the value of the ExtrinsicCurvature in different ways by changing the value of the 4D metric g_, using Setup or g_ itself as explained in its help page.

Besides being indexed with two indices, ExtrinsicCurvature accepts the following keywords as an index:

definition: when passed alone, ExtrinsicCurvature returns its 4D definition in terms of the UnitNormalVector and the spacetime metric g_. When passed with space indices, it returns its expression in terms of them spatial components of g_.

matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in ExtrinsicCurvature[]) returns a Matrix that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of ExtrinsicCurvature. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting Matrix takes into account the character of the indices.

nonzero: returns a set of equations, with the left-hand side as a sequence of two positive numbers identifying the element of $\mathrm{\γ\_\_i,j}$ and the corresponding value on the right-hand side. Note that this set is actually the output of the ArrayElems command when passing to it the Matrix obtained with the keyword matrix.

The %ExtrinsicCurvature command is the inert form of ExtrinsicCurvature, so it represents the same tensor but entering it does not result in performing any computation. To perform the related computations as if %ExtrinsicCurvature were ExtrinsicCurvature, use value.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{Setup}\left(\mathrm{spacetimeindices}\,\mathrm{spaceindices}\right)$

$\left[\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\,\mathrm{spacetimeindices}=\mathrm{greek}\right]$

$\mathrm{ExtrinsicCurvature}\left[\mathrm{definition}\right]$

${\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}=-\frac{{ℒ}_{\mathit{n}}\left({\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}\right)}{2}$

$\mathrm{UnitNormalVector}\left[\mathrm{definition}\right]$

${\mathit{n}}_{\mathrm{\mu }}=-\mathbf{\alpha }{▿}_{\mathrm{\mu }}\left(t\right)$

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{gamma3\_}\left[\mathrm{lineelement}\right]$

$\frac{{R_r}^2{\mathbf{\ⅆ}\left(r\right)}^2}{1+2E}+R^2{\mathbf{\ⅆ}\left(\mathrm{\theta }\right)}^2+R^2{\sin \left(\mathrm{\theta }\right)}^2{\mathbf{\ⅆ}\left(\mathrm{\phi }\right)}^2$

$\mathrm{ExtrinsicCurvature}\left[\right]$

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

$\mathrm{ExtrinsicCurvature}\left[\mathrm{~i},j,\mathrm{matrix}\right]$

${\mathbf{{\rm K}}}_{\phantom{}\phantom{i}j}^{\phantom{}i\phantom{j}}=\left[\begin{array}{ccc}-\frac{\frac{{\partial }^{2}R}{\partial t\partial r}}{R_r}& 0& 0\\ 0& -\frac{\stackrel{\mathbf{.}}{R}}{R}& 0\\ 0& 0& -\frac{\stackrel{\mathbf{.}}{R}}{R}\end{array}\right]$

$\mathrm{\%ExtrinsicCurvature}\left[1,1\right]$

${\mathbf{{\rm K}}}_{1,1}$

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

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

$\mathrm{\%ExtrinsicCurvature}\left[\mathrm{\mu },\mathrm{\nu }\right]-\mathrm{\%ExtrinsicCurvature}\left[\mathrm{\nu },\mathrm{\mu }\right]$

${\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}-{\mathbf{{\rm K}}}_{\mathrm{\nu },\mathrm{\mu }}$

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

$0$

$\mathrm{ExtrinsicCurvature}\left[\mathrm{`~`}\right]$

${\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{\left(-1-2E\right)\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)}{{R_r}^3}& 0& 0& 0\\ 0& -\frac{\stackrel{\mathbf{.}}{R}}{R^3}& 0& 0\\ 0& 0& -\frac{{\csc \left(\mathrm{\theta }\right)}^2\stackrel{\mathbf{.}}{R}}{R^3}& 0\\ 0& 0& 0& 0\end{array}\right]$

$\mathrm{ExtrinsicCurvature}\left[\mathrm{nonzero}\right]$

${\mathbf{{\rm K}}}_{\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\,1\right)=-\frac{R_r\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)}{1+2E}\,\left(2\,2\right)=-R\stackrel{\mathbf{.}}{R}\,\left(3\,3\right)=-R{\sin \left(\mathrm{\theta }\right)}^2\stackrel{\mathbf{.}}{R}\right\}$

$\mathrm{ExtrinsicCurvature}\left[\mathrm{`~`},\mathrm{nonzero}\right]$

${\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\,1\right)=\frac{\left(-1-2E\right)\left(\frac{{\partial }^{2}R}{\partial t\partial r}\right)}{{R_r}^3}\,\left(2\,2\right)=-\frac{\stackrel{\mathbf{.}}{R}}{R^3}\,\left(3\,3\right)=-\frac{{\csc \left(\mathrm{\theta }\right)}^2\stackrel{\mathbf{.}}{R}}{R^3}\right\}$

$\mathrm{Christoffel}\left[\mathrm{definition}\right]$

${\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\frac{{\partial }_{\mathrm{\nu }}\left(g_{\mathrm{\alpha },\mathrm{\mu }}\right)}{2}+\frac{{\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)}{2}-\frac{{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\mu },\mathrm{\nu }}\right)}{2}$

$\mathrm{Christoffel3}\left[\mathrm{definition}\right]$

${\mathbf{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\frac{{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\alpha }}\mathrm{\beta }}{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}\left({\partial }_{\mathrm{\lambda }}\left({\mathbf{\gamma }}_{\mathrm{\beta },\mathrm{\kappa }}\right)+{\partial }_{\mathrm{\kappa }}\left({\mathbf{\gamma }}_{\mathrm{\beta },\mathrm{\lambda }}\right)-{\partial }_{\mathrm{\beta }}\left({\mathbf{\gamma }}_{\mathrm{\kappa },\mathrm{\lambda }}\right)\right)}{2}$

$\mathrm{Ricci3}\left[\mathrm{\mu },\mathrm{\nu }\right]$

${\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }}$

$=\mathrm{convert}\left(\,\mathrm{g\_}\right)$

${\mathit{R}}_{\mathrm{\mu },\mathrm{\nu }}={\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\sigma }}}^{\phantom{\mathrm{\mu }}\mathrm{\sigma }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\nu }}\mathrm{\tau }}\left(\frac{{\partial }_{\mathrm{\alpha }}\left({\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\α1}}}^{\phantom{}\mathrm{\alpha },\mathrm{\α1}}\right){\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\α2}}}^{\phantom{\mathrm{\sigma }}\mathrm{\α2}}{\mathbf{\gamma }}_{\mathrm{\tau }\phantom{\mathrm{\α3}}}^{\phantom{\mathrm{\tau }}\mathrm{\α3}}\left({\partial }_{\mathrm{\α3}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α2}}\right)+{\partial }_{\mathrm{\α2}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α3}}\right)-{\partial }_{\mathrm{\α1}}\left({\mathbf{\gamma }}_{\mathrm{\α2},\mathrm{\α3}}\right)\right)}{2}+\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\α1}}}^{\phantom{}\mathrm{\alpha },\mathrm{\α1}}{\partial }_{\mathrm{\alpha }}\left({\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\α2}}}^{\phantom{\mathrm{\sigma }}\mathrm{\α2}}\right){\mathbf{\gamma }}_{\mathrm{\tau }\phantom{\mathrm{\α3}}}^{\phantom{\mathrm{\tau }}\mathrm{\α3}}\left({\partial }_{\mathrm{\α3}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α2}}\right)+{\partial }_{\mathrm{\α2}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α3}}\right)-{\partial }_{\mathrm{\α1}}\left({\mathbf{\gamma }}_{\mathrm{\α2},\mathrm{\α3}}\right)\right)}{2}+\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\α1}}}^{\phantom{}\mathrm{\alpha },\mathrm{\α1}}{\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\α2}}}^{\phantom{\mathrm{\sigma }}\mathrm{\α2}}{\partial }_{\mathrm{\alpha }}\left({\mathbf{\gamma }}_{\mathrm{\tau }\phantom{\mathrm{\α3}}}^{\phantom{\mathrm{\tau }}\mathrm{\α3}}\right)\left({\partial }_{\mathrm{\α3}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α2}}\right)+{\partial }_{\mathrm{\α2}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α3}}\right)-{\partial }_{\mathrm{\α1}}\left({\mathbf{\gamma }}_{\mathrm{\α2},\mathrm{\α3}}\right)\right)}{2}+\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\α1}}}^{\phantom{}\mathrm{\alpha },\mathrm{\α1}}{\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\α2}}}^{\phantom{\mathrm{\sigma }}\mathrm{\α2}}{\mathbf{\gamma }}_{\mathrm{\tau }\phantom{\mathrm{\α3}}}^{\phantom{\mathrm{\tau }}\mathrm{\α3}}\left({\partial }_{\mathrm{\alpha }}\left({\partial }_{\mathrm{\α3}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α2}}\right)\right)+{\partial }_{\mathrm{\alpha }}\left({\partial }_{\mathrm{\α2}}\left({\mathbf{\gamma }}_{\mathrm{\α1},\mathrm{\α3}}\right)\right)-{\partial }_{\mathrm{\alpha }}\left({\partial }_{\mathrm{\α1}}\left({\mathbf{\gamma }}_{\mathrm{\α2},\mathrm{\α3}}\right)\right)\right)}{2}-\frac{{\partial }_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\chi }}}^{\phantom{}\mathrm{\alpha },\mathrm{\chi }}\right){\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\psi }}}^{\phantom{\mathrm{\alpha }}\mathrm{\psi }}{\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\omega }}}^{\phantom{\mathrm{\sigma }}\mathrm{\omega }}\left({\partial }_{\mathrm{\omega }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\psi }}\right)+{\partial }_{\mathrm{\psi }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\omega }}\right)-{\partial }_{\mathrm{\chi }}\left({\mathbf{\gamma }}_{\mathrm{\omega },\mathrm{\psi }}\right)\right)}{2}-\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\chi }}}^{\phantom{}\mathrm{\alpha },\mathrm{\chi }}{\partial }_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\psi }}}^{\phantom{\mathrm{\alpha }}\mathrm{\psi }}\right){\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\omega }}}^{\phantom{\mathrm{\sigma }}\mathrm{\omega }}\left({\partial }_{\mathrm{\omega }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\psi }}\right)+{\partial }_{\mathrm{\psi }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\omega }}\right)-{\partial }_{\mathrm{\chi }}\left({\mathbf{\gamma }}_{\mathrm{\omega },\mathrm{\psi }}\right)\right)}{2}-\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\chi }}}^{\phantom{}\mathrm{\alpha },\mathrm{\chi }}{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\psi }}}^{\phantom{\mathrm{\alpha }}\mathrm{\psi }}{\partial }_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\omega }}}^{\phantom{\mathrm{\sigma }}\mathrm{\omega }}\right)\left({\partial }_{\mathrm{\omega }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\psi }}\right)+{\partial }_{\mathrm{\psi }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\omega }}\right)-{\partial }_{\mathrm{\chi }}\left({\mathbf{\gamma }}_{\mathrm{\omega },\mathrm{\psi }}\right)\right)}{2}-\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\chi }}}^{\phantom{}\mathrm{\alpha },\mathrm{\chi }}{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\psi }}}^{\phantom{\mathrm{\alpha }}\mathrm{\psi }}{\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\omega }}}^{\phantom{\mathrm{\sigma }}\mathrm{\omega }}\left({\partial }_{\mathrm{\omega }}\left({\partial }_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\psi }}\right)\right)+{\partial }_{\mathrm{\psi }}\left({\partial }_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\chi },\mathrm{\omega }}\right)\right)-{\partial }_{\mathrm{\chi }}\left({\partial }_{\mathrm{\tau }}\left({\mathbf{\gamma }}_{\mathrm{\omega },\mathrm{\psi }}\right)\right)\right)}{2}+\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\α7},\mathrm{\beta }}}^{\phantom{}\mathrm{\α7},\mathrm{\beta }}{\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\α8}}}^{\phantom{\mathrm{\sigma }}\mathrm{\α8}}{\mathbf{\gamma }}_{\mathrm{\tau }\phantom{\mathrm{\α9}}}^{\phantom{\mathrm{\tau }}\mathrm{\α9}}\left({\partial }_{\mathrm{\α9}}\left({\mathbf{\gamma }}_{\mathrm{\α7},\mathrm{\α8}}\right)+{\partial }_{\mathrm{\α8}}\left({\mathbf{\gamma }}_{\mathrm{\α7},\mathrm{\α9}}\right)-{\partial }_{\mathrm{\α7}}\left({\mathbf{\gamma }}_{\mathrm{\α8},\mathrm{\α9}}\right)\right){\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\kappa }}}^{\phantom{}\mathrm{\alpha },\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\alpha }}\mathrm{\lambda }}{\mathbf{\gamma }}_{\mathrm{\beta }\phantom{\mathrm{\upsilon }}}^{\phantom{\mathrm{\beta }}\mathrm{\upsilon }}\left({\partial }_{\mathrm{\upsilon }}\left({\mathbf{\gamma }}_{\mathrm{\kappa },\mathrm{\lambda }}\right)+{\partial }_{\mathrm{\lambda }}\left({\mathbf{\gamma }}_{\mathrm{\kappa },\mathrm{\upsilon }}\right)-{\partial }_{\mathrm{\kappa }}\left({\mathbf{\gamma }}_{\mathrm{\lambda },\mathrm{\upsilon }}\right)\right)}{4}-\frac{{\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\α4},\mathrm{\beta }}}^{\phantom{}\mathrm{\α4},\mathrm{\beta }}{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\α5}}}^{\phantom{\mathrm{\alpha }}\mathrm{\α5}}{\mathbf{\gamma }}_{\mathrm{\sigma }\phantom{\mathrm{\α6}}}^{\phantom{\mathrm{\sigma }}\mathrm{\α6}}\left({\partial }_{\mathrm{\α6}}\left({\mathbf{\gamma }}_{\mathrm{\α4},\mathrm{\α5}}\right)+{\partial }_{\mathrm{\α5}}\left({\mathbf{\gamma }}_{\mathrm{\α4},\mathrm{\α6}}\right)-{\partial }_{\mathrm{\α4}}\left({\mathbf{\gamma }}_{\mathrm{\α5},\mathrm{\α6}}\right)\right){\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\zeta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\zeta }}{\mathbf{\gamma }}_{\mathrm{\beta }\phantom{\mathrm{\iota }}}^{\phantom{\mathrm{\beta }}\mathrm{\iota }}{\mathbf{\gamma }}_{\mathrm{\tau }\phantom{\mathrm{o}}}^{\phantom{\mathrm{\tau }}\mathrm{o}}\left({\partial }_{\mathrm{o}}\left({\mathbf{\gamma }}_{\mathrm{\iota },\mathrm{\zeta }}\right)+{\partial }_{\mathrm{\iota }}\left({\mathbf{\gamma }}_{\mathrm{o},\mathrm{\zeta }}\right)-{\partial }_{\mathrm{\zeta }}\left({\mathbf{\gamma }}_{\mathrm{\iota },\mathrm{o}}\right)\right)}{4}\right)$

$\mathrm{Christoffel3}\left[\mathrm{definition}\right]$

${\mathbf{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\frac{{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\alpha }}\mathrm{\beta }}{\mathbf{\gamma }}_{\mathrm{\mu }\phantom{\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}\left({\partial }_{\mathrm{\lambda }}\left({\mathbf{\gamma }}_{\mathrm{\beta },\mathrm{\kappa }}\right)+{\partial }_{\mathrm{\kappa }}\left({\mathbf{\gamma }}_{\mathrm{\beta },\mathrm{\lambda }}\right)-{\partial }_{\mathrm{\beta }}\left({\mathbf{\gamma }}_{\mathrm{\kappa },\mathrm{\lambda }}\right)\right)}{2}$

$\mathrm{Riemann3}\left[\mathrm{~mu},\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta },\mathrm{definition}\right]$

${\mathit{R}}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}}={\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\kappa }}}^{\phantom{}\mathrm{\mu },\mathrm{\kappa }}{\mathbf{\gamma }}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}{\mathbf{\gamma }}_{\mathrm{\alpha }\phantom{\mathrm{\sigma }}}^{\phantom{\mathrm{\alpha }}\mathrm{\sigma }}{\mathbf{\gamma }}_{\mathrm{\beta }\phantom{\mathrm{\tau }}}^{\phantom{\mathrm{\beta }}\mathrm{\tau }}{R}_{\mathrm{\kappa },\mathrm{\lambda },\mathrm{\sigma },\mathrm{\tau }}+{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\beta }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\beta }}}{\mathbf{{\rm K}}}_{\mathrm{\nu },\mathrm{\alpha }}-{\mathbf{{\rm K}}}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\alpha }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\alpha }}}{\mathbf{{\rm K}}}_{\mathrm{\nu },\mathrm{\beta }}$

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

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

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

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

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

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

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

$\mathrm{Ricci}\left[\right]$

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

$\mathrm{Ricci3}\left[\right]$

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

$\mathrm{Ricci3}\left[i,j,\mathrm{matrix}\right]$

${\mathit{R}}_{i,j}=\left[\begin{array}{ccc}\frac{2m}{r^2\left(2m-r\right)}& 0& 0\\ 0& \frac{m}{r}& 0\\ 0& 0& \frac{{\sin \left(\mathrm{\theta }\right)}^{2}m}{r}\end{array}\right]$

$\mathrm{Ricci3}\left[\mathrm{~i},j,\mathrm{matrix}\right]$

${\mathit{R}}_{\phantom{}\phantom{i}j}^{\phantom{}i\phantom{j}}=\left[\begin{array}{ccc}-\frac{2m}{r^3}& 0& 0\\ 0& \frac{m}{r^3}& 0\\ 0& 0& \frac{m}{r^3}\end{array}\right]$

$\mathrm{ExtrinsicCurvature}\left[\right]$

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

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

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

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

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

[5] Arnowitt, R., Dese, S., Misner, C.W., The Dynamics of General Relativity, Chapter 7 in Gravitation: an introduction to current research (Wiley, 1962), https://arxiv.org/pdf/gr-qc/0405109v1.pdf