The gamma3_[mu, nu], displayed as ${\mathrm{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}$, is a computational representation for a 4D tensor whose space components are the space 3D metric of a hypersurface defined by some global function $\mathrm{{\rm T}}\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,t\right)=\mathrm{constant}$. It is assumed that the hypersurface is spacelike, i.e. the corresponding 3D metric is definite positive with signature (+ + +).

gamma3_[mu, nu] is a purely spatial tensor that projects 4D tensors into the 3D hypersurface, resulting in purely spatial tensors, all of whose components are equal to zero when one of its indices is contravariant and has a timelike value (i.e. contravariant 0, represented by ~0). gamma3_[mu, nu] is defined in terms of the spacetime metric g_ and the UnitNormalVector $n_{\mathrm{\mu }}$ as

Note that, because the hypersurface is spacelike, the UnitNormalVector is timelike, i.e. one whose covariant space components are equal to 0.

With one index contravariant and the other covariant, ${\mathbf{\gamma }}_{\mathrm{\nu }}^{\mathrm{\mu }}$ projects an arbitrary 4D contravariant vector into the 3D hypersurface. To project higher rank tensors into the spatial hypersurface, each free index has to be contracted in this way. Thus, a projector normal to the hypersurface, that you can define to compute with it using Define, is given by

Besides using the 4D metric g_ to raise and lower indices of a 4D tensor, if the 4D tensor is purely spatial you can as well raise and lower its 4D free indices by contracting them with the gamma3_ (that is also a purely spatial 4D tensor). And as with any other tensors whose space parts are also 3D tensors (i.e. when replacing the 4D spacetime free indices by 3D space indices), their space indices are raised and lowered using the 3D metric gamma3_ with space indices (not the 4D metric g_ with space indices).

This projector projects an arbitrary 4D covariant vector into a corresponding timelike vector.

In the 3+1 formulation, it is relevant the values of the Lapse and Shift, that you can set using Setup and its lapseandshift keyword - for that purpose see setting the Lapse and Shift using Setup.

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). With any of these two signatures, the space components of the 4D gamma3_[mu, nu] are related to the space components of the 4D spacetime metric $g_{\mathrm{\mu },\mathrm{\nu }}$ by

When the spacetime indices of gamma3_ assume integer values, it is expected they are between 0 and 3, and the  corresponding value of gamma3_ is returned. The value 0 always points to the position of the timelike component, so with the signatures (- - - +) and (+ + + -) indexing with 0 is the same as indexing with 4, while with the signatures (+ - - -) and (- + + +) it is the same as indexing with 1.

An important distinction is made between the 3D spatial metric represented by Physics:-gamma_, related to the infinitesimal distance measured by one observer at a fixed point in space, expressed in terms of the contravariant spatial components of the metric g_ as ${\mathrm{\gamma }}_{}^{i,j}=g_{}^{i,j}$ (see [1], sec.(84)), and the 3D spatial metric of a hypersurface defined by a fixed constant value of t, the time, represented by Physics:-ThreePlusOne:-gamma3_, related to the infinitesimal spatial distance on this hypersurface, the distance measured by a normal (Eulerian) observer (whose velocity is perpendicular to the hypersurface), and so can be obtained from the covariant 4D one by restricting it to constant time surfaces, namely ${\mathbf{\gamma }}_{i,j}=g_{i,j}$

The definition of Physics:-ThreePlusOne:-gamma3_ arises naturally when defining simultaneity in the general theory of relativity as the possibility of synchronizing clocks located at different points in space (ref [1], sec. (84)). As is well known, for that synchronization to be possible, all the components $g_{0,j}$ of the spacetime metric must be equal to zero, and that condition can always be achieved, in any gravitational field, by an appropriate choice of coordinates (ref [1], sec. (84) and (97)). If, in addition, $g_{0,0}=1$, the reference system is synchronous. In this referential the time lines are geodesics in the 4D spacetime and normal to the hypersurface $t=\mathrm{constant}$, and the 4D infinitesimal interval is written as.

Under a time rescaling $t\text{'}=G\left(t\right)$ and a general transformation of the space coordinates ${\left(x\text{'}\right)}^i=F^i\left(x^j\,t\right)$, and taking into account ${\mathbf{\gamma }}_{i,j}=g_{i,j}$, this infinitesimal interval changes into

where $\mathbf{\alpha }$ is the Lapse and $\mathrm{\β\_\_i}$ is the Shift, and the new time lines $t=\mathrm{constant}$ are no longer orthogonal to the 3D hypersurface whose geometry is described by $\mathrm{\γ\_\_i,j}$. Instead, the UnitNormalVector is. In this decomposition of the infinitesimal interval into its time and space parts, the 4-coordinate degrees of freedom are represented by the freedom in choosing the Lapse and the Shift. (Conversely, one can depart from this general form of the infinitesimal interval and construct a synchronous reference system, where $g_{0,0}=1$ and $g_{0,j}=0$, by solving a related Hamilton-Jacobi equation see [1] sec.(97)).

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 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 3D metric gamma3_ in different ways by changing the value of the 4D metric g_, using Setup or g_ itself as explained in its help page. For example, among others, the simplest way to set the Schwarzschild metric and related coordinates is to enter g_[sc]

Besides being indexed with two indices, gamma3_ accepts the following keywords as an index:

definition: when passed alone, gamma3_ 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_.

determinant: when passed alone, gamma3_ returns the determinant of the  all-covariant $\mathrm{\γ\_\_i,j}$ . If this keyword is passed together with indices, that can be covariant or contravariant, the resulting determinant takes into account the character of the indices.

line_element: (synonym: lineelement) when passed alone, gamma3_ returns the 3D line element for the 4D current metric expressing the differentials of the coordinates using d_.

matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in gamma3_[]) 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 gamma3_. 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.

In turn, some predefined values for the 4D spacetime metric can be set by indexing the metric with a name or a portion of it - see g_ - and in that way you can indirectly set the values of the 3D gamma3_ metric.

The %gamma3_ command is the inert form of gamma3_, so it represents the same tensor but entering it does not result in performing any computation. To perform the related computations as if %gamma3_ were gamma3_, use value.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}={\mathit{n}}_{\mathrm{\mu }}{\mathit{n}}_{\mathrm{\nu }}+g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{gamma3\_}\left[i,j,\mathrm{definition}\right]$

${\mathbf{\gamma }}_{i,j}=g_{i,j}$

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

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

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

${\mathbf{\ⅆ}\left(x\right)}^2+{\mathbf{\ⅆ}\left(y\right)}^2+{\mathbf{\ⅆ}\left(z\right)}^2-{\mathbf{\ⅆ}\left(t\right)}^2$

$\mathrm{gamma3\_}\left[\mathrm{line\_element}\right]$

${\mathbf{\ⅆ}\left(x\right)}^2+{\mathbf{\ⅆ}\left(y\right)}^2+{\mathbf{\ⅆ}\left(z\right)}^2$

$\mathrm{gamma3\_}\left[\mathrm{\mu },\mathrm{\nu }\right]-\mathrm{gamma3\_}\left[\mathrm{\nu },\mathrm{\mu }\right]$

$0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{gamma3\_}\left[\mathrm{line\_element}\right]$

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

$\mathrm{gamma3\_}\left[\mathrm{\mu },\mathrm{\nu },\mathrm{matrix}\right]$

${\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{{R_r}^2}{1+2E}& 0& 0& 0\\ 0& R^2& 0& 0\\ 0& 0& R^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& 0\end{array}\right]$

$\mathrm{gamma3\_}\left[\mathrm{`~`}\right]$

${\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{1+2E}{{R_r}^2}& 0& 0& 0\\ 0& \frac{1}{R^2}& 0& 0\\ 0& 0& \frac{{\csc \left(\mathrm{\theta }\right)}^2}{R^2}& 0\\ 0& 0& 0& 0\end{array}\right]$

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

$\frac{{R_r}^2{R}^4{\sin \left(\mathrm{\theta }\right)}^2}{1+2E}$

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

${\mathbf{\gamma }}_{\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\,1\right)=\frac{{R_r}^2}{1+2E}\,\left(2\,2\right)=R^2\,\left(3\,3\right)=R^2{\sin \left(\mathrm{\theta }\right)}^2\right\}$

$\mathrm{gamma3\_}\left[\mathrm{`~`},\mathrm{nonzero}\right]$

${\mathbf{\gamma }}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\,1\right)=\frac{1+2E}{{R_r}^2}\,\left(2\,2\right)=\frac{1}{R^2}\,\left(3\,3\right)=\frac{{\csc \left(\mathrm{\theta }\right)}^2}{R^2}\right\}$

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

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

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

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

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

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

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

${\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)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{\%gamma3\_}\left[1,1\right]$

${\mathbf{\gamma }}_{1,1}$

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

$-\frac{r}{2m-r}$

[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.