As is the case of all the tensors of the ThreePlusOne package, D3_[mu] is a 4D spacetime tensor with special meaning (acting) on a 3D hypersurface specified by the values of the Lapse and Shift. The D3_[mu] command is a computational representation for the covariant differential operator in that hypersurface. Its space components also form a tensor in the 3D hypersurface, so D3_ can be used with 3D space indices, for instances as in D_[i](A[j]).

On the screen, D3_, is displayed as the corresponding D_ command, but in black $\mathrm{\▿\_\_\μ}$, instead of in blue $\mathrm{\▿\_\_\μ}$.

Note that the space components of - say - D3_[mu](A[nu](x, y, z, t)) are not necessarily the same as the space components of the 4D tensor D_[mu](A[nu](x, y, z, t)). Instead, the relationship between these two is given by

where ${\mathbf{\gamma }}_{\mathrm{\mu }}^{\mathrm{\alpha }}$ is the gamma3_, a 4D tensor, the spatial metric induced in the 3D hypersurface by the 4D g_ metric, also a projection operator that projects 4D tensors into the 3D hypersurface, resulting in purely spatial 4D tensors, all of whose components are equal to zero when their indices are contravariant and any of them has a timelike value (i.e. contravariant 0).

D3_ can be used as well without an index, as in D3_(A) displayed as $\mathit{▿}\left(A\right)$, in which case it represents the total differential in curvilinear coordinates, and the output comes automatically expanded as D3_[mu](A) * D3_(X[~mu]), where X[~mu] is the corresponding spacetime vector.

The %D3_ command is the inert form of D3_, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

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

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

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

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

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

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

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

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

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

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

$\mathrm{Setup}\left(\mathrm{coordinatesystems}=\mathrm{cartesian}\right)$

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

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

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

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

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

${\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\varnothing$

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

${\mathbf{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\varnothing$

$\mathrm{D3\_}\left(X\left[\mathrm{~nu}\right]\right)$

$\mathbf{\ⅆ}\left(X_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)$

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

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

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

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

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

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

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

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

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

$\mathrm{D3\_}\left[j\right]\left(\mathrm{\Phi }\left(X\right)\right)$

${\mathbf{\gamma }}_{j\phantom{\mathrm{\mu }}}^{\phantom{j}\mathrm{\mu }}{\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\left(X\right)\right)$

$\mathrm{D3\_}\left(\mathrm{\Phi }\left(X\right)\right)$

${\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\left(X\right)\right)\mathbf{\ⅆ}\left(X_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)$

$\mathrm{Define}\left(A\left[j\right]\right)$

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

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

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

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

$\mathrm{CompactDisplay}\left(A\left[j\right]\left(X\right)\right)$

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

$\mathrm{D3\_}\left(A\left[\mathrm{~j}\right]\left(X\right)\right)$

$\left({\mathit{▿}}_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{j}}^{\phantom{}j}\right)\right)\mathbf{\ⅆ}\left(X_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)$

$\mathrm{D3\_}\left[j\right]\left(A\left[\mathrm{~j}\right]\left(X\right)\right)$

${\mathit{▿}}_j\left(A_{\phantom{}\phantom{j}}^{\phantom{}j}\right)$

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

${\mathbf{\gamma }}_{j\phantom{\mathrm{\alpha }}}^{\phantom{j}\mathrm{\alpha }}{\mathbf{\gamma }}_{\mathrm{\beta }\phantom{j}}^{\phantom{\mathrm{\beta }}j}\left({\partial }_{\mathrm{\alpha }}\left(A_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\beta }}\mathrm{\alpha },i}^{\phantom{}\mathrm{\beta }\phantom{\mathrm{\alpha },i}}{A}_{\phantom{}\phantom{i}}^{\phantom{}i}\right)$

$\mathrm{D3\_}\left[j\right]\left(A\left[\mathrm{~j}\right]\left(X\right)\right)$

${\mathit{▿}}_j\left(A_{\phantom{}\phantom{j}}^{\phantom{}j}\right)$

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

${\mathbf{\gamma }}_{j\phantom{\mathrm{\alpha }}}^{\phantom{j}\mathrm{\alpha }}{\mathbf{\gamma }}_{\mathrm{\beta }\phantom{j}}^{\phantom{\mathrm{\beta }}j}\left({\partial }_{\mathrm{\alpha }}\left(A_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\beta }}\mathrm{\alpha },i}^{\phantom{}\mathrm{\beta }\phantom{\mathrm{\alpha },i}}{A}_{\phantom{}\phantom{i}}^{\phantom{}i}\right)$

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

$\frac{{\left(A_{\phantom{}\phantom{1}}^{\phantom{}1}\right)}_r\left(2m-r\right)r+m{A}_{\phantom{}\phantom{1}}^{\phantom{}1}+{\left(A_{\phantom{}\phantom{2}}^{\phantom{}2}\right)}_{\mathrm{\theta }}\left(2m-r\right)r+2A_{\phantom{}\phantom{1}}^{\phantom{}1}\left(2m-r\right)+{\left(A_{\phantom{}\phantom{3}}^{\phantom{}3}\right)}_{\mathrm{\phi }}\left(2m-r\right)r+\cot \left(\mathrm{\theta }\right)A_{\phantom{}\phantom{2}}^{\phantom{}2}\left(2m-r\right)r}{\left(2m-r\right)r}$

$\mathrm{\%D3\_}\left[i\right]\left(\mathrm{gamma3\_}\left[j,k\right]\right)$

${\mathit{▿}}_i\left({\mathbf{\gamma }}_{j,k}\right)$

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

$0$

$\mathrm{Setup}\left(\mathrm{metric}=\left\{\left(1\,1\right)=\mathrm{f\_\_1}\left(X\right)\,\left(1\,4\right)=\mathrm{f\_\_14}\left(X\right)\,\left(2\,2\right)=\mathrm{f\_\_2}\left(X\right)\,\left(2\,4\right)=\mathrm{f\_\_24}\left(X\right)\,\left(3\,3\right)=\mathrm{f\_\_3}\left(X\right)\,\left(3\,4\right)=\mathrm{f\_\_34}\left(X\right)\,\left(4\,4\right)=\mathrm{f\_\_4}\left(X\right)\right\}\right)$

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

$\mathrm{Coordinates:}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]\mathrm{.\; Signature:}\left(\mathrm{+\; +\; +\; -}\right)$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{f\_\_1}\left(X\right)& 0& 0& \mathrm{f\_\_14}\left(X\right)\\ 0& \mathrm{f\_\_2}\left(X\right)& 0& \mathrm{f\_\_24}\left(X\right)\\ 0& 0& \mathrm{f\_\_3}\left(X\right)& \mathrm{f\_\_34}\left(X\right)\\ \mathrm{f\_\_14}\left(X\right)& \mathrm{f\_\_24}\left(X\right)& \mathrm{f\_\_34}\left(X\right)& \mathrm{f\_\_4}\left(X\right)\end{array}\right]$

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

$\left[\mathrm{metric}=\left\{\left(1\,1\right)=\mathrm{f\_\_1}\left(X\right)\,\left(1\,4\right)=\mathrm{f\_\_14}\left(X\right)\,\left(2\,2\right)=\mathrm{f\_\_2}\left(X\right)\,\left(2\,4\right)=\mathrm{f\_\_24}\left(X\right)\,\left(3\,3\right)=\mathrm{f\_\_3}\left(X\right)\,\left(3\,4\right)=\mathrm{f\_\_34}\left(X\right)\,\left(4\,4\right)=\mathrm{f\_\_4}\left(X\right)\right\}\right]$

$\mathrm{D3\_}\left[\mathrm{\mu }\right]\left(\mathrm{gamma3\_}\left[\mathrm{~mu},\mathrm{~nu}\right]\right)$

$0$

$\mathrm{ds2}≔x^2{\mathrm{dx}}^2+y^2{\mathrm{dy}}^2+z^2{\mathrm{dz}}^2+xy\mathrm{dx}\mathrm{dy}-{\mathrm{dt}}^2$

$\mathrm{ds2}≔x^2{\mathrm{dx}}^2+xy\mathrm{dx}\mathrm{dy}+y^2{\mathrm{dy}}^2+z^2{\mathrm{dz}}^2-{\mathrm{dt}}^2$

$\mathrm{Setup}\left(\mathrm{coordinates}=\mathrm{cartesian}\,\mathrm{metric}=\mathrm{ds2}\right)$

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

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

$\mathrm{Coordinates:}\left[x\,y\,z\,t\right]\mathrm{.\; Signature:}\left(\mathrm{+\; +\; +\; -}\right)$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}{x}^2& \frac{xy}{2}& 0& 0\\ \frac{xy}{2}& y^2& 0& 0\\ 0& 0& z^2& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]$

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

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

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\,\mathrm{metric}=\left\{\left(1\,1\right)=x^2\,\left(1\,2\right)=\frac{xy}{2}\,\left(2\,2\right)=y^2\,\left(3\,3\right)=z^2\,\left(4\,4\right)=\mathrm{-1}\right\}\right]$

$\mathrm{D3\_}\left[\mathrm{\mu }\right]\left(\mathrm{gamma3\_}\left[\mathrm{~mu},\mathrm{~nu}\right]\right)$

$0$

$\mathrm{simplify}\left(\mathrm{SumOverRepeatedIndices}\left(\mathrm{D3\_}\left[\mathrm{\mu }\right]\left(\mathrm{gamma3\_}\left[\mathrm{~mu},\mathrm{~1}\right]\right)\right)\right)$

$0$

[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.