D3_ - Maple Help

Physics[ThreePlusOne][D3_] - indexed covariant differential operator with respect to curvilinear space coordinates acting on a 3D hypersurface

 Calling Sequence D3_[mu](A) D3_[mu](A, [X]) D3_(A) D3_(A, [X])

Parameters

 mu - a name representing an integer number between 1 and the spacetime dimension or a space index running from 1 to the dimension - 1, can also be the number itself or preceded by ~ to represent a contravariant index A - any mathematical expression or relation between expressions, or a set or list of them, or an rtable [X] - a list of differentiation variables to which the index j refers

Description

 • 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

${\mathbit{▿}}_{{\mathrm{\mu }}}\left({{A}}_{{\mathrm{\nu }}}\right)={\mathbf{\gamma }}_{{\mathrm{\mu }}}^{{\mathrm{\alpha }}}{\mathbf{\gamma }}_{{\mathrm{\nu }}}^{{\mathrm{\beta }}}{{\mathbit{▿}}}_{{\mathrm{\alpha }}}\left({{A}}_{{\mathrm{\beta }}}\right)$

 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 $\mathbit{▿}\left({A}\right)$, 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.
 • Besides the definition in terms of D_ above, the covariant D3_[mu] = $\mathrm{▿__μ}$ can also be defined in terms of the 3D Christoffel3 symbols, in the same way the covariant 4D D_[mu] is defined in terms of the 4D Christoffel symbols, so by replacing Christoffel by Christoffel3. The resulting expression is also valid when restricting the values of the 4D free and repeated indices entering this definition to the 3D space.
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$$\mathrm{with}\left(\mathrm{ThreePlusOne}\right)$
 ${}{}\mathrm{_______________________________________________________}$
 $\mathrm{Setting}{}\mathrm{lowercaselatin_is}{}\mathrm{letters to represent}{}\mathrm{space}{}\mathrm{indices}$
 $\mathrm{Defined as 4D spacetime tensors}{}\left(\mathrm{see ?Physics,ThreePlusOne}\right){,}{\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{▿}}_{{\mathrm{\mu }}}{,}{\mathbf{\Gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{\mathbf{\beta }}_{{\mathrm{\mu }}}{,}{\mathbit{n}}_{{\mathrm{\mu }}}{,}{\mathbit{t}}_{{\mathrm{\mu }}}{,}{\mathbf{Κ}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}$
 $\mathrm{Changing the signature of spacetime from}{}\left(\mathrm{- - - +}\right){}\mathrm{to}{}\left(\mathrm{+ + + -}\right){}\mathrm{in order to match the signature customarily used in the ADM formalism}$
 ${}{}\mathrm{Systems of spacetime coordinates are:}{}{}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 ${}{}\mathrm{_______________________________________________________}$
 $\left[{\mathrm{ADMEquations}}{,}{\mathrm{Christoffel3}}{,}{\mathrm{D3_}}{,}{\mathrm{ExtrinsicCurvature}}{,}{\mathrm{Lapse}}{,}{\mathrm{LapseAndShiftConditions}}{,}{\mathrm{Ricci3}}{,}{\mathrm{Riemann3}}{,}{\mathrm{Shift}}{,}{\mathrm{TimeVector}}{,}{\mathrm{UnitNormalVector}}{,}{\mathrm{gamma3_}}\right]$ (1)
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (2)

Set up a coordinate system to work with - the first one to be set is automatically taken as the differentiation variables for D_ and so also for D3_

 > $\mathrm{Setup}\left(\mathrm{coordinatesystems}=\mathrm{cartesian}\right)$
 ${}{}\mathrm{Systems of spacetime coordinates are:}{}{}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 $\left[{\mathrm{coordinatesystems}}{=}\left\{{X}\right\}\right]$ (3)

When Physics is initialized, the default spacetime metric is of Minkowski type. You can see the metric querying Setup, as in Setup(metric);, or directly entering the metric g_ with no indices

 > $\mathrm{g_}\left[\right]$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{rrrr}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]\right)$ (4)

Check the nonzero components of Christoffel: since this is a Minkowski spacetime, there are none, then $\mathrm{▿__j}=\mathrm{∂__j}$

 > $\mathrm{Christoffel}\left[\mathrm{nonzero}\right]$
 ${{\mathrm{\Gamma }}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}{\varnothing }$ (5)

The nonzero components of Christoffel3 are also none

 > $\mathrm{Christoffel3}\left[\mathrm{nonzero}\right]$
 ${\mathbf{\Gamma }}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{=}{\varnothing }$ (6)

Note the difference in color: the 4D tensors are displayed in blue while the corresponding tensors of ThreePlusOne are displayed in black.

Because the Christoffel symbols are all equal to zero, the covariant derivative is equal to the standard derivative expressed using the d_ operator, both for D_ and D3_

 > $\mathrm{D3_}\left(X\left[\mathrm{~nu}\right]\right)$
 ${\mathbf{ⅆ}}{}\left({{X}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}\right)$ (7)

Set the scenario as a Schwarzschild spacetime in spherical coordinates; you can do this entering Setup(metric = Schwarzschild) or in the simpler form taking advantage of abbreviations and directly using the spacetime metric g_ command

 > $\mathrm{g_}\left[\mathrm{sc}\right]$
 ${}{}\mathrm{_______________________________________________________}$
 ${}{}\mathrm{Systems of spacetime coordinates are:}{}{}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}{}\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}{}{}{}\left\{X=\left(r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right)\right\}$
 ${}{}\mathrm{The Schwarzschild metric in coordinates}{}{}{}\left[r{,}\mathrm{\theta }{,}\mathrm{\phi }{,}t\right]$
 $\mathrm{Parameters:}{}\left[m\right]$
 $\mathrm{Signature:}{}\left(\mathrm{+ + + -}\right)$
 ${}{}\mathrm{_______________________________________________________}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}-\frac{r}{2{}m-r}& 0& 0& 0\\ 0& {r}^{2}& 0& 0\\ 0& 0& {r}^{2}{}{\mathrm{sin}{}\left(\mathrm{θ}\right)}^{2}& 0\\ 0& 0& 0& \frac{2{}m-r}{r}\end{array}\right]\right)$ (8)

The covariant derivative of a scalar is always equal to the $\partial$ derivative

 > $\mathrm{D3_}\left[j\right]\left(\mathrm{\Phi }\left(X\right)\right)$
 ${\mathbf{\gamma }}_{{j}\phantom{{\mathrm{\mu }}}}^{\phantom{{j}}{\mathrm{\mu }}}{}{{\partial }}_{{\mathrm{\mu }}}{}\left({\mathrm{\Phi }}{}\left({X}\right)\right)$ (9)

The covariant differential of a scalar function

 > $\mathrm{D3_}\left(\mathrm{\Phi }\left(X\right)\right)$
 ${{\partial }}_{{\mathrm{\mu }}}{}\left({\mathrm{\Phi }}{}\left({X}\right)\right){}{\mathbf{ⅆ}}{}\left({{X}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)$ (10)

For illustration purposes Define an arbitrary 3D space tensor  $A$; to indicate that this is a 3D tensor, define it with its space indices explicit, for example

 > $\mathrm{Define}\left(A\left[j\right]\right)$
 $\mathrm{Defined objects with tensor properties}$
 $\left\{{{A}}_{{j}}{,}{\mathbit{▿}}_{{\mathrm{\mu }}}{,}{{▿}}_{{\mathrm{\mu }}}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{,}{{\mathrm{\sigma }}}_{{\mathrm{\mu }}}{,}{{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{\mathbf{\beta }}_{{\mathrm{\mu }}}{,}{{C}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{{\partial }}_{{\mathrm{\mu }}}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbf{\gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\gamma }}}_{{i}{,}{j}}{,}{{\mathrm{\Gamma }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{\mathbf{\Gamma }}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}}{,}{{G}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\epsilon }}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{R}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{,}{\mathbit{t}}_{{\mathrm{\mu }}}{,}{\mathbf{Κ}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathbit{n}}_{{\mathrm{\mu }}}{,}{{X}}_{{\mathrm{\mu }}}\right\}$ (11)

Use a compact display for this function to avoid redundant repeated display of its functionality plus having derivatives displayed indexed by the differentiation variables

 > $\mathrm{CompactDisplay}\left(A\left(X\right)\right)$
 ${A}{}\left({X}\right){}{\mathrm{will now be displayed as}}{}{A}$ (12)

Now we have, for the total 3D covariant differential and the 3D covariant derivative, respectively,

 > $\mathrm{CompactDisplay}\left(A\left[j\right]\left(X\right)\right)$
 ${A}{}\left({X}\right){}{\mathrm{will now be displayed as}}{}{A}$ (13)
 > $\mathrm{D3_}\left(A\left[\mathrm{~j}\right]\left(X\right)\right)$
 ${{\mathrm{D3_}}}_{{\mathrm{μ}}}{}\left({{A}}_{{\mathrm{~j}}}{}\left({X}\right){,}\left[{X}\right]\right){}{\mathrm{d_}}{}\left({\left({X}\right)}_{{\mathrm{~mu}}}\right)$ (14)
 > $\mathrm{D3_}\left[j\right]\left(A\left[\mathrm{~j}\right]\left(X\right)\right)$
 ${{\mathrm{D3_}}}_{{j}}{}\left({{A}}_{{\mathrm{~j}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (15)

This D3_ covariant derivative is expressed in terms of the Christoffel3 symbols in the same way the covariant derivative D_ is expressed in terms of the 4D Christoffel. To see that you can use expand or convert to d_

 > $\mathrm{convert}\left(,\mathrm{d_}\right)$
 ${{\mathrm{gamma3_}}}_{{j}{,}{\mathrm{~alpha}}}{}{{\mathrm{gamma3_}}}_{{\mathrm{β}}{,}{\mathrm{~j}}}{}\left({{\mathrm{d_}}}_{{\mathrm{α}}}{}\left({{A}}_{{\mathrm{~beta}}}{}\left({X}\right){,}\left[{X}\right]\right){+}{{\mathrm{Christoffel}}}_{{\mathrm{~beta}}{,}{\mathrm{α}}{,}{\mathrm{ν}}}{}{{A}}_{{\mathrm{~nu}}}{}\left({X}\right)\right)$ (16)

Set the spacetime metric g_ by giving the square of the spacetime interval

 > $\mathrm{ds2}≔{x}^{2}{\mathrm{dx}}^{2}+{y}^{2}{\mathrm{dy}}^{2}+{z}^{2}{\mathrm{dz}}^{2}+xy\mathrm{dx}\mathrm{dy}-{\mathrm{dt}}^{2}$
 ${\mathrm{ds2}}{≔}{{x}}^{{2}}{}{{\mathrm{dx}}}^{{2}}{+}{x}{}{y}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{{y}}^{{2}}{}{{\mathrm{dy}}}^{{2}}{+}{{z}}^{{2}}{}{{\mathrm{dz}}}^{{2}}{-}{{\mathrm{dt}}}^{{2}}$ (17)
 > $\mathrm{Setup}\left(\mathrm{coordinates}=\mathrm{cartesian},\mathrm{metric}=\mathrm{ds2}\right)$
 ${}{}\mathrm{Systems of spacetime coordinates are:}{}{}{}\left\{X=\left(x{,}y{,}z{,}t\right)\right\}$
 ${}{}\mathrm{_______________________________________________________}$
 $\mathrm{Coordinates:}{}\left[x{,}y{,}z{,}t\right]{}\mathrm{. Signature:}{}\left(\mathrm{+ + + -}\right)$
 ${}{}\mathrm{_______________________________________________________}$
 ${{\mathrm{g_}}}_{{\mathrm{μ}}{,}{\mathrm{ν}}}{=}\left(\left[\begin{array}{cccc}{x}^{2}& \frac{1}{2}{}x{}y& 0& 0\\ \frac{1}{2}{}x{}y& {y}^{2}& 0& 0\\ 0& 0& {z}^{2}& 0\\ 0& 0& 0& -1\end{array}\right]\right)$
 ${}{}\mathrm{_______________________________________________________}$
 $\mathrm{_______________________________________________________}$
 $\left[{\mathrm{coordinatesystems}}{=}\left\{{X}\right\}{,}{\mathrm{metric}}{=}\left\{\left({1}{,}{1}\right){=}{{x}}^{{2}}{,}\left({1}{,}{2}\right){=}\frac{{x}{}{y}}{{2}}{,}\left({2}{,}{2}\right){=}{{y}}^{{2}}{,}\left({3}{,}{3}\right){=}{{z}}^{{2}}{,}\left({4}{,}{4}\right){=}{-1}\right\}\right]$ (18)

The generalized divergence of a vector in curvilinear coordinates

 > $\mathrm{D3_}\left[j\right]\left(A\left[\mathrm{~j}\right]\left(X\right)\right)$
 ${{\mathrm{D3_}}}_{{j}}{}\left({{A}}_{{\mathrm{~j}}}{}\left({X}\right){,}\left[{X}\right]\right)$ (19)
 > $\mathrm{expand}\left(\right)$
 ${\mathbf{\gamma }}_{{j}\phantom{{\mathrm{\alpha }}}}^{\phantom{{j}}{\mathrm{\alpha }}}{}{\mathbf{\gamma }}_{\phantom{{}}\phantom{{j}}{\mathrm{\beta }}}^{\phantom{{}}{j}\phantom{{\mathrm{\beta }}}}{}\left({{\partial }}_{{\mathrm{\alpha }}}{}\left({{A}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}}^{\phantom{{}}{\mathrm{\beta }}}{}\left({X}\right)\right){+}{{\mathrm{\Gamma }}}_{\phantom{{}}\phantom{{\mathrm{\beta }}}{\mathrm{\alpha }}{,}{\mathrm{\nu }}}^{\phantom{{}}{\mathrm{\beta }}\phantom{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}}{}{{A}}_{\phantom{{}}\phantom{{\mathrm{\nu }}}}^{\phantom{{}}{\mathrm{\nu }}}{}\left({X}\right)\right)$ (20)
 > $\mathrm{TensorArray}\left(\right)$
 $\frac{\left({\mathrm{diff}}{}\left({{A}}_{{\mathrm{~1}}}{}\left({X}\right){,}{x}\right)\right){}{x}{}{y}{}{z}{+}{{A}}_{{\mathrm{~1}}}{}\left({X}\right){}{y}{}{z}{+}\left({\mathrm{diff}}{}\left({{A}}_{{\mathrm{~2}}}{}\left({X}\right){,}{y}\right)\right){}{x}{}{y}{}{z}{+}{{A}}_{{\mathrm{~2}}}{}\left({X}\right){}{x}{}{z}{+}\left({\mathrm{diff}}{}\left({{A}}_{{\mathrm{~3}}}{}\left({X}\right){,}{z}\right)\right){}{x}{}{y}{}{z}{+}{{A}}_{{\mathrm{~3}}}{}\left({X}\right){}{x}{}{y}}{{x}{}{y}{}{z}}$ (21)

To compute with a representation for D3_ without actually performing the operation, use the inert form %D3_. To afterwards perform the operation use value. For example, the covariant derivative of the 3+1 metric gamma3_ is equal to 0

 > $\mathrm{%D3_}\left[i\right]\left(\mathrm{gamma3_}\left[j,k\right]\right)$
 ${{\mathrm{%D3_}}}_{{i}}{}\left({{\mathrm{gamma3_}}}_{{j}{,}{k}}\right)$ (22)
 > $\mathrm{value}\left(\right)$
 ${0}$ (23)
 > 

References

 [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.

Compatibility

 • The Physics[ThreePlusOne][D3_] command was introduced in Maple 2017.