tensor(deprecated)/Christoffel1 - Help

tensor

 Christoffel1
 compute the Christoffel symbols of the first kind

 Calling Sequence Christoffel1(D1g)

Parameters

 D1g - rank three tensor_type of character [-1,-1,-1] representing the partial derivatives of the COVARIANT metric tensor. The components of D1g must be defined using the index/cf1 indexing function (see below) which takes care of the symmetry in the first two indices of the first partials of the metric tensor (due to the symmetry of the metric).

Description

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][Christoffel] and Physics[Christoffel] instead.

 Specifically,

${{\mathrm{D1g}}_{\mathrm{comts}}}_{i,j,k}≔{g}_{ij,k}$

 • The resultant tensor_type, Cf1 say, of this routine is the Christoffel symbols of the first kind, indexed as shown below:

${{\mathrm{Cf1}}_{\mathrm{compts}}}_{i,j,k}≔\left[ij,k\right],$

 where $\left[\mathrm{ij},k\right]$ is in conventional notation.
 • D1g, the partials of the covariant metric should be obtained using the function tensor[d1metric] once the metric itself is known.
 • Indexing Function: Because of the symmetry in the first two indices of the Christoffel symbols of the first kind, the array of the calculated symbols use the index/cf1 indexing function.  This function indexes an array of rank 3 so that it is automatically symmetric in its first two indices.  Use of this indexing function decreases the number of symbols that must be assigned and stored to the number of independent symbols.
 • Simplification: This routine uses the tensor/Christoffel1/simp routine to carry out simplification of each independent Christoffel symbol of the first kind.  By default, it is initialized to the tensor/simp function.  It is recommended that the tensor/Christoffel1/simp routine be customized to suit the particular needs of the problem at hand.
 • This function is part of the tensor package, and so can be used in the form Christoffel1(..) only after performing the command with(tensor) or with(tensor, Christoffel1).  The function can always be accessed in the long form tensor[Christoffel1](..).

Examples

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][Christoffel] and Physics[Christoffel] instead.

Define the coordinate variables and the covariant metric under the Schwarzchild metric.

 > $\mathrm{with}\left(\mathrm{tensor}\right):$
 > $\mathrm{coord}≔\left[t,r,\mathrm{th},\mathrm{ph}\right]:$
 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},\mathrm{sparse},1..4,1..4\right):$
 > ${\mathrm{g_compts}}_{1,1}≔1-\frac{2m}{r}:$${\mathrm{g_compts}}_{2,2}≔-\frac{1}{{\mathrm{g_compts}}_{1,1}}:$
 > ${\mathrm{g_compts}}_{3,3}≔-{r}^{2}:$${\mathrm{g_compts}}_{4,4}≔-{r}^{2}{\mathrm{sin}\left(\mathrm{th}\right)}^{2}:$
 > $g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{g_compts}\right)\right)$
 ${g}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cccc}{1}{-}\frac{{2}{}{m}}{{r}}& {0}& {0}& {0}\\ {0}& {-}\frac{{1}}{{1}{-}\frac{{2}{}{m}}{{r}}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}& {0}\\ {0}& {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-}{1}{,}{-}{1}\right]\right]\right)$ (1)
 > $\mathrm{D1g}≔\mathrm{d1metric}\left(g,\mathrm{coord}\right):$
 > $\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{D1g}\right)$
 ${\mathrm{Cf1}}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}{\mathrm{ARRAY}}{}\left({\mathrm{cf1}}{,}\left[{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}\right]{,}\left[\left({1}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{2}\right){=}{-}\frac{{m}}{{{r}}^{{2}}}{,}\left({1}{,}{1}{,}{3}\right){=}{0}{,}\left({1}{,}{1}{,}{4}\right){=}{0}{,}\left({1}{,}{2}{,}{1}\right){=}\frac{{m}}{{{r}}^{{2}}}{,}\left({1}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{2}{,}{3}\right){=}{0}{,}\left({1}{,}{2}{,}{4}\right){=}{0}{,}\left({1}{,}{3}{,}{1}\right){=}{0}{,}\left({1}{,}{3}{,}{2}\right){=}{0}{,}\left({1}{,}{3}{,}{3}\right){=}{0}{,}\left({1}{,}{3}{,}{4}\right){=}{0}{,}\left({1}{,}{4}{,}{1}\right){=}{0}{,}\left({1}{,}{4}{,}{2}\right){=}{0}{,}\left({1}{,}{4}{,}{3}\right){=}{0}{,}\left({1}{,}{4}{,}{4}\right){=}{0}{,}\left({2}{,}{1}{,}{1}\right){=}\frac{{m}}{{{r}}^{{2}}}{,}\left({2}{,}{1}{,}{2}\right){=}{0}{,}\left({2}{,}{1}{,}{3}\right){=}{0}{,}\left({2}{,}{1}{,}{4}\right){=}{0}{,}\left({2}{,}{2}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{2}\right){=}\frac{{m}}{{\left({-}{r}{+}{2}{}{m}\right)}^{{2}}}{,}\left({2}{,}{2}{,}{3}\right){=}{0}{,}\left({2}{,}{2}{,}{4}\right){=}{0}{,}\left({2}{,}{3}{,}{1}\right){=}{0}{,}\left({2}{,}{3}{,}{2}\right){=}{0}{,}\left({2}{,}{3}{,}{3}\right){=}{-}{r}{,}\left({2}{,}{3}{,}{4}\right){=}{0}{,}\left({2}{,}{4}{,}{1}\right){=}{0}{,}\left({2}{,}{4}{,}{2}\right){=}{0}{,}\left({2}{,}{4}{,}{3}\right){=}{0}{,}\left({2}{,}{4}{,}{4}\right){=}{-}{r}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}{,}\left({3}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{1}{,}{2}\right){=}{0}{,}\left({3}{,}{1}{,}{3}\right){=}{0}{,}\left({3}{,}{1}{,}{4}\right){=}{0}{,}\left({3}{,}{2}{,}{1}\right){=}{0}{,}\left({3}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{2}{,}{3}\right){=}{-}{r}{,}\left({3}{,}{2}{,}{4}\right){=}{0}{,}\left({3}{,}{3}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{2}\right){=}{r}{,}\left({3}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{4}\right){=}{0}{,}\left({3}{,}{4}{,}{1}\right){=}{0}{,}\left({3}{,}{4}{,}{2}\right){=}{0}{,}\left({3}{,}{4}{,}{3}\right){=}{0}{,}\left({3}{,}{4}{,}{4}\right){=}{-}{{r}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{th}}\right){}{\mathrm{cos}}{}\left({\mathrm{th}}\right){,}\left({4}{,}{1}{,}{1}\right){=}{0}{,}\left({4}{,}{1}{,}{2}\right){=}{0}{,}\left({4}{,}{1}{,}{3}\right){=}{0}{,}\left({4}{,}{1}{,}{4}\right){=}{0}{,}\left({4}{,}{2}{,}{1}\right){=}{0}{,}\left({4}{,}{2}{,}{2}\right){=}{0}{,}\left({4}{,}{2}{,}{3}\right){=}{0}{,}\left({4}{,}{2}{,}{4}\right){=}{-}{r}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}{,}\left({4}{,}{3}{,}{1}\right){=}{0}{,}\left({4}{,}{3}{,}{2}\right){=}{0}{,}\left({4}{,}{3}{,}{3}\right){=}{0}{,}\left({4}{,}{3}{,}{4}\right){=}{-}{{r}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{th}}\right){}{\mathrm{cos}}{}\left({\mathrm{th}}\right){,}\left({4}{,}{4}{,}{1}\right){=}{0}{,}\left({4}{,}{4}{,}{2}\right){=}{r}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}{,}\left({4}{,}{4}{,}{3}\right){=}{{r}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{th}}\right){}{\mathrm{cos}}{}\left({\mathrm{th}}\right){,}\left({4}{,}{4}{,}{4}\right){=}{0}\right]\right){,}{\mathrm{index_char}}{=}\left[{-}{1}{,}{-}{1}{,}{-}{1}\right]\right]\right)$ (2)

The user may also view the result using the tensor package function displayGR.