Geodesics computes and attempts to solve the geodesic equations for the spacetime metric g_. These equations are parametrized by an affine parameter $\mathrm{\tau }$ typically representing the proper time for a timelike curve, or distance for a spacelike curve. These geodesic equations computed are

where $X_{}^{\mathrm{\mu }}$ is the SpaceTimeVector and ${\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\nu }}^{\mathrm{\mu }}$ are the Christoffel symbols. As it is the case of the other commands in Physics, Geodesics operates in tensorial notation. To perform the same computation using differential forms notation see DifferentialGeometry[Tensor][GeodesicEquations].

When a name - say tau is provided as first argument, it will be used to represent the parameter for the geodesic equations; otherwise, the symbol $\mathrm{\tau }$ itself is used.

By default, Geodesics computes only the equations, without attempting solving them. To request compute and solve the equations in one step, pass the option output = solutions. When Geodesics or dsolve fail in finding a solution for the equations, you can still look for particular solutions of different forms that may be of interest using the symmetry commands of PDEtools like FunctionFieldSolutions and PolynomialSolutions, or InvariantSolutions used together with its various options for particularizing solutions.

The ODE system returned by default by Geodesics is returned as a list of equations and automatically simplified to a canonical form (reduced involutive form - RIF and DifferentialAlgebra) using differential algebra techniques. The order of the equations in the list corresponds to a total degree, obtained with an orderly ranking (see details in casesplit). Simplifying the ODE system this way is a frequently desired but in some cases expensive mathematical computation. To avoid it, pass the optional argument integrabilityconditions = false, in which case Geodesics will return faster, a set of equations, not a list, and with no differential simplification.

When computing just the geodesic equations, it also possible to request them in tensor notation, as an algebraic equation, instead of as a set or list of equations for the tensorial components. Although this kind of output always looks just like the definition for the geodesic equations shown above, the algebraic equation returned actually contains the computable objects like the SpaceTimeVector and Christoffel symbols, and can be manipulated in different ways, including getting from it all the equations for its components.

When the metric g_ depends on parameters, either symbols or functions of spacetime variables, the  equations computed with Geodesics are expected to be valid for arbitrary values of these parameters. It is sometimes of interest, however, to investigate the different kinds of solutions that may exist for different particular values of these parameters. To perform such an investigation, use the optional argument parameters = ... where the right-hand-side is a set or a list with the parameters. The system of geodesic equations will then be split into cases with respect to these parameters using differential algebra techniques. As frequently happens when splitting into cases, the resulting lists of equations may involve inequations as well. Note: passing parameters automatically forces simplifying the ODE system to a canonical (reduced involutive RIF) form regardless of the value of the keyword integrabilityconditions.

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

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

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

$\mathrm{Setup}\left(\mathrm{coordinates}=X\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

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

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

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

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

$\mathrm{Geodesics}\left(\mathrm{tensornotation}\right)$

$\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}{X}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\tau }\right)=0$

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

$\left[\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{x4}\left(\mathrm{\tau }\right)=0\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{x3}\left(\mathrm{\tau }\right)=0\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{x2}\left(\mathrm{\tau }\right)=0\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{x1}\left(\mathrm{\tau }\right)=0\right]$

$\mathrm{Geodesics}\left(\mathrm{output}=\mathrm{solutions}\right)$

$\left\{\mathrm{x1}\left(\mathrm{\tau }\right)=\mathrm{c\_\_1}\mathrm{\tau }+\mathrm{c\_\_2}\,\mathrm{x2}\left(\mathrm{\tau }\right)=\mathrm{c\_\_3}\mathrm{\tau }+\mathrm{c\_\_4}\,\mathrm{x3}\left(\mathrm{\tau }\right)=\mathrm{c\_\_5}\mathrm{\tau }+\mathrm{c\_\_6}\,\mathrm{x4}\left(\mathrm{\tau }\right)=\mathrm{c\_\_7}\mathrm{\tau }+\mathrm{c\_\_8}\right\}$

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

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

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

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

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

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

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

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

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

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

$\mathrm{Geodesics}\left(\mathrm{tensornotation}\right)$

$\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}{X}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\tau }\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\alpha },\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\alpha },\mathrm{\nu }}}\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}{X}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(\mathrm{\tau }\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}{X}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(\mathrm{\tau }\right)\right)=0$

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

$\left[\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{\theta }\left(\mathrm{\tau }\right)=\frac{\sin \left(\mathrm{\theta }\left(\mathrm{\tau }\right)\right)\cos \left(\mathrm{\theta }\left(\mathrm{\tau }\right)\right){\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)}^{2}r\left(\mathrm{\tau }\right)-2\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\theta }\left(\mathrm{\tau }\right)\right)}{r\left(\mathrm{\tau }\right)}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{\phi }\left(\mathrm{\tau }\right)=-\frac{2\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)\left(r\left(\mathrm{\tau }\right)\cot \left(\mathrm{\theta }\left(\mathrm{\tau }\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\theta }\left(\mathrm{\tau }\right)\right)+\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)}{r\left(\mathrm{\tau }\right)}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}r\left(\mathrm{\tau }\right)=\frac{4{\left(m-\frac{r\left(\mathrm{\tau }\right)}{2}\right)}^2\left(\cos \left(\mathrm{\theta }\left(\mathrm{\tau }\right)\right)-1\right)\left(\cos \left(\mathrm{\theta }\left(\mathrm{\tau }\right)\right)+1\right){r\left(\mathrm{\tau }\right)}^3{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)}^2-4{\left(m-\frac{r\left(\mathrm{\tau }\right)}{2}\right)}^2{r\left(\mathrm{\tau }\right)}^3{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\theta }\left(\mathrm{\tau }\right)\right)}^2+4{\left(m-\frac{r\left(\mathrm{\tau }\right)}{2}\right)}^{2}m{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}t\left(\mathrm{\tau }\right)\right)}^2-m{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)}^2{r\left(\mathrm{\tau }\right)}^2}{\left(2m-r\left(\mathrm{\tau }\right)\right){r\left(\mathrm{\tau }\right)}^3}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}t\left(\mathrm{\tau }\right)=-\frac{2m\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}t\left(\mathrm{\tau }\right)\right)}{r\left(\mathrm{\tau }\right)\left(-2m+r\left(\mathrm{\tau }\right)\right)}\right]$

$G\left[\mathrm{\theta }=\frac{\mathrm{\pi }}{2}\right]≔\mathrm{eval}\left(\,\mathrm{\theta }\left(\mathrm{\tau }\right)=\frac{\mathrm{\pi }}{2}\right)$

$G_{\mathrm{\theta }=\mathrm{\pi }\frac{1}{2}}≔\left[0=0\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}\mathrm{\phi }\left(\mathrm{\tau }\right)=-\frac{2\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)}{r\left(\mathrm{\tau }\right)}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}r\left(\mathrm{\tau }\right)=\frac{-4{\left(m-\frac{r\left(\mathrm{\tau }\right)}{2}\right)}^2{r\left(\mathrm{\tau }\right)}^3{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)}^2+4{\left(m-\frac{r\left(\mathrm{\tau }\right)}{2}\right)}^{2}m{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}t\left(\mathrm{\tau }\right)\right)}^2-m{\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)}^2{r\left(\mathrm{\tau }\right)}^2}{\left(2m-r\left(\mathrm{\tau }\right)\right){r\left(\mathrm{\tau }\right)}^3}\,\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}t\left(\mathrm{\tau }\right)=-\frac{2m\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}r\left(\mathrm{\tau }\right)\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}t\left(\mathrm{\tau }\right)\right)}{r\left(\mathrm{\tau }\right)\left(-2m+r\left(\mathrm{\tau }\right)\right)}\right]$

$\mathrm{dsolve}\left(G\left[\mathrm{\theta }=\frac{\mathrm{\pi }}{2}\right]\right)\left[1..2\right]$

$\left[\left\{r\left(\mathrm{\tau }\right)=6m\right\}\,\left\{\mathrm{\phi }\left(\mathrm{\tau }\right)=\mathrm{c\_\_2}\mathrm{\tau }+\mathrm{c\_\_3}\right\}\,\left\{t\left(\mathrm{\tau }\right)=-6\sqrt{6}m\mathrm{\phi }\left(\mathrm{\tau }\right)+\mathrm{c\_\_1}\,t\left(\mathrm{\tau }\right)=6\sqrt{6}m\mathrm{\phi }\left(\mathrm{\tau }\right)+\mathrm{c\_\_1}\right\}\right],\left[\left\{r\left(\mathrm{\tau }\right)=\mathrm{c\_\_4}\right\}\,\left\{\mathrm{\phi }\left(\mathrm{\tau }\right)=\mathrm{c\_\_2}\mathrm{\tau }+\mathrm{c\_\_3}\right\}\,\left\{t\left(\mathrm{\tau }\right)=\int \frac{\sqrt{mr\left(\mathrm{\tau }\right)}\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)r\left(\mathrm{\tau }\right)}{m}\ⅆ\mathrm{\tau }+\mathrm{c\_\_1}\,t\left(\mathrm{\tau }\right)=\int -\frac{\sqrt{mr\left(\mathrm{\tau }\right)}\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}\mathrm{\phi }\left(\mathrm{\tau }\right)\right)r\left(\mathrm{\tau }\right)}{m}\ⅆ\mathrm{\tau }+\mathrm{c\_\_1}\right\}\right]$

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