Simulating an orbit around a spherical mass
The curvature of space-time around a spherical mass is described by the Schwarzschild metric.
![Rendered by QuickLaTeX.com c^2 d\tau^2 = \left(1 - \frac{2GM}{c^2 r} \right) c^2 dt^2 - \left(1 - \frac{2GM}{c^2 r} \right)^{-1} dr^2 - r^2 \left(d \theta^2 + sin^2 \theta d \varphi^2 \right)](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-7013b3f3261446fccf5a8b01db1c7b1d_l3.png)
We provide an Excel file simulating the orbit of a test particle around the spherical mass using a straightforward forward Euler simulation.
Download link: 20161106_geodesic_line_simulation_riemann_schwarzschild_2d_v3
The simulation is performed in a 2-dimensional space where
![Rendered by QuickLaTeX.com \theta = \pi/2](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-6151bc46680fe7c6214c9e6fd4cc8790_l3.png)
![Rendered by QuickLaTeX.com c=1](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-8bc0f94c08cf058d316895e763d34082_l3.png)
![Rendered by QuickLaTeX.com d\tau^2 = \left(1 - \frac{2GM}{r}\right) dt^2 - \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 - r^2 d \varphi^2](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-b5bda41e1887cd42750a74267691c9c7_l3.png)
or
![Rendered by QuickLaTeX.com d\tau^2 = \left[ \begin{matrix} dr & d \varphi & dt \end{matrix} \right] \left[ \begin{matrix} - \left(1 - \frac{2GM}{r}\right)^{-1} & 0 & 0 \\ 0 & - r^2 & 0 \\ 0 & 0 & \left(1 - \frac{2GM}{r}\right) \end{matrix} \right] \left[ \begin{matrix} dr \\ d \varphi \\ dt \end{matrix} \right] = g_{ij} dx^i dx^j](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-1f57e1d3b974797a6d315e3b7f1a6908_l3.png)
where the right hand side uses Einstein notation.
Given coördinates and first order derivatives
at time
, the 2nd order derivatives feeding the simulation can be derived using the geodesic equation:
![Rendered by QuickLaTeX.com \frac{d^2x^{\mu}}{dq^2} + \Gamma^{\mu}_{\nu\lambda} \frac{dx^{\nu}}{dq} \frac{dx^{\lambda}}{dq} = 0](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-fca8cd31f0cb3e685f28c968f3485198_l3.png)
where
![Rendered by QuickLaTeX.com \Gamma](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-4f420945e64069f30b66c3d17e2f98ac_l3.png)
![Rendered by QuickLaTeX.com \Gamma^m_{ij} = \frac{1}{2}g^{ml} \left( \frac{\delta g_{il}}{\delta x^j} + \frac{\delta g_{lj}}{\delta x^i} - \frac{\delta g_{ji}}{\delta x^l} \right)](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-72a6a905dbb9f859370862777f68bbc3_l3.png)
again using Einstein notation and with
![Rendered by QuickLaTeX.com g^{ml}](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-20b8d1a7e35fb11d6c0dde563906f9f8_l3.png)
![Rendered by QuickLaTeX.com g](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png)
The exact matrices
![Rendered by QuickLaTeX.com \Gamma](https://ivan.goethals-jacobs.be/wp-content/ql-cache/quicklatex.com-4f420945e64069f30b66c3d17e2f98ac_l3.png)
An example is provided below and can be simulated in the Excel by setting the starting values to and
and the simulation step size to
.
![Simulation with G=M=1, at a distance of about 25 times the Schwarzschild radius. Perihelium-shift is clearly visible.](https://ivan.goethals-jacobs.be/wp-content/uploads/2016/12/Orbit_schwarzschild_001-300x286.png)