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A Mathematica package that provides a set of functions for performing coordinate-based tensor calculations with a focus on general relativity and black holes in particular.

Example usage

The package has extensive documentation and tutorials. Below we give a few simple examples of the package in action.

Defining the metric

All calculations using GeneralRelativityTensors require a metric with explicit values. You can define your own metric and there are also a range of useful build in metrics, i.e.,

g = ToMetric["Kerr"]

Other build in metrics include "Minkowski", "Schwarzschild", "ReissnerNordstrom", “KerrNewman”, etc (see the documentation of ToMetric[] for more details).

To view the components of any tensor you can use the TensorValues[] function. For example, with the metric defined above, TensorValues[g] returns

\[\left( \begin{array}{cccc} \frac{a^2 \sin ^2(\theta )-a^2+2 M r-r^2}{a^2 \cos ^2(\theta )+r^2} & 0 & 0 & -\frac{2 a M r \sin ^2(\theta )}{a^2 \cos ^2(\theta )+r^2} \\ 0 & \frac{a^2 \cos ^2(\theta )+r^2}{a^2-2 M r+r^2} & 0 & 0 \\ 0 & 0 & a^2 \cos ^2(\theta )+r^2 & 0 \\ -\frac{2 a M r \sin ^2(\theta )}{a^2 \cos ^2(\theta )+r^2} & 0 & 0 & \frac{\sin ^2(\theta ) \left(\left(a^2+r^2\right)^2-a^2 \sin ^2(\theta ) \left(a^2-2 M r+r^2\right)\right)}{a^2 \cos ^2(\theta )+r^2} \\ \end{array} \right) \nonumber\]

Defining tensors

The real power of the package comes from forming and manipulating tensors. Tensors are created using the ToTensors[] command. A few things to note:

The following command defines a tensor $t_a$ on the metric $g$ above where all the components are functions of $r$.

t1 = ToTensor[{"NewTensor", "t"}, g, {f1[r], f2[r], f3[r], f4[r]}, {-a}]

The contravariant version, $t^a$, can be accessed via t[a]. As with the metric, the values of any tensor can be explicitly computed using TensorValues[].

Common tensors

Some common tensors are build in. For instance ChristoffelSymbol[g, ActWith -> Simplify] will compute the ChristoffelSymbols, $\Gamma^\alpha_{\beta\gamma}. The ActWith option means the Simplify command will be applied to each component before returning the result. Useful built in common tensors include: RiemmannTensor, RicciTensor, RicciScalar, Einstein Tensor, KretschmannScalar, WeylTensor and many more.

Merging tensors

Another way to construct tensors is by merging other tensors using the command MergeTensors[]. As an example you can construct the Einstein tensor from the Ricci tensor, scalar and the metric via the commands:

g = ToMetric["Schwarzschild"]
ricT = RicciTensor[g]
ricS = RicciScalar[g]
einExpr = ricT[-a, -b] - g[-a], -b] ricS/2

The variable einExpr is not yet a tensor, but rather the sum and product of three different Tensors. These can be combined using the MergeTensors, e.g.,

einS = MergeTensors[einSExpr, {"EinsteinSchwarzschild", "G"}, ActWith -> Simplify]

This returns $G_{ab}$. You can explicitly then check that the Schwarzschild solution is a vacuum solution via TensorValues[EinS] which returns

{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}

More examples

The above just scratches the surface of what this package can do. Check the documentation for more details.

Authors and contributors

Seth Hopper, Barry Wardell