This article describes how one can use worksheets to guide undergraduate students through the process of finding solutions to specific cases of the Einstein equation of general relativity. The worksheets provide expressions for a metric's Christoffel symbols and Ricci tensor components for fairly general metrics. Students can use a worksheet to adapt these expressions to specific cases where symmetry or other considerations constrain the metric components' dependencies, and then use the worksheet's results to reduce the Einstein equation to a set of simpler differential equations that they can solve. This article illustrates the process for both a diagonal metric and a metric with one off-diagonal element.

## References

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See, for example, the computer tools “GRTensor II” (<http://grtensor.phy.queensu.ca>), “Maxima” (<http://maxima.sourceforge.net>), or “Ricci” (<http://www.math.washington.edu/~lee/Ricci/>). See also the list at <https://en.wikipedia.org/wiki/Tensor_software>, though this may not be up to date.

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One can find a variety of general relativity simulation programs (as well as other resources) at <http://www.compadre.org/relativity/search/search.cfm?gs=1374&SS=1384&b=1>.

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See supplementary material at http://dx.doi.org/10.1119/1.4939908, which includes PDF versions of the Diagonal Metric Worksheet and the Off-Diagonal Metric Worksheet, the Xojo source code, and a PDF listing of the source code. To run the computer code, download the Xojo IDE (free) from <http://xojo.com>. This program extensively uses a Xojo feature called “operator overloading,” which allows for the definition of what the +, −, and $*$ operators do when they appear between variables pointing to instances of the Expression class. Please note that the program's output still required quite a bit of human post-processing to yield the Off-Diagonal Metric Worksheet. Also note that the program uses “G” where this paper and the worksheet use “H” (which we substituted in the post-processing stage because “G” could easily be confused with the universal gravitational constant).

14.

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17.

Ref. 7, problem P23.1 (page 276).

18.

Ref. 7, problem P23.6 (page 278).

19.

Ref. 7, pp. 292–302, 306.

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Ref. 7, pp. 376–378.

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Ref. 7, p. 271ff.

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Ref. 3, pp. B-4, B-12 through B-15.

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Ref. 7, pp. 180–182.

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Ref. 15 (Hobson

*et al*.), p. 256.© 2016 American Association of Physics Teachers.

2016

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