The GRTensor package, developed by K. Lake, P. Musgrave and D. Polleny, has long provided the general relativity community with an excellent collection of command for symbolic computations in general relativity. This functionality is also now available within the DifferentialGeometry package and its subpackage Tensor, as well as within the Physics package.

This worksheet provides a quick introduction to General Relativity computations using DifferentialGeometry.

with(DifferentialGeometry): with(Tensor): with(Tools):

DGsetup([t, r, theta, phi], M1, verbose);

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[t\,r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{M1}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{M1}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{M1}\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{M1}\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{M1}\,1\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{M1}\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{M1}\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{M1}\,1\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{frame\; name:\; M1}$

g := evalDG( A(r) *dt &t dt + B(r)* dr &t dr + r^2*(dtheta &t dtheta + sin(theta)^2* dphi &t dphi));

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{M1}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,A\left(r\right)\right]\,\left[\left[2\,2\right]\,B\left(r\right)\right]\,\left[\left[3\,3\right]\,r^2\right]\,\left[\left[4\,4\right]\,r^2{\sin \left(\mathrm{\θ}\right)}^2\right]\right]\right]\right)$

Ein := EinsteinTensor(g);

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{M1}\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{\left(\mathrm{diff}\left(B\left(r\right)\,r\right)\right)r\+{B\left(r\right)}^2-B\left(r\right)}{A\left(r\right)r^2{B\left(r\right)}^2}\right]\,\left[\left[2\,2\right]\,\frac{\left(\mathrm{diff}\left(A\left(r\right)\,r\right)\right)r-A\left(r\right)B\left(r\right)\+A\left(r\right)}{A\left(r\right){B\left(r\right)}^2{r}^2}\right]\,\left[\left[3\,3\right]\,\frac{1}{4}\frac{2\left(\mathrm{diff}\left(\mathrm{diff}\left(A\left(r\right)\,r\right)\,r\right)\right)A\left(r\right)B\left(r\right)r-\left(\mathrm{diff}\left(B\left(r\right)\,r\right)\right)\left(\mathrm{diff}\left(A\left(r\right)\,r\right)\right)A\left(r\right)r-{\left(\right)}^{}}{}\right]\right]\right]\right)$