# Schwarzschild Curvature

## Initialization

Run the following code to initialize $\LaTeX$ output and load some macros, either by clicking on "Evaluate" or by typing Shift+Enter.

## Code

Now enter a line element in the box below, adapting the given code as needed. First declare any parameters or functions, then provide a list of coordinates using the MakeC command as shown below (note the double parentheses). Finally, use the Makeg command to enter both the signature (number of $-$ signs) and the coefficients $h^i$ of an orthonormal basis $\sigma^i=h^i\,dx^i$ (no sum) of 1-forms in orthogonal coordinates. The result should be the line element in tensor notation.

Components are displayed below by default in an orthonormal frame ($e$). To compare with the standard literature, you can instead display the results in a coordinate basis ($f$) by uncommenting the MakeF command below.

Use the command MakeGam() to compute and display the connection 1-forms $\omega^i{}_j=\Gamma^i{}_{jk}\sigma^k$ for an orthonormal basis. (You can then list the nonzero connection coefficients $\Gamma^i{}_{jk}$ with the command nab.display().)

Use the command MakeOm() to compute and display the curvature 2-forms $\Omega^i{}_j=\frac12R^i{}_{jkl}\sigma^k\wedge\sigma^l$ for an orthonormal basis. (You can then print specific curvature components $R^i{}_{jkl}$ with the command riem[i,j,k,l]).

Use the command MakeRic() to compute the components of the Ricci tensor $R_{ij}=R^m{}_{imj}$, then use the command ric.up(g,1)[:] to raise an index and display the result. (These are the components of the Ricci vector-valued 1-form $\mathbf{\vec{R}}=R^i{}_j\sigma^j\mathbf{\hat{e}}_i$.)

Compute and display the components of the Ricci scalar $R=R^i{}_i$.

Use the command MakeEin() to compute the components of the Einstein tensor $G_{ij}=R_{ij}-\frac12g_{ij}R$, then use the command ein.up(g,1)[:] to raise an index and display the result. (These are the components of the Einstein vector-valued 1-form $\mathbf{\vec{G}}=G^i{}_j\sigma^j\mathbf{\hat{e}}_i$.)