kernel(scalar_backend="mathematica")

{t,r,\theta,\phi}::Coordinate;
{\mu,\nu,\rho,\sigma,\kappa,\lambda}::Indices(values={t,r,\phi,\theta}, position=fixed).
rs::LaTeXForm("r_s").
g_{\mu\nu}::Metric;
g^{\mu\nu}::InverseMetric;
\partial{#}::PartialDerivative.

\Delta::Depends(r);
\Sigma::Depends(r);

kerr:= { g_{t t}            = - (1 - rs r / \Sigma),
         g_{t \phi}         = - rs a r \sin(\theta)**2 / \Sigma,
         g_{\phi t}         = - rs a r \sin(\theta)**2 / \Sigma,
         g_{r r}            = \Sigma / \Delta,
         g_{\theta \theta } = \Sigma,
         g_{\phi \phi}      = ( r**2 + a**2 + rs a**2 r \sin(\theta)**2 / \Sigma ) \sin(\theta)**2 };

complete(kerr, $g^{\mu\nu}$);

DeltaSigma:= { \Delta = r**2 - 2 m r + a**2, \Sigma = r**2 + a**2 \cos(\theta)**2 };

substitute(kerr, DeltaSigma)
simplify(kerr);
display(kerr)

# import cdb.relativity as gr
# 
# ch = gr.christoffel_from_metric()
# evaluate(ch, kerr, rhsonly=True);