# Solution of Einstein's equations in a vacuum

+1 vote

Hello people.
I'm trying to solve Einstein's equations in vacuo for a generic line element, I can get to the equations G _ {\ alpha \ beta}, but I can't get the values for exp (\ mu) and \ exp (nu). If anyone knows thank you.

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{r,t,\phi,\theta}::Coordinate;
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\chi,\gamma}::Indices(values=t,r,\phi,\theta},position=fixed);

\partial{#}::PartialDerivative;
g_{\mu\nu}::Metric;
g^{\mu\nu}::InverseMetric;

v::Depends(r);
u::Depends(r);

ss:={g{t t} = \exp(v),
g
{r r} =-\exp(u),
g_{\theta\theta} = r2,
g_{\phi\phi} = r
2 \sin(\theta)**2}.
complete(ss,$g^{\mu\nu}$);

ch := \Gamma^{\mu}{\nu\rho} = 1/2 g^{\mu\sigma} (
\partial
{\rho}{g{\nu\sigma}}
+\partial
{\nu}{g{\rho\sigma}}
-\partial
{\sigma}{g_{\nu\rho}});
evaluate(ch,ss,rhsonly=True);

rm:= R^{\rho}{\sigma\mu\nu} = \partial{\mu}{\Gamma^{\rho}{\nu\sigma}}
-\partial
{\nu}{\Gamma^{\rho}{\mu\sigma}}
+\Gamma^{\rho}
{\mu\lambda}\Gamma^{\lambda}{\nu\sigma}
-\Gamma^{\rho}
{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma};
substitute(rm,ch);
evaluate(rm,ss,rhsonly=True);

rc:= R{\sigma\nu} = R^{\rho}{\sigma\rho\nu};
substitute(rc, rm)
evaluate(rc, ss, rhsonly=True);

src:= R = g^{\sigma\nu}R_{\sigma\nu};
substitute(src,rc);
evaluate(src,ss,rhsonly=True);

Eins := G{\sigma\nu} = R{\sigma\nu}-1/2g_{\sigma\nu}R;
substitute(Eins,rc);
substitute(Eins,src);
evaluate(Eins,ss,rhsonly=True);

from cdb.core.component import *
from cdb.core.manip import *

gtt = getcomponent(Eins,$t,t$)[1];
grr = get
component(Eins,$r,r$)[1];
gth = getcomponent(Eins,$\theta,\theta$)[1];
gph = get
component(Eins,$\phi,\phi$)[1];

from cdb.sympy.solvers import *

eq1:= @(gtt)-@(grr);

from sympy import *