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I am investigating the metric $ds^2 = -dt^2 + (1+C)dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2, $ where $C$ is a constant. This intuitively seems like flat space but actually has a non-zero Kret …

Does anyone know whether this metric has been studied before or if it has a proper name? $$ds^2 = -dt^2 + e^{2At} dx^2 + e^{2Bt} dy^2 + e^{2Ct} dz^2$$ i.e. a de Sitter metric which has a different e …

I am trying to introduce a tortoise coordinate for a modified Schwarzschild metric $$\mathrm{d}s^2=\left(1-\frac{2M\mathop{}\!\mathrm{erf}(r)}{r}\right) \mathrm{d}t^2 + \left(1-\frac{2M\mathop{}\!\ma …

You raise tensors using your metric tensor. For flat spacetime, this is the Minkowski metric $\eta_{\mu\nu}$. You must contract the Minkowski metric with one of the indices of your tensor in order to …

I am trying to derive the weak-field Schwarzschild metric, but starting from the same form as Schwarzschild: $ds^2=-(1+2\Phi(r))dt^2+(1-2\Psi(r))dr^2 +r^2 d\Omega^2$ which has $R=-2\partial_r^2 \Phi …