What is happening when a metric function changes sign?

I currently worked on a type of modified Tolman-Oppenheimer-Volkoff (TOV) equation. Usually people have this equation from a static spherically symmetric metric that has this form: $$ds^2= -C(r) dt^2+\frac{dr^2}{1-2Gm(r)/r}+r^2(d\theta^2+\sin^2\theta d\varphi^2).$$ From what I understand from the textbooks, e.g. Spacetime & Geometry by Carroll, the usual TOV equation will describe the inner metric of a massive spherical body with radius $$R$$ and mass $$M$$, i.e., the exterior metric should be the usual Schwarzchild metric ($$C\to 1-2GM/r$$ and $$m\to M$$ as $$r\to R$$). This inner metric also should not changes sign, i.e., $$C>0$$ and $$1-2Gm/r > 0$$ for $$r\in (0,R]$$.

Now my question is this: What is happening in physical sense when $$C<0$$ or $$1-2Gm/r < 0$$?

FYI, my problem is the following. After running the numerical calculation for the modified TOV equation, it gives me weird answer even though the numerical solutions satisfy the boundary condition, i.e., the pressure of an ideal fluid vanishes at the surface $$p(r=R)=0$$. I obtain that $$C>0$$ for $$r\in (0,R]$$ but $$1-2Gm/r\leq 0$$ for $$r\in (0,r']$$ with $$r' (its value depends on the numerical calculation).

I tried reading the more advanced books, e.g. General Relativity by Wald, but I still don't find the answer.

1 Answer

The Einstein equations never allow the metric to change signature (the number of eigenvalues of the metric tensor of a given sign) up to curvature singularities. What you see in the Schwarzschild metric at $$r=2M$$ is a coordinate singularity and you can see that the signature is the same above and below this point, it is just that the $$r$$ coordinate plays the role of time inside the event horizon. Physically this means that observers "at rest at infinity" (observers very far from the black hole that do not see the black hole as moving) will see funny things happening at $$r=2M$$ because $$t$$ is the time that runs on their clocks. For example, they will see objects redden and dim as they fall towards $$r=2M$$ and eventually become undetectable. However, from the perspective of the infalling objects, nothing too special happens at $$r=2M$$ apart from the fact that they become unable to communicate with observers and objects at $$r>2M$$.

But back to your problem: what you are describing sounds like an artifact in the numerical code. The Einstein equations should always resist changing the sign of $$1-2Gm(r)/r$$ while not changing the sign of $$C(r)$$ by a singularity. Numerical error is large near singularities, so maybe you managed to just integrate through it by using a too large step, or maybe there is a typo in the equations. Sometimes you get funny behavior when non-physical equations of state are used (speed of sound is large than speed of light), but I never heard about nonphysical equations of state being able to change metric signature.