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I examine parametric stability of interior Schwarzschild solution by analyzing its circular geodesics. The picture below shows the result of solving corresponding geodesics equation (Is black hole event horizon a transcendent tachyon?). Depending on combination of the reduced Schwarzschild and geodesics curvature radii, $\alpha\equiv r_S/R$ and $r_0\equiv r_0/R$, there are four distinct regions divided by three curves in the diagram.

My question is how to interpret the region $\text{IV}$? It represents the spacetime "behind" the hyperspace defined by the equation $g_{00}(r,\alpha)=0$, however there are no circular geodesics there.

enter image description here

JanG
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1 Answers1

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I don't know what to make of your "$\rm r_0=\frac{r_0}{R}$", but the radius of the sphere must be larger than $\rm \frac{9}{8} r_s$. If it's smaller the interior Schwarzschild solution isn't valid since homogenous and static density is not possible with smaller spheres (that would require imaginary pressure), so if your $\rm R$ is the sphere's radius then your $\alpha$ needs to be smaller than $\frac{8}{9}=0.\dot{8}$, everything above is unphysical.

Yukterez
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