If I understand correctly neutron stars are so dense that general relativistic effects are not negligible anymore. Does this mean that the volume inside of neutron stars is bigger than we would expect because of the resulting spacetime curvature?


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Yes, if you look at the $g_{rr}$ term of the interior Schwarzschild metric for homogenous spheres you'll see that $\surd |g_{rr}|={\rm d}R/{\rm d}r>1$, so the proper volume (measured with stationary rulers) is larger than that of an euclidean sphere of the same cirumference.

For the earth the proper radius $R$ is larger by $\rm +1.5 \ mm$ than the coordinate radius $r$, and the proper volume by $\rm +4.5e11 \ m^3$ compared to the euclidean sphere. For a neutron star it depends on its specific radius and also the spin, which makes the metric different than Schwarzschild.


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