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Yukterez
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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 differentdifferent than Schwarzschild.

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.

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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Yukterez
Yukterez
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Yes, if you look at the $g_{rr}$ term of the inner Schwarzschild metricinterior Schwarzschild metric for homogenous spheres you'll see that $\surd g_{rr}={\rm d}R/{\rm d}r>1$$\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 neutonneutron star it depends on its specific radius and also the spin, which makes the metric different than Schwarzschild.

Yes, if you look at the $g_{rr}$ term of the inner 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 radius is larger by $\rm +1.5 \ mm$ and the volume by $\rm +4.5e11 \ m^3$ compared to the euclidean sphere. For a neuton star it depends on its specific radius and also the spin, which makes the metric different than Schwarzschild.

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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Yukterez
Yukterez
  • 13.1k
  • 2
  • 32
  • 63

Yes, if you look at the $g_{rr}$ term of the inner 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 radius is larger by $\rm +1.5 \ mm$ and the volume by $\rm +4.5e11 \ m^3$ compared to the euclidean sphere. For a neuton star it depends on its specific radius and also the spin, which makes the metric different than Schwarzschild.