Undergrad Ricci Scalar For Astronomical Body

[dsaun777]

What would be a rough estimate for the Ricci scalar curvature of an astronomical object like the sun? Assuming the sun is a perfect fluid and you are calculating the rest frame of the sun, only the density component would be factored in. Assuming the sun is roughly 2*10³⁰ kg. Please just make very simplified assumptions, I am just looking for an estimate in terms of m^-2. Is it just the Einstein gravity constant times the energy density?

 

[PeterDonis]

What would be a rough estimate for the Ricci scalar curvature of an astronomical object like the sun?

There is no such thing as "the" Ricci scalar curvature for a large object. The Ricci scalar is a quantity at a particular event in spacetime, not a global quantity.

A rough estimate of the Ricci scalar at a particular point in a perfect fluid is $(8 \pi G / c^4) ( \rho c^2 + 3 p )$, where $\rho$ is the density and $p$ is the pressure. So you can get a rough "average" value for a large body by using average values of $\rho$ and $p$. For most bodies, like the Sun, $p$ is so small compared to $\rho c^2$ that it can be ignored. So an "average" estimate would be $(8 \pi G / c^2) \rho_\text{average}$. The average density is $M / (4 \pi R^3 / 3)$, so the "average" Ricci scalar would be $6 G M / R^3 c^2$.

 

[dsaun777]

The average density is $M / (4 \pi R^3 / 3)$, so the "average" Ricci scalar would be $6 G M / R^3 c^2$.

Yes, that is what I thought. Thanks.