# How much is time slowed down inside a planet or star?

An answer to What would be the rate of acceleration from gravity in a hollow sphere? states "that according to General Relativity time passes more slowly inside a hollow massive sphere than it does outside".

How much does time slow down by? Is it enough to require adjustments for, say, predicting how Earth's molten core behaves or nuclear reactions in the heart of the Sun?

• The Earth is only a hollow sphere in Jules Verne. – hobbs May 23 '15 at 4:33
• – user6760 May 23 '15 at 6:25
• @hobbs In real life it is an infinite series of concentric hollow spheres, which each follow the same rules. – Asher May 23 '15 at 6:54

That discussion is about a spherical shell and does not apply to the Earth nor to the sun. In the latter situations assuming uniform mass distribution we know that time is dilated by $t(\infty)/t(r)=\sqrt{g_{00}}=\sqrt{1 - \frac{2GM}{rc^2}}$, where $r$ is the distance from the center.
For a spherical SHELL however we have to go back and forth between the GR and its Newtonian limit. Since the Earth's and Sun's gravity is weak we can approximate $\sqrt{g_{00}} = \sqrt{1 - 2\Phi(r)}$, now the potential $\Phi(r)$ outside the shell is the same as before $\Phi_{\text{out}}(r)=GM/rc^2$, however inside the shell it is $\Phi_{\text{in}}(r) = GM/Rc^2 =$const.
There fore the different in passage of time between a position $r$ from the center and infinitely far away is $$t(\infty)/t(r) = \sqrt{1 - 2\Phi(r)}=\left\{\begin{array}{cc} \sqrt{1 - \frac{2GM}{Rc^2}} =\text{const}\qquad & \forall r\leq R \\ \sqrt{1 - \frac{2GM}{rc^2}} \qquad & \forall r\geq R \end{array}\right.$$