general relativity - Time Dilation inside a hollow shell

Assuming I have a hollow shell with total mass $M$ and radius $r$. On the surface, the gravitational time dilation would be

$$\tau=t \cdot \sqrt{1-\frac{v_{esc}^2}{c^2}}$$

where

$$v_{esc} = \sqrt{\frac{2 \cdot G \cdot M}{r}}$$

but inside the shell there would be no gravitational field (Newton's shell theorem and Birkhoff's theorem).

But still, the escape velocity required to escape to infinity would be the same as on the surface, since inside the shell you could move without any accelerating or decelerating forces acting on you until you reach the surface, from where you would get pulled backwards.

So is the time dilation inside the hollow shell relative to a field free observer at infinity

1. zero (I assume it's not) or


2. the same as on the surface (my best guess), or


3. something completely different?

I found some not really duplicate but related threads on the interior of black holes which did not really focus on the math, but I am more behind the calculations in terms of $M$ and $r$.

Answer

For an asymptotically flat metric, the proper time measured by a "stationary" observer (defined here as one whose path through spacetime only has changing $t$, and no changing spatial coordinates) is $$ d \tau = \sqrt{ - g_{tt}} dt, $$ where $g_{tt}$ is the time-time component of the metric. For a "weak" gravitational field, this works out to be $$ g_{tt} \approx - \left( 1 + \frac{2 \Phi}{c^2} \right), $$ where $\Phi$ is the gravitational potential, defined such that $\Phi \to 0$ as $r \to \infty$. Thus, $$ d \tau = \sqrt{ 1 + \frac{2 \Phi}{c^2}} dt. $$ In this form, it is pretty obvious that the time dilation factor is the same everywhere inside the shell, since $\Phi$ is a constant inside a hollow shell (compare the electrostatic equivalent if you're not convinced of this.)

Note that your formula, in terms of the escape velocity, is equivalent to this one if you define the escape velocity at any point as "the velocity for which the object's total energy is zero." (Zero total energy means, of course, that the particle can escape to infinity.) In this case, we have $$ \frac{1}{2} m v_\text{esc}^2 + m \Phi = 0 \quad \Rightarrow \quad v_\text{esc}^2 = - 2 \Phi $$ and your result above is recovered. In this interpretation, the "escape velocity" from inside a hollow sphere would be the same as the escape velocity from the surface: if we launch a projectile inside the shell, it will travel with constant velocity until it reaches the surface of the shell; and if we open a little porthole in the shell at that point for the projectile, it's as if we launched it from the surface with that same velocity.