Time dilation occurs inside a hollow spherical shell
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General relativity discussions and applications of Birkhoff's theorem demonstrate that gravitational time dilation occurs inside a hollow spherical mass due to the Schwarzschild metric solutions in the cavity.
# Time Dilation inside a hollow shell
Tags: general-relativity, black-holes, metric-tensor, symmetry, time-dilation
- Score: 18
- Views: 1687
- Answers: 2
- Answered: yes
- Asked by: Yukterez (15103 rep)
- Asked: 2016-01-16
- Edited: 2021-09-22
- Site: physics
## Question
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
zero (I assume it's not) or
the same as on the surface (my best guess), or
something completely different?
I found some not really duplicate but related threads on the interior of black holes which did not really
# Is spacetime flat inside a spherical shell?
Tags: general-relativity, gravity, spacetime, metric-tensor, curvature
- Score: 75
- Views: 10914
- Answers: 2
- Answered: yes
- Asked by: Leos Ondra (2325 rep)
- Asked: 2012-11-07
- Edited: 2016-06-18
- Site: physics
## Question
In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the distance.
What is the result of general theory of relativity? Is the spacetime flat inside (given the fact that orbit of Mercury rotates I don't think so)? How is signal from the cavity redshifted to an observer at infinity?
## Answers
### Answer by Qmechanic (score: 136 [ACCEPTED])
Here, we will only answer OP's two first questions (v1). Yes, Newton's Shell Theorem generalizes to General Relativity as follows. The Birkhoff's Theorem states that a spherically symmetric solution is static, and a (not necessarily thin) vacuum shell (i.e. a region with no mass/matter) corresponds to a radial branch of the Schwarzschild solution
$$\mathrm{d}s^2~=~-\left(1-\frac{R}{r}\right)c^2 \mathrm{d}t^2
+ \left(1-\frac{R}{r}\
Everything we examined (2) — 1 independent source
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