Considering that the gravitational field of a spherical shell is rectified in its interior, can I consider that even then the time inside it is dilated with respect to a distant clock?

Imagine the following situation:

A massive spherical shell located in the cosmos away from everything. Inside it has a watch (A). On the outside of the shell is a second watch (B). Far from this shell is the third watch (C). These clocks were initially synchronized, but the second clock (B) already shows a significant delay compared to the third clock (C).

I would like to know the time that A indicates. I think the watch (A) inside the spherical shell is slow, and always, it will indicate the same time as the second watch (B), because there is a rectified gravitational field internally with the same intensity value as that field located near the external surface.

I would like to know: Time A = Time B, or Time A = Time C?

  • $\begingroup$ These clocks were initially synchronized GR and synchronizing clocks is problematic as explained on Wikipedia. $\endgroup$ – StephenG Jul 1 '18 at 5:39

The solution to your question is time A = time B < time C.

The reason is that since there is no mass inside the hollow region, the Schwarzschild radius is zero.

The metric in the hollow region is flat Minkowski space.

Time ticks still differently inside the shell then outside the shell.

Because where $\Phi$ is the gravitational potential, defined such that $\Phi \to 0$ as $r \to \infty$. Thus, $$ d \tau = \sqrt{ 1 + \frac{2 \Phi}{c^2}} dt. $$

The time dilation formula is the same everywhere inside the shell.

You can understand why time is ticking slower inside the shell is that when you send a photon from inside the shell, it will be redshifted.

So although the spacetime is flat inside the shell, time still ticks slower, because it depends on the gravitational potential. An that is not zero inside the shell.

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  • $\begingroup$ If we change clocks by rulers, can I say that the ruler A = ruler B < ruler C? $\endgroup$ – João Bosco Oct 10 '18 at 22:07
  • $\begingroup$ Can you answer this last question I asked? It's important to me. $\endgroup$ – João Bosco Oct 10 '19 at 2:32
  • $\begingroup$ @JoãoBosco you can use photon clocks as rulers, and then the photon ticks one as it moves between the mirrors, and if more time passes, the mirrors are farther, thus the ruler is shorter. in that case, ruler C is shorter. $\endgroup$ – Árpád Szendrei Oct 10 '19 at 2:36
  • $\begingroup$ I think that, if a ruler C is far from a massive body, it is always larger than a ruler B located next to it. Am I wrong? $\endgroup$ – João Bosco Oct 11 '19 at 1:56
  • $\begingroup$ @JoãoBosco GR does cause time dilation, but it does not cause length contraction. physics.stackexchange.com/questions/388961/… $\endgroup$ – Árpád Szendrei Oct 11 '19 at 2:12

As was pointed out by @Arpad Szendrei gravitational time dilation depends on the gravitational potential. To answer the question we make use of Newton’s Shell Theorem according to which the mass $M$ of the shell can be thought as to be concentrated at its center for any point outside the shell.

Thus the potential outside is - $GM/R$, where $R$ is the distance to the center. Potential at any point inside - $GM/r$, with $r$ representing the radius of the shell. From this, it follows that for a distant observer time dilation is the same on the surface of the shell and everywhere inside it.

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