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?
Answer
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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