Time Dilation in a Galaxy: Calculating Mass Effects

[sha1000]

TL;DR Summary

Time dilation as a function of distance from the center of galaxy.

Hello everyone

- The gravitational force near the edge of the galaxy at point A (see attached image) can be calculated by assuming that all the galactic mass is located in the center of the galaxy.
- In order to calculate the gravitational force in the middle of the galaxy (point B) we take into account only the inner volume mass and we can neglect all the mass in outer volume.

Can we apply the same logic for the gravitational time dilation?
What is the mass which I must use for the calculation of the time dilation at the point B? Do I need to take into account only the inner mass?

Thank you in advance.

 

[PeroK]

Gravitational time dilation across the galaxy must be fairly negligible - is it not? In general, the problem is not easy. See, for example:

https://physics.stackexchange.com/q...ide-a-massive-non-uniform-non-rotating-sphere

 

[Dale]

Summary:: Time dilation as a function of distance from the center of galaxy.

The gravitational force near the edge of the galaxy at point A (see attached image) can be calculated by assuming that all the galactic mass is located in the center of the galaxy.

Are you thinking of galaxies that are roughly spherical? I don’t think this is true for disk shaped galaxies.

 

[sha1000]

Are you thinking of galaxies that are roughly spherical? I don’t think this is true for disk shaped galaxies.

Thanks for the response.

Actually the question is not really about the galaxies. But more about the time dilation in general. As another example we can take a planet.

To calculate the time dilation near the surface of the Earth we can use the equation:

For any radius "outside" the planet: M = total mass of the Earth.

But what mass must be considered to calculate the time dilation in the "middle of the Earth" or "near the center"?
Is it only the inner volume mass, like in the case of the gravitational force?

P.S. Let us consider that the density of the Earth is constant.

 

[PeroK]

Thanks for the response.

Actually the question is not really about the galaxies. But more about the time dilation in general. As another example we can take a planet.

To calculate the time dilation near the surface of the Earth we can use the equation:
View attachment 289027

For any radius "outside" the planet: M = total mass of the Earth.

But what mass must be considered to calculate the time dilation in the "middle of the Earth" or "near the center"?
Is it only the inner volume mass, like in the case of the gravitational force?

That's the same question that was asked in the link I gave. The reference there is to Wald, section 6.2 for the "interior" metric. It gets complicated, even for a perfect fluid.

 

[sha1000]

That's the same question that was asked in the link I gave. The reference there is to Wald, section 6.2 for the "interior" metric. It gets complicated, even for a perfect fluid.

If I understand correctly the complication arises from the compressibility issues etc.

But my question is about oversimplified cases.

For example, inside a sphere: one need to take into account only the inner volume mass for the calculation of the gravitational force.
So is it same for the time dilation? (Without considering mass distribution, pressure, compressibility etc).

 

[PeroK]

If I understand correctly the complication arises from the compressibility issues etc.

But my question is about oversimplified cases.

For example, inside a sphere: one need to take into account only the inner volume mass for the calculation of the gravitational force.
So is it same for the time dilation? (Without considering mass distribution, pressure, compressibility etc).

The simple answer is no, it's not the same for gravitational time dilation.

The result you quote is the shell theorem for Newtonian gravity. And, where the Newtonian solution is a close approximation of the GR solution, it's logical that the shell theorem applies in GR as well.

Gravitational time dilation, however, is fairly negligible for these cases where the Newtonian approximation applies. The redshift from the surface of the Sun is very small and gravitational time dilation across the galaxy must be almost negligible. Moreover, the galaxy is not a sphere in the first place, so any calculation that is not simply a gross approximation is going to be difficult. The calculations you propose will be in that category of gross approximations.

If you want to go beyond this, then (as evidenced by the reference to Wald's GR book) an analytic solution for the metric inside a region of mass (whether that is the interior of a planet, star or galaxy) gets very complicated even for the simplest case of a perfect fluid.

 

[Dale]

To calculate the time dilation near the surface of the Earth we can use the equation:

Yes, but the only reason we can use that equation is because the Earth is roughly spherical and doesn't rotate much. That equation wouldn't hold for a disk galaxy.