How deep in a gravitational well are we?

[Agerhell]

As proven experimentally clocks tick slower deep in a gravitational well and the difference in energy levels between atomic/molecular quantum state also becomes smaller deep in a gravitational well. This is sometimes known as "gravitational time dilation" and "gravitational redshift" I think.

When you move away from the gravitational well of the Earth clocks starts to tick somewhat faster. Then if you move away from the sun clocks starts to click even faster as you move away from the sun. This is basically all experimentally proven.

But what about the next level? How deeply embedded are we in the gravitational field of the milky way?

Can you take it another level and say that we have a local cluster of galaxies contributing to another level of redshift, and how much of a gravitational redshift is that?

Have these numbers been found experimentally/calculated?

I know that you also have redshift due to velocities and "space expansion" which complicates things.

 

[Grinkle]

https://worldbuilding.stackexchange.com/questions/18955/how-long-is-a-day-in-intergalactic-space

This person calculates about 1/billionth of a second per day second (sorry, should have read it more carefully).

That number is Earth only, subsequent discussion estimates solar system / galactic gravity effects.

 

[phinds]

As proven experimentally clocks tick slower deep in a gravitational well ...

Well, no, they do not. They APPEAR to tick slower to an observer at a higher point in the gravity well, but the clocks themselves are ticking at one second per second. If you bring one back up out of a gravity well, it will have made fewer ticks than the one that stayed higher but that doesn't mean it ticked at a different rate, just that it took a different path through spacetime and made a different number of ticks, not that it had a different rate of ticks.

 

[Agerhell]

https://worldbuilding.stackexchange.com/questions/18955/how-long-is-a-day-in-intergalactic-space

This person calculates about 1/billionth of a second per day second (sorry, should have read it more carefully).

That number is Earth only, subsequent discussion estimates solar system / galactic gravity effects.

Well obviously for the Earth and the Sun you can just plug in the known values for the distances and the masses into the known formula for gravitational time dilation and get the results. I was wondering about the next level, how aggregations of stars in our galaxy, our galaxy in total, clusters of galaxy etc, might affect the gravitational time dilation/ gravitational redshift.

If we take the formula for gravitational time dilation which might be written:

$$T_{coordinate}=\frac{T_{proper}}{\sqrt{1-\frac{2GM}{rc^2}}}$$

We see that it resemble the formula for orbital velocity for an object in circular motion due to gravity in coordinate time:

$$v=\sqrt{GM/R}$$

So whenever astronomers claimed to have found how fast we revolve around the galaxy barycenter, I see it is reported as 828 000 kilometers per hour, or that our galaxy or a cluster of galaxies are moving with a certain velocity maybe it is acceptable to approximate the gravitational redshift using:

$$T_{coordinate}=\frac{T_{proper}}{\sqrt{1-\frac{2v^2}{c^2}}}$$

I was not really looking for a formula though.