Time dilation from galactic gravitational mass

[strangerep]

Wouldn't the change in gravitational potential be significantly different for a 3D sphere (where the mass is evenly distributed in 3-space and as we move in 3 dimensions, the calculation is ^3) vs a flat surface (where the mass isn't evenly distributed in 3-space and so it's ^2)?

You can get a feel for the difference by just looking at the Newtonian case -- where the non-relativistic gravitation (Poisson) equations are solved for various mass distributions. This is explained in considerable detail at Bovy's Galactic Dynamics Online Book, especially ch8 et. seq.

 

[Ibix]

I wouldn't call it "solid". Not at all. See the Deur thread over in BTSM.

In the context of this thread I was merely meaning that it was a fleshed-out mathematical proposal from a serious physicist, as opposed to the "hey wouldn't it be cool if..." idea in this thread which can be dismissed with an order of magnitude calculation. I suspect I'm not qualified to have an opinion on Deur's idea's plausibility.

 

[strangerep]

In the context of this thread I was merely meaning that it was a fleshed-out mathematical proposal from a serious physicist, [...]

I don't think it's "fleshed out".

But I hope other people here with good proficiency in GR will take a closer look at it, and offer their well-considered opinions on this point.

 

[Vanadium 50]

The fact that "dark matter" is proportional to the visible mass is

This is not a fact. The mass-to-light ratio for galaxies varies by quite a bit: more than a factor of about 5.

 

[malawi_glenn]

This is not a fact. The mass-to-light ratio for galaxies varies by quite a bit: more than a factor of about 5.

Another missuse of "proportionality" :(

 

[Vanadium 50]

Well, you can always divide any two quantities A by B and say "look, they are proportional!" Just not sure where that gets you.

 

[Musant]

So the claim is that even with a difference in mass density on the order of 10^8 between the region of the core of the galaxy and the solar neighborhood for example, the difference in time dilation and length expansion is negligible. Hard to believe.

 

[PeterDonis]

Hard to believe.

You don't have to "believe". You can do the math and see. Or you could just read the earlier posts in this thread where the math is already done.

 

[PeroK]

So the claim is that even with a difference in mass density on the order of 10^8 between the region of the core of the galaxy and the solar neighborhood for example, the difference in time dilation and length expansion is negligible. Hard to believe.

Perhaps you've watched Interstellar too many times.

 

[Musant]

You don't have to "believe". You can do the math and see. Or you could just read the earlier posts in this thread where the math is already done.

The time dilation factor I had seen previously, and the one I usually consider is simply SQRT(1-2GM/(rc^2)).

Question for you: In a time dilated reference frame where the time units of c are dilated, shouldn't one also expand units length such that c remains constant? If so, wouldn't the radius term in the expression for time dilation factor be a function rather than a constant?

 

[PeterDonis]

The time dilation factor I had seen previously, and the one I usually consider is simply SQRT(1-2GM/(rc^2)).

This is for a single spherically symmetric gravitating body. As was already pointed out earlier in the thread (which, btw, is almost two years old), a galaxy does not even come close to meeting that description.

In a time dilated reference frame where the time units of c are dilated

What does this even mean?

 

[Musant]

This is for a single spherically symmetric gravitating body. As was already pointed out earlier in the thread (which, btw, is almost two years old), a galaxy does not even come close to meeting that description.


What does this even mean?

I apologize if that wasn't clear. The units of c are, for example, in meters per second. Time dilation affects the length of a second. We assume that light always travels the same number of meters per second in a vacuum. So if the second is longer, then light would travel a greater number of meters in that longer interval. It follows that when time is dilated, length should also 'dilate' in order that the constant c remain constant. But, I'm happy to be shown the error in that logic. Thank you for your courteous replies.

 

[PeterDonis]

Time dilation affects the length of a second.

No, it doesn't. Time dilation amounts to a comparison of arc lengths along different timelike worldlines. Those arc lengths are measured in the same seconds along both worldlines.

I'm happy to be shown the error in that logic.

Your starting premise is wrong. See above.

 

[Musant]

No, it doesn't. Time dilation amounts to a comparison of arc lengths along different timelike worldlines. Those arc lengths are measured in the same seconds along both worldlines.


Your starting premise is wrong. See above.

But you might find that the two things 'amount' to the same.

 

[PeterDonis]

But you might find that the two things 'amount' to the same.

No, you won't.

At this point I am closing the thread since we are already off topic, and it's two years old anyway. @Musant, I would strongly encourage learning more about how relativity actually works.