# Time dilation from galactic gravitational mass

• I
• marcosdb
In summary, there is a negligible relativistic effect between the center of the galaxy and the outside, which cannot explain the size of the "delta" in galactic acceleration. Effects of speeds of this order magnitude are measurable in experiments, but are of a similar order of magnitude to the effects of gravitational time dilation on the applicable scales.

#### marcosdb

TL;DR Summary
Why doesn't time dilation explain all the changes in acceleration? Why is dark matter necessary?
I've been thinking about how rotational speeds don't fall off high distances from galactic centers, for which dark matter is generally an explanation for the increase in acceleration

Speed = distance / time

But time is relative

What "time" is used in these calculations?

Wouldn't time be moving "faster" for stars distant from galactic center?

Why is that not enough to explain the "delta" in acceleration? (which is not truly a delta; the object would be moving at a constant speed, but would appear to us, to be accelerating as it moves away from the gravitational mass & time for it speeds up)?

marcosdb said:
Why is that not enough to explain the "delta" in acceleration?
Because the effect you are talking about is far too small, only a few parts per million in a typical galaxy.

The tangential speed is too small to have any relativistic effects.

malawi_glenn said:
The tangential speed is too small to have any relativistic effects.
Effects of speeds of this order magnitude are measurable in experiments, and are also of a similar order of magnitude to the effects of gravitational time dilation on the applicable scales. The problem, as I said in post #2, is that this is far too small compared to the effects that dark matter is postulated to explain.

malawi_glenn
PeterDonis said:
Effects of speeds of this order magnitude are measurable in experiments,
Yeah maybe ~100 km/s is measurable since its about c/3000

To make sure I understand:

Are you saying that the tangential speed of the star itself is too small, or that the change in gravitational potential (from the inside of the galaxy to the outside) is too small to have any relativistic effects?

I was referring to the latter (i.e. the change in gravitational potential if we assume that the star started at the center of the galaxy, where gravitational potential is high, and moved away from galactic center) - are you saying that delta in gravitational potential is still too small to cause significant time dilation?

marcosdb said:
Are you saying that the tangential speed of the star itself is too small, or that the change in gravitational potential (from the inside of the galaxy to the outside) is too small to have any relativistic effects?
They both have relativistic effects. Just not anywhere near large enough to explain the size of the effect we see.

Maybe its just semantics of my behalf that caused this confusion.
In principle, when you walk down the sidewalk at 3 m/s there is still a relativistic effect, albeit extremely small.

marcosdb said:
To make sure I understand:

Are you saying that the tangential speed of the star itself is too small, or that the change in gravitational potential (from the inside of the galaxy to the outside) is too small to have any relativistic effects?
"Relativistic effects" is vague, and whether any particular effects are detectable by our measurements depends on how accurate our measurements are.

However, your question is about galaxy rotation curves, and the effects of both tangential speed and gravitational time dilation in galaxies are far too small to explain galaxy rotation curves based on just the matter that is visible to us.

ohwilleke
Gotcha, that would make sense; do you know if there's research/thesis/paper/etc. on the subject, with numbers?

i.e. where they calculated the time dilation delta between the center of the galaxy vs the outside of the galaxy, and the delta in time was negligible?

It seems like, given the mass of a galaxy, that time in the center of the galaxy would be going at least 2X that of the outside, no?

That's just a off-the-cuff guess, of course, but would be interested to read up on a paper that has examined this & found it to be negligible.

Going to add my own +1 here as I'm also very interested in discussing this, and @sha1000 did a far better job at asking the question than me

I would be really curious to know a rough estimate here of the type of time dilation we'd see

Is it really insignificant, when you consider a galaxy is far more massive than a black hole?

Or would you see a significant difference in time from a star at Point B vs Point A? (i.e. 2X or above?)

It wouldn't have to be a perfect estimation, just an approximate one, to know whether it's significant or not

Last edited by a moderator:
Won't me replying to it there cause it to be bumped up?

The author has some really nice diagrams there which nicely demonstrate what I'm trying to say, and it seems that question went unanswered

I'm happy to discuss here instead, whichever is preferred

Something tells me you don't believe us! Do you have any comprehension of the sophistication of models of galaxy dynamics? You really think that all physicists researching this simply forgot gravitational time dilation?

You should start looking online for papers and texts on galaxy dynamics. And yes, I know you won't understand them. But at least you might realise that astrophysicists are not just a bunch of bozos who don't understand even the most basic physics.

MidgetDwarf, Vanadium 50, Motore and 3 others
I'm a big believer in:
1. Being able to show receipts - even for a complex subject, there should be some rough estimate which can explain this to laymen (Einstein's explanation of relativity is a great example)
2. That people who have been looking at something for a long time can miss something quite obvious (myself, as a software engineer, included)
As such, I think that any claim of "it's insignificant" would have plenty of data behind it (papers where the effect was estimated to be 0.0001 or whatever)

It seems strange to make that the claim "it's so complex, we can't even have a simple rough estimate formula for it" and "it's insignificant" can be made at the same time.

The former is needed for the latter, no?

Motore
marcosdb said:
As such, I think that any claim of "it's insignificant" would have plenty of data behind it
Why? There are TONS of things that are insignificant. Do you think people are going to write papers on them?

A lot of papers/studies are written/done about the effect of something being insignificant

When considering a hypothesis (one like "dark matter is the likely cause of mass/velocity we can't account for"), I'd expect in that hypothesis a list of things which could explain but don't, with the work showing how each of the things, added up, still aren't enough to account for it

Wouldn't you?

marcosdb said:
It seems strange to make that the claim "it's so complex, we can't even have a simple rough estimate formula for it" and "it's insignificant" can be made at the same time.
The speed of the Sun around the galactic centre is about ##2.3 \times 10^5 \ m/s##. That's about ##0.001 c##. The gamma factor is ##1.000000294##. That does not, therefore, entail significant relativistic time dilation.

Likewise, you can calculate the gravitational time dilation to be similarly insignificant.

PeroK said:
Likewise, you can calculate the gravitational time dilation to be similarly insignificant.
The Sun is about ##10^{21} m## from the galactic centre. We can take the mass of the galaxy as ##1.5 \times 10^{12}## solar masses. The time dilation factor (roughly because the galaxy is not a sphere) is approximately ##0.999997767##. That again means that nothing significant is happening with time dilation.

The point is that all this is already in the computer models. The important stuff is estimating mass distribution within the galaxy, because that does significantly effect the rotation curves (whether you include a ##\pm 10^{-6}## relativistic correction or not.

 ​

marcosdb said:
Won't me replying to it there cause it to be bumped up?
That's not the point. The point is that there is no discussion going on in that thread now; it ended more than a year ago. You don't post in other people's threads just to bump them. You post in other people's threads if you have something useful to add to the discussion, which you can't if there is no discussion going on any longer.

marcosdb said:
The author has some really nice diagrams there which nicely demonstrate what I'm trying to say
You can certainly reference or quote from posts in the other thread here in this one.

Moderator's note: Some posts from the earlier thread that the OP referenced have been moved to this one. Please continue discussion here.

malawi_glenn
marcosdb said:
I think that any claim of "it's insignificant" would have plenty of data behind it
Physicists don't bother going to all the trouble of publishing papers on items where the calculation, even a rough one, is so simple that they can assign it to undergraduates as a homework problem. This calculation certainly falls into that category.

A rough order of magnitude calculation is sufficient, and if you're not satisfied with the ones already posted, because you're wondering if it makes a significant difference that they only looked at time dilation out where the Sun is instead of at the center of the galaxy, it's easy to show that that doesn't make a significant difference.

As already noted, we don't have a general exact solution for the metric in the interior of a massive body, or for the metric inside a massive system like a galaxy (particularly as a galaxy is a disk, not a sphere). But for a rough order of magnitude calculation, we can use an exact solution we do have, namely, the one for a spherically symmetric mass of constant density. This solution can be found in most GR textbooks, and was actually discovered by Schwarzschild, soon after he discovered his solution for a spherically symmetric vacuum. An expression for the metric can be found here:

https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric

What we want is the metric coefficient ##g_{tt}## at ##r = 0##, i.e., at the center of the body, since the square root of that metric coefficient gives us the gravitational time dilation factor. This is:

$$\frac{9}{4} \left( 1 - \frac{2 G M}{c^2 R} \right)$$

where ##M## is the total mass and ##R## is the radius at the surface. Without even calculating this, we can see at once that this value is just ##9/4## times the value of ##g_{tt}## at the surface. This means the time dilation factor at the center will be the square root of ##9/4##, or ##3/2##, times the time dilation factor at the surface. In other words, it is of the same rough order of magnitude as the time dilation factor at the surface. And that, all by itself, is sufficient to ground the statements that multiple people have already made in this thread about this effect being much too small to account for galaxy rotation curves by itself.

Bandersnatch, PeroK and marcosdb
Thanks so much Peter, much appreciated!

I'll dig in to better understand, but first a quick question from the things I did understand:

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)?

Motore and PeroK
marcosdb said:
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)?
The exact behavior would be different, but it would still be the same rough order of magnitude.

Also, the quantity of interest is not gravitational potential, it's the time dilation factor. The radial dependence of that is different from the radial dependence of the gravitational potential.

In short, you can't just wave your hands and use Newtonian intuitions and expect to get useful answers.

marcosdb
PeroK said:
ikewise, you can calculate the gravitational time dilation to be similarly insignificant.
You can.
I can.
But can he?

PeterDonis said:
particularly as a galaxy is a disk, not a sphere
True for the stars. Substantially less true for the mass. (i.e. :no dark disk")

True for the stars. Substantially less true for the mass. (i.e. :no dark disk")
In the dark matter models, yes, the dark matter distribution is more spherical than the visible matter distribution.

Also, closer to the center of the galaxy, the distribution even of visible matter (stars) is more spherical.

The bulge is an odd beast, for sure.

It's perhaps worth noting that people do seriously consider the possibility that other GR effects are responsible for "dark matter" galaxy rotation curves. Time dilation won't cut it, but there's an awful lot of rotating mass in a galaxy and we know that rotating masses can (in some circumstances) "frame drag" and cause free fall paths to co-rotate. It's not inconceivable that this kind of effect could cause galaxies to spin faster than Newton would suggest. There was a relatively recent solid proposal along these lines by a man called Deur. The jury is still very much out on that one (the maths is not simple, even by GR professionals' standards), but my point is that "what we call dark matter is actually just an unaccounted for GR effect" is very much under consideration. Time dilation isn't in the running, though.

Last edited:
malawi_glenn, marcosdb and PeroK
Ibix said:
It's perhaps worth noting that people do seriously consider the possibility that other GR effects are responsible for "dark matter" galaxy rotation curves. Time dilation won't cut it, but there's an awful lot of rotating mass in a galaxy and we know that rotating masses can (in some circumstances) "frame drag" and cause free fall paths to co-rotate. It's not inconceivable that this kind of effect could cause galaxies to spin faster than Newton would suggest. There was a relatively recent solid proposal along these lines by a man called Deur. The jury is still very much out on that one (the maths is not simple, even by GR professionals' standards), but my point is that "what we call dark matter is actually just an unaccounted for GR effect" is very much under consideration. Time dilation isn't in the running, though.
Thanks Ibix!

The fact that "dark matter" is proportional to the visible mass is why I had an inkling that it seems highly likely that it's just a side-effect of matter we're not accounting for

I am curious, though, why time dilation couldn't be one of these pieces (I agree, it may not be the only one)

Above, Peter points out that at most, it's 3/2, "same order of magnitude"

But the delta between the theoretically predicted values & observed values are also same order of magnitude (e.g. a factor of 4 in Messier 33 spiral galaxy)

So it seems that time dilation could actually push the predicted value a lot closer, and would also explain the proportionality to visible matter

Perhaps time dilation + cogravitation, together, get quite close?

Will lookup proposal by Deur, thank you!

weirdoguy and PeroK
marcosdb said:
The fact that "dark matter" is proportional to the visible mass is why I had an inkling that it seems highly likely that it's just a side-effect of matter we're not accounting for
I don't think it is. There are dark matter-free galaxies and cases where the halo is mis-aligned with the visible matter.
marcosdb said:
I am curious, though, why time dilation couldn't be one of these pieces (I agree, it may not be the only one)

Above, Peter points out that at most, it's 3/2, "same order of magnitude"
It's way too small. @PeterDonis was saying the real value of time dilation must be the same order of magnitude as his "everything is spherical" approximation, not that it was similar to the magnitude of dark matter effects. You can plug a galaxy mass and radius into his formula and see this easily enough.
marcosdb said:
So it seems that time dilation could actually push the predicted value a lot closer, and would also explain the proportionality to visible matter

Last edited:
PeterDonis
marcosdb said:
Peter points out that at most, it's 3/2, "same order of magnitude"
You need to read more carefully. What I actually said is that the gravitational time dilation at the center of a spherically symmetric massive body of constant density is 3/2 of the gravitational time dilation at the surface of that body. And we already know, from calculations that others had already posted in this thread, that the gravitational time dilation at the surface of such a body is many orders of magnitude too small to account for galaxy rotation curves. Therefore the gravitational time dilation at the center is also many orders of magnitude too small to account for galaxy rotation curves.

PeroK
Ibix said:
There are dark matter-free galaxies
Well, maybe. Two have been identified, right next to each other, and the "DM Free" inference depends on the distance, and the distance a) is questionable, or at least arguable (as determined by people actually arguing about it in the literature), and b) correlated between the two examples.

Further, the history of how these galaxies managed to be stripped of their DM is far from cleart: the story is not that they never had any, the story is that they used to have some but now it's gone. However, this interaction has not disrupted the stellar population. You need to have this happen via multiple "glancing blows" in interactions with other galaxies.

I don't think this is a settled issue, and am pretty sure everybody would like to see more examples discovered.

phinds, PeroK and Ibix
Ibix said:
It's perhaps worth noting that people do seriously consider the possibility that other GR effects are responsible for "dark matter" galaxy rotation curves. Time dilation won't cut it, but there's an awful lot of rotating mass in a galaxy and we know that rotating masses can (in some circumstances) "frame drag" and cause free fall paths to co-rotate. It's not inconceivable that this kind of effect could cause galaxies to spin faster than Newton would suggest. There was a relatively recent solid proposal along these lines by a man called Deur.
I wouldn't call it "solid". Not at all. See the Deur thread over in BTSM.

Ibix said:
The jury is still very much out on that one (the maths is not simple, even by GR professionals' standards), [...]
Well, I reckon it's not worth anything. Try getting Tully-Fisher out of Deur's method...

Moreover, these flat rotation curve effects happen in regimes of very low velocity, and very weak fields (hence low acceleration) -- which is why Newtonian gravity is used in much of standard galactic dynamics theory. But the observed RC phenomena could more accurately be called "sub-Newtonian" -- meaning slower+weaker than we're normally familiar with in terrestrial and solar gravitation. Imho, going from Newtonian up to SR+GR is headed in the wrong direction.

Ibix and PeroK
marcosdb said:
I am curious, though, why time dilation couldn't be one of these pieces
In the low-acceleration regime, the observations point to a scale-invariant equation of motion governed by an acceleration scale constant (i.e., different from the usual EoM in Newtonian gravity, which follows the non-uniform scaling implied by Kepler's 3rd law), together with the well-known Tully-Fisher law. (Newbies tend to ignore, or be unaware of, the importance of the latter phenomenon.)

See this MOND paper which explains why these two pieces of phenomena severely restrict the ways that one might fruitfully modify the Newtonian (or even GR) equations of motion.

PeroK