Ok, I get that one of the main arguments for dark matter is that observed gravitational interactions on the scale of the size of galaxies just don’t work without it being included.
However, the stars positions we see are of course not where they are now, we are seeing the far side of the galaxy 100,000 years ago, those stars have all moved and in turn the gravitational interactions between them are all different. The same would be true of any galaxy we can observe, we are seeing each star at a significantly different position from it’s current position.

Also as I understand it the n-body problem in general relativity is actually insolvable, even with super computer simulations. So I guess my question is, how do we actually know that gravitational interactions on a galactic scale are “wrong” and that we need dark matter to explain them when we can’t actually observe or calculate accurate star positions because of the time differences due to the size of galaxies?

I am only a layman interested in cosmology, so I may be completely misunderstanding something here…

Any single object in a galaxy is influenced by the total mass of everything else. While the movement of and thus influence from any single star is impossible to calculate perfectly, the overall distribution of mass in a galaxy isn’t changing significantly.

Well, sure, you can’t practically calculate the exact orbit of each star in a distant galaxy. Actually, even calculating the exact orbit of nearby objects in our Solar System is tricky enough, particularly for long periods of time. I believe special circuits have been built in attempts to calculate the orbit of Pluto out as far as possible, to see if the Solar System is stable.

But this is not necessary to make the conclusions on which the existence of dark matter is based. First of all, you are calculating very weak gravitational interactions at a fairly coarse level, so you don’t need GR, Newtonian gravity is just fine.

Secondly, you would tend to do things like calculate the expected orbital period of stars a given distance from a galaxy’s center, given the distribution of other stars you see. You wouldn’t even attempt to do that using the actual positions of all the 10^11 stars. You’d just assume all those stars create an average time-invariant gravitational field in which your star of interest moves. This is called a “mean field” approximation, and it’s done all the time to try to approximate the orbits of bodies in many-body systems. It works very well when there are a lot of other bodies with a distribution that doesn’t change over time. So the actual precise positions of all the stars doesn’t matter – you just need the average distribution, as if all the other stars were smeared out in a great big unmoving unchanging star-paste.

Finally, you would assume that the distribution of stars doesn’t change very fast, only on a timescale comparable to the lifetime of the galaxy. The difference in light travel times across the galaxy is very small by comparsion (100,000s of years versus 100 millions of years). So the fact that you don’t see the stars all at the same time isn’t important – it’s close enough.

Thanks for detailed answer Carl Pham, however I don’t understand how you can make the assumption that instantaneous Newtonian gravity is adequate for these calculations, when we don’t have any actual examples we can measure to prove that.
You seem to be begging the question, assuming that a Newtonian model is sufficient, even though the Newtonian model does not make correct predictions that match the observed data…

Let’s for the sake of argument start with your assumption that, to accurately calculate the gravitational effects of the galaxy on any single star, we have to compute the influence of each and every star on our single star, and we have to take into consideration gravity not being instantaneous and all our sources of gravity moving. This is of course impossible. Each galaxy contains too many stars do deal with in this manner. So how can we reasonably simplify?

Well we observe that galaxies appear to be relatively stable entities. We observe them in a few basic forms, and rarely observe them falling apart, and when we do it’s obvious it’s due to a collision, not due to galaxies being prone to erratic behaviour. This means that although we can’t calculate the gravitational influence from each and every star and do so based on their position a few tens of thousands of years ago, we can reasonably divide the galaxy into regions large enough that tens of thousands of years ago the region would have had very nearly the same average distribution of stars, and thus the same gravitational influence, even though the actual stars aren’t all the same and definitely aren’t in the same position.

Then we get the wrong results and the astrophysicists scratch their heads and come up with the dark matter hypothesis. Your layman quibble with this appears to be “but are you sure it wouldn’t come out right if you could do a more detailed analysis, including time?” As someone with a basic physics education I don’t see how that would be a hypothesis worth pursuing, but it’s entirely possible it’s been included in at least some galaxy simulations.

We can however do some calculations that place an upper bound on the error introduced by assuming Newtonian gravity. Everything done is an approximation to reality here, even using GR would be an approximation because we would still be averaging across millions of stars. What matters is not that the approximations are “wrong” but that we understand what bounds we can place on result of the approximations, and know that they do not lead to wild errors. So yes, Newtonian gravity is “wrong” but it isn’t wrong enough to matter, and it is feasible to explicitly calculate a useful bound on the error.

The diameter of our galaxy is about 100,000ly. But the rotational period is about 250,000,000y. So even a quick comparison of the numbers - that gravity transits the galaxy about 250 times as fast as the time to rotate would suggest things are not going to be wildly out.

ok well I don’t have enough background to debate the issue any further . Personally I think it would be more “interesting” for it to turn out that we were simply wrong about the way that gravity behaves on a galactic scale than for the answer to be dark matter . 95 percent of the universe being invisible an inert dark matter just seems such a waste to me

The sorts of calculations that are needed to show the existence of dark matter are quite simple. You don’t need to track the motions of individual objects in a galaxy, because the overall density profile of the galaxy remains constant. At any given time, for instance, some stars are entering the core and some are leaving it, but the number entering is approximately the same as the number leaving, and so we can at a very good approximation say that the core is of constant mass. Further, the core of a galaxy is very close to being spherically symmetric, so for any object outside of the core, we can treat the core as being a single point mass. The disk of a galaxy is more complicated, of course, but not all that much more complicated, and can also be approximated well by a simple stationary model.

That said, though, there are some sorts of orbital calculations for which you do need to account for individual objects. How do models of those situations deal with the time delay? The simple answer is that they don’t. The more complicated answer, though, is that they don’t need to: There are some very subtle and difficult-to-understand effects in general relativity that (so long as gravity is the only relevant force acting between objects) exactly cancel out the effects of the time delay. Proving that this is so is extremely difficult, but once one very smart theorist has proven it, everyone else can easily take advantage of it.