0

This description of the relationship between general relativity and Newtonian gravity looks pretty good. It also seems to have gotten a lot of upvotes, so I assume it reflects mainstream thought on the topic. However, additional assumptions are required to get from one to the other, and I am not sure the possible problems this may cause have been adequately explored anywhere.

If we used those same assumptions to attempt predicting the behavior of gravity at galactic or cosmological scales, would it still be an accurate theory? If not (which I am assuming is the case), are we not dealing with two incompatible theories instead of one?

Sorry if this sounds at all confused or vague, hopefully it can be improved by some comments. If it helps, this question is partly motivated by the discussion here in which everyone seems to agree that Newtonian mechanics is not at all compatible with general relativity. However, some participants seem to think it is interesting while the others don't. My initial motivation (which lead me to that discussion) was wondering whether Newtonian gravity could really be derived from (as opposed to approximate) general relativity, which is something I heard/read somewhere and just accepted until now.

Community
  • 1
Livid
  • 1,096

3 Answers3

1

Here are some facts that might clear up your confusion:

Firstly, Newtonian gravity is a theory that is only valid in certain regimes, while general relativity is valid for a much wider range of situations. Hence, it is logical that, in order to obtain a Newtonian description of gravity, we need to consider a special situation, namely exactly the type of situation where this description is adequate. In my answer to the post you linked, I gave the assumptions under which Newtonian gravity appears as a special case from general relativity.

Secondly, the fact that, given certain special situations, the general relativistic description of gravity turns into the good old Newtonian one, does not mean that the general relativistic framework is not much more broadly applicable. In particular, those special assumptions used to get a Newtonian description are not assumptions that always hold true in the general relativistic description of gravity. In cosmological scenario's, the predictions of general relativity are not the same as those of Newtonian gravity. Hence, there is no problem with applying general relativity to describe other scenario's, which Newtonian gravity does not adequately describe.

Danu
  • 16,576
  • 10
  • 71
  • 110
1

For the purposes of this answer, I will consider the universe to be correctly described by General Relativity (possibly coupled to classical EM fields and matter) in its full form. That is, I will ignore quantum mechanics and any corrections to GR that we might find in the future.

There seems to be some confusion with words, so let me explain my understanding. To derive A from B means that B implies A in a mathematical sense. No more, no less. Sometimes people use the word more loosely, but what can you do. With this definition, it's not possible to derive Newtonian gravity from GR, because they have different predictions. For example, Newtonian gravity predicts that time flows at the same rate everywhere while GR does not. More practically, the Newtonian theory and GR predict different rates of precession of the perihelion of Mercury. Deriving Newton from Einstein would mean that Newton is correct (since Einstein is correct), and that is not the case.

You can derive things using assumptions. For example, if we assume that the Sun is perfectly spherical and is surrounded by empty space, we can deduce that the metric appropriate to the solar system is the Schwarzschild one. Whether our assumptions are correct or not is a different question; the Sun is not spherical and space is not empty, but if they were then the Schwarzschild metric would be the one to use.

Usually our assumptions are only approximately correct, as in the example above. The Schwarzschild metric works pretty well but not perfectly; to be more precise we could begin by using the Kerr metric instead, and then try to account for perturbations due to the planets and dust floating around.

A limiting case is related but different. This is where Newtonian gravity comes into play. We can expand the GR metric and curvature tensors as a power series around the flat metric; as long as we keep the full series this is the exact metric. Now, the Newtonian limit arises when we make the assumption that we're going to use this metric for small curvatures and velocities. In this case, keeping only the linear term of the series is a good approximation to the full metric. But this linearized metric is not an exact solution of the Einstein equations, only an approximate one. This approximation will be good as long as our assumptions are fulfilled.

TL;DR: We're free to make as many assumptions as we want and derive things; our assumptions hold in the real world with varying degrees of accuracy. A limiting case is different: you modify an equation assuming that your new, simpler equation will be a good approximation to the old one.

Javier
  • 28,811
1

The equivalence (or limit) of GR to Netwonian gravity happens for low curvature (equivalently weak gravitational field) and for low speeds relative to c.

You said that the fact that gravity is instantaneous in Newtonian gravity but travels at c in GR makes them incompatible. No, that is exactly what equivalent for low speeds means. It means that v/c is very small. That is just like saying c is very large compared to the speeds of interest in whatever case you are trying to apply it to. That the rate at which anything related to the predictions or observation is much smaller than the speed of the gravity perturbations, i.e., for that limiting case or approximation (or whatever similar words you want to use to mean the same thing) it is like the speed at which gravity propagates is infinite.

When the first measurements for GR took place like that for the perihelion of mercury and others, people derived the equations for GR, and saw those differences (like the perihelion) as the only observables they could maybe then measure for that case (I.e., orbits around a spherical object) were Nettonina orbits plus the very small effect of mercury's perihelion. As time went on other measurements of other predictions (call them cases if you will) were measured, always compatible.

In the 1960's or early 1970's there was some interest in parametrizing the approximations of GR, or perhaps more accurately in expanding GR in terms in such a way that one could parametrize what level of approximation (or assumption) one was making. It led to the PPN treatment, parametrized post Newtonian formalism, where one could expand GR in terms of deviations from Netonian gravity, and one could see what was one approximating and to what extent. Sort of like expanding special relativity as a power series in terms of powers of v/c. See it at https://en.m.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism

It's very instructive. It is used nowadays to be able to follow which approximation one is using for very strong gravity, such as near Black Holes, because in the strong gravity (and v/c gets closer and closer to 1), it has not been possible to solve in all the needed cases fully the nonlinear GR equations, and so numerical methods using the PPN formalism are used. If you look up the papers out of the LIGO collaboration that detected gravitational waves last year and this year, you'll see a description of which orders of the PPN formalism they had to do for different calculations.

Hoe this helps

Bob Bee
  • 14,246