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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.

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Livid
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    Newtonian gravity is the low energy/weak field limit (small speeds, small curvature) of GR. This is a fact. In the post you linked first the necessary assumptions are made within the frame work of GR to get to Newtonian gravity. It can be derived from GR as shown under the assumptions made, which are valid in a certain limit, in which Newtonian gravity holds. – N0va Oct 21 '16 at 23:30
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    You seem confused about what a limiting case is. You can derive Newtonian gravity from GR by assuming certain quantities become arbitrarily small. Then, for finite but small values of those quantities, you can use Newtonian gravity as an approximation for what GR says. This is entirely analogous to replacing $\sin x$ with $x$ for small $x$. – knzhou Oct 21 '16 at 23:31
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    Newtonian gravity couldn't possibly be derived from GR because GR is correct (except for quantum stuff) and the Newtonian theory is not. It is an approximation valid in certain cases. – Javier Oct 21 '16 at 23:42
  • @M.J.Steil and knzhou Thanks, I do not disagree with anything you said, and may indeed be confused about the use of "limiting case". I am concerned that making assumptions A to recapitulate Newtonian gravity, but assumptions B to compare GR to larger scale observation is somehow inconsistent. Eg, if you assume the universe is described by the metric is "almost flat", will you not get different results when predicting the behavior of galaxies, galaxy clusters, etc? – Livid Oct 21 '16 at 23:45
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    @Livid Sorry, I still don't see the problem. You can use different limiting cases / approximations in different places. The core theory is still GR. – knzhou Oct 21 '16 at 23:51
  • @knzhou I think the difference is that I consider including additional assumptions to be different than terms dropping out of equations because they are very small. Edit: The latter is what comes to mind when I read "limiting case". – Livid Oct 21 '16 at 23:53
  • @Javier To me, your comment seems to be in conflict with the others here. I think it may come down to different interpretations of terminology like derive and limiting case. You may be in a better position to figure that out than me, but I think similarly. Just that gravity travels instantaneously in Newtonian mechanics and at c in GR would seem to make them incompatible in my mind. As an approximation though, sure. – Livid Oct 22 '16 at 00:07

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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
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  • Thanks, in that case can you give any idea of how many "special cases" it is possible to derive from the GR core theory? I am wondering if it is surprising that a correspondence with Newtonian gravity pops out, or is it more along the lines of what I have heard regarding string theory where 10^500 different "special cases" are said to be possible. – Livid Oct 22 '16 at 00:33
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    @Livid: You seem to think that a "special case" means something like a "version" of the theory. As I said before, it doesn't; within GR Newtonian gravity is an approximation. Personally I don't think "special case" is a good phrase to use in this context. – Javier Oct 22 '16 at 00:39
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    You're confusing some things here, Livid. The "special limits" of string theory are of a very different nature than the special case of GR that leads to the description that matches Newtonian gravity. There is one, unique theory of general relativity. It is a beautiful theory, and it is not at all surprising that it must reproduce Newtonian predictions in the regimes where Newtonian physics applies. That's because we know Newtonian physics works in those cases, so any good theory must be able to reproduce it. – Danu Oct 22 '16 at 00:40
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    Now, there is a wholly different issue in string theory: String theory is typically defined as a theory that "lives" in 10 or 11 spacetime dimensions. In order to produce a 4-dimensional theory, one needs to "compactify" (you can find more info about this on the site!) the extra dimensions. This can be done in many different ways. That's what people talk about when they talk about $10^{500}$ different possibilities of realizing a string theory as a 4-dimensional theory. In particular, there is not a single, unique way of realizing string theory as a theory in 4 dimensions!! – Danu Oct 22 '16 at 00:42
  • @Danu I think I understand now though, what is being called "assumptions" refers to parameter choices. Does that sound right? Even in that case, if you can add an arbitrary number of assumptions (eg parameter p =[ 0, .5 , 1] and parameter q = [10^-20, 10, 10^20], etc) at some point the theory becomes so flexible it can fit any data or recapitulate any prior successful theory. I am just wondering how susceptible GR is to this scenario. – Livid Oct 22 '16 at 01:04
  • @Livid Aha! Now I understand. You come from a stat/ML background, and you think GR is like some general method, like neural nets. – knzhou Oct 22 '16 at 01:38
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    A somewhat better analogy is, GR is a specific neural net, with fixed coefficients. (Physicists, as a group, have "learned" these coefficients by performing physics experiments and figuring out what equations describe the world.) Newtonian mechanics is the specific neural net you get if you want to run the thing more cheaply and slice off half the neurons in the intermediate layers. – knzhou Oct 22 '16 at 01:39
  • @knzhou You are somewhat right and somewhat wrong. When bringing up the flexibility I was coming from that background. But I recognize that GR is an analytical model deduced from a few postulates which is far superior to using pattern recognition techniques in terms of computational efficiency, constraints, and understandability. Ie it is a real theory. The analogy is still good though. – Livid Oct 22 '16 at 02:58
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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
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  • I think this is a good explanation, but still I am uncomfortable with treating the assumption "the sun/earth/etc is a sphere" at the same level as assuming things about the universe as a whole. Note: As I commented above, it was helpful to me to conceptualize these "assumptions" as arbitrary (ie theoretically unconstrained) parameter choices. – Livid Oct 22 '16 at 01:20
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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
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  • I have run some Newtonian gravity solar system simulations and can tell you that the results are qualitatively accurate and roughly quantitatively accurate when using instantaneous gravity. Introducing a propagation time causes all sorts of problems. Maybe that would not happen when using a small enough timestep though (ie if it is an integration artifact). That is why I was thinking that difference between the two theories is a huge deal. – Livid Oct 22 '16 at 03:01
  • I've never done those, but now I can see why you'd worry about the gravity speed. The force equations should be good enough, we barely were able to measure mercury's perihelion change, others were too small. I assume Newton is normally used without the propagation time for the gravity (i.e., using instantaneous positions, rather than 'retarded' ones). When you say 'quantitatively accurate', yes, I'd try to figure out first what how much the time step might affect. I'd try to also do a rough estimate of the gravity time, and the different force vector direction. – Bob Bee Oct 22 '16 at 04:01
  • And there could be other factors that are more important to the accuracy. See the wiki article (sorry if you're already way past that) at https://en.m.wikipedia.org/wiki/Numerical_model_of_the_Solar_System. Don't know if you are doing one body with the sun, or the N body problem. I'm sure if anybody tried to do the N body relativistically they'd start from Newtonian, hard enough numerically. For the Black Holes, only 2 bodies, it was hard as they got closer, as the horizons touched each other it became real complicated, calculations diverging till they figured out how. – Bob Bee Oct 22 '16 at 04:16