# I Is it possible to reduce or modify Einstein's Field Equations so they exactly mirror Newtonian gravity behavior?

#### Herbascious J

Summary
A bit of an odd question but, I thought it worth asking.
Is it possible to reduce and/or modify the EFE so that they make the exact same prediction as Newton's law of gravity? I am wondering if the slight differences in prediction from these two mathematical approaches can be identified at a particular place in the EFE or if it's the geometrical nature of GR that is expressing itself. Can some of the variables simply be omitted, or set to zero, or something like that, that would then allow the equations to exactly mirror Newton, while retaining the Geometrical warping of spacetime? Obviously these equations would be incorrect, but the point I am curious about is the mechanics of how the predictions diverge from one another. Ty.

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#### PeterDonis

Mentor
Is it possible to reduce and/or modify the EFE so that they make the exact same prediction as Newton's law of gravity?
Not quite. Newtonian gravity can be given a geometric formulation (it's called Newton-Cartan theory [1]), but this does not take the form of a single field equation that can be viewed as a modification or reduction of the EFE.

#### PeterDonis

Mentor
while retaining the Geometrical warping of spacetime
No. In Newtonian gravity, there is no spacetime; space and time are separate and don't "mix" together.

#### Herbascious J

Thank you for the responses. The Newton Cartan theory lead is great.

#### Ibix

Slightly different approach - relativity contains Newtonian gravity as a low speed weak field limit. You write the metric tensor $g_{\mu\nu}$ as the flat spacetime metric $\eta_{\mu\nu}$ plus a perturbation $h_{\mu\nu}$. As long as the elements of $h$ are small then you can neglect second order terms and the field equations reduce to the time-time equation (you kind of have to work in coordinates where that makes sense in order to think of $h$ as small in a coherent way), and this reduces to Poisson's equation for Newtonian gravity.

So by treating gravity as a weak tensor field on a flat background spacetime, you get Newton.

#### PeterDonis

Mentor
relativity contains Newtonian gravity as a low speed weak field limit
More precisely, GR contains Newtonian gravity as a low speed, weak field, static limit--the field itself cannot change with time. In GR terms, this means the spacetime geometry has to be static. (The most common static geometry used in this limit is the Schwarzschild geometry.)

You write the metric tensor $g_{\mu\nu}$ as the flat spacetime metric $\eta_{\mu\nu}$ plus a perturbation $h_{\mu\nu}$. As long as the elements of $h$ are small then you can neglect second order terms
What you are describing here is linearized GR, which is less restrictive than the Newtonian limit. The most common application of linearized GR is in studying gravitational waves.

As above, the Newtonian limit includes not only that the elements of $h$ are small, but that everything is moving slowly compared to the speed of light and that the field does not change with time.

#### Herbascious J

and that the field does not change with time
Would an example of a changing field be something like two large bodies passing close to one another, or a body spinning?

#### PeterDonis

Mentor
Would an example of a changing field be something like two large bodies passing close to one another, or a body spinning?
Any scenario with multiple gravitating bodies will be a changing field.

The field of a single spinning body, if it's spinning slowly enough, can be approximated by the field of a static (non-spinning) body, and the parts of the spinning body can be treated as slow moving objects in the static field, so the Newtonian approximation can still work for this case. But the "spinning slowly enough" condition can be very strict if you have accurate enough measurements; for example, now that Gravity Probe B has measured the frame dragging due to the Earth's rotation (which is a non-Newtonian effect), the Earth cannot be treated as spinning slowly enough with that measurement accuracy.

#### Ibix

As above, the Newtonian limit includes not only that the elements of $h$ are small, but that everything is moving slowly compared to the speed of light and that the field does not change with time.
I actually thought that "static" was a condition, but convinced myself it couldn't be - I should have checked.

But I find I have a question. Purely Newtonian models of the solar system (clearly a non-static system) were good enough to spot the anomalous precession of Mercury which stems from violation of the weak field restriction, but nothing from the static spacetime restriction. So presumably the solar system is "more nearly static" than it is "nearly weak field". Is there any way to characterise how bad the Newtonian approximation is in a given circumstance? Something like the degree to which non-circular orbits fail to close, or the gravitational wave power output?

#### PeterDonis

Mentor
the solar system (clearly a non-static system)
It is in GR terms if you consider all the planets, as well as the Sun, as massive objects, i.e., sources of gravity. But you can get a pretty good approximation by only considering the Sun as a source of gravity and all the planets as test objects moving in the Sun's Schwarzschild spacetime geometry. (The Sun does rotate, but so slowly that we can't observe any effects of that rotation on the spacetime geometry.) And if you then take the Newtonian approximation of that model, that is still enough to spot the perihelion precession of Mercury as a deviation from the model (see below).

were good enough to spot the anomalous precession of Mercury
"Spot" in the sense that the Newtonian model made predictions which were detectably different from actual observations, yes. The reason for that is that the Newtonian model does not include the extra perihelion precession of Mercury; that is a GR effect that does not show up in the Newtonian approximation.

Is there any way to characterise how bad the Newtonian approximation is in a given circumstance?
Sure, just carry the approximation further, to higher order, and evaluate the higher order terms--for example, compare the magnitude of higher order terms in $h$ to the magnitude of higher order terms in $v / c$.

#### PeterDonis

Mentor
presumably the solar system is "more nearly static" than it is "nearly weak field"
I'm not sure that's true. Doing a more complete Newtonian calculation for the solar system easily shows that the barycenter of the solar system is displaced from the center of the Sun, and the displacement varies with time; this is a measure of the degree to which the solar system is not static. I believe this was known before the extra perihelion precession of Mercury was known.

#### Ibix

I'm not sure that's true. Doing a more complete Newtonian calculation for the solar system easily shows that the barycenter of the solar system is displaced from the center of the Sun, and the displacement varies with time; this is a measure of the degree to which the solar system is not static. I believe this was known before the extra perihelion precession of Mercury was known.
Sure - but that's kind of what I'm getting at. Despite Newton only being accurate for the static case, it works well enough in a pretty obviously non-static case that its predictions are indistinguishable from GR up to the precision of Gravity Probe B. Whereas violating the weak field restriction is detectable with the precision available 100+ years ago.

What is it that's large enough that Newton doesn't predict Mercury's orbit well enough? That's the effect of space-space terms in $h$ (I think I can get away with coordinate-dependent language because "$h$ is small" is already coordinate dependent to some extent) and depends on Mercury's orbital radius compared to the Schwarzschild radius. But what's not large enough that Newton does predict everything else to high precision even though the assumption of staticness isn't really justified?

Perhaps I just need to expand in orders of $h$ and its derivatives as you suggest and look for the scale of coefficients.

#### PeterDonis

Mentor
Despite Newton only being accurate for the static case, it works well enough in a pretty obviously non-static case that its predictions are indistinguishable from GR up to the precision of Gravity Probe B. Whereas violating the weak field restriction is detectable with the precision available 100+ years ago.
Keep in mind what "non-static" actually means for this particular scenario (the solar system): it means having multiple sources of the gravitational field. In Newtonian mechanics, although there is no known closed form solution for more than two bodies (except for some special cases), it is easy to do it numerically: just sum the forces due to each source. How accurate this will be, compared to a full GR solution, will depend on how large gravitational nonlinearity is--i.e., how much the true solution differs from a simple superposition of the solutions due to each individual gravitating body. And nonlinearity, heuristically, is going to go like "weak field squared"--i.e., the largest nonlinear term will be something like the square of $h$, whereas the weak field corrections themselves go like $h$. So in a situation like the solar system, I think we should expect the Newtonian approximation to be much more accurate for things like calculating the barycenter than for things like the perihelion precession of Mercury.

#### haushofer

Not quite. Newtonian gravity can be given a geometric formulation (it's called Newton-Cartan theory [1]), but this does not take the form of a single field equation that can be viewed as a modification or reduction of the EFE.

It can, but it takes a singular limit on the metric. I'll check the reference for it; it's in the thesis i mentioned.

Staff Emeritus
exact same prediction
The only way this will ever happen is with the exact same equations.

If you want to argue that close enough is good enough, you need to say how close is close enough.

#### Herbascious J

The only way this will ever happen is with the exact same equations.

If you want to argue that close enough is good enough, you need to say how close is close enough.
For this discussion, I mean exact. I am more interested in the mathematical mechanics than the measurability or accuracy. In fact, I would expect Newton's equation to fall out of the math, and I guess that's what I was kind of driving at. Trying to understand if there is some way to modify variables in the EFE so that Newton's equation is left at the end, but perhaps it would be in some scalar or vector form (the only reason I say that is to allow for the resultant equation to be geometrical somehow, but alas I am out of my element here). It seems like the Newton-Cartan theory is that solution? I understand the static, low velocity, low mass approach as well, and how that would approach Newton. As these parameters would reduce to zero that maybe the EFE themselves have a Newton skeleton in them? It's this barebones skeleton equation so-to-speak that I am curious if it is exactly Newton or not. This discussion has been great and very interesting to read so thank you and I think I feel like my question is addressed and I can have a rudimentary feel for the nature of the math. At this point I expect one needs to learn the math.

Edit: I would even be curious to learn if changing the speed of light, perhaps to infinity, would have some effect. Whether extreme modifications like that would shine some light on the fundamentals or not.

Staff Emeritus
or this discussion, I mean exact.
Then this will not happen, because if exact is exact, you need the same equations. Furthermore, it won't matcvh reality.

#### haushofer

See e.g. Dautcourt's paper

Newton-Cartan is derived there from a singular ("c-->oo") limit of the EFE.

#### PAllen

More precisely, GR contains Newtonian gravity as a low speed, weak field, static limit--the field itself cannot change with time. In GR terms, this means the spacetime geometry has to be static. (The most common static geometry used in this limit is the Schwarzschild geometry.)

What you are describing here is linearized GR, which is less restrictive than the Newtonian limit. The most common application of linearized GR is in studying gravitational waves.

As above, the Newtonian limit includes not only that the elements of $h$ are small, but that everything is moving slowly compared to the speed of light and that the field does not change with time.
I don’t agree with this. Consider the Einstein-Infeld-Hoffman equations for approximating n-body motion in GR. These are based on the first order post Newtonian approximation. In the limit of c approaching infinite, they become exactly the Newtonian differential equation. Thus, there is a well defined Newtonian limit to a nonstatic GR situation.

#### m4r35n357

See e.g. Dautcourt's paper
Unless I have failed to understand the UI, I don't think that paper is available to most readers of this site (like me).

#### haushofer

Unless I have failed to understand the UI, I don't think that paper is available to most readers of this site (like me).

It was a quick response by telephone, this link should work ;)

#### haushofer

Summary: A bit of an odd question but, I thought it worth asking.

Is it possible to reduce and/or modify the EFE so that they make the exact same prediction as Newton's law of gravity? I am wondering if the slight differences in prediction from these two mathematical approaches can be identified at a particular place in the EFE or if it's the geometrical nature of GR that is expressing itself. Can some of the variables simply be omitted, or set to zero, or something like that, that would then allow the equations to exactly mirror Newton, while retaining the Geometrical warping of spacetime? Obviously these equations would be incorrect, but the point I am curious about is the mechanics of how the predictions diverge from one another. Ty.

So the answer is: you can write down "EFE-like covariant field equations", which are a geometric and general- covariant generalization of the Poisson equation. The same can be done for the geodesic equation.

However, the metrical structure is NOT the same in these EFE's. The absoluteness of time prevents one to write down a full-blown rank-4 metric for spacetime. Instead you have one spatial metric, and one temporal one. This makes Newton-Cartan theory very tricky compared to good old GR. But at the level of equations of motion, this is all just Newtonian gravity. Newtonian spacetime then consists of two metric, one temporal and one spatial, and its curvature is what we call Newtonian gravity.

Mathematicians will frown their eyebrows probably by the notion of "singular metrics", but that's another story ;)

#### Ibix

This makes Newton-Cartan theory very tricky compared to good old GR.

#### haushofer

It's a familiar theme. Look e.g. at the Poincare algebra, and compare it to the Bargmann algebra. The reason is that Galilei- and Poincare-relativistic theories are by the correspondence principle related by contractions, whether on the algebra side (Inönü-Wigner contractions) or on the equations of motion (see Dautcourts paper).

"Is it possible to reduce or modify Einstein's Field Equations so they exactly mirror Newtonian gravity behavior?"

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