# Can Newtonian Gravity be Thought of as a Theory of Curved Time?

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• Grasshopper
GR expert to understand what you're getting into.In summary, the article argues that Newtonian gravity can be thought of as a theory of curved time, and provides a few sources to back this up.f

#### Grasshopper

Gold Member
TL;DR Summary
A few sources I've seen have argued this. I would like to hear the thoughts of physicists
Before I attempt to delve into the math of tensors and curved spacetime, I'm hoping to get a more general intuitive grasp of things. As such, I'm parsing through a lot of lower level articles on these topics, and several that I've come across have argued that Newtonian gravity can be thought of as a theory of curved time (as opposed to the curved spacetime of general relativity).

(1) Is there any validity to thinking of Newtonian gravity this way?
(2) If so, could you elaborate why it is so?

Thanks as always.

A couple of sources where I've seen this claim in:

http://einsteinrelativelyeasy.com/index.php/dictionary/102-Newtonian-limit

http://www.gravityfromthegroundup.org/pdf/timecurves.pdf

I wrote this as undergrad, but I'm okay with any level of math as long as it's explained. I know this is general relativity, so I'd expect to see some EFEs.

One other thing I've heard (but forgot to put in the OP) is that the notion that Newtonian gravity can be thought of as a theory of curved time isn't strictly speaking true, but instead applies to an approximation of Newtonian gravity found in general relativity when various parameters approach Newtonian gravity in a limit (I think it might be called "weak field approximation" - and in fact it may have been here that I read it, but it's been a good while, so unfortunately I can't recall specifically).

I don't see how this is possible in principle.

Newton's universe is based on a fixed, absolute time. Essentially, gravitational events permeate the universe instantly.

If you want to curve time, that's fine - but it's no longer Newton's theory - it's something new. (And I suspect it only name drops Newton in an attempt make it seem a slightly less jagged pill for skeptics to swallow.)

Just my 2c.

One other thing I've heard (but forgot to put in the OP) is that the notion that Newtonian gravity can be thought of as a theory of curved time isn't strictly speaking true, but instead applies to an approximation of Newtonian gravity found in general relativity when various parameters approach Newtonian gravity in a limit (I think it might be called "weak field approximation" - and in fact it may have been here that I read it, but it's been a good while, so unfortunately I can't recall specifically).
It's called the Static Weak Field Metric. It has the same field equation as Newtonian gravity. This completely expresses Newtonian gravity in geometric terms. It may, however, include the non-Newtonian effect of gravitational time dilation.

It's an approximation of GR. not Newtonian gravity.

Grasshopper
If you only consider things moving slowly and insist that the metric be flat spacetime plus a small perturbation, then you find that you can neglect almost all the terms in the Einstein Field Equations. The surviving terms are in the ##tt## component (which is to say that spatial curvature is negligible) and are Poisson's equation, which is the vector calculus form of Newtonian gravity.

So I wouldn't say that Newtonian gravity is a theory of time curvature. Rather, I'd say Newtonian gravity is an approximation to GR that drops out of the ##tt## component of the equations in the limit that spatial curvature is negligible and matter distribution is very slowly changing (or not changing at all, strictly).

You can express Newtonian gravity in geometric terms, a theory known as Newton-Cartan gravity. I don't know much about it, but there's an Insight if you want to look.

PeterDonis, Grasshopper and vanhees71
Grasshopper
Thanks for the replies. What I'm getting out of this is that you can approximate general relativity for certain parameters that make it almost turn it into what Newtonian gravity as a theory of curved time would be, but not quite, since other components remain, even if they are negligible (and even still, there would be gravitational time dilation). Additionally, Newtonian gravity can be re-worked as a spacetime curvature theory, but without a single spacetime interval (NC Theory).

is that reasonable or do I need to re-read everything posted here again? ;)

(I suspect I'll be re-reading everything anyway)

What I'm getting out of this is that you can approximate general relativity for certain parameters that make it almost turn it into what Newtonian gravity as a theory of curved time would be,
I don't think that's a helpful view. You do get Poisson's equation out of the non-spatial term of GR, but in Newtonian gravity you interpret that as a potential energy implying a force, which is a rather different thing. I think the "time curvature" thing is to emphasise that the "rubber sheet" diagrams of gravity are barking up the wrong tree - what they show is spatial curvature, which is utterly negligible in every day situations.

I would just say that you can recover the maths of Newtonian gravity in the weak field low speed limit of GR. I'd avoid getting into philosophical discussion of what that means Newtonian gravity "really is" - you'd almost certainly have to revisit such a claim when we get a quantum theory of gravity anyway. The truth is that Newtonian gravity was an earlier, cruder model. Our more precise model explains when the earlier model will and won't work because you can see what assumptions you have to make to recover the maths of that earlier theory.

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Thanks for the replies. What I'm getting out of this is that you can approximate general relativity for certain parameters that make it almost turn it into what Newtonian gravity as a theory of curved time would be, but not quite, since other components remain, even if they are negligible (and even still, there would be gravitational time dilation). Additionally, Newtonian gravity can be re-worked as a spacetime curvature theory, but without a single spacetime interval (NC Theory).

is that reasonable or do I need to re-read everything posted here again? ;)

(I suspect I'll be re-reading everything anyway)

I think you should just get going with learning GR. These things will become clear once you've learned the relevant material. I find that until you've learned something properly these discussions can never make total sense. The only way really to learn about curved spacetime, coodinate-free descriptions, covariant derivative, tensors etc. is to get stuck into the subject!

vanhees71 and Ibix
If you look up Newton-Cartan theory, you'll see that there's only spacetime curvature in the time-space components of the Riemann tensor. I.e. only R_{i0j0} is non-zero.

vanhees71
But to answer the question directly in the OP, I'd say "no". Spacetime, with time absolute, is described by a product TxS, with T representing a 1-dimensional manifold. The intrinsix curvature of such a 1-dim. manifold is per definition zero. So "the curvature of time" is per definition zero.

In Newton-Cartan theory curvature only shows up when you consider closed loops with directions in both space and time.

PeterDonis
But to answer the question directly in the OP, I'd say "no". Spacetime, with time absolute, is described by a product TxS, with T representing a 1-dimensional manifold. The intrinsix curvature of such a 1-dim. manifold is per definition zero. So "the curvature of time" is per definition zero.

In Newton-Cartan theory curvature only shows up when you consider closed loops with directions in both space and time.
I would take a practical view. Can one define a spacetime geometry and some kinematic principle (e.g. particles follow geodesics) and obtain the precise predictions of Newtonian gravity?

Can one define a spacetime geometry and some kinematic principle (e.g. particles follow geodesics) and obtain the precise predictions of Newtonian gravity?

No. The best you can do, which is what Newton-Cartan theory does, is define a spacetime that has two metrics, a three-dimensional space metric and a one-dimensional "time metric" (the term "metric" is kind of a misnomer in that case, basically for the reason @haushofer gives), which are distinct and don't have any interplay with each other. You cannot define a spacetime that has only one metric that applies to all four dimensions.

cianfa72
No. The best you can do, which is what Newton-Cartan theory does, is define a spacetime that has two metrics, a three-dimensional space metric and a one-dimensional "time metric" (the term "metric" is kind of a misnomer in that case, basically for the reason @haushofer gives), which are distinct and don't have any interplay with each other. You cannot define a spacetime that has only one metric that applies to all four dimensions.
I thought you could? Hartle covers this in his book Gravity, section 6.6. Using the static weak field metric, he concludes that Newtonian gravity can be expressed in geometric terms in curved spacetime. He claims the same field equation:
$$\nabla^2 \Phi = 4\pi G \mu$$

Using the static weak field metric, he concludes that Newtonian gravity can be expressed in geometric terms in curved spacetime.

I don't have the book, so I don't know what exactly he claims, but the field equation itself does not imply that there must be a spacetime metric in Newtonian gravity. It only implies that there must be a space metric, since the space geometry is the one that has the nabla operator as a derivative operator.

Of course there is a spacetime metric in the underlying GR solution being used (the Schwarzschild metric), but that just means there is a spacetime metric in the weak field approximation in GR--as long as you don't throw away too many terms in your approximation. But in getting to the field equation you give, you do have to throw away too many terms--you have to throw away the ones that contain the relativistic effects, like relativity of simultaneity, that imply that the only metric present is a spacetime metric--that it is impossible to have just a space metric as Newtonian gravity does.

Another way of saying all this is that the field equation you give, like Newtonian equations in general, is not invariant under local Lorentz transformations. It's invariant under local Galilean transformations. But the latter are not compatible with the existence of a spacetime metric. They are only compatible with the existence of a space metric.

haushofer
I thought you could? Hartle covers this in his book Gravity, section 6.6. Using the static weak field metric, he concludes that Newtonian gravity can be expressed in geometric terms in curved spacetime. He claims the same field equation:
$$\nabla^2 \Phi = 4\pi G \mu$$
Yes, and you can describe this with a connection and Riemann tensor, R^i_{0i0} (see MTW). But if you want to derive this connection from a metrical structure, you need two metrics. The reason is simple: there is no flat spacetime metric invariant under the Galilei group, as you can check.

Of course, there is the temptation to add the temporal and spatial metric, but the resulting metric would take you out of the non-relativistic realm for the reason above.

Another way of seeing: the Newtonian limit is a singular limit on the metric, such that the terms you usually neglect in the Newtonean limit are strictly zero. See

G. Dautcourt, “On the Newtonian limit of General Relativity”

.

Another way of saying all this is that the field equation you give, like Newtonian equations in general, is not invariant under local Lorentz transformations. It's invariant under local Galilean transformations. But the latter are not compatible with the existence of a spacetime metric. They are only compatible with the existence of a space metric...
...and a temporal metric ;)

There is a whole lecture on this with all the mathematical details explaining why we can interpret Newtonian spacetime as curved:

PeroK
There is a whole lecture on this with all the mathematical details explaining why we can interpret Newtonian spacetime as curved

There are several different statements being discussed in this thread, which it is important to keep conceptually distinct:

(1) We can define a connection and a curvature tensor on spacetime, considered as a 4-D manifold, for Newtonian gravity. This is what is meant by "Newtonian spacetime is curved" in references that have been given.

(2) We cannot define a spacetime metric in Newtonian gravity on spacetime, considered as a 4-D manifold. We can only define a space metric and a time metric that are separate. So the connection and curvature tensor that we can define, as in #1 above, are not derived from a spacetime metric as they are in GR.

(3) There is no "theory of curved time" in Newtonian gravity. The only nonzero components of the curvature tensor as in #1 above have space indexes as well as time indexes.

vanhees71
Well, I'm glad I got some clarification on that. It does disappoint me that so many sources so flippantly ignore what appear to be relevant details when they make the claim discussed here. Why even bother saying it when the audience likely doesn't know what a tensor is in the first place?

Regardless, I think maybe I need to move out of the simple algebra and calculus representation of special relativity so I can finally jump into general relativity. I've never been comfortable with tensors, because it's hard for me to see them as anything other than a list of numbers, but it seems there's no other way than to just cut my teeth.

Well, the essential thing about tensors is that they don't care about your coordinates. Hence changing coordinates/your basis means that your components change the other way around. Being defined as linear maps, this means that new components are linear combi's of the old components.

The rest is just detail

vanhees71