# Gravity and the curvature of space-time

## Main Question or Discussion Point

This is something I have pondered for some while... it is so obvious that there must be an answer and is probably a silly question, but I haven't found an answer yet... so....

Gravity is a consequence of the localised curvature of space. According to Relativity, space (space-time) is curved. So:

Does the general curvature of space cause a gravitational force (or an anti-gravitational force where curvature is "convex") over inter-galactic and greater distances?

(If it did, then where localised gravitational forces decrease with distance, the anti-gravitational force created by the curvature of space would increase with distance, would they not?)

I realise this question is probably a consequence of a complete misunderstanding on my part, but it would be interesting to know why

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Simon Bridge
Homework Helper
This is something I have pondered for some while... it is so obvious that there must be an answer and is probably a silly question, but I haven't found an answer yet... so....
A question not asked for fear of looking silly is even sillier...

Gravity is a consequence of the localised curvature of space. According to Relativity, space (space-time) is curved. So:
Minor tweaks:
"local curvature" not "localized curvature" ...
In GR gravity is a pseudo-force ... it's presence is indistinguishable from an accelerating reference frame.
We can describe it using curved coordinate systems.

Does the general curvature of space cause a gravitational force (or an anti-gravitational force where curvature is "convex") over inter-galactic and greater distances?
Yes ... you'll recall that gravity is a long-range force.
The idea is that distant objects don't need to know about each other to gravitate towards each other. No "action at a distance" needed.

(If it did, then where [local] gravitational forces decrease with distance, the anti-gravitational force created by the curvature of space would increase with distance, would they not?)
No.
Consider an analogy - the electric field due to a positive charge decreases with distance - does that mean the field due to negative charges gets stronger with distance?

What happens is that space-time gets flatter with distance - less curved. A convex surface can get less convex towards the edges just like a concave surface may get less concave.

Aside:
I used to say, "there's no such thing as a silly question." but some people took that as a challenge... <twitch>

Thank you. I am just a layman with an interest, not a physicist, so my question was born out of genuine ignorance!

My misunderstanding of gravity comes from the often used 2D stretched tablecloth with a heavy object in the middle creating a "dump" into which surrounding objects are attracted.... combined with the description of the Universe as "saddle shaped" or at least "curved".... so by combining the two I get the local curvature of space (from a massive object) creating a gravitational force that would decrease with distance from the object, but a gravitational force caused by the curvature of the Universe itself that would increase with distance between objects.

Aside: There is nothing more dangerous than a little knowledge!

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An interesting read, thank you. The curvature of space/time is clearly far more complicated than the Pringle described in popular science!

Simon Bridge
Homework Helper
Thank you. I am just a layman with an interest, not a physicist, so my question was born out of genuine ignorance!
It's about normal - don't worry.
The first thing you learn in science is that everyone is ignorant.

My misunderstanding of gravity comes from the often used 2D stretched tablecloth with a heavy object in the middle creating a "dump" into which surrounding objects are attracted....
Yes - and that has mislead a lot of people ... it's an analogy only - and works best, when it does at all, when you don't factor in large-scale curvature.

There is nothing more dangerous than a little knowledge!
If a little knowledge is a dangerous thing, where is he with so much as to be out of trouble?

pervect
Staff Emeritus
This is something I have pondered for some while... it is so obvious that there must be an answer and is probably a silly question, but I haven't found an answer yet... so....

Gravity is a consequence of the localised curvature of space. According to Relativity, space (space-time) is curved. So:
As others have noted, it's a curvature of space-time, and not just space. The difference is important!.

Does the general curvature of space cause a gravitational force (or an anti-gravitational force where curvature is "convex") over inter-galactic and greater distances?
What gravity does _locally_ is cause a tidal force, a small relative acceleration that's proportional to distance. Locally, you really can't talk about the absolute accleration - in general we are considering two bodies, both of which are in "free fall", and locally we look at their relative acceleration.

If you study some of AT's diagrams, or various other sources about "geodesic deviation , such as MTW's "parable of the ant", you can get a better feeling for how objects moving along geodesics can and do appear to accelerate away from each other in the presence of curvature.

When one talks about inter-galactic distances, and greater distances, one is not talking locally. Local means "small" - intergalactic distances are in general, conceptually large distances, not small ones.

What you do when you talk about "gravity" over such large distances is that you sum together all the local contributions to the relative acceleration. As a relatively subtle point, the summation issue becomes rather tricky, and it turns out it's better to view gravity as the local tidal force, and avoid worrying about how to do the sum.

But for the broad overview, I'll omit these subtle, but important points, and talk in purely Newtonian terms, where there is only one way to do the sum.

The local description of gravity is a tidal force. The Newtonian formula for tidal force would be 2GM/r^3 in the radial direction (modulo sign conventions).

The global description of gravity (which works best in Newtonian theory) is the integral of the above. When you integrate 2GM/r^3, you get -GM/r^2, which is the Newtonian force. So the "global" newtonian force is caused by the summation (or integral) of the tidal forces.

A.T.
The local description of gravity is a tidal force.
I would start even more local, where the tidal force becomes negligible, so the Equivalence Principle applies. In this local description gravity is either an inertial force or the consequence of curve-linear coordinates in accelerated frames.

The next step is to consider the entire area including the gravity source (e.g. planet) where the tidal forces are not negligible and cause different movement of free fallers, at different positions.

Then the global view over intergalactic distances that describes the curvature of the universe.

Thanks for all the replies. I have done some reading and now understand the basic idea of general relativity. Very exciting stuff.... have even downloaded an accelerometer on my phone so I can watch myself accelerate away from the earth at 9.8ms/s.... and know why this is happening! !!

However. .. this just begs another question from an ignorant layman.... it would seem that gravity isn't really a force at all, not like the other three fundamental forces..... so why all the fuss about gravity not being as strong as the other forces? For gravity to have a stronger effect mass would have to warp space-time more, which means particles would have to interact more strongly with the Higgs field I presume.

So the question is, why does anyone expect gravity to be roughly equal to the other fundamental forces when it isn't a force at all?

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As to my original question, perhaps I could ask it in a more relativity friendly format:

If space time is curved convexly over intergalactic distances then presumably the geodesic path of matter on one side will be diverging from the geodesic path of matter on the other, causing the galaxies to move away from each other.... ???

Or is that just bollocks because curvature of space-time is exclusively the result of local mass and space-time is essentally flat unless affected by mass? Is the idea that the Universe is saddle (or Pringle) shaped just a myth?

WannabeNewton
So the question is, why does anyone expect gravity to be roughly equal to the other fundamental forces when it isn't a force at all?
What does "equal" even mean here? You have to make your question much more precise I'm afraid.

Also you're trying to compare a classical theory of gravity (general relativity), relying on a classical dynamical field (the metric tensor ##g_{\mu\nu}##), with quantum theories relying on quantum fields (e.g. ##A^{\mu}## in QED). Also the term "force" as applied to QFT is probably better termed "interaction". What we really care about in the end, to put it in very loose terms, are the transition amplitudes for various interactions. Electric interactions between fermions (via virtual photons) can give rise to "attractions" and "repulsions" so the interaction can also be termed a "force" but the underlying mechanism involves transition amplitudes; in Newtonian mechanics the underlying mechanism literally involves a 3-vector that we call force.

Simon Bridge
Homework Helper
Thanks for all the replies. I have done some reading and now understand the basic idea of general relativity. Very exciting stuff.... have even downloaded an accelerometer [app] on my phone so I can watch myself accelerate away from the earth at 9.8ms/s.... and know why this is happening! !!
I'd check that direction - which way does gravity act again?

However. .. this just begs another question from an ignorant layman.... it would seem that gravity isn't really a force at all, not like the other three fundamental forces..... so why all the fuss about gravity not being as strong as the other forces?
What fuss?

Comment on the relative strength of the forces is just for comparison.
The term "force" is still used when it is convenient even though we understand Newtonian forces, these days, as an emergent effect of fundamental interactions like WannabeNewton was saying.

We don't have to throw out the old models just because they are wrong.

So the question is, why does anyone expect gravity to be roughly equal to the other fundamental forces when it isn't a force at all?
I don't think anyone expect gravity to be equal to the others. Can you provide an example so we know what you mean?

As to my original question, perhaps I could ask it in a more relativity friendly format:

If space time is curved convexly over intergalactic distances then presumably the geodesic path of matter on one side will be diverging from the geodesic path of matter on the other, causing the galaxies to move away from each other.... ???
They'd move away from the center of that "local bulge". On a Universal scale, however, that would imply some sort of center to Everyting. This we don't have.

Or is that just bollocks because curvature of space-time is exclusively the result of local mass and space-time is essentially flat unless affected by mass?
A local antigravitational bulge like you describe would imply the presence of a lot of negative energy by the center. This translates to mass that has a very large amount of the square-root of minus one in it. It's not clear what this means or if it is physically possible.

Is the idea that the Universe is saddle (or Pringle) shaped just a myth?
Without knowing where you got this image from, there is no way to tell.
Sounds like a fanciful artistic description from a pop-sci show.
i.e. nonsense.

pervect
Staff Emeritus
As to my original question, perhaps I could ask it in a more relativity friendly format:

If space time is curved convexly over intergalactic distances then presumably the geodesic path of matter on one side will be diverging from the geodesic path of matter on the other, causing the galaxies to move away from each other.... ???
Basically the idea that the galaxies follow geodesic paths through space-time is correct, (you seem to be familiar with the concept of geodesics) and the idea that the geometry of space-time thus determines whether or not these paths converge is mostly correct.

However, the curvature of space-time affects the relative acceleration between geodesics (and because galaxies follow geodesic paths, the relative acceleration between galaxies). It does not directly control the relative velocity between galaxies.

Thus, in the absence of dark energy or a cosmological constant, the galaxies would be moving apart, but accelerating towards each other, so that the universal expansion was slowing down. Note that the acceleration would be in the opposite direction of the velocity in this instance.

Latest observations suggest that dark energy is present, so the universal expansion is (rather suprisingy) accelerating.

A.T.
have even downloaded an accelerometer on my phone so I can watch myself accelerate away from the earth at 9.8ms/s.
I'd check that direction - which way does gravity act again?
The direction is correct. An accelerometer measures proper acceleration, which points upwards when standing on the Earth's surface.

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Simon Bridge
Homework Helper
The direction is correct. An accelerometer measures proper acceleration, which points upwards when standing on the Earth's surface.
Oh yes - if I make a bubble-acceleromter the acceleration is in the direction the bubble moves - which would be up if I point it vertically.

As above...
Direction of gravity

OK. In Newtonian gravity a force of attraction exists between all bodies in relation to their mass so the force of gravity acts to accelerate me towards the Earth. However as I understand it, in relativity the mass of the Earth warps space-time so that my geodesic converges with that of the Earth, and the Earth is therefore pushing up on me, forcing me up and away from my geodesic line (and to a far lesser extent I am forcing the Earth down and away from its geodesic line). As the Earth is pushing up on my feet and up and away from my natural geodesic, does that not mean I am being accelerated away from the Earth rather than towards it??

Scratch that. I just realised I am talking bollocks. Converging geodesics "are" gravity, therefore gravity is accelerating me towards the Earth... the Earth is simply resisting that force. OK.

Relative weakness of gravity

?? It is my understanding that gravity "should" be roughly equal in strength to the other three fundamental forces and that some explanation for its weakness was necessary.... Is that not part of the appeal of string theory... that gravity would permeate the hidden dimensions and that this would explain its relative weakness?

My question was why this should be an issue since according to Relativity, Gravity isn't a real force at all, merely a consequence of the warping of space-time...... however I now realise my error (as pointed out by WannabeNewton)... I am doing a pick-and-mix of two theories. (Unfortunately I didn't understand anything else WannabeNewton said!)

I do not see why a saddle shaped Universe should imply a centre. The possibility that the Universe is saddle shaped is not new, however there is recent evidence from the Cosmic Background Radiation that this might possibly be close to the true "shape" of the Universe.

So my question is, how much could the shape of the Universe account for the expansion? Could a Universe which is becoming more curved as it expands account for the acceleration of the expansion of the Universe? Presumably not or someone who actually knows what they are talking about would have spotted it long ago... I was just wondering why it wouldn't work.

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Simon Bridge
Homework Helper
"[large scale structure of space-time] can be visualized as having a saddle-like negative geometry, where parallel lines would diverge."
... this is what they mean by a "saddle geometry".
"saddle shaped" is some editors idea of a good headline - ignore.

This is the problem of trying to picture (3+1)D space-time geometry in 2D terms.

Also notice that the research all this is based on is still fairly tentative - look at all the caveats.

So my question is, how much could the shape of the Universe account for the expansion? Could a Universe which is becoming more curved as it expands account for the acceleration of the expansion of the Universe?
... this is a common enough question: generally - maybe the observed accelerating expansion is an artifact of some large-scale geometry thingy?

iirc: a less than flat universe just means there is not enough mass/energy to halt the expansion ... how could that lead to an acceleration?

I think the main schools of thought are either that the current ideas about gravity are very good and predict some extra stuff - dark energy etc - or that they are not good and need to be modified.
http://www.fromquarkstoquasars.com/dark-matter-or-modified-gravity-two-competing-theories-battle/

A.T.
Direction of gravity

OK. In Newtonian gravity a force of attraction exists between all bodies in relation to their mass so the force of gravity acts to accelerate me towards the Earth. However as I understand it, in relativity the mass of the Earth warps space-time so that my geodesic converges with that of the Earth, and the Earth is therefore pushing up on me, forcing me up and away from my geodesic line (and to a far lesser extent I am forcing the Earth down and away from its geodesic line). As the Earth is pushing up on my feet and up and away from my natural geodesic, does that not mean I am being accelerated away from the Earth rather than towards it??

Scratch that. I just realised I am talking bollocks.
Actually you are quite right. You just have to distinguish between proper acceleration (what an accelerometer measures) and coordinate acceleration (velocity change in some coordinate system).

Converging geodesics "are" gravity, therefore gravity is accelerating me towards the Earth... the Earth is simply resisting that force. OK.
Earth's Gravity causes downwards coordinate acceleration of free fallers in the rest frame of the Earth, while their proper acceleration is zero.

The contact force from the Earth causes upwards proper acceleration of things standing on the surface, while their coordinate acceleration is zero in the rest frame of the Earth.