# Thread: Gravitation cancelling in curved space?

1. Hi everyone,
From what I understand, gravitation propagates along the dimensions of space. So in a curved space it will not propagate in an isotropic way ( same in all directions I mean ).
If we consider 2 sources of gravitation A and B, could it be that A attracts B from 2 opposite directions if space makes a loop between A and B?
In other words, the 2 curvatures of spacetime produced by A on B would have opposite directions, and could even cancel out, in which case there would be no gravitational effect at all?
There is also the possibility of gravity loops where A would attract B several times from the same direction.
Just out of curiosity, is such a scenario theoretically possible? I guess it would take a very strong gravitationnal field, I don't know exactly.
Nic.

2.

3. Originally Posted by Nic321
Hi everyone,
From what I understand, gravitation propagates along the dimensions of space. So in a curved space it will not propagate in an isotropic way ( same in all directions I mean ).
If we consider 2 sources of gravitation A and B, could it be that A attracts B from 2 opposite directions if space makes a loop between A and B?
In other words, the 2 curvatures of spacetime produced by A on B would have opposite directions, and could even cancel out, in which case there would be no gravitational effect at all?
There is also the possibility of gravity loops where A would attract B several times from the same direction.
Just out of curiosity, is such a scenario theoretically possible? I guess it would take a very strong gravitationnal field, I don't know exactly.
Nic.
It is rather difficult to have opinions on this for there seems there are two minds as to whether the Universe is finite or infinite.
To me it feels like all the concepts I have had for the last 50 years have to change.

4. Hi Robittybob1,
It also depends on what we call universe. Our universe may be finite, but inside an infinite multiverse, whatever the kind of multiverse that is.
If our universe is finite, maybe that kind of loop of gravity could eventually happen, I don't know.
I was wondering for instance if that could not explain the acceleration of the expansion of the universe. During the first billions years each mass of the universe would have attracted every other mass from one direction. The propagation of gravitation from the other side ( a loop around the universe ) would not have been able to reach these masses because of the very fast expansion. After billions of years, the expansion slowed down so the propagation coming at the speed of light would have been able to catch up the other masses "from behind". This would have had the effect or reducing the attraction effect of gravity, thus leading to the acceleration of the expansion.
I doubt it would be the real explanation to the acceleration of the expansion, but maybe that kind of scenario could happen in some circumstances, I don't know I was just wondering.

5. Because of the superposition theorem, and because real gravity fields are spherical, and there is no "anti-gravity," only opposite or contra-gravity, cancellation can only be local, not global even if two diametrically opposed fields are superposed. The superposition will only yield a zero field at one point; all other points will be complex. This is the nature of two intersecting spheres.

6. In other words, the 2 curvatures of spacetime produced by A on B would have opposite directions, and could even cancel out, in which case there would be no gravitational effect at all?
The presence of a non-Minkowskian metric is connected to but not identical to net acceleration. Conversely, just because there is no net acceleration present ( e.g. at Lagrange points ) does not mean space-time is Minkowskian there; such points still have a gravitational potential which is different than a reference point far away ( at infinity ). For example, if you were to place a clock at the Lagrange point between two very massive bodies, even if the clock remains stationary there, it will tick at a different rate as compared to a reference clock very far away. Another example would be space-time in the interior of a cavity within a thin shell of matter - space-time is locally flat there ( no tidal accelerations ), yet clocks placed inside such a cavity are still gravitationally time dilated as compared to reference clocks at infinity.

Mathematically, such small regions are described by metrics which are locally isometric but not identical to the Minkowski metric - Minkowski space-time being the asymptotic limit at infinity.

There is also the possibility of gravity loops where A would attract B several times from the same direction.
I'm afraid I am not sure what you mean by this.

7. Originally Posted by Schneibster
Because of the superposition theorem, and because real gravity fields are spherical, and there is no "anti-gravity," only opposite or contra-gravity, cancellation can only be local, not global even if two diametrically opposed fields are superposed. The superposition will only yield a zero field at one point; all other points will be complex. This is the nature of two intersecting spheres.
Well I believe that in some cases gravity can be repulsive if there is a cosmological constant. I believe inflation is a case like that.
But other than that, concerning the case what I was talking about, yes the energy of the source is positive and gravitation is only attractive. So indeed the exact cancellation would only happen at the intersection of the 2 spheres I guess, which would be a circle.
The other points where the 2 curvatures would superimpose ( where the propagation of gravity from different directions would manage to meet ), there would be only a partial cancellation.

8. The presence of a non-Minkowskian metric is connected to but not identical to net acceleration. Conversely, just because there is no net acceleration present ( e.g. at Lagrange points ) does not mean space-time is Minkowskian there; such points still have a gravitational potential which is different than a reference point far away ( at infinity ). For example, if you were to place a clock at the Lagrange point between two very massive bodies, even if the clock remains stationary there, it will tick at a different rate as compared to a reference clock very far away. Another example would be space-time in the interior of a cavity within a thin shell of matter - space-time is locally flat there ( no tidal accelerations ), yet clocks placed inside such a cavity are still gravitationally time dilated as compared to reference clocks at infinity.
Mathematically, such small regions are described by metrics which are locally isometric but not identical to the Minkowski metric - Minkowski space-time being the asymptotic limit at infinity.
Would it be true to say that space in the interior of a cavity or at the Lagrange point between 2 masses is "stretched" and that it is the stretching of space that causes the time dilation?
Let's take the analogy of a squared net. If one puts a mass in the net, the net stretches in the direction of the mass, and a certain curvature is created.
If on another hand one pulls on the 4 sides of the squared net equally, the net is also stretched isotropically ( same in all directions ), so without curvature.
In the case of space, it would be the stretching of space (which would be linked to the potential ) that would cause the time dilation.
In the case of the empty shell, space is stretched by the shell. In the case of a Lagrange point between 2 masses, space is stretched by the 2 masses.
In the case of a mass, space in stretched by the mass, but in the direction of the mass, so with a curvature, like the mass in the net.
The acceleration would be caused by the curvature, not by the the stretching itself. In the case of an empty shell or at a Lagrange point, there is no curvature, so no acceleration.
I know that the analogy of the net is not perfect, because space a sort of dynamic structure, unlike a net. There might be others differences but I just used this analogy to get my point across.
Does it make sense to look at it like that?
I'm afraid I am not sure what you mean by this.
Let's say you have 2 masses A and B in a curved space, meaning that the dimensions of space make a loop. Then the propagation of the gravitation from A would follow the loop, and could do several loops. So B would receive the gravitation from A several times, although with decreasing intensity at each loop.
If for instance gravity is transmitted by gravitons, the gravitons emitted from one mass would circle around the curved space any number of time until they are intercepted by another mass.

9. Originally Posted by Nic321
Originally Posted by Schneibster
Because of the superposition theorem, and because real gravity fields are spherical, and there is no "anti-gravity," only opposite or contra-gravity, cancellation can only be local, not global even if two diametrically opposed fields are superposed. The superposition will only yield a zero field at one point; all other points will be complex. This is the nature of two intersecting spheres.
Well I believe that in some cases gravity can be repulsive if there is a cosmological constant. I believe inflation is a case like that.
Yes, but that would require exotic matter; and inflation requires a strong negative lambda for the "slow roll." The only other way for it to exist is that CC accumulates as space expands, because it is a global parameter of spacetime; you get a little bit of lambda for every little bit of expansion. But ultimately, as you have sensed, this is immaterial to this particular conversation.

Originally Posted by Nic321
But other than that, concerning the case what I was talking about, yes the energy of the source is positive and gravitation is only attractive. So indeed the exact cancellation would only happen at the intersection of the 2 spheres I guess, which would be a circle.
Or a point if they are tangent. That's actually the precise scenario I was speaking of. I hadn't thought of the circular region that would result if they interpenetrated.

Originally Posted by Nic321
The other points where the 2 curvatures would superimpose ( where the propagation of gravity from different directions would manage to meet ), there would be only a partial cancellation.
Yes.

10. Originally Posted by Nic321
Would it be true to say that space in the interior of a cavity or at the Lagrange point between 2 masses is "stretched" and that it is the stretching of space that causes the time dilation?
It's rather more like a mesa - the top of a mesa mountain is perfectly flat, but at a different altitude than the surrounding land. Likewise space-time - the interior of the cavity is flat, but at a different gravitational potential than a reference point at infinity.
I'd be careful with the "stretching", since mathematically the Riemann tensor is identically zero everywhere in the interior of the cavity.

Does it make sense to look at it like that?
I personally wouldn't choose this particular analogy, but so long as you are clear about its limitations, then yes, I does make sense.

Let's say you have 2 masses A and B in a curved space, meaning that the dimensions of space make a loop.
I am still not sure what you mean by "loop". I see no loop in the space-time between two massive bodies.

11. Originally Posted by Markus Hanke
It's rather more like a mesa
Great analogy.

12. It's rather more like a mesa - the top of a mesa mountain is perfectly flat, but at a different altitude than the surrounding land. Likewise space-time - the interior of the cavity is flat, but at a different gravitational potential than a reference point at infinity.
I'd be careful with the "stretching", since mathematically the Riemann tensor is identically zero everywhere in the interior of the cavity.
Ok, but that's why I made the distinction between the stretching of space and the curvature. The stretching represents the potential, and has no effect on time dilation and acceleration. In the case where space is stretched equally from all direction, like a cavity, space is stretched but flat, and there is no curvature of space-time. The Riemann tensor is zero inside the cavity.
In cases where there is a curvature of space however, space-time is curved and there is an acceleration ( the Riemann tensor is not null).
It is an analogy of course, but from what I understand, space changes shape around a mass, it is like "pulled" by the mass or "stretched" in the direction of the mass. In the case of a mass, contrarily to the cavity, there is not only a stretching of space but also a curvature, so that creates is a curvature of space-time and an acceleration.
The point is, the potential must be something physical, it must be linked to the fabric of space. The acceleration is more linked to the gradient of the potential if you will( which says if space is curved or not ). By saying all this I am just trying to give general ideas.

I am still not sure what you mean by "loop". I see no loop in the space-time between two massive bodies.
I was refering to a loop in space, not space-time, in a case where space is curved.
Say you have 2 points A and B. If space is curved, you can go from A to B by from different directions.
It would be the same for the propagation of gravitation itself, because gravitation propagates along the directions of space (well I guess). So the gravitation of A would make a loop inside the curved space.
Another example. If for instance the universe is finite, you can take a spaceship and accelerate in one direction, sooner or later you get back to your starting point. Whether you are a spaceship or gravitons propagating at the speed of light, your trajectory will be the same. You might even have a gravitational effect on yourself. ( I am not saying that's what happens it's just an example )

13. Yes, but that would require exotic matter; and inflation requires a strong negative lambda for the "slow roll." The only other way for it to exist is that CC accumulates as space expands, because it is a global parameter of spacetime; you get a little bit of lambda for every little bit of expansion. But ultimately, as you have sensed, this is immaterial to this particular conversation.
If gravitation makes a loop and "comes back", it could be seen a a form of negative energy maybe, because it would pull from behind.
I don't know how that kind of loopback could be modeled in general relativity and the kind of metric it would produce.

14. Originally Posted by Nic321
The stretching represents the potential, and has no effect on time dilation and acceleration.
Be careful here, because gravitational potential does have an effect on time dilation.

The point is, the potential must be something physical, it must be linked to the fabric of space.
The link is the metric tensor, which differs inside cavity from the Minkowski tensor at infinity, even though both regions are flat. I really can't think of any very good visualisation for this, though.

Say you have 2 points A and B. If space is curved, you can go from A to B by from different directions.
That depends very much on how it is curved, and also on the global topology of the space in question, specifically on whether it is singly or multiply connected.

If for instance the universe is finite, you can take a spaceship and accelerate in one direction, sooner or later you get back to your starting point.
Ok, this would work for trajectories in space, but it doesn't work for world lines in space-time, because you can travel only at a finite speed.

15. Originally Posted by Nic321
I don't know how that kind of loopback could be modeled in general relativity and the kind of metric it would produce.
The only topological construct I am aware of that comes even close to what you describe is called a closed time-like curve. This is a hypothetical construct which appears in certain black hole solutions which possess angular momentum; however, such things would be hidden behind event horizons.

16. Originally Posted by Nic321
The point is, the potential must be something physical, it must be linked to the fabric of space. The acceleration is more linked to the gradient of the potential if you will( which says if space is curved or not ).
Originally Posted by Markus Hanke
Originally Posted by Nic321
The stretching represents the potential, and has no effect on time dilation and acceleration.
Be careful here, because gravitational potential does have an effect on time dilation.
Indeed. Gravitational potential is all about time dilation. And it's not about curvature, either. One can define gravitational potential for a constantly accelerated frame of reference in flat spacetime. Curvature is not directly about acceleration, gravitational potential, or time-dilation, as these can be manifested in flat spacetime. It is about the tidal effect.

17. Originally Posted by Nic321
The stretching represents the potential, and has no effect on time dilation and acceleration.
Be careful here, because gravitational potential does have an effect on time dilation.
Oups, my bad, I wanted to say "The stretching represents the potential, and has no effect on acceleration."
The stretching DOES have an effect on time dilation, that's actually the point. It's the curvature that has no effect on time dilation.
The point is, the potential must be something physical, it must be linked to the fabric of space.
The link is the metric tensor, which differs inside cavity from the Minkowski tensor at infinity, even though both regions are flat. I really can't think of any very good visualisation for this, though.
Ok, what I call the "stretching" is what would be described in the metric tensor.
You don't necessarily agree with the notion of "stretching", but we know that space can be curved. If space can be curved, it seems to me that it can be pulled or stretched, it is malleable.
This being said it doesn't necessarily mean it is something rigid that is stretched, it is maybe something more dynamic, or more emergent.
Say you have 2 points A and B. If space is curved, you can go from A to B by from different directions.
That depends very much on how it is curved, and also on the global topology of the space in question, specifically on whether it is singly or multiply connected.
Well, I guess space would have an unusual shape in that scenario.
Maybe that would happen close to the center of a black hole where space is very curved. If a mass gets close to the center, its gravitational field would not be isotropic around it.
It would be distorted by the curvature of space. The falling mass close to the center would produce more gravitational effect in the direction of the center because space is curved in that direction.
Ok, this would work for trajectories in space, but it doesn't work for world lines in space-time, because you can travel only at a finite speed.
Ok, so you think that gravity propagation would not follow the curvature of space?

18. Hi KJW,
Indeed. Gravitational potential is all about time dilation. And it's not about curvature, either. One can define gravitational potential for a constantly accelerated frame of reference in flat spacetime. Curvature is not directly about acceleration, gravitational potential, or time-dilation, as these can be manifested in flat spacetime. It is about the tidal effect.
I thought that curvature was the cause of the acceleration. Could you give an example where there is curvature and no acceleration?

19. Originally Posted by Nic321
If space can be curved, it seems to me that it can be pulled or stretched, it is malleable.
Ok, I don't know how to reply to this without getting technical.
The theory of GR uses Riemann geometry to model gravitation. What this means is that we have a pseudo-Riemannian manifold that has a connection and a metric defined on it. The principle invariants of a connection are curvature and torsion; Einstein's GR uses a very specific connection, the Levi-Civita connection, the defining characteristic of which is that torsion vanishes everywhere, leaving only curvature. A concept such as "stretching" does not really fit into this picture - how would you relate this to the covariant derivative ?

Ok, so you think that gravity propagation would not follow the curvature of space?
"Propagating gravity" would be gravitational waves - and yes, these do indeed follow the background curvature of space-time. Not only that, but they also couple to the background curvature, and if that curvature is large enough you get highly non-linear effects such as damping, backscatter, and "tails" moving at subliminal speeds. You also get self-interactions.

I thought that curvature was the cause of the acceleration.
Space-time curvature leads to geodesic deviation, i.e. relative acceleration of test particles. Curved world-lines in flat space-time represent accelerated frames of reference, which is associated with gravitational potential, as KJW has pointed out.
The upshot is that you can have gravitational potential even in completely flat space-times.

20. Gee it's complicated, I am going to think about your post. From what you are explaining, the idea of stretching seems to be too simplistic...

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