How gravity arises from spacetime curvature?

[weio]

Hi there

I've been trying to find out how spacetime curvature actually produce gravity, but all i can find is articles using math, which is far too advanced for me to understand. Can anyone give me a theoretical explanation to how the gravity arise from the spacetime curvature according to the theory of general relativity?

Much appreciated.

weio

 

[LURCH]

Actually, general relativity does not so much propose that gravity is "caused buy" a curvature of space-time. This would be somewhat like stating that waves are caused by Little Hills moving across the surface of water. Rather, it would be more accurate to say that GR and describes gravity as being a curvature of space-time. The actual cause is not known.

 

[Gza]

Maybe I'm off base here, but I do think that GR gives an interpretable cause of gravity. That would be in the form of matter or energy causing the warping of space, thereby giving rise to the phenomenon we call gravity.

 

[zoobyshoe]

I can't make heads or tails of the "warped space-time" explanations either. Space is nothing. How can you warp nothing? Mathematically, I guess.

If I think of space as actually being occupuied by some sort of aether, then I can start to visualize this being "warped" but I have trouble still, since the Earth is a sphere and I can't visualize a spherical "warp".

The notion of trying to go straight but ending up going in a curved path, more or less makes sense by analogy to many things, but that's is the only aspect of these descriptions that I can get a handle on.

 

[Gza]

The notion of trying to go straight but ending up going in a curved path, more or less makes sense by analogy to many things, but that's is the only aspect of these descriptions that I can get a handle on.

You're not alone. I'd be mighty suspicious of anyone who told me they had a solid physical grasp on these concepts. Einstein couldn't even make the leap and he came up with this stuff!

 

[zeta101]

zoobyshoe, in my opinion (its my opinion since i haven't actually ever read this anywhere) space is not nothing, or rather space-time is not nothing. There is something to space-time, something that allows it to be warped. I'm sure you've seen the popular digram of a bowling ball (which represents, say a planet, or a star like our sun) placed on a flat stretched rubber sheet, the bowling ball causes the sheet to warp under its weight.

Obviously this is a 'messy' analogy, but try not to dwell on that. Picture how a rubber sheet would warp in the presence of this bowling ball...this is in TWO dimensions, now try and picture this in 3D, its hard, like picturing many things in 3D. But normalling if you see the 2D case you can accept the 3D case, and just accept its hard for you to visualise.

Also, for weio, I'm sure there is an answer to your question, but it might not be fundamental enough to satisfy you. Having not studied GR yet I can't help you, but you might want to think about the analogy I gave for zoobyshoe about the bowling ball on the rubber sheet, since anything placed on the sheet, like another ball will roll towards the bowling ball due to the curvature, but again, this is a very messy analogy, its not a DIRECT analogy, but its some kind of starting point i suppose.

 

[NateTG]

What happens is that in non-euclidean geometries, the notion of 'straight line' gets a bit wierd.

Imagine an ant walking along the surface of a cylinder. Even though the ant is walking along a 'locally straight line' (a geodesic), it's walking along something that we (as outsiders) might percieve as a curved line. Similarly, if the ant is walking in a locally straight line on a moving turtable, although the ant is moving in a locally straight line, the path it describes is not straight.

Classically, it's assumed that space-time is euclidean (or flat), so a 'locally straight' line is also a straight line. And that gravity is a force that perturbs motion that would otherwise go along this straight line. Einstein postulated that the path that an object takes under the influence of gravity is the 'locally straight' path. This is intemately related to the notion that gravitational and inertial massess are the same. Since we can describe the paths that objects take under the influence of gravity, we can plot them, and describe the 'shape' of space-time.

It's also worth noting that for GR to work, you have to imagine a 4D warped space (which would be embedded in 5-space) and not a 3D space, since GR includes time as one of the dimensions.

 

[LURCH]

I agree with Zeta that the famous "rubber sheet" analogy is the best place to start. The tough part is, as has been stated, excepting the extra dimension. The rubber sheet has two dimensions until a mass is introduced, warping it in a third direction the "up and down" that direction. The marble trying to roll past [in front or behind] the bowling ball or [to the left or the right] of it, will be drawn toward it. This is because the sheet has been warped in the "up and down" that direction.

In moving this phenomenon to 3-D, we see the same behavior exhibited by massive objects in space. It is simply that extra dimension is added, because an object trying to fly past a planet (for example) would have its course altered whether it tried to go [in front of or behind], [to the left or the right], or [above or below] the planet. This is because those three dimensions are curved in a fourth direction, which we cannot see nor even visualize. We can only observe the behavior of objects and logically deduce that this fourth direction exists.

 

[jcsd]

Actually how the curvature of spacetime causes gravity is not the most difficult to conceive part of GR as it's based on suprisingly simple logic.

In SR the worldline of an object in an inertial refernce frame is straight, the worldline of an object that is undergong accelartion is not straight. Gravity cause all objects whether they be photons, protons spaceships or space llamas to accelarate, not only that a gravitational field will cause all objects to accelerate by the same amount. So if we imagine a spacetime continuum containing a gravitational field there will be areas (infact due to the infinite reach of gravity all areas will be affected) where all worldlines curve and not only that they will curve in a simlair manner, depednt only the path that they take. In order for it's worldline not be curved an object has to accelerate, consequently we find the 'shortest' path through spacetime is no longer the straight path but the curved path. This is exactly like manifolds with intrinsic curvature in differential geometry! For example imagine the surface of a sphere, this is a manifold with intrinsic curvature, the shortest paths along it's surfaces are curves traced by great circles (a fact that has long been known by sailors when traveling across the globe). Therefore we can mdoel our gravity in our spacetime as curvature.

 

[zoobyshoe]

zoobyshoe, in my opinion (its my opinion since i haven't actually ever read this anywhere) space is not nothing, or rather space-time is not nothing. There is something to space-time, something that allows it to be warped.

Someone quoted Einstein in the Relativity forum, in the past month or two, as having said something to the effect that GR doesn't stand unless there is something along the lines of an aether. (He wasn't talking about the classical notion of the aether, of course, whose properties were proven to be impossible.) Someone else found quotes by Einstein that said things along the same lines last summer, that there must be something we might call an aether. That is vague information, but I think if you keep your eyes open you may run into these remarks by Einstein.

To the extent that I think of space as the presence of something that might be called an aether it provides something that could be warped but that really only produces more questions about how to visualized this warpage or curvature.

 

[jcsd]

You don't need an aether to explain curvature. Ask yourself this question: should we really assume that spacetime is Minowskian or space is Euclidian?

 

[zoobyshoe]

You don't need an aether to explain curvature. Ask yourself this question: should we really assume that spacetime is Minowskian or space is Euclidian?

The problem that Weio and I are having is not with the explanation per se but with all the suggested explanation/visualizations. Once you put a bowling ball on a rubber sheet or an ant on a cylinder you suggest that there is a very physical, Newtonian, way to visualize gravity despite not intending to.

As I said the temptation to visualize an aether that is being warped or curved is strong, but ends up just raising more questions. What I said about Einstein apparently suspecting something like an aether was a vaguely supporting remark to zeta101s opinion that "space is not nothing", and didn't directly apply to the gravity visualization issue.

To the extent that all explanation/visualizations rule out a lot of other possible speculations about the nature of gravity, they are extremely useful. It is clear that gravity isn't some form of magnetism, for instance, or electric attraction.

 

[jcsd]

The basic thing to realize is that the way that gravity changes the worldlines in spacetime is analgous to a curved (pseudo-Riemannian) manifold, so we don't need to actually have some sort of physical background.

 

[Imparcticle]

I asked this same question a while ago and I was told that bodies of mass in space (say a planet) had a stream of energy comming from it that warped space-time, thus causing the phenomenon of gravity.

 

[selfAdjoint]

Well that's not a good description. The best mantra is "Space tells matter and energy how to move, and matter and energy tell space how to curve." Both energy in the form of mass (by [tex]e=mc^2[/tex]) and other kinds of energy determine the curvature of spacetime, according to Einstein's field equations. And the curvature defines what is natural motion (geodesics).

There is no "streaming" of energy; a rock out in space has energy just because it has mass, and that energy causes spacetime to curve a little bit in its neighbothood.

 

[zoobyshoe]

The basic thing to realize is that the way that gravity changes the worldlines in spacetime is analgous to a curved (pseudo-Riemannian) manifold, so we don't need to actually have some sort of physical background.

"Manifold: d: a topological space in which every point has a neighborhood that is homeomorphic to the interior of a sphere in Euclidian space of the same number of dimensions."

-Merriam Websters Collegiate Dictionary 10th edition

Is this what you mean by "manifold"?

 

[jcsd]

Yep. Just think of a manifold as something which is locally Euclidian.

 

[zoobyshoe]

Yep. Just think of a manifold as something which is locally Euclidian.

Speak to me of these neighborhoods. Each point has a neighborhood homeomorphic to the interior of a sphere.

Why is it "to the interior of a sphere" rather than "to a sphere". If we remove the shell of a sphere, the shape that remains is just another sphere. Or, does this mean empty space with a spherical boundary?

In any event, since all the points have these neighborhoods, all the neighborhoods must overlap into kind of solid, no?

 

[jcsd]

For most curves that we use an individual point on that curve is simlair to a straight line, i.e. it has a gradient whose value is a real number. this becomes obvious when you 'zoom in' on any part of such a curve it becomes more and more like a starightline. A manifold is simalir is this respect to the curve as every point on an n-dimensional manifold is simlair to an n-dimensional Euclidan space, a fact that becomes more obvious when you 'zoom in' on any part.

 

[Gokul43201]

Does a point mass create curvature that is spherically symmetric ?

 

[zoobyshoe]

For most curves that we use an individual point on that curve is simlair to a straight line, i.e. it has a gradient whose value is a real number. this becomes obvious when you 'zoom in' on any part of such a curve it becomes more and more like a starightline. A manifold is simalir is this respect to the curve as every point on an n-dimensional manifold is simlair to an n-dimensional Euclidan space, a fact that becomes more obvious when you 'zoom in' on any part.

Which is a nice expansion on what you said about manifolds being locally Euclidian, and which I think I grasp.

However, I'm interested in these spheres, or interiors of spheres that comprise the neighborhoods of any point. Here's why: it struck me that if any point has a neighborhood homeomorphic to the interior of a sphere, then any motion in any direction would bring the moving object, be it micro or macroscopic, up against what, in euclidian terms, would be the interior wall of a hollow sphere, such that the easiest route by which to coninue forward would also include an increasingly downward pointing force, as you passed farther through the shell.

Since each point has such a sphere for its neighborhood, the whole of space would be packed with them, Since they are homeomorphic to the interior of a sphere only, not the exterior, you would only encounter downward and away curves, no upward and toward you curves.

Even if you're not moving, your molecules are vibrating and your electrons are orbiting, all encountering these same interior spheres.

Is this what these spherical neighborhoods are about?

 

[selfAdjoint]

The spheres (balls would be a better term) being talked about here are OPEN, meaning they don't have the boundary as part of their definition. The manifold is covered with overlapping spherical neighborhoods, and when you travel far enough, you don't run into a wall, you just cross into another neighborhood (in fact you spend some time in the overlap, first), so at that time your in more than one neighborhood).

Consider the Earth. Let one neighborhood be the northern hemisphere, extended say 10 miles south of the equator. And let the other neighborhood be the southern hemisphere. They overlap along that 10 mile strip. And each one of them is topologically an open disk - which is a solid 1 dimensional sphere (i.e. a filled in circle), but only the interior of it.

 

[zoobyshoe]

I see there is no boundary included in the definition. However "interior" of a sphere, (or ball) is specified. The neighborhoods of every point are homeomorphic (the same shape as) to the the interior of a sphere (which can only be as opposed to the exterior) of a euclidian space of the same # of dimensions.

I am trying to establish how these add up to gravity. Perhaps a good question to ask is where does each point lie with respect to its corresponding interior-of-a sphere neighborhod. Are the points always at the center? (I have been assuming they are.)

 

[KingNothing]

In response to the rubber street analogy:

When I put this into 3d, I tend to come up with a slightly different analogy:

I think of dropping a marble into some liquid, let's say, water. As it goes, no matter where it goes (although I assume it would go down), it displaces the liquid, and in a way, warps it to accompany the added volume.

I guess in that case it's volume, and with the ball its mass, but that's the best I can really do for now.

 

[zoobyshoe]

In response to the rubber street analogy:

When I put this into 3d, I tend to come up with a slightly different analogy:

I think of dropping a marble into some liquid, let's say, water. As it goes, no matter where it goes (although I assume it would go down), it displaces the liquid, and in a way, warps it to accompany the added volume.

I guess in that case it's volume, and with the ball its mass, but that's the best I can really do for now.

I've tried to explore something like this as well to vsualize the curvature of space/time. What would happen to a fluid whose volume could not change if a sphere were introduce into it. In classical situations the total pressure of the fluid would rise. Suppose that with gravity the increased fluid pressure remains localized and doesn't redistribute throughout the universe. Every mass would carry a local pressure gradient around with it: the closer you got the higher the pressure.

I don't see where that would add up to gravity. It seems like the opposite would result: all these local high pressure systems trying to stay equidistant from all others. This sounds more like electric charge than gravity, and it's also an aether theory.


So, I thought, how could I turn it around such that a marble in a fluid carried a pressure vacancy around with it rather than a pressure density. If we start off with an aether and concieve of mass or energy as localized aether densities: coagulations or flocculations of aether, then mass or energy would travel around with a halo consisting of an aether vaccuum.

In order for some aether to get denser at any given point, it would have to leave an aether vacancy around it. More aether would migrate into the vacancy leaving an even larger vacany in its wake. In the center the aether compacts simply because it has been accelerated together.

A planet, say, goes through space with this halo of aether vacancy around it. Is would constantly be vaccuuming in more aether and the halo would be maintained. Any mass that enters the pressure gradient represented by the halo would be pushed to the surface of the planet where the pressure is always less.

So that is where trying to imagine something like a marble in a fluid got me. I think it is quite far away from GR.

 

[KingNothing]

Let's just call it MR.

 

[Mikado]

I think there is a fundamental fact that was omitted in the rubber sheet analogy. The bowling balls on the sheet warp the sheet because gravity acts upon its weight. For the analogy to hold there must be a 'force' that acts upon masses for it to create curvature->gravity. This force got to be proportional to the mass, hence a simple acceleration. Which would mean that everything in this world has the same acceleration through this extra dimension. I think that is one big problem to be examinating.

Another thing, if there is a sort of eather, would light be a wave traveling on it. Because if it does, then the definition of eather and the curvature can only be very constrained since light must always travel at the same speed on it.

 

[zoobyshoe]

Let's just call it MR.

What's that, Marble Relativity?

 

[zoobyshoe]

I think there is a fundamental fact that was omitted in the rubber sheet analogy. The bowling balls on the sheet warp the sheet because gravity acts upon its weight.

Someone started a thread about this a while back called something like "Explaining gravity with gravity" which expressed doubt that using a phenomenon to explain that very same phenomenon was helpful.

Another thing, if there is a sort of eather, would light be a wave traveling on it. Because if it does, then the definition of eather and the curvature can only be very constrained since light must always travel at the same speed on it.

All I can say about any aether that might actually exist is that it can't have the same properties as the classical aether was supposed to have.

 

[Michael F. Dmitriyev]

Gravitation this is manifestation of action of Absolute Time at the Space. The space thus is curved. Therefore force of gravity is subjected to nonlinear dependence on distance between objects.
Gravity force equal to G (4pi) M1*M2/(4pi)r^2 indeed.
Here (4pi)r^2 is a square of sphere with radius r.
It is determined by 4-dimension geometry of the space.