My problem with the relativity representation on gravity.

[pervect]

I am confused. The rubber sheet model is frequently used in conjunction with discussions about GR, in which it's used to show the concept of "space/time" being distorted in the presence of mass. Which bit of the model represents time and how does the model portray anything other than a 'simple', classical 2D potential well? If time is involved then an animation could be misleading (we don't normally animate simple xy graphs). Perhaps I am just having a problem with interpretation.

I think this is a poor representation.

So do most people here.

Add me to the list of people here (many of which are science advisors, and/or PF mentors) who think the diagram is unusually poor.

The reasons for the poorness have mostly been discussed, and a few better alternatives have been mentioned.

So while you may or may not be misinterpreting it, even if you did interpret it correctly, it would, unfortunately, do little to actually aid you in understanding general relativity.

Amongst diagrams I can somewhat recommend are the "parable of the apple", which you can find in http://www.eftaylor.com/pub/chapter2.pdf, which is a publically downloadable chapter from "Exploring Black Holes" by E F Taylor. You can find a similar diagram in Misner, Thorne, and Wheeler's textbook "Gravitation". In fact, you can see it on the front cover :-).

https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20

minus the explanatory text - and unfortunately it doesn't make that much sense without the explanation.

 

[Dale]

The rubber sheet model is frequently used in conjunction with discussions about GR, in which it's used to show the concept of "space/time" being distorted in the presence of mass. Which bit of the model represents time

None.

how does the model portray anything other than a 'simple', classical 2D potential well?

It doesn't.

 

[sophiecentaur]

Well, now I can rest easy at nights chaps.

 

[ikjyotsingh]

Once again, the rubber sheet analogy is ALWAYS used to represent the Schwarzschild solution. It is not used for any other solution in G.R. You can only have an isolated mass bending spacetime in a static, spherically symmetric setting, this spacetime is then asymptotically flat. If you take the Schwarzschild vacuum metric, and take constant time slices and embed that into cylindrical coordinates, you get the equation of a surface, which is a paraboloid, which represents the "dimple" in spacetime that is caused by an isolated mass. Any particle moving in the vicinity of this mass moves along a geodesic assuming there are no non-gravitational forces.

The rubber sheet analogy (which is a mathematically correct analogy), only applies to the Schwarzschild metric.

 

[ikjyotsingh]

Rubber sheet model

I am confused. The rubber sheet model is frequently used in conjunction with discussions about GR, in which it's used to show the concept of "space/time" being distorted in the presence of mass. Which bit of the model represents time and how does the model portray anything other than a 'simple', classical 2D potential well? If time is involved then an animation could be misleading (we don't normally animate simple xy graphs). Perhaps I am just having a problem with interpretation.

Hello. As I explained in my other post, the rubber sheet model is not a general model in G.R., and is a visualization of only ONE solution in G.R., a solution that is static and spherically symmetric, that is, it has one time-like and 3 space-like Killing vectors, and by Birkhoff's theorem, this can only be the Schwarzschild metric.

Time plays no role in THIS class of solutions, as the solution is static, and asymptotically flat, more precisely, it is time-symmetric.

In general, you cannot show the concept of spacetime, as spacetime is a 4D pseudo-Riemannina manifold, these are impossible to visualize in the way we are used to visualizing geometry. You must embed these manifolds in Euclidean geometry to get a good visualization of them.

Also, more generally, the idea of evolving a spacetime in time is not trivial, the whole field of numerical relativity and 3+1 dynamical relativity is built upon this. One must essentially break apart the space-time symmetry to consider a foliation of spatial slices moving in time. It is how the curvature of those spacelike slices evolve in "time" that determine the dynamics of the model. In the Schwarzschild/rubber sheet visualization, each spatial slice is constant in time, which is what you would expect.

Hope this helps.
Ikjyot Singh Kohli

 

[ikjyotsingh]

It is about embedding diagrams

Yes, that is my point. Even if they show a correct graphic the description is not correct.

You seem to think that this is a discussion about embedding diagrams. It is not. It is a discussion about the rubber sheet analogy. Even if the rubber sheet analogy is presented with an accurate graphic of an embedding diagram, it remains a deeply flawed analogy, and the embedding diagram is irrelevant to the analogy.

This is a discussion about embedding diagrams, no one seems to realize it, that's why there is so much confusion! The rubber sheet analogy which is an Euclidean embedding of the Schwarzschild vacuum metric only applies in this metric. You cannot have the rubber sheet analogy in any other context in G.R. It only applies for a static, spherically symmetric, vacuum, and asymptotically flat metric. Embed the S-metric into Euclidean space and you get the rubber sheet, there is simply no other context in G.R. which can produce the rubber sheet analogy.

The Schwarzschild solution, perhaps due to the prediction of black holes is the most popular solution in G.R. (other than FLRW), that's why these rubber sheet diagrams have taken on so much momentum, but they only apply in the S-metric case.

The idea then, is not flawed at all, and the embedding diagram is where the analogy comes from!

 

[Dale]

This is a discussion about embedding diagrams, no one seems to realize it, that's why there is so much confusion! The rubber sheet analogy which is an Euclidean embedding of the Schwarzschild vacuum metric only applies in this metric. You cannot have the rubber sheet analogy in any other context in G.R. It only applies for a static, spherically symmetric, vacuum, and asymptotically flat metric. Embed the S-metric into Euclidean space and you get the rubber sheet, there is simply no other context in G.R. which can produce the rubber sheet analogy.

The Schwarzschild solution, perhaps due to the prediction of black holes is the most popular solution in G.R. (other than FLRW), that's why these rubber sheet diagrams have taken on so much momentum, but they only apply in the S-metric case.

The idea then, is not flawed at all, and the embedding diagram is where the analogy comes from!

A rubber sheet supporting a weight doesn't deform into Flamm's paraboloid. It deforms approximately into a gravity well. The rubber sheet analogy is simply not an embedding diagram of any spacetime, even the Schwarzschild spacetime.

 

[ikjyotsingh]

Rubber shet

A rubber sheet supporting a weight doesn't deform into Flamm's paraboloid. It deforms approximately into a gravity well. The rubber sheet analogy is simply not an embedding diagram of any spacetime, even the Schwarzschild spacetime.

Anything supporting a mass in G.R. necessarily deforms into Flamms' paraboloid. I have a challenge for you then, in this regard. Find a solution of the Einstein Field equations that supports a mass other than the S-metric!

 

[Dale]

Anything supporting a mass in G.R. necessarily deforms into Flamms' paraboloid.

No, that is not true. A spring supporting a mass certainly doesn't even remotely deform into a Flamms paraboloid. On the surface of the Earth where the weak field limit is appropriate a rubber sheet deforms to first order into a Newtonian potential well, not a Flamm's paraboloid. Here is an explanation:

http://en.wikipedia.org/wiki/Gravity_well#The_rubber-sheet_model

I have a challenge for you then, in this regard. Find a solution of the Einstein Field equations that supports a mass other than the S-metric!

The point is that the deformation in a rubber sheet is not a solution to the EFE.

 

[ikjyotsingh]

Rubber Sheet model

No, that is not true. A spring supporting a mass certainly doesn't even remotely deform into a Flamms paraboloid. On the surface of the Earth where the weak field limit is appropriate a rubber sheet deforms to first order into a Newtonian potential well, not a Flamm's paraboloid. Here is an explanation:

http://en.wikipedia.org/wiki/Gravity_well#The_rubber-sheet_model

The point is that the deformation in a rubber sheet is not a solution to the EFE.

How would you even model a spring supporting a mass in T_{ab}? I don't know what that has to do with anything.

So, according to your wikipedia article, spacetime is being modeled by a physical rubber sheet, of course that's nonsense. In all my years of studying G.R., I have never considered "rubber sheet" to mean an actual rubber sheet. This won't work in general, but not for the reasons listed in this forum thus far. The deformations in the rubber sheet are given by the Poisson's equation which is a second-order ELLIPTIC partial differential equation. Such PDEs are acausual and thus would not exist in a pseudo-Riemannian manifold. This is actually why the rubber sheet analogy considering an actual, physical rubber sheet would fail.

 

[Dale]

So, according to your wikipedia article, spacetime is being modeled by a physical rubber sheet, of course that's nonsense. In all my years of studying G.R., I have never considered "rubber sheet" to mean an actual rubber sheet.

Many people get fooled by that nonsense. I am glad that you weren't, but that is why we get so many questions here on this topic.

 

[EskWIRED]

Here is a youtube video from someone who took the effort to do a more reasonable job. We could poke holes in this one also, but at least he saw the problems with the standard explanations and tried to do a better job. I think he succeeded.

I like what he did.

I've been wanting to see an interactive program that would allow you to draw such an array. Ideally, it would allow you to alter the "mass" and the size of the center attractor, and to see the effect on the gridlines due to the changes.

Even better would be the ability to add another, small mass, and see it go into orbit (or not) depending on its velocity and direction. I'd love to model black holes using software like that.

 

[1977ub]

Isn't it a bit misleading for all [not all but most] of the video's captions to refer to "space" as being curved by gravity? After all, it is not objects which are distorted or bent by the presence of the planet, but space-TIME which is curved, inclining freely *moving* objects to veer toward the Earth unless held aloft.


"...space all around an object is bent...space is bent toward the objects..."

 

[A.T.]

I am confused. The rubber sheet model is frequently used in conjunction with discussions about GR, in which it's used to show the concept of "space/time" being distorted in the presence of mass.

The rubber sheet with dents in conjunction with GR shows just the spatial distortion described by the Schwarzshild metric. It looks similar to potential wells:

http://en.wikipedia.org/wiki/Gravity_well#Gravity_wells_and_general_relativity

But it has nothing to do with them. The spatial curvature is irrelevant to objects at rest in space, like an apple that starts to fall.

 

[stevendaryl]

I agree with you 100% They are doing exactly what you say. They are explaining gravity in term of gravity. It’s stupid, annoying, and does not explain anything.

I don't think it's that bad. The trampoline model works like this: You have a bowling ball sitting on a trampoline, causing the trampoline's surface to become warped, so it's no longer flat. Then you roll a marble across the surface of the trampoline. Its path isn't straight, but is curved.

General Relativity can be thought of as describing two different effects: (1) How does matter and energy affect spacetime curvature, and (2) How does spacetime curvature affect the motion of particles (and more generally, other non-gravitational physical phenomena)? The trampoline model helps to understand effect number (2), but not effect number (1). As far as I know, there is no help for understanding effect number (1).

The part that too me is the hardest to understand from the trampoline model is that in GR, the warping is to spacetime, not space alone.

 

[A.T.]

Isn't it a bit misleading for all of the video's captions to refer to "space" as being curved by gravity? After all, it is not objects which are distorted or bent by the presence of the planet, but space-TIME which is curved, inclining freely *moving* objects to veer toward the Earth unless held aloft.


"...space all around an object is bent...space is bent toward the objects..."

The captions are correct. And that's why this pictures are just as useless as the rubber sheet: it shows ony spatial distortion. Making it 3D doesn't make it better. Eventually worse. I doubt that those distorted 3D grids have anything to do with the spatial Schwarshild metric. Probably just some artists vague idea of distortion.

But the main problem is, they omit the time dimension.

 

[sophiecentaur]

The captions are correct. And that's why this pictures are just as useless as the rubber sheet: it shows ony spatial distortion. Making it 3D doesn't make it better. Eventually worse. I doubt that those distorted 3D grids have anything to do with the spatial Schwarshild metric. Probably just some artists vague idea of distortion.

But the main problem is, they omit the time dimension.

Perhaps a two dimensional projection of a 2D space plus time? Can't get my brain around what it could look like but we can handle 2D presentation of 3D images fairly well.

 

[A.T.]

Perhaps a two dimensional projection of a 2D space plus time? Can't get my brain around what it could look like but we can handle 2D presentation of 3D images fairly well.

For radial fall you just need 1 spatial dimension. That is what is used in the links i posted before. If you want 2+1 curved space time, you need two diagrams.

 

[ikjyotsingh]

I just want to know why everyone here is interested in visualizing spacetime?

 

[Dale]

Spacetime is fairly important in GR.

 

[ikjyotsingh]

Umm yeah

Spacetime is fairly important in GR.

Spacetime is fairly important in GR.

Umm yeah, but visualizing it is such a feeble task. You can only visualize it where an appropriate embedding exists, which according to my memory only exists for Bianchi I, V, and IX models, in addition to Schwarzschild family of metrics.

Why don't you talk about visualizing it in terms of how a pseudo-Riemannian manifold is constructed as a Hausdorff atlas of charts rather than all this diagram nonsense. At least the atlas of charts is correct both mathematically and physically, and provides an interpretation for which there can be no confusion.

 

[1977ub]

The captions are correct.

Are you saying that "space is bent" near the planet?

 

[WannabeNewton]

Why don't you talk about visualizing it in terms of how a pseudo-Riemannian manifold is constructed as a Hausdorff atlas of charts rather than all this diagram nonsense. At least the atlas of charts is correct both mathematically and physically, and provides an interpretation for which there can be no confusion.

This doesn't help you visualize curvature, it is a standard textbook depiction of the requirement of diffeomorphic transitions between overlapping coordinate charts of a maximal atlas. It doesn't help one visualize the physics of relativity in curved space - time in any way. Note however that I'm not saying one MUST have a way of visualizing the physics in GR; it is silly to ask for such visualizations in full generality because we are talking about pseudo - riemannian 4 - manifolds and of course we can't visualize such things innately.

 

[A.T.]

Are you saying that "space is bent" near the planet?

http://en.wikipedia.org/wiki/Schwarzschild_metric#Flamm.27s_paraboloid

But this is not what describes gravity. It describes some of the differences between Newton and GR. To understand how Newtons gravity is modeled in GR you have to include the time dimension. See the links I posted on page 1.

 

[wacki]

The best visualisation of curved space-time I came across are the three papers in

https://www.physicsforums.com/showthread.php?t=381683

see post #7 (from A.T.)

they are mathematically sound (at least for me) and it gave me a bit of intuition.
I think one can't hope for more than what's presented in those papers.

 

[Passionflower]

The spatial curvature is irrelevant to objects at rest in space, like an apple that starts to fall.

What do you mean by 'an object at rest in space'?

 

[pervect]

What do you mean by 'an object at rest in space'?

Wald has a definition on pg 288 of "General Relativity"

In a static spacetime the notion of "staying in place" is well defined; it means following an orbit of the Killing field [itex]\xi^a[\itex].

It might be a good idea to add that the Killing field is hypersurface orthogonal, but I didn't see Wald mention that explicitly

 

[WannabeNewton]

It might be a good idea to add that the Killing field is hypersurface orthogonal, but I didn't see Wald mention that explicitly

Page 119. To paraphrase, a space - time $(M,g_{ab})$ is stationary if there exists a one parameter group of isometries on the space - time such that the orbits of the group action are time - like curves. Furthermore, the space - time is static if there exists a space - like hypersurface orthogonal to the orbits. By Frobenius' theorem this is equivalent to the condition that $\xi _{[c}\triangledown _{b}\xi _{a}] = 0$ where $\xi ^{a}$ is the time - like killing vector field. Of course if we had coordinates $(t,x^1,x^2,x^3)$ on some region of this static space - time, with the coordinate vector field of the first coordinate having the usual interpretation as a "time direction" of this coordinate system, then the more intuitive notion of hypersurface orthogonal would be that $\xi ^{a}\propto \triangledown ^{a}t$ which leads to the other condition anyways via a computation.

 

[Passionflower]

Wald has a definition on pg 288 of "General Relativity"

Where does he talk about something 'at rest in space'?

Wald is talking about an object in a static asymptotically flat vacuum space-time that maintains a fixed coordinate distance from the Schwazschild radius. That is obviously not the same as 'at rest in space'.

In relativity objects simply have no velocity wrt space.

 

[WannabeNewton]

A particle is at rest (stationary) in a static space - time if it follows an orbit of the space - like hypersurface orthogonal time - like killing field i.e. $u^{a} = \frac{\xi ^{a}}{(-\xi^{b} \xi _{b})^{1/2}}$ where as noted above, $\xi ^{a}$ satisfies $\xi _{[c}\triangledown _{b}\xi _{a]} = 0$ and $u^{a}$ is the 4 - velocity of the particle. Simple as that. This is more general than the special case of the Schwarzschild space - time so I have no idea why you are even bringing that up.

 

[Passionflower]

A particle is at rest (stationary)...

Stationary is not the same as 'at rest is space'.

Anyone who thinks that an object can be at rest in space does clearly not understand the principle of relativity.

Space is not something one can compare one's velocity against.

 

[WannabeNewton]

So what is your proof that we cannot define rest with respect to the orbits of the time - like killing vectors of the static space - time? They are certainly a geometric aspect of the space - time.

I can even "abstract" this to stationary axisymmetric space - times. In $(t,\phi,x^2,x^3)$ coordinates I define the family of locally non rotating observers to be the observers who are 'at rest' with respect to the $t = \text{const.}$ hypersurfaces i.e. whose 4 - velocity is given by $u^a = \alpha \triangledown ^{a}t$. The angular momentum is given by $L = g_{ab}u^{a}\psi^{b} = \alpha g_{ab}\triangledown ^{a}t\psi^{b} = \alpha g_{ab}g^{ac}\triangledown _{c}t \psi^{b} = \alpha \delta^{c}_{b}\triangledown _{c}t \psi^{b}$ where $\psi^{a}$ is the space - like killing field whose integral curves are closed. In the coordinate basis this is of course just $L = \alpha \delta^{\mu}_{\nu}\triangledown _{\mu}t \psi^{\nu} = \alpha \delta^{0}_{1} = 0$. Are you telling me it's just a coincidence that my defining the notion of being locally non - rotating as following an orbit of $\triangledown ^{a}t$ actually gave $L = 0$? Pray tell.

 

[Passionflower]

Not sure what you are trying to show with all your math, I am well aware of the meaning of static and stationary spacetimes.

I remain of the opinion that the concept of being "at rest in space" is nonsense in relativity, in fact it is even nonsense in Galilean relativity.

Things can be at rest wrt to other things but not wrt to space.

And one more time: Wald does not use the phrase 'at rest in space'.

 

[WannabeNewton]

The quote by Pervect never made any claim about being at absolute rest in space. Of course that is nonsensical. It just said that given a space - time possessing a certain one parameter isometry group, we can utilize the orbits of the group action of said isometry group to define a notion of rest with respect to the orbits. In the above example of defining locally non rotating observers in stationary axisymmetric space - times notice how I said an observer is 'at rest' with respect to the spatial hypersurfaces $t = \text{const.}$ if his 4 - velocity $u^{a} = \alpha \triangledown^{a} t$ (the proportionality scalar field $\alpha$ is just the normalization); this isn't alien from galilean relativity wherein we describe observers in collinear uniform motion with respect to one another. This example was meant to show that we can deduce physical properties of the observer with precisely that definition of locally non rotating e.g. the fact that the observer's angular momentum vanishes, as shown above.

I don't recall me nor Pervect claiming absolute rest. Do you agree with the above however?

 

[Passionflower]

The quote by Pervect never made any claim about being at absolute rest in space. Of course that is nonsensical. It just said that given a space - time possessing a certain isometry group, we can utilize the orbits of the group action of said isometry group to define a notion of rest with respect to the orbits. I don't recall me nor Pervect claiming absolute rest. Do you agree with the above however?

Certainly things can be at rest wrt to other things in curved spacetime except of course when a spacetime is non-stationary. I also see no issues with considering things at rest wrt certain coordinate values, for instance a Schwarzschild radius or a shell with a given r-value.