ST Does Spacetime Actually Curve?

[ginarific]

Simply put, does space actually curve or is it only the perception of space that curves? I've always thought space actually curved, but I'm not sure if this question is even valid since empty space isn't tangible in any real way.

(This is in reference to both length contraction and the curvature of spacetime due to large masses.)

 

[elfmotat]

Define "actually."

GR is a geometrical theory which predicts that particles will follow geodesics through a curved spacetime, and the curvature is determined by the stress-energy tensor. As far as we can tell, the predictions of GR are accurate.

Is this what you mean? Does the statement that "the predictions of a curved-spacetime theory are accurate" satisfy you?

 

[Tea Jay]

If you call it space time, instead of space, it works better...and, the curve is related to gravity on the concave sides.

 

[ginarific]

Right, this is what I meant by "I'm not entirely sure if my question is valid".

With the curvature of spacetime, spacetime is "curved" due to the effect of gravity. However, what about length contraction for an object moving close to c? The "space" or the "amount of absence" between points A and B doesn't actually change, only the object's perception of it changes.

I guess I'm trying to reconcile those two ideas, because in one, it seems that space really does curve (for all particles) while for the other, it is only an apparent change.

 

[netheril96]

Right, this is what I meant by "I'm not entirely sure if my question is valid".

With the curvature of spacetime, spacetime is "curved" due to the effect of gravity. However, what about length contraction for an object moving close to c? The "space" or the "amount of absence" between points A and B doesn't actually change, only the object's perception of it changes.

I guess I'm trying to reconcile those two ideas, because in one, it seems that space really does curve (for all particles) while for the other, it is only an apparent change.

"Space" and "perception of space" is the same thing. In GR only spacetime is an independent and absolute entity, whereas space and time depend on the observer.

 

[maverick_starstrider]

Right, this is what I meant by "I'm not entirely sure if my question is valid".

With the curvature of spacetime, spacetime is "curved" due to the effect of gravity. However, what about length contraction for an object moving close to c? The "space" or the "amount of absence" between points A and B doesn't actually change, only the object's perception of it changes.

I guess I'm trying to reconcile those two ideas, because in one, it seems that space really does curve (for all particles) while for the other, it is only an apparent change.

The amount of contraction depends on YOUR relative velocity remember. It's not intrinsically shrinking. However, I'm afraid I'm a little fuzzy as to what you mean by "actually" curves. At the most general level what space-time curvature MEANS is that the shortest distance between two points (i.e. event A occurs at place x_A,y_A,z_A and a time t_A and event B occurs at x_B,y_B,z_B, and a time t_b) is no longer a straight line through space time. If you can imagine space-time as a discrete grid (which it's not), like the pixels (or voxels) of a computer image then space curvature means that the actual size of the voxels closer to a source of energy/mass are different than those further away. Thus the path of minimal length through my discrete spacetime is no longer so simple. However, from your perspective you're simply going in a straight line and have no idea you're being contracted.

 

[dm4b]

You may have picked the wrong forum to ask this question. I tried in two separate posts and my questions were either avoided with semantical arguments, or I was told I was thinking about it wrong.

Interestingly enough, about 1 month later I was reading a Roger Penrose text where he said exactly what I was thinking about this. So, at least he agrees with me.

Point is, gravity does have distinct differences when compared to the other three forces, and these differences mainfest, in general, as something we call spacetime curvature. Indeed, that is the interpretational description given to the mathematics involved.

Now, if you're asking does spacetime curve like if you took a piece of paper and started bending it or rolling it up. The answer is no. This is not what GR addresses. This is called extrinsic curvature and addresses a surace being bent into and embedded within a higher dimensional space.

Now, if you started stretching the piece of paper and mathematically described that without any reference to a higher dimensional space, this would be an example of intrinsic curvature. This is what general relativity addresses.

Now, is this what spacetime REALLY does? I don't know, and I don't think anybody truly knows. I doubt it. Empty space is clearly different than a piece of paper. In addition, time is also involved, as it is spacetime that "curves" as a whole.

But, this curvature does effect the path of photons, and the perihelion of mercury because the field couples to itself, frame dragging and all the rest. There is a list of "properties" the other three forces don't exhibit.

So, I guess you can call this "curvature" and say spacetime really does "curve". But, getting a complete mental picture of what is really going on I think is something that still eludes us.

 

[bapowell]

You may have picked the wrong forum to ask this question. I tried in two separate posts and my questions were either avoided with semantical arguments, or I was told I was thinking about it wrong.

GR describes gravitational dynamics as manifested through the geometric properties of the spacetime manifold. We can perceive the gravitational dynamics through observations (photons following curved paths, orbits of celestial bodies). If the gravity is "real", then so must be the geometry. Unless you want to start getting into whether or not what observers perceive actually conveys the reality of the thing being perceived and so on. But then that's a philosophical discussion, not a physical one. That's probably why the good people on this forum were unable to help you.

 

[PeterDonis]

Interestingly enough, about 1 month later I was reading a Roger Penrose text where he said exactly what I was thinking about this. So, at least he agrees with me.

Do you have a reference?

Point is, gravity does have distinct differences when compared to the other three forces,

I assume you mean the fact that all objects "fall with the same acceleration" in a gravitational field, or, equivalently, that inertial mass and gravitational mass are exactly equal. This is the aspect that makes the physics of gravity amenable to an interpretation as spacetime curvature.

and these differences mainfest, in general, as something we call spacetime curvature. Indeed, that is the interpretational description given to the mathematics involved.

Yes, an "interpretational description". It is not the only possible interpretational description. That's why it's not really possible to answer the OP's question. If there is more than one interpretation of the same physics, asking which interpretation is "real" is not going to get a clear answer.

Also, you *can* interpret the other forces as curvature, if you are willing to expand the "space" that you are interpreting as being curved. See below.

Now, if you're asking does spacetime curve like if you took a piece of paper and started bending it or rolling it up. The answer is no. This is not what GR addresses. This is called extrinsic curvature and addresses a surace being bent into and embedded within a higher dimensional space.

Actually, GR does address this; you can talk about the extrinsic curvature of a particular spatial "slice" taken out of a spacetime, as embedded in the spacetime as a whole. Different slices can have different extrinsic curvatures, even though the spacetime as a whole has the same intrinsic curvature.

But, this curvature does effect the path of photons, and the perihelion of mercury because the field couples to itself, frame dragging and all the rest. There is a list of "properties" the other three forces don't exhibit.

Other than the equality of inertial and gravitational mass, what properties do you think the other forces don't exhibit that gravity does? Other fields can couple to themselves: gluons do directly (there are interactions involving 3 gluons and no other particles), and photons and W/Z bosons do indirectly (e.g., photons can scatter off other photons by forming virtual electron-positron pairs). "Frame dragging" has analogues in magnetism (in fact one name for it is "gravitomagnetism").

Also, as I said above, the other forces can also be interpreted as curvature, if you just add some dimensions in addition to the usual four of spacetime. The simplest example is electromagnetism: it can be interpreted as curvature in a fifth dimension that is something like having a tiny circle attached to each point of spacetime. Make the circle a multi-dimensional manifold and you can include the other forces (weak and strong) as well.

 

[Antiphon]

Space really does curve like this.

If there's no gravity someplace and you make a large triangle, you can add up the angles in space and they'll add up to 180 degrees.

If you do this someplace with gravity present you may get another angle. This will happen even though all three sides appear perfectly strait.

Edit: note that the presence of an electromagnetic field does *not* create this effect even though the forces are astronomically larger.

 

[bapowell]

Also, as I said above, the other forces can also be interpreted as curvature, if you just add some dimensions in addition to the usual four of spacetime. The simplest example is electromagnetism: it can be interpreted as curvature in a fifth dimension that is something like having a tiny circle attached to each point of spacetime. Make the circle a multi-dimensional manifold and you can include the other forces (weak and strong) as well.

Looks like you're referring to Kaluza-Klein here. While a stroke of genius, it doesn't hold up to experiment due to the introduction of an additional scalar degree of freedom associated with the compact 5th dimension that isn't observed in nature. The other gauge forces -- electromagnetism, weak, and strong nuclear -- can be interpreted as curvatures, but not of ordinary space. Instead, the forces correspond to curvatures in the abstract symmetry space associated with each force: U(1) for EM, SU(2) for weak, and SU(3) for the strong force. This is rather far afield from the OP; I mention it here only for completeness.

 

[maverick_starstrider]

Space really does curve like this.

If there's no gravity someplace and you make a large triangle, you can add up the angles in space and they'll add up to 180 degrees.

If you do this someplace with gravity present you may get another angle. This will happen even though all three sides appear perfectly strait.

Edit: note that the presence of an electromagnetic field does *not* create this effect even though the forces are astronomically larger.

Electromagnetic fields don't curve spacetime now? I don't know if it's detectable but they carry energy.

 

[Antiphon]

Electromagnetic fields don't curve spacetime now? I don't know if it's detectable but they carry energy.

They gravitate by their energy. But as the previous post mentioned, it doesn't work out to interpret the EM forces as a space time curvature.

 

[dm4b]

Do you have a reference?

Well, Penrose is far from the only guy to point out the uniqueness of gravity (as I was originally trying to do) so I'm sure you can find tons of references (including any standard GR textbook that talks about gravity not being a force, but rather as a feature of space-time, which also points out its uniqueness in this regard!)

However, one of the more delightful to read descriptions is in Penrose's book Shadows of the Mind, Section 4.4, where he talks about causality and light-cone tilting, something that becomes very evident in highly “curved” space-times. I'll quote some of it here that elucidate this point, but the entire section is a good read.

“The reason for this is that gravity actually influences the causal relationships between space-time events, and it is the only physical quantity that has this effect. Another way of phrasing this is that gravity has the unique capacity to 'tilt' light cones. No physical field other than gravity can tilt light cones, nor can any collection whatever of non-gravitational physical influences”

….

“The foregoing remarks illustrate the fact that “tilting' of light cones, i.e. the distortion of causality, due to gravity, is not only a subtle phenomenon, but a real phenomenon …. Nothing known in physics other than gravity can tilt the light cones, so gravity is something that is simply different from all other known forces and physical influences, in this very basic respect”

(Italics: Penrose emphasis; Bold: my emphasis)
----------------------------------

As far as extrinsic vs intrinsic curvature, sure you can use the math to talk about extrinsic curvature, but to quote Wald:

“Our space-time manifold M with space time metric g_uv is not naturally embedded (at least as far as we know) in a higher dimensional space. Thus, we wish to develop an intrinsic notion of curvature ... ”

So, yeah extrinsic curvature can be “pulled out” of the the math. But, does describing gravity utilizing General Relativity require it? Does General Relativity predict that our Universe is embedded in a higher dimensional space? No. This is why I said it does not address it. (i.e there are no useful testable hypotheses). On the other hand, the notion of intrinsic curvature, in light of GR, does lead to solid testable descriptions of certain phenomenon that, if you wish, could be said to be caused by space-time “curvature”.

You can say a lot with math. If I don't mind putting the Earth at the center of the solar system, I can describe the planetary motions with epicycles, but this does not represent reality. Is this the same with adding extra dimensions to re-interpret the other three forces as curvature. Is it useful and are there any known tests to exhibit the reality of the idea in question? Maybe there is. I'm unfamilar with that idea. If there is not, I'm not sure it's pertinent to the question at hand?

The OP is titled “does space actually curve”. I think GR, as pointed out by Penrose, has plenty of evidence to support the idea that there is a uniqueness to the reality of gravity, part of which can be called space-time curvature.

Good luck with visualizing it though ;-)

 

[dm4b]

Unless you want to start getting into whether or not what observers perceive actually conveys the reality of the thing being perceived and so on. But then that's a philosophical discussion, not a physical one. That's probably why the good people on this forum were unable to help you.

Hi bapowell,

I wasn't trying to get into that. In my last post, I mention that section within Penrose's book, which I quote a very minimal amount of. What's contained within that part of the book, is part of what I was getting at.

I know different people have different ideas on where physics ends and philosophy starts. In my opinion, this all still falls well within the bounds of physics.

 

[PeterDonis]

Looks like you're referring to Kaluza-Klein here.

Not quite. That was an early effort along the same lines, but as you say, it didn't hold up to experiment. I was referring to what you talk about next:

The other gauge forces -- electromagnetism, weak, and strong nuclear -- can be interpreted as curvatures, but not of ordinary space. Instead, the forces correspond to curvatures in the abstract symmetry space associated with each force: U(1) for EM, SU(2) for weak, and SU(3) for the strong force.

I agree this isn't the same as "space" itself curving, but I thought it was worth mentioning that the general idea of "curvature" is not limited to gravity.

 

[PeterDonis]

“The reason for this is that gravity actually influences the causal relationships between space-time events, and it is the only physical quantity that has this effect. Another way of phrasing this is that gravity has the unique capacity to 'tilt' light cones. No physical field other than gravity can tilt light cones, nor can any collection whatever of non-gravitational physical influences”

Yes, this qualifies as a property that gravity has that no other interaction has. And I'm pretty sure that the presence of tilting of the light cones is equivalent to the presence of tidal gravity, which means that tilting of the light cones can be considered as equivalent to spacetime curvature. If that qualifies as spacetime curvature being "real", then yes, spacetime curvature is real. But it still depends on interpreting the tilting of the light cones as evidence of spacetime curvature, instead of, say, evidence of a "physical field" that can tilt light cones; notice that this latter term is the one Penrose uses.

 

[Passionflower]

It is often said that the rubber sheet analogy is not very good to demonstrate gravity however I personally find the 'tilting of light cones' analogy far worse.

 

[Naty1]

It's useful to remember that two things affect "spacetime curvature": relative velocity [of an observer] and gravitational potential. But only the latter type of curvature affects gravity. If relative velocity really affected gravity, we could observe a fast particle turn into a black hole. [That's a separate topic of several very detailed discussions in these forums.]


Here is how Drgreg of these forums [uniquely] explained a way to visualize 'spacetime curvature', quoted from a post of several years ago:

"... let's restrict our attention to 2D spacetime, i.e. 1 space dimension and 1 time dimension, i.e. motion along a straight line. …
In the absence of gravitation, an inertial frame corresponds to a flat sheet of graph paper with a square grid. If we switch to a different inertial frame we "rotate" to a different square grid, but it is the same flat sheet of paper. (The words "rotation" and "square" here are relative to the Minkowski geometry of spacetime, which doesn't look quite like rotation to our Euclidean eyes, but nevertheless it preserves the Minkowski equivalents of "length" (spacetime interval) and "angle" (rapidity).)

If we switch to a non-inertial frame [an accelerated observer] but still in the absence of gravitation), we are now drawing a curved grid, but still on the same flat sheet of paper. Thus, relative to a non-inertial observer, an inertial object seems to follow a curved trajectory through spacetime, but this is due to the curvature of the grid lines, not the curvature of the paper which is still flat.

When we introduce gravitation, the paper itself becomes curved. (I am talking now of the sort of curvature that cannot be "flattened" without distortion. The curvature of a cylinder or cone doesn't count as "curvature" in this sense.) Now we find that it is impossible to draw a square grid to cover the whole of the curved surface. The best we can do is draw a grid that is approximately square over a small region, but which is forced to either curve or stretch or squash at larger distances. This grid defines a local inertial frame, where it is square, but that same frame cannot be inertial across the whole of spacetime.

So, to summarize, "spacetime curvature" refers to the curvature of the graph paper, regardless of observer, whereas visible curvature in space is related to the distorted, non-square grid lines drawn on the graph paper, and depends on the choice of observer..."

 

[PeterDonis]

It's useful to remember that two things affect "spacetime curvature": relative velocity [of an observer] and gravitational potential. But only the latter type of curvature affects gravity. If relative velocity really affected gravity, we could observe a fast particle turn into a black hole. [That's a separate topic of several very detailed discussions in these forums.]

I don't think this is the standard usage of the term "spacetime curvature". In the standard terminology of GR, spacetime curvature is tidal gravity; it isn't affected by relative velocity and it isn't the same as gravitational potential. In the terminology of DrGreg's post that you quoted, the intrinsic curvature of the sheet of paper, which makes the grid curve or stretch or squash, is measured by us as tidal gravity.

 

[Passionflower]

I don't think this is the standard usage of the term "spacetime curvature". In the standard terminology of GR, spacetime curvature is tidal gravity; it isn't affected by relative velocity and it isn't the same as gravitational potential. In the terminology of DrGreg's post that you quoted, the intrinsic curvature of the sheet of paper, which makes the grid curve or stretch or squash, is measured by us as tidal gravity.

Well, I think yes and no :)

Name me one solution for which there exists inertial acceleration and no tidal effects that satisfies the EFEs.

If there is none then, modus tollens, when we have solution for which there exists inertial acceleration there must be tidal effects and thus spacetime curvature. :)

 

[Naty1]

I don't know the detailed mathematics well enough to have an opinion so I'd be happy to utilize a consensus opinion, if one can be reached.

oops, maybe 'that ain't going to happen.'
I found this quote in my notes from pervect. MTW :

… nowhere has a precise definition of the term “gravitational field” been
given --- nor will one be given. Many different mathematical entities are
associated with gravitation; the metric, the Riemann curvature tensor, the
curvature scalar … Each of these plays an important role in gravitation
theory, and none is so much more central than the others that it deserves the name “gravitational field.”

here
https://www.physicsforums.com/showthread.php?t=156168# Post # 12

 

[Naty1]

Back to the OP:

I'm not sure if this question is even valid since empty space isn't tangible in any real way.

It is a valid question...but I would not go down the 'tangible' nor 'real' road; 'tangible' to begin with has no scientific meaning. Is entropy any more tangible?? Or time? it's really an irrelevant issue. One way to think about such things is are they measureable...actually or in theory...that helps pin stuff down.

 

[PeterDonis]

Name me one solution for which there exists inertial acceleration and no tidal effects that satisfies the EFEs.

I think if you try to come up with a precise definition of "inertial acceleration", it will turn out to be equivalent to tidal gravity, since the only way you can really tell that "inertial acceleration" is present is by its spatial variation. (If the "acceleration due to gravity" were the same everywhere, how would you measure it?)

 

[Naty1]

The thread I posted above, from 2007, discusses some aspects of PeterDonis' perspective.

Good points above...I really like the curvature and tilting of light cones..

 

[lugita15]

The other gauge forces -- electromagnetism, weak, and strong nuclear -- can be interpreted as curvatures, but not of ordinary space. Instead, the forces correspond to curvatures in the abstract symmetry space associated with each force: U(1) for EM, SU(2) for weak, and SU(3) for the strong force. This is rather far afield from the OP; I mention it here only for completeness.

Where I can find more information on this? It seems really interesting. Can gravity be thought of as the curvature of the Lie group of symmetries under diffeomorphisms?

 

[bapowell]

Where I can find more information on this?

Peskin and Schroeder has a section on the geometry of gauge invariance, as does Cheng and Li.

It seems really interesting. Can gravity be thought of as the curvature of the Lie group of symmetries under diffeomorphisms?

Gravity is thought of as the curvature of real space (on which the diffeo group acts), not as a curvature of the group space itself. I am not an expert on gauging gravity, but my understanding is that yes, the group of diffeomorphisms is the symmetry group of gravity. The affine connection in GR that facilitates parallel transport along geodesics in real space is analogous to the vector potential in electromagnetism, which serves the same function in U(1) space. Following the program outlined in the above texts, one finds that it is possible to derive gauge field analogs of the main formal geometric structures found in ordinary differential geometry. This is a beautiful result. Ultimately, the mathematics of gauge theories is rooted in the study of fiber bundles on manifolds. Studying these theories from this vantage point is exciting because you get to see in an clear way how gauge interactions and classical GR arise within the same mathematical framework.

 

[lugita15]

Peskin and Schroeder has a section on the geometry of gauge invariance, as does Cheng and Li. Gravity is thought of as the curvature of real space (on which the diffeo group acts), not as a curvature of the group space itself. I am not an expert on gauging gravity, but my understanding is that yes, the group of diffeomorphisms is the symmetry group of gravity. The affine connection in GR that facilitates parallel transport along geodesics in real space is analogous to the vector potential in electromagnetism, which serves the same function in U(1) space. Following the program outlined in the above texts, one finds that it is possible to derive gauge field analogs of the main formal geometric structures found in ordinary differential geometry. This is a beautiful result. Ultimately, the mathematics of gauge theories is rooted in the study of fiber bundles on manifolds. Studying these theories from this vantage point is exciting because you get to see in an clear way how gauge interactions and classical GR arise within the same mathematical framework.

I was thinking more about quantum gravity, where we have a spin-2 gauge-bosonic field, and you have diffeomorphism invariance as a gauge symmetry. Then would gravity be representable as curvature of the diffeomorphism group?

 

[bapowell]

I was thinking more about quantum gravity, where we have a spin-2 gauge-bosonic field, and you have diffeomorphism invariance as a gauge symmetry. Then would gravity be representable as curvature of the diffeomorphism group?

I'm not sure that the distinction between quantum and classical gravity changes things. As I said, I'm not an expert on understanding gravity as a gauge theory (and hopefully someone else can chime in); however, I don't think it's correct to think of gravity as the curvature of the Lie group itself. In gauge theories, the field strength tensor describes the curvature of the connection -- the space of forms that transform under the adjoint representation of the gauge group. So it's the curvature of the space of gauge fields, not of the Lie group manifold itself.

 

[Islam Hassan]

Spacetime curvature was a very elegant (and accurate: in predicting other physical observations than itself) proposition until 2004 with the launch of Gravity Probe B:

http://en.wikipedia.org/wiki/Gravity_Probe_B

This experiment finally provided the physical proof that spacetime does indeed curve via measurement of spacetime curvature's geodetic and frame-dragging effects in the vicinity of the earth.

IH