Curvatures in space-time: actual reality or mathematical concept?

[tris_d]

As far as I know, experiment cannot distinguish such things. Note that also "flat space" is a mathematical term; it has no meaning that I know of in a physical sense (how can empty space be literally "flat"?; that's like saying that it is "green"!).

I was using term used in cosmology, by flat they mean 3D Euclidean space. They think they could measure the curvature of the universe by examining some differences in temperature of background radiation. I say extrinsic curvature would be impossible to detect in any case, we would need to step into higher dimension, but in our realm all those extrinsic curves would be straight lines. So I have no idea what in the world did they hope to measure, but they did measure it, and it turns out the curvature is zero. Heh.

 

[harrylin]

I was using term used in cosmology, by flat they mean 3D Euclidean space. They think they could measure the curvature of the universe by examining some differences in temperature of background radiation. [..]

Right - and what "they" mean with "3D Euclidean space" is described in the link that I gave; it's a measurement relationship that depends on how an operator handles and defines his measurement tools. In particular the "space" (more precisely, Einstein calls it "space data" and "space co-ordinates") of a rotating disc is "non-Euclidean". Perhaps that is not exactly what you mean with that expression. :tongue2:

 

[PeterDonis]

Is that about charge can be both positive and negative while gravity only attracts?

No, it's about what I said: gravity obeys the equivalence principle, electromagnetism does not. Put another way, all objects have the same ratio of inertial to gravitational mass; but different objects have different ratios of charge to mass.

And while we at it, is there any reason why we could not propose electric or magnetic fields are perhaps some curvatures in space as well?

Not in ordinary spacetime, no. But you can interpret the math of EM fields, and of the weak and strong interactions as well, as describing curvature in an abstract space. That probably deserves a separate thread, though, and not in this forum (the Quantum Physics forum would probably be a better place).

First time I hear about. Tell me everything. What's the problem for people to accept?

I personally don't have a problem accepting it, so I'm the wrong person to ask.

That confuses me. Is space in GM considered "empty" or "physical"?

Yes. It's empty because it has zero stress-energy. It's physical for the reasons given in my response to Q-reeus about whether spacetime is a "physical medium".

(Btw, you keep saying "space" instead of "spacetime". They're not the same thing; see my comment at the end of this post about what is and is not flat.)

Are you saying there is no way to explain that with flat space?

As Q-reeus pointed out, you can view the quasi-static part of the spacetime curvature as being due to a field on an unobservable flat spacetime background as well. But as I pointed out, if you adopt that interpretation, the quasi-static part of the field does not propagate; it's just there.

At least the whole universe needs not to be curved for that. Current data suggests we live in infinite flat universe.

It suggests that we live in an infinite *spatially* flat universe. That does *not* mean the *spacetime* of the universe is flat. It isn't; it's curved, but the curvature (in the standard FRW coordinates in which the spatial slices are flat) is all in the time dimension.

 

[tris_d]

I don't see the problem there. For our purposes, the curvature of a pseudo riemannian 4 - manifold is well - defined and computed on local coordinate charts. The EFEs then give the relation between the curvature and the stress - energy tensor.

Are these curvatures around some mass radial or circular? Could you say where some curve starts, where it ends, where it curves and how much it curves? And what happens to space, does it get stretched and squashed?

 

[Q-reeus]

Q-reeus: "Well if it is just mathematical abstraction in GR then GR is severely self-inconsistent since it's eq'ns predict gravitational waves as purely a disturbance of spacetime curvature"

I am labeling this "DS2Qr(1)": Do you have any solid evidence for this supposed self-inconsistency, or is this just another of your unsubstantiated assertions? By solid evidence I mean either a rigorous derivation according to the accepted mathematical rules of GR that leads to a self-contradiction, or a mainstream scientific reference describing the inconsistency you allege. A non-rigorous description of a scenario that you are unable to analyze does not qualify as evidence.

I suspect not and await your usual response, perhaps we can label it "Qr2DS(1)" and save ourselves a lot of typing in the future.

I'll save you even more typing - just read again that above passage you quoted - bold emphasis added. Read it in context of post I was addressing. There is nothing to deal with here - you are barking up the wrong tree.

I can also provide DS2Qr(2): That is not solid evidence, it is up to you to provide evidence to support your claim, not up to me to provide evidence against it. Please come back when you actually have some evidence.

Have no idea what all this 'DS2Qr(2)' code is supposed to mean. And btw, it would pay to tread a bit lightly rather than being aggressive. You should know that sort of thing can come home to roost. Let you off very lightly here Fact is if you were honest there would have been a simple admission in #15 you got it plain wrong in #5.

 

[Q-reeus]

I think you're either misremembering or misinterpreting--and to be fair, these issues are not easy to disentangle given all that has been said about them by various experts in the field. To briefly restate the key points from earlier discussions:
(1) Neither "static" nor "propagating" curvature contributes to the SET; *no* aspect of gravity contributes to the SET. (I've always been consistent about this. Also, I should note that I am assuming a zero cosmological constant; including that would just get us into a further argument about whether the cosmological constant should be viewed as an "aspect of gravity" or not, depending on which side of the EFE we choose to put it on. I don't think we need to go there.)

Thanks for that clarification which I will strive to remember clearly - unlike obviously faulty recollection of previous occasion.

(2) If one really, really wants to salvage some kind of "contribution of gravity to total energy", one can define various pseudo-tensors that allow one to do that. However, they are all pseudo-tensors, meaning they are dependent on how you slice up spacetime into space and time, and they can only be consistently defined in certain kinds of spacetimes. None of this changes #1 above.

(3) One can also, in certain kinds of spacetimes, define "total energy at a given time" in different ways, which can sometimes give different results. In the case of a spacetime containing gravitational waves, such as a binary pulsar spacetime, one can show that the ADM energy and the Bondi energy are different, and the difference between them is standardly viewed as "energy carried away by gravitational waves". But this energy cannot be localized; it can only be "seen" when you look at the global properties of the spacetime. None of this changes #1 above either.

Got the picture. Which imo is still a big problem for GR conceptually. What is the bedrock justification for such conditionally allowing an energy content but never a gravitating energy content to gravitational field? There is afaik no argument that say field 'pressure' can cancel field energy. So what gives here? Equivalence principle makes it all too problematic to define field energy - so hang it, just excise it altogether. Null results for Lunar etc. Nordtvedt effect imo suggests strongly there has been an unjustified sweeping under the rug.

Then in GR, curvature is "curvature of an actual physical medium" by your definition, and *not* just "curvature of spatial and temporal relationships", since GR predicts that gravitational waves can exist. What's the problem? (Also see further comments at the end of this post on spacetime as a "physically stressed medium".)

See above comments.

AFAIK there isn't any such alternate formulation that works for cosmology. But there is a lot of literature out there that I have not read, so I may be missing something. Any relevant links would be welcome.

Following article has been mentioned here at PF before: http://xxx.lanl.gov/abs/gr-qc/9912003 ,there are various others from same author there at arXiv. A recent one by what may be termed a more mainstream theoretician is http://arxiv.org/abs/1209.3511. Suggest a Google search if one wants to chase up more.

To me, curved spacetime *is* a "physically stressed medium" that can transport energy. But looking at it that way dives below the level that GR addresses; it involves trying to come up with a more fundamental theory that underlies GR, such as string theory or loop quantum gravity. AFAIK it is generally accepted that GR is not a "fundamental" theory in this sense; it is a low-energy approximation to some other more fundamental theory. So I don't expect GR to tell me *how* curved spacetime can be a physically stressed medium; I need the more fundamental theory to do that. Unfortunately we don't have any way to probe the structure of spacetime at a small enough distance scale to investigate this sort of thing experimentally.

Right, but to me this is an admission there is this conceptual shortfall in GR. As I wrote in earlier post - ether banished in SR but seems to return in GR as some kind of elephant in the living room.

 

[tris_d]

Curvature is here a mathematical characteristic of the "world of events" that is called space-time.

Compare: http://www.bartleby.com/173/17.html
http://en.wikipedia.org/wiki/Spacetime (see the intro + "Spacetime in general relativity")

As a matter of fact, Einstein and others also used the expression "gravitational field". But "field" is a very vague, abstract concept; its original meaning is merely "area" or "zone".

Is there any diagram depicting these geodesics in 3D?

 

[Dale]

I'll save you even more typing - just read again that above passage you quoted - bold emphasis added.

OK, so to be clear you are not claiming that an inconsistency exists in GR?

Have no idea what all this 'DS2Qr(2)' code is supposed to mean. And btw, it would pay to tread a bit lightly rather than being aggressive. You should know that sort of thing can come home to roost. Let you off very lightly here Fact is if you were honest there would have been a simple admission in #15 you got it plain wrong in #5.

DS2Qr(2) is meant to be short hand for DaleSpam to Q-reeus standard response number 2.

I didn't get #5 wrong, I addressed what I saw as the content of the question, and you focused on the obvious fact that it was not directly related to the context of the question. I said exactly what I meant and there was nothing incorrect in what I said. It certainly is possible that I misunderstood phinds' intent, but at the most that would have made #5 irrelevant, not wrong.

There wasn't any dishonesty, and whatever you may say about my "aggressiveness", you are the one went immediately to casting character aspersions here by calling me dishonest.

 

[WannabeNewton]

Are these curvatures around some mass radial or circular? Could you say where some curve starts, where it ends, where it curves and how much it curves? And what happens to space, does it get stretched and squashed?

The curvature is determined by the mass - energy distribution. It doesn't have to be a fixed spherical mass; we can, for example, have dust. Once you know what the metric tensor is you can compute the christoffel symbols and, in principle, solve the geodesic equation for the geodesics on the manifold, given initial conditions, which will of course determine the future behavior of the geodesics assuming the solutions are well behaved. You can also use geodesics to describe the curvature at points on a riemannian manifold; this is called sectional curvature if you want to look that up. I'm not sure how to characterize "stretched" or "squashed".

 

[PeterDonis]

What is the bedrock justification for such conditionally allowing an energy content but never a gravitating energy content to gravitational field? There is afaik no argument that say field 'pressure' can cancel field energy.

You've got things backwards. None of the things you are talking about are part of the basic conceptual foundation of GR. As I've said many times, that is the EFE. The EFE says unequivocally that, as my #1 said, gravity does not appear in the SET. *That* is the only "bedrock justification" you need. GR simply does not treat the stuff you are talking about as fundamental; the questions you are asking are questions about a particular conceptual scheme you want to overlay on GR, not about GR itself. See next comment.

So what gives here? Equivalence principle makes it all too problematic to define field energy - so hang it, just excise it altogether.

GR does not say this. GR says that there is no *need* to "define field energy" at all. You can explain all gravitational phenomena without it; just use the EFE, in which "field energy" does not even appear. All this worrying about "field energy" is *your* problem, not GR's.

Nordtvedt effect imo suggests strongly there has been an unjustified sweeping under the rug.

The best evidence we have indicates that there is no Nordvedt effect, so I don't understand why you think an effect that doesn't exist shows some problem with GR, since GR predicts that it does not exist.

Following article has been mentioned here at PF before: http://xxx.lanl.gov/abs/gr-qc/9912003 ,there are various others from same author there at arXiv. A recent one by what may be termed a more mainstream theoretician is http://arxiv.org/abs/1209.3511.

I'll look at these in more detail when I have a chance, but on a quick skim they don't seem to cover cosmology at all; they only talk about asymptotically flat cases like the spacetime around a single gravitating body or a small system of bodies like the binary pulsar. This is the same limitation I've seen in all treatments of this subject: the assumption of a flat background spacetime (even if it's unobservable) forces the actual observed solution to be asymptotically flat.

Right, but to me this is an admission there is this conceptual shortfall in GR.

It's only a conceptual shortfall if you expect GR to be a complete fundamental theory. By that criterion, we have never had a scientific theory without a conceptual shortfall.

 

[Q-reeus]

Q-reeus: "One glaring problem for me is that gravitational field is allowed to have an ambiguously defined energy content but not allowed to act as it's own further source, despite the insistence that all other forms of stress-energy must contribute. Ask an expert why and good luck getting a sensible answer."
I am not aware of it. What theory, what equation are you referring to?

Sorry - forgot about your #33. Check out here Go down to the very first equation and then read the following few lines after that.

Q-reeus: "The fact of gravitational waves as per binary pulsar data imo screams out one of two things - curvature of an actual physical medium on geometrical formulation of gravity, or physical field propagation through flat spacetime on field formulation of gravity. But definitely not just curvature of spatial and temporal relationships."
I don't know about that either. What did we measure and what was the reading?

There are lots of articles on this but here's one.

 

[surajt88]

After I saw the Einstein Rings, I was convinced that spacetime curvature was "real".

 

[Q-reeus]

OK, so to be clear you are not claiming that an inconsistency exists in GR?

Of course I'm claiming an inconsistency exists - but not as per your misconstrued quotes!

DS2Qr(2) is meant to be code for DaleSpam to Q-reeus standard response number 2.

Stick to plain english - there is less typing in the end.

I didn't get #5 wrong, I addressed what I saw as the content of the question, and you focused on the obvious fact that it was not directly related to the context of the question. I said exactly what I meant and there was nothing incorrect in what I said. It certainly is possible that I misunderstood phinds' intent, but at the most that would have made #5 irrelevant, not wrong. There wasn't any dishonesty, and your insinuation that there was is typical of your own hypocritical aggressiveness.

You may choose to persist in that line, but the thread is there for all to read and judge. Your answer in #5 was quite misleading given the entirely clear context of what it was addressing and best to admit as such.

 

[Q-reeus]

Q-reeus: " What is the bedrock justification for such conditionally allowing an energy content but never a gravitating energy content to gravitational field? There is afaik no argument that say field 'pressure' can cancel field energy."

You've got things backwards. None of the things you are talking about are part of the basic conceptual foundation of GR. As I've said many times, that is the EFE. The EFE says unequivocally that, as my #1 said, gravity does not appear in the SET. *That* is the only "bedrock justification" you need. GR simply does not treat the stuff you are talking about as fundamental; the questions you are asking are questions about a particular conceptual scheme you want to overlay on GR, not about GR itself. See next comment.

I don't accept having gotten things backwards above at all. I asked for the conceptual justification. You simply stated the GR formal position without offering any explanation for why in GR gravity is excised from being SET source term.

Q-reeus: "So what gives here? Equivalence principle makes it all too problematic to define field energy - so hang it, just excise it altogether."

GR does not say this. GR says that there is no *need* to "define field energy" at all. You can explain all gravitational phenomena without it; just use the EFE, in which "field energy" does not even appear. All this worrying about "field energy" is *your* problem, not GR's.

Owing to typically extreme weakness of contribution, no seeming need to include up to present levels of observational test. I believe Nordtvedt results actually do undermine that position - more below on that.

Q-reeus: "Nordtvedt effect imo suggests strongly there has been an unjustified sweeping under the rug."
The best evidence we have indicates that there is no Nordvedt effect, so I don't understand why you think an effect that doesn't exist shows some problem with GR, since GR predicts that it does not exist.

Please - don't stoop to excising part of a quote that completely changes it's import. The bit you left out: "Null results for Lunar etc. Nordtvedt effect..."
It's precisely the null result(s) that imo strongly implies, if not outright demands, that gravity does gravitate: null results imply m_a = m_p = m_i, with all m's owing to gravitational binding energy which I trust you agree is inclusive of gravitational field itself.

 

[tris_d]

In any case, sticking to geometric concept, how do you explain energy-momentum transport in GW's without it directly implying a physically 'stressed' medium of some sort?

I have no idea what are you guys are talking about, but when you said "medium" I thought of what Lorenz commented about Einstein's theory.

- Lorentz on his side continued to use the aether concept. In his lectures of around 1911 he pointed out that what "the theory of relativity has to say ... can be carried out independently of what one thinks of the aether and the time". He commented that "whether there is an aether or not, electromagnetic fields certainly exist, and so also does the energy of the electrical oscillations" so that, "if we do not like the name of "aether", we must use another word as a peg to hang all these things upon."

 

[Dale]

Of course I'm claiming an inconsistency exists - but not as per your misconstrued quotes!

:rofl: OK.

Insert DaleSpam to Q-reeus standard response #1 (DS2Qr(1))

I eagerly await your solid evidence for whatever inconsistency you are claiming exists, and until such evidence is produced I will refrain from any hint of construing anything about the details of said inconsistency so as to avoid any possible future misconstruing on my part.

 

[harrylin]

Is there any diagram depicting these geodesics in 3D?

You seem to refer to a website that I cannot see (cafedots.com?) and on a topic that I did not discuss.

ADDENDUM: suddenly I see the picture now, I think that you corrected it; it is as "3D" as can be done on a 2D screen, and the word "geodesic" seems misplaced.

 

[Q-reeus]

:rofl: OK.

Insert DaleSpam to Q-reeus standard response #1 (DS2Qr(1))

I eagerly await your solid evidence for whatever inconsistency you are claiming exists, and until such evidence is produced I will refrain from any hint of construing anything about the details of said inconsistency so as to avoid any possible future misconstruing on my part.

Just read my #52. If you cannot figure from that what I consider to be an inconsistency in GR, just forget it as 'irrelevancy' if that helps.

 

[harrylin]

[..] And what happens to space, does it get stretched and squashed?

GR only discusses the metrical qualities of space, just as those of the rotating disk (did you read it? I doubt that you had time for that!)

According to the theory, if you bring a stick near the Earth and lay it on the ground then its length will be unaltered; but if you hold it up (not including effects from weight), the stick will be slightly shortened.

 

[tris_d]

You seem to refer to a website that I cannot see (cafedots.com?) and on a topic that I did not discuss.

Sorry, fixed it now. You gave me a link where there was a diagram of "two-dimensional analogy of spacetime distortion." Is there anything similar depicted in 3D?

 

[tris_d]

Sum of angles in a triangle! Greater than 180 degrees = +ve curvature, less than 180 degrees = -ve curvature. Note that this can be explained also using a non-geometric field theory = 'effective' curvature.

As I understand extrinsic curvatures, which is what I think you are talking about, would not be detectable in the realm itself, only from the higher dimensional space.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_variable/index.html

 

[harrylin]

Sorry, fixed it now. You gave me a link where there was a diagram of "two-dimensional analogy of spacetime distortion." Is there anything similar depicted in 3D?

I had next also amended my answer; and I don't know a 3D site. But it's not important.

[..] I thought of what Lorenz commented about Einstein's theory.

- Lorentz on his side continued to use the aether concept. In his lectures of around 1911 he pointed out that what "the theory of relativity has to say ... can be carried out independently of what one thinks of the aether and the time". He commented that "whether there is an aether or not, electromagnetic fields certainly exist, and so also does the energy of the electrical oscillations" so that, "if we do not like the name of "aether", we must use another word as a peg to hang all these things upon."

Right. That referred to what Einstein later renamed "special relativity".
Einstein admitted in 1920 (but also before and after) that general relativity suggests some kind of an ether that you might call "space". In earlier answers I tried to explain that such a "space" is not to be confused with the "space" component of "space-time"; the first "space" has a physical (or even metaphysical) meaning, and the second "space" has a geometrical/mathematical meaning.
- http://en.wikisource.org/wiki/Ether_and_the_Theory_of_Relativity

Note: if you are like me, then you may have to carefully read it at least three times in order to correctly understand it.

 

[PeterDonis]

As I understand extrinsic curvatures, which is what I think you are talking about

He's not. He's talking about intrinsic curvature, which can be detected by measurements made within the manifold, without any reference to an embedding in a higher-dimensional space.

 

[tris_d]

The point is, as several people here mentioned, that one can map phenomena to a flat background - which is just as much the fruit of our imagination as a curved one.

I always imagined 'space' as abstract thing, a ruler to help us make sense of things and put them in perspective. A "container" and mathematical construct against which we make measurements, a conceptual tool by which we relate and understand, not actual reality.

Fiddling with "space matrix" is like fiddling with numbers on your measuring tape. It's supposed to be constant, a reference. I think we need some linear and uniform 'space' to serve as underlying "reference grid", even if the space itself can indeed curve, stretch and whatever.

 

[tris_d]

He's not. He's talking about intrinsic curvature, which can be detected by measurements made within the manifold, without any reference to an embedding in a higher-dimensional space.

I'm not sure how it matter then. Can that help us distinguish whether some volume of space can actually be curved?

 

[harrylin]

I always imagined 'space' as abstract thing, a ruler to help us make sense of things and put them in perspective. A "container" and mathematical construct against which we make measurements, a conceptual tool by which we relate and understand, not actual reality.

Fiddling with "space matrix" is like fiddling with numbers on your measuring tape. It's supposed to be constant, a reference. I think we need some linear and uniform 'space' to serve as underlying "reference grid", even if the space itself can indeed curve, stretch and whatever.

Those two statement are hard to match I'm afraid: we can only use rulers which will be affected by gravitation as I explained in post #57. Consequently we do not have such a reference except in theory far away from heavy bodies. For such descriptions, be on the lookout for such qualifications as "non-local frame", "at a far distance", "observer in deep space" etc.

PS: you received a lot of food for thought that should keep you occupied for a day except if you are a supergenius without a job. Before discussing further, please go through it.

 

[PeterDonis]

According to the theory, if you bring a stick near the Earth and lay it on the ground then its length will be unaltered; but if you hold it up (not including effects from weight), the stick will be slightly shortened.

Shortened relative to what? There is a germ of something valid in what you're saying, but it needs to be described very carefully to avoid a number of common pitfalls.

For an example of one common pitfall, the one you appear to have fallen into, consider the following thought experiment: I take two sticks that, when floating freely far away from all gravitating bodies, are the same length, the same material composition, etc. I bring them both near the Earth. I place one stick horizontally and one stick vertically. After correcting for the effects of weight, both sticks will be the *same* length; the vertical one will *not* be shorter than the horizontal one.

What *will* be true, in this thought experiment, is the following: if we measure the circumference of a circle at the radius (from the center of the Earth) the horizontal stick, which we'll assume is also the radius from the center of the Earth of the bottom of the vertical stick, we will find that that circumference is some number times the measured length of the stick (either one since they're both the same measured length). If we then measure the circumference of a circle at the radius from the center of the Earth of the top of the vertical stick, we would expect the following relationship to hold between the two circumferences and the stick length $L$:

$$L = \frac{C_{top}}{2 \pi} - \frac{C_{bottom}}{2 \pi}$$

However, we will find that the above relationship does *not* hold; in fact, what we will find is this:

$$L > \frac{C_{top}}{2 \pi} - \frac{C_{bottom}}{2 \pi}$$

In other words, there is "more distance" in between the two circles than would be expected from the formulas of Euclidean geometry. But this is a global property of the spacetime (more precisely, of the particular set of spatial slices we have cut out of the spacetime, the ones that are slices of "constant time" to static observers). It is not something you can observe by comparing horizontal and vertical measurements of otherwise identical objects.

 

[tris_d]

Curvature is definitely measurable. It is tidal effects. So I guess that curvature is "actual" by this definition but not "actual reality" by the previous definition.

I'm interested. Can you tell me what did we measure and what was the reading?

 

[PeterDonis]

I'm not sure how it matter then. Can that help us distinguish whether some volume of space can actually be curved?

Yes. That's the point I was making when I said that intrinsic curvature can be detected by measurements made purely within the manifold.

 

[Dale]

Just read my #52. If you cannot figure from that what I consider to be an inconsistency in GR, just forget it as 'irrelevancy' if that helps.

Insert DaleSpam to Q-reeus standard response #2 (DS2Qr(2))

I certainly agree with the classification as "irrelevancy". If you ever do get some good evidence supporting your claim then I will be glad to reclassify and discuss.

 

[harrylin]

Shortened relative to what? There is a germ of something valid in what you're saying, but it needs to be described very carefully to avoid a number of common pitfalls.

For an example of one common pitfall, the one you appear to have fallen into,

Shortened according to a far away reference, as I indicated (I could have been more specific but it was aimed at tris who surely understood it); I took it from Einstein's 1916 paper who discussed the effect from the gravitational field. Do you think that he fell in a trap, or that there was "a germ of something valid in what he was saying" about GR?

PS: I explained the metric meaning to tris with my reference to Einstein's explanation based on the rotating disk. I hope that he will read it. And as you verbally contradict Einstein, I'll cite him to avoid misunderstanding:

"The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational field, when we place it radially in the field."

 

[pervect]

I think "curvatures" can not really exist in some 3D spatial volume as actual geometrical feature without reference frame against which it would be curved against. Actual curves would require actual reference frame and "empty space" contains nothing, so for it to contain some actual topology seem to be direct contradiction.

If you can explain some phenomena by supposing space is actually curved that I can not explain with the concept of potentials, gradients or fields, then I will submit space is actually curved as the best explanation.

This is a bit demanding. Let's take an example. Would you agree that the surface of the Earth is curved?

If you could come up with some combination of "fields and gradients and whatnot" to allow you to navigate on it successfully, would you suddenly declare that the Earth's surface was "not curved", and join the Flat Earth society?

I think it's probably rather more productive to focus on the math - which you didn't ask about, deciding to ask about the more phiilosohpihcal stuff first (which tends to lead to long, rambling, endless discssions ...

As far as the math goes, you'll need the concept of smooth curves connecting points. This may actually be the hardest part of the math, but it's easy enough to accept without getting into the precise defiitions of manifolds, topological spaces, and all that - especially if you're not too fussy.

You'll also need the concept of length, specifically the length of one of these curves.

Once you have those two down, that's really all you need to define curvature. You don't need "reference frames" and you don't need to worry about whether empty space is empty, half-empty, full, has polka-dots, or whatever. It's all irrelevant to what you do need.

Given this much, you can define a set of special curves, which are the shortest curves connecting two points in space. These are the geodesics (or rather a subset of them).

And you can follow in the steps of Einstein, and start to draw little quadrilaterals from these geodesics - and make them square, by making all four sides equal, and the diagonals equal as well.

Einstein's discussion of curvature is online at http://www.bartleby.com/173/24.html

Then, when you compare the length of the diagonal to the length of the side of the square, you'll have your first indication of what "intrinsic curvature" is about. If you're on a plane, this ratio will be sqrt(t). If you're on a curved surface, like the Earth, you'll find that the length of the diagonals is not precisely the sqrt(2), but slightly different, getting further and further away from being sqrt(2) the larger you draw your figure.

The textbook mathematical treatment is perhaps a little more involved than this, but not really a whole lot. Generally in a textbook treatment one will at some point introduce the notion of parallel transport in fuller treatment. But you can think of parallel transport as the appication of a geometrical construction (Schild's ladder) based on "equal distances". So it's very helpful, and will eventually be needed, but you can get it from the notion of distances.

So to sum it all up - distances define geometry, and geometry defines curvature. That's really all there is to it.

 

[Dale]

I'm interested. Can you tell me what did we measure and what was the reading?

Sure, we measure tidal gravity all the time in oil exploration and other similar things: http://en.wikipedia.org/wiki/Gravity_gradiometry

In addition, you could easily do more direct kinematic measurements of the changing distance between two free-falling objects, although I don't know of anyone who has done that.

 

[tris_d]

To me, curved spacetime *is* a "physically stressed medium" that can transport energy. But looking at it that way dives below the level that GR addresses; it involves trying to come up with a more fundamental theory that underlies GR, such as string theory or loop quantum gravity. AFAIK it is generally accepted that GR is not a "fundamental" theory in this sense; it is a low-energy approximation to some other more fundamental theory. So I don't expect GR to tell me *how* curved spacetime can be a physically stressed medium; I need the more fundamental theory to do that. Unfortunately we don't have any way to probe the structure of spacetime at a small enough distance scale to investigate this sort of thing experimentally.

This is what I said is contradiction: "physically stressed medium" does not equal "empty space", medium implies some "substance". How is that "physical medium" you speak of different from 'aether' concept?

 

[micromass]

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