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

[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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