By the way, in GR, where does the need for curvature come from ?

[lalbatros]

I understand why it is so desirable to be able to write all the laws of physics by the same rule in any system of coordinates.

I also nearly understand that the equivalence principle leads to the need of curved spacetime.

But how to make that as obvious as possible?

Thanks,

Michel

 

[HallsofIvy]

Without "forces" in the classical sense, everything moves in "straight lines" (geodesics). You need to curve space-time in order to make those geodesics give the motion we actually observe,

 

[pervect]

The usual textbook explanation is this:

Suppose we have two observers at different heights in a static gravitational field. Then gravitational time dilation (which follows from the equivalence principle alone) indicates that the higher clock is ticking faster.

If you interpret this situation geometrically, it can be shown that you must have a quadrilateral where all sides are parallel, but a pair of opposite sides has a different length.

This argument doesn't actually show that space-time has a non-zero curvature tensor. It does indicate that space-time has non-unity metric coefficients, or non-zero Christoffel symbols.

 

[RandallB]

I nearly understand that the equivalence principle leads to the need of curved spacetime.

Don’t confuse Space-Time curves with GR, they apply to SR analogies. Curves or warping in GR is significantly more complex than that.
GR uses at least 4 dimensions that have no direct relation to what we see as our 3 xyz dimensions. Such that by simply standing still and allowing time to pass requires that we must be moving in all of 4 of those GR dimensions; call them ABCD if you like.
The warps or curves in those 4D’s as we move there cause us to feel force (gravity) in our 3D space, like on your bathroom scale in the AM, as it helps your maintain your standing still position without moving down though the floor. It is our movement in and the warped structure of that 4D space that gives us the perception of our 3D world including how time passes as we see our reality. And the mass we see here in our 3D world can be directly related to how that GR 4D space structure is warped.

Some (like Kaluza–Klein) even add a 5th dimension to GR (perhaps a ‘version’ of time there NOT related to our own), in any case the movements there do not allow for what we see as time to run backwards.

WHY do we need such a ridiculous idea and theory?
To explain gravity as an action at a distance without anything actually spanning that distance to facilitate classical actions and reactions.

So why not just ditch GR and use the Standard Model idea of exchanging graviton particles that Particle Theories like QM and others can use. There we don’t have ‘action at a distance’ because of the gravitons to account for it.
Would be nice but, no one has found a graviton, and on large scales GR works while Particle Theories do not. Just as on small scales Particle Theories work and GR does not.

GO FIGURE - which in fact lots of folks are trying very hard to do exactly that.

 

[cesiumfrog]

GR is based on the concept of identifying gravity with acceleration: an apple dangles from it's stem not because some "gravity force" is pulling the apple downwards, but because the entire tree is accelerating upwards.

Even though orchards here in the southern hemisphere are accelerating in opposite directions to those in the northern hemisphere, they never get further apart from one another. Curved space-time reconciles that.

 

[RandallB]

Curved space-time reconciles that.

No it doesn’t.
Curved Space-Time is just a nice term (more applicable to working SR problems of time dilation than gravity) that is much over used in simplistic explanations intended for the lay public. Frankly I think that audience could be treated better.

GR is based on concepts of math and geometry that are much more advanced and complex than that. If it was a simple as “Curved Space-Time” it wouldn’t have taken Einstein ten years to sort the theory out.

 

[Los Bobos]

In "Die Grundlage der allgemeinen Relativitätstheorie (Annalen der Physik, Band 49, 1916)" Einstein points out the following.
At first he is explaining at the equivalence of the acceleration and gravitation. Here he points out that as the ray of light seems to curved in the accelerated coordinate system (think of flashlight and elevator) the same has to hold also in presence of gravitation field (as the effects are locally same). Now, this is kind of a good hint if you already know about differential geometry. But not enough as such.

The real reason comes when he examines two coordinate systems K and K', such that K' has a path which is a circle in K-system, starting at K. Now, always K measure the length of K':s path to be something times pi. And now for the sweet stuff. Because the radius of the K':s path is not affected by Lorentz contraction (always perpendicular to speed) but the length is, K' does not always get "something times pi" for the length of his path. And as you know this holds in non-euclidean geometry. And as acceleration (system in circular motion is accelerated system) is equivalent with gravitation, in presence of gravitational field the geometry is not euclidean.

The above is not in any ways a word-to-word translation, as you probably understood :).

 

[cesiumfrog]

No it doesn’t.
Curved Space-Time is just a nice term (more applicable to working SR problems of time dilation than gravity) that is much over used in simplistic explanations intended for the lay public. Frankly I think that audience could be treated better.

GR is based on concepts of math and geometry that are much more advanced and complex than that. If it was a simple as “Curved Space-Time” it wouldn’t have taken Einstein ten years to sort the theory out.

So maybe I left out the field equations. But then, didn't Einstein himself consider their precise form something of an interim solution?

I'm not sure whether you're criticising the term I used or the content of my statement. I think my choice of words is fair not just because the OP asked the question in those terms but because "curved space-time" seems like a pretty good way of describing "a continuum that has a normal space-time nature (from special relativity) locally at every point, but which is not connected together in the most trivial manner (so, in half as many dimensions, it may have more in common with the surface of a sphere or dougnut than with the surface of a bench)".

And what more obvious motivation for a non-euclidean manifold can there be, but to make parrallel lines fail to remain equally separated whilst particularly curved lines succeed? I haven't studied the history for a while, but I suspect Einstein made the link pretty quickly (it seems like the kind of idea you can't really work to, but which might suddenly flash into your mind) and needed to spend a long time working with Grossmann to bring the math up to speed.

 

[Chris Hillman]

Two textbooks which should answer all your curvature questions

Hi again, Michel,

You will probably want to read in MTW, Gravitation, about Schild's ladder (section 10.3), geodesic deviation (section 11.2), and about curvature generally in Riemannian/Lorentzian geometry and its applications in gtr. Don't be frightened off by the "weighty" appearance of this beautiful book! (Assuming of course, that you are not already using it.) It is not nearly as forbidding as some like to claim.

Another textbook, which is much shorter and which may present a more friendly appearance, is Gravitational Curvature by Theodore Frankel. (Unfortunately, out of print). This book features an unusually detailed explanation of several interesting and important ways to understand curvature and its central role in gtr.

RandallB:

1. Wherever you said "dimensions", I think you meant to say "directions", or even better, "coordinate basis vector fields".

2. When you said "simply standing still and allowing time to pass requires that we must be moving in all of 4 of those GR dimensions", you probably shouldn't have. I don't think it would be easy to turn this into a sensible statement which corresponds to how gtr actually models gravitational phenomena.

3. Somewhere in there, I think you are (possibly deliberately, trying to simplify things for the original poster) conflating two distinct concepts which can be associated with a "timelike congruence" (the world lines of an idealized family of observers whose world lines smoothly fill up a region of spacetime without any mutual intersections; such a congruence is defined by giving a timelike vector field, not neccessarily a timelike geodesic vector field but without loss of generally a timelike UNIT vector field). To wit: first, frame fields (aka "orthonormal basis of unit vector fields") and, second, a slicing of some region of spacetime into spatial hyperslices.

The first allows us to describe any tensor, say the Riemann tensor, with respect to a frame associated with some family of observers. (Some authors speak of "physical components" of tensors, in contrast to the components with respect to a coordinate basis, which in general have no operational significance in physics.)

In particular, we can employ the Bel decomposition to break the Riemann tensor (in a four dimensional Lorentzian manifold) into three pieces, the electrogravitic tensor (which corresponds in gtr to the tidal tensor), the magnetogravitic tensor (which in gtr describes such effects as spin-spin forces, if any), and the topogravitic tensor (which corresponds fairly closely to what you would probably consider "spatial curvature"). This decomposition is independent of physics, but plays a key role in gtr. Note that a "nonspinning geodesic frame" is the closest we can get in a curved spacetime, according to gtr, to Lorentz frames. (This means: a frame field in which the timelike unit vector field yields an geodesic congruence, and the three spatial unit vector fields have vanishing Fermi derivatives with respect to the timelike vector field; see MTW.)

The second only works for "irrotational" timelike vector fields (aka "hypersurface orthogonal" vector fields or congruences). If so, we can define the three-dimensional Riemann curvature tensor of each hyperslice (its content is fully captured by the three-dimesional Ricci tensor). This can be computed from the spacetime curvature together with the irrotational congruence; MTW has a very detailed discussion of this, but the same material is well covered in more recent books, such as the one by Eric Poisson, A Relativist's Toolkit, Cambridge University Press, 2004.

One often sees statements the effect that gtr models gravitation as a curvature effect, and indeed in some sense IDENTIFIES "the gravitational field" with the Riemann curvature tensor. Such statements are correct, as far as they go. However, maybe you were only saying that it would be misleading to state that "this is all there is to general relativity". I don't think anyone said that here, but I would probably agree that claims of the form "that is all there is too gtr" invariably appear misleading when you take a closer look.

You did make one point with which I certainly agree (indeed, MTW and other textbooks stress this point): strictly speaking, the notion of covariant differentiation belongs to the domain of Riemannian (Lorentzian) geometry, and is particularly closely associated with the differential geometry of curves in (not neccessarily curved!) manifolds. As such, it is almost unavoidable when discussing scenarios involving accelerating observers in special relativity (as does the notion of frame fields!), but it does not neccessarily have anything to do with gravitation, or even to physics.

Cesium frog: I know what you mean, and your remark is essentially correct, I think, but be careful! You might have overlooked an important point: it is not easy to define directions GLOBALLY in curved spacetimes (that is, to define a specific unit vector field, in this case a spacelike unit vector field, on a given manifold, using only geometric properties of the metric tensor or derived tensor such as the curvature tensor). Sometimes you can appeal to geometric features which particular Lorentzian spacetimes may possess, such as Killing vector fields or distinguished null directions, to well-define some "distinguished direction" at each event. In this case, you are appealing to the coordinate vector field @/@r (clearly I need to learn how to obtain some mathematical markup around here, if this is possible!), but this is not a Killing vector field in the Schwarzschild vacuum solution.

But about the nifty things which congruences can do only in curved manifolds: if you don't already know about the beautiful Clifford congruence on S^3, you will love this! Even better, a good place to find a nice picture of this congruence is in one of the best short introductions to gtr, a lovely expository article called "The Geometry of the Universe", by none other than Roger Penrose. I made some attempt to get this on-line, but AFAIK it is currently only available in a (very nice!) out of print book, or rather two editions by different publishers, both out of print. The cheaper one was Mathematics Today, edited by Lynn Arthur Steen, Vintage, 1980. Be careful: there is a third book with the same title which is completely different!

So what is this Clifford congruence? Well, a picture is worth a thousand words, but it consists of a family of great circles on S^3 which twist around each other at just the right rate to balance the geodesic convergence with the divergence due to twisting--- think of two skew lines in euclidean space--- so that they maintain constant distance in a suitable sense. This congruence has many lovely interpretations in mathematics, e.g. the "Hopf fibration" and the geometry of the algebra of quaternions. For our purposes, it is a beautiful illustration, in Riemannian geometry, of the notion of the vorticity of a vector field. (I can't resist adding that these great circles, aka "Hopf circles", also define a family of nested flat two dimensional tori, the "Hopf tori", which turn up in gtr, for example in the geometry of certain "pp-waves".)

Los Bobos: I don't think we want to get into a discussion of either the history or the philosophy of gtr, but it rather appalling how much ink has been spilled over the passage you quoted from Einstein's 1916 exposition of his then new theory of gravitation! In particular, this passage later assumed new significance in the seemingly endless controversy (more imaginary than real) over the so-called "Ehrenfest paradox" (be wary of recent papers and arXiv eprints on suchissues, which vary widely in quality).

Chris Hillman

 

[RandallB]

RandallB:
you said "dimensions", I think you meant “directions" or "coordinate basis vector fields".

I think you are (possibly deliberately, trying to simplify things for the original poster)

"orthonormal basis of unit vector fields"
spatial hyperslices.
Riemann tensor,
"physical components" of tensors,
( four dimensional Lorentzian manifold)
electrogravitic tensor ( corresponds in gtr to the tidal tensor),
Fermi derivatives with respect to the timelike vector field;
"irrotational" timelike vector fields (aka "hypersurface orthogonal" vector fields or congruences).

...gtr models gravitation as a curvature effect...maybe you were only saying that it would be misleading to state that "this is all there is to general relativity". I don't think anyone said that here,

The idea that GR is as simple as Space-Time curvature is implied allover the place. So no I wasn’t trying to simplify things for the original poster, just trying to get the OP to understand that GR is NOT as simple as the popular explanation “Curved Space-Time”!

What GR describes in 4D can not easily by described in our 3D world – note how we even have problems with what “D” is! I say “dimension” you say "directions" or "coordinate basis vector fields". How about using “independent measurements of change”. Even there goes to the debate on the need for at least one more dimension/directions to gauge those changes against that can only move in one direction. Or time added to that GR 4D reality for a total of 5D (KK-GR, 1020’s, & GR views). And since our time is based on the curving of the 4D’s, there is little chance “GR time” could ever be related to our “3D version of time”, hence IMO the valid claim by many that GR is “background independent” (Smolin, etc.) Even a description of what a 4D dimension/direction is, that can be understood by our 3D perception, is near impossible.

The fact the GR is complex is only reinforced by the many terms, a few of which you used, that are needed to describe it.

So maybe only IMO, and as Einstein saw his old teacher's idea, Space-Time is a simple analogy to view SR time and distance measurements by. The curves it inspired in the 4D world of GR to respond to the idea of “the equivalence principle” are much different than and more complex than simple “curves in space time”.

 

[Chris Hillman]

The benefits of standardized terminology

Hi, RandallB,

Long experience shows that in discussions of any subject in applied mathematics, such as mathematical physics, it is best to try to use terms which have previously and firmly established technical meanings in the standard sense, in order to avoid needless and exponentially cascading confusion.

In the case at hand, the number of vectors in a basis for a (finite dimensional) vector space V can be taken as the dimension of V. This definition appeals to the theorem that changing to another basis never changes this number: it is a numerical invariant of V, so we can regard it as a property of V itself, not of our representation of V.

Analogously, we can represent a linear operator P on V as a matrix, with respect to some basis. Changing to a new basis will change the matrix representing P, but the trace will be invariant, so we can regard the trace of an operator as a property of the operator itself, not of our representation of it.

Another analogy: we can write down the line element in some coordinate chart covering and then regard this a locally representing the metric tensor defining a Lorentzian manifold (assuming some underlying and not yet fully specified structure as a smooth manifold). If we transform to another coordinate chart, the appearance of the line element changes. But in any chart we can compute the Riemann tensor and the eigenvalues of this tensor. These eigenvalues will appear to assume a different functional form when we change coordinates, but only because we have relabled events in the domain of our original chart. It might well happen the eigenvalues satisfy some algebraic relationship throughout the domain, e.g. they might have the form 2,2,-1,-1,-1,-1 times some scalar function (as happens for the Schwarzschild vacuum and many other solutions). This will be an invariant property of our Lorentzian manifold itself, not a of our representation of it.

Chris Hillman

 

[RandallB]

Long experience shows …… it is best to try to use terms which have previously and firmly established technical meanings in the standard sense, in order to avoid needless and exponentially cascading confusion.

I think your agreeing with me…
I certainly agree with you -- that “Space-Time Curvature” is hopelessly lost as a term that can be used between people with an expectation that both will understand each other. There is no “standardized terminology” that defines a common understanding of it and how it should be understood.

Even the simple requirement of understanding how it relates to 4 or 5 Dimensions as defining; “dimensions”, “directions”, “coordinate basis vector fields” “independent measurements of change” or etc. is not clearly understood and visualized in classical/layman’s terms. To further apply ‘eigenvalues’ and various ‘forms’ defined within various ‘manifolds’ or ‘domains’ only complicates the issue as different views can have various ways of applying (or not) those terms.

So the point is “Space-Time Curvature” is a nice Pop-Science term to generalize a lot of different ideas when addressing a Lay audience.
But it is not useful as a foundation or “standardized term” than can be counted on to mean the same thing to many different users as a tool to visualize Multiple (greater than our local 3 +1) Dimensional Theories like GR, QM, BM MWI etc. (Note: I take the Uncertainty Principle in OQM as an additional mathematical dimension)
On its own, “Curved Space-Time” is inadequate to establish a common understanding or representations of the possible reality of those theories, that can be shared between 3D Classical beings like us. As you point out, something more complex to understand those theories is required.
RB