I Euclidean geometry and gravity

[PeterDonis]

stresses ... from change of proper geometry.

You still haven't shown the math of how you pick these out from other stresses. And, as I pointed out in a post some time ago, and reinforced with my post #100 just now, I don't think that's even possible. The proper acceleration of each worldline in the congruence that describes the object is a single thing. There's no way to pick out what part of it is due to "stresses from change of proper geometry" and what part is due to other stuff. And there can't be any "other" stresses that aren't accounted for by the proper acceleration.

 

[PAllen]

I think their claim is more limited than that. De Sitter spacetime has a timelike KVF, and a Killing motion using that KVF can certainly be Born rigid.

What their equation (34) shows is that, in a spacetime which is not exactly de Sitter, but also contains other stress-energy besides the cosmological constant, the particular congruence they describe is not rigid; it has nonzero expansion. That in itself doesn't show that it's impossible to find a rigid congruence; the rest of that section appears to be trying to show that, but I'm not sure I understand the argument being presented there.

Look later in the paper. They show a wider class of possible born rigid objects when the slices are not flat. But, in any case, homogeneity is nowhere near enough to allow a a rigid congruence.

 

[pervect]

Regarding this, I have the following question: Is it possible to embed a flat (Euclidean) two-dimensional surface in a curved, three-dimensional space?

I have not read anything definitive on that, but my current thinking is that it's not possible in general, but it may be possible in specific cases.

The usual approach is not about embeddings. Instead, one starts with some curved manifold, and specifies a pair of vectors at this point. The pair of vectors pick out a specific two dimensional subspace of the higher (in general) dimensional tangent space.

The tangent space is an important concept in differential geometry. In two dimensions, one might envision a sphere as a two dimensional manifold, then one can visualize the tangent space of the sphere at some point as a plane tangent to the sphere at that point. We can see from this example then that the tangent space is different from the manifold - points in the tangent space are not points on the manifold (except perhaps for one shared point, if one uses the diagram).

Tangent spaces exist in higher dimensions than the easy-to-visualize case of the tanget plane to a sphere, and since we are talking about a higher dimensional case, we'll need to use a higher dimensional tangent space.

My argument basically is that if you can't find a pair of vectors that makes the sectional curvature vanish at some point, you can't possibly embed a plane at that point. I believe you are doomed before you start.

Being able to find such a pair of vectors does not create an embedded tangent plane as you are requesting - it's more like an ininitesimallyu small version of such an embedded plane, it is only flat in an infinitesimal region. "Growing" it to a finite size is another problem - my intiuition suggests it won't always be possible but I don't have a good argument.

Since this is the result of my thinking (which isn't as great as it used to be sadly), don't regard this post with the same confidence as you would from reading something in a textbook. (You were already not doing that, I imagine, but I'll say it anyways0.

There is also the possibility of miscommuniation do to not-shared concepts. I am basically assuming we have a manifold, with a pre-existing metric that represents "distances" between nearby points on the manifold, a concept that we are not allowed to change. Perhaps this assumption can be eased, I don't know, I'm too used to making it to think seriously about not making it. I'm further making an assumption about not only about the metric, but what's called the "connection". I'm using what's called the Levi-Civita connection, which has the prpoerty that "straight lines" are the shortest distance between two points.

If I were more of a mathemtician, I probably wouldn't be talking about distances, I'd be talking about connections. I'm very used to computing curvature from the metric, but a sneaky little recollection is telling me that it's actually computed from the connection. But I don't want to try to explain connection.

To give a counter-example, we can succesfully make flat maps of the curved earth, but they won't be to scale. If we redefine distance between points to match what's drawn on the map, then suddenly the earth is flat, not round.

And if we are sufficiently deep in mathematics, we can talk about curvature without mentioning distances at all. But I'm not used to doing that. I'm more along the lines of "distances determine geometry".

 

[PAllen]

My argument basically is that if you can't find a pair of vectors that makes the sectional curvature vanish at some point, you can't possibly embed a plane at that point. I believe you are doomed before you start.

This can't be right. By that logic, you can't embed a 2-sphere in Euclidean 3-space since it has positive scalar curvature and every such plane will have zero curvature. Thus, this appears to be irrelevant to the problem.

I would say the obvious place to start would be the Gauss-Codazzi equations.

Also, there is the possibility of non-smooth embeddings, which are much easier. Two examples of surfaces that cannot be smoothly embedded to Euclidean 3-space but can be embedded without this restriction are the flat torus and a surface of constant negative curvature.

 

[PAllen]

No, I never assumed the object will remain rigid during the transport towards the mass, which would be the equivalent to spinning up the disc.

In fact, I say the exact opposite: The object will definitely change its proper geometry during the transport.

Ok, sorry I misunderstood that.

But once it has arrived, we can keep it hovering as a static object that remains rigid, which would be the equivalent of spinning at a constant rate.

In both scenarios the proper geometry of the object changes from Euclidean (when far from the mass / non-spinning) to negatively curved (when near the mass / spinning).

This is where I start to differ, and we may never agree. An object is a world tube, and all separations to time plus 3-geometry are a matter of convenience. Essential physics is contained in the world tube as whole.

In both scenarios we can apply smoothly distributed external forces needed for the proper acceleration of all object parts, which eliminates stresses related to the non-inertial motion. But the stresses related to the change of proper geometry will remain.

Here, I see what I think your are getting at (more below) but don't think it is useful (or clear to others) to think so much in terms of stresses induced by change of proper geometry. More on this below.

I disagree. The proper geometry of an object, measured with rulers at rest to that object, has very much a physical meaning. It's this geometry that determines how much rigid material you would need to build that object in its current state, for example: static near the big mass or spinning with high circumferential speed.

If you build an already spinning spoke wheel from pre-fabricated segments, you will need more segments for the circumference, than for a non-spinning version of the same radius. This is not a matter of convention.

Here is where I see what you are getting at, I think. I would describe it more as follows:

An initially rigid body forced to undergo a motion (either through changing spacetime or something like change of rotation) incompatible with rigidity will deform. When in equilibrium in a new rigid state, it would be the same as an identically constructed object with no history. However, it would typically not be the same as a conceptually similar object would be constructed in the new location/state. This gets fuzzy, but I will clarify by example.

I think my slowness to understand this point was thinking in terms of homogeneous materials. It became much easier for me to see when dropping this bias. Consider your example of a spoked wheel being spun up (I think the motion toward a BH is essentially similar, which is why I initially proposed the spinning disc comparison). Imagine the rim is extremely stiff and the spokes are thin malleable metal. Then, spinning up the disc will result in the rim moving inward and spokes buckling. Once spun up, this is no different from 'an identically constructed object' but it is certainly not how you normally construct a spinning spoked wheel. Alternatively, imagine the spokes are stiff and strong while the rim is a light stretched rubber band. Then, when spun up, you would find the rubber band was tightly stretched. Again you could build it this way, but if you're following a 'standard recipe' and building spinning, you wouldn't do so.

What I think is central here, is that the resulting matter distribution is 'unexpected'. But I think it would be hard to remove my scare quotes and rigorously define this. It is 'relative' to a notion of a construction recipe.

 

[A.T.]

Here, I see what I think your are getting at (more below) but don't think it is useful (or clear to others) to think so much in terms of stresses induced by change of proper geometry.

It might be clearer to look at the changes in proper geometry directly, rather than via the stresses. See below.

Alternatively, imagine the spokes are stiff and strong while the rim is a light stretched rubber band. Then, when spun up, you would find the rubber band was tightly stretched.

Very good example. Here, we could even replace the rubber circumference band with telescopic elements with markings on them, which instead of becoming stressed, merely indicate their current proper length for everyone to see. So you can spin up the wheel safely, and directly observe how its proper geometry changes.