I Euclidean geometry and gravity

[PAllen]

I think it is important to recall points made in some other threads about the limitations imposed by Herglotz-Noether : they apply to a 3 parameter congruence, i.e. one with nonzero volume in a spatial slice. For a strictly 2-d surface, you can find all sorts of motion plus rotation that preserve rigid positioning within the 2-surface, because it can freely change shape (embedding) within the containing spacetime, while maintaining intrinsic metric rigidity. In effect, it can behave like a non-stretchable fabric, allowing many 'rigid' motions not possible for an object with thickness. Once you add nominal thickness, any bending will necessarily produce expansion and compression.

 

[PAllen]

What I know and don't know as of this point in the discussion:

1) You can have rigid body (congruence) of parametrized by varying r, $\theta$, and a small $\phi$ thickness. This can exist at any place outside the horizon of an SC BH, and be of any size whatsoever (as long as it is all outside the horizon).

2) If you inquire as to physical plausibility, the limiting factors are simply the ability to resist the compressive stress of being held in position radially, as well as the tidal gravity stress. Both of these are encoded in the proper acceleration of the congruence world lines.

What I still don't know is what happens if you consider the motion of one such 'body' very far from the BH, radially, slowly, until it is relatively close to the BH horizon. Is it possible for this to be a rigid motion?

The obvious congruence to test would be the Fermi-Normal congruence built from a world line stationary, far away until t1, then slowly (very small rate of change of proper acceleration) decreases in r to e.g. 2 times the SC radius, then remains stationary at the new radius.

A first question to ask is whether the 'stationary' congruence in (1) above is Fermi-Normal. If not, then there are major parts of SR rigid motion theory that don't carry over to GR>

 

[PeterDonis]

the real content of the theorem is what cases are possible. I have not been able to access discussions of this.

I'll see if I can find any references in the previous threads we had here. At least one of them, IIRC, dealt with a scenario in Schwarzschild spacetime.

 

[PeterDonis]

The GP constant position congruence is the same as the SC congruence.

Yes, agreed. But...

The only thing that differs in GP coordinates is that the congruence is sliced differently for surfaces of constant time. These slices are definitely not orthogonal to the congruence

They're not orthogonal to the static congruence (the Killing congruence), yes. They are orthogonal to the free-falling Painleve congruence. I was using the latter orthogonality to illustrate how a congruence can be hypersurface orthogonal, with the orthogonal hypersurfaces being flat, but still not satisfy the H-N theorem (because the congruence has nonzero expansion).

The congruence remains hypersurface orthogonal (meaning there exists a non-intersecting family of slices orthogonal to the congruence), irrespective of simultaneity choice for coordinate construction.

Yes, agreed.

 

[pervect]

Consider two separate objects held static against gravity by external forces. Now you connect them with a beam of negligible mass. The stress in that beam can obviously be zero, because it doesn't make a difference if it's there or not. You can continue this with many such objects to build a truss structure where all the beams are stress free.

But if you build it in flat space, and bring it into curved space, you will never get the beams stress free, no matter what forces you apply to the nodes.

My intuition leads to a similar conclusion, but I think we would need to clarify what a "beam" is in this context. I am not sure I have a solid enough definition of what mathematical properties a beam has to make a mathematical argument about why this should be so. The beam being elastic (so that a strain causes a stress) is a start, but the idea of going from a curved space to a flat space is sort of fuzzy. My thinking is that the beam consists of atoms, which attract and repel each other based on their distances apart, but that's not a formal mathematical definition, and it's still rather fuzzy about what we are actually doing to this arrangment of presumed atoms to make it transition from a curved space to a flat one. Are we moving the colleciton of atoms / beam? If so, does it matter how we move it? Can we say for sure that if we move it "slowly" the details of exactly what path we move it and where we apply the force to make it move don't matter?

I suppose my best attempt at justification would be to say that there is probably no born-rigid (in the restrictive first sense) that moves the beam, which implies that the distance between atoms is disturbed. But this still seems far from rigorous.

 

[PAllen]

What I still don't know is what happens if you consider the motion of one such 'body' very far from the BH, radially, slowly, until it is relatively close to the BH horizon. Is it possible for this to be a rigid motion?

I suppose my best attempt at justification would be to say that there is probably no born-rigid (in the restrictive first sense) that moves the beam, which implies that the distance between atoms is disturbed. But this still seems far from rigorous.

I agree this is the fundamental question. I lean in the same direction as you (@pervect ). [This would mean that Herglotz-Noether extended to GR is substantively different in that a whole class of possible rigid motions in SR don't typically exist in GR].

However, I think looking at this in terms of stresses is not right. Consider that case of spinning up a disc in SR. You can write a plausible congruence representing the motion of 'atoms' from a state of constant rotation, then increasing rotation, then constant rotation at a higher rate. The initial stage and final stage will be rigid, the transition will not be rigid because it can't be. It is not as if there are extreme stresses on the object, or that it somehow 'could' be rigid if strong enough. Each world line will have a smaller proper acceleration during the initial state, increasing during the transition, and then constant and larger at the end. These proper accelerations are what determine the stresses in the material. The change in mutual relative position of points is just something that happens because it must. These geometric changes will affect underlying fields as well, so I don't think they are associated with real stresses. Spinning up a disc is clearly possible and has no dramatic consequences unless you spin it up to the point where it flies apart.

So assuming the rigid nominally thick paper, oriented in an equatorial plane, is moved radially from very far away to near the BH horizon, the position of atoms will change with respect to each other during the the transition (because they must). At the beginning and end, it will be rigid (equilibrium). I claim there will be no unusual stresses during the transition - just those associated with increasing compressive stress from resisting free fall, and additional stress from increasing tidal gravity. At the end, it will be no different from a paper constructed in the higher curvature region.

 

[PeterDonis]

a whole class of possible rigid motions in SR don't typically exist in GR

In a generic curved spacetime, yes, because the two types of rigid motions allowed by the H-N theorem each depend on the spacetime having some kind of symmetry. In Schwarzschild spacetime, there is at least one such symmetry, the timelike KVF.

 

[PAllen]

Another thought on this is to emphasize that spatial geometry is simply a choice of simultaneity, with the same conventionality as in any other case. The paper near the BH would be measured as having Flamm paraboloid geometry by a collection of stationary instruments. On the other hand, measured by a 'rain' of instruments at free fall from infinity, it would appear to be Euclidean flat. The invariant statement (assuming post #66 and others that have proposed the same are correct) is that the motion described cannot be rigid.

 

[A.T.]

The only such stresses on a rigid object, whatever its history, are those of tidal gravity.

If the external support forces are distributed smoothly throughout the body and locally match the force need to hover, then tidal gravity is handled as well, and the stresses can, in principle, approach zero.

But I disagree with the 'whatever its history' part. I think it does matter if:
a) the object was constructed hovering stress-free near the mass, using the above forces
b) the object was constructed stress-free far away, and then moved to hover near the mass, using the above forces

I think that in case b) it's not possible to make the stresses negligible, even in principle. And the reason is this:

The paper near the BH would be measured as having Flamm paraboloid geometry by a collection of stationary instrumen

The geometry of the paper, measured by instruments at rest to the paper, is the paper's intrinsic geometry. Changing the paper's intrinsic geometry, from Euclidean to Flamm, will introduce stresses, that cannot be removed by any smoothly applied external forces.

 

[PAllen]

If the external support forces are distributed smoothly throughout the body and locally match the force need to hover, then tidal gravity is handled as well, and the stresses can, in principle, approach zero.

In my view, this is not a sensible approach. What you have is a collection of independent rockets, not a body. In a body, in any normal context, stresses come from body’s resistance to external forces. As soon as you replace your independent rockets with inter atomic forces, you have stress.

But I disagree with the 'whatever its history' part. I think it does matter if:
a) the object was constructed hovering stress-free near the mass, using the above forces
b) the object was constructed stress-free far away, and then moved to hover near the mass, using the above forces

I think that in case b) it's not possible to make the stresses negligible, even in principle.

I disagree with this. Once the body is in its new position, the only thing needed to hold each element in place is a force to produce the proper acceleration of the element world line. There is no place for a memory effect. You try to get around this by proposing massless struts between the body elements with properties you specify to produce your conclusion, but they are irrelevant. Really, if you add them, they have to part of the same congruence with the same properties as the rest of the body, and then your argument falls apart.

 

[A.T.]

There is no place for a memory effect.

I think there is.

Every beam/strut has a relaxed proper length, and the current stress in it depends on how the current proper length deviates from the stress-free proper length. The combination of all current proper beam lengths defines the current proper geometry of a static object.

If the object is built stress-free far away from any mass, then its beams are only all stress-free if the object's proper geometry is Euclidean.

If you then place that object hovering near a big mass, the object's proper geometry cannot stay Euclidean. So the beams cannot all be stress-free, no matter how smoothly you support the structure against gravity.

 

[Tomas Vencl]

If the external support forces are distributed smoothly throughout the body and locally match the force need to hover, then tidal gravity is handled as well, and the stresses can, in principle, approach zero.

Yes, but are not the tidal gravity forces just those originating from the difference between flat and curved spacetime ?

 

[PAllen]

I think there is.

Every beam/strut has a relaxed proper length, and the current stress in it depends on how the current proper length deviates from the stress-free proper length. The combination of all current proper beam lengths defines the current proper geometry of a static object.

If the object is built stress-free far away from any mass, then its beams are only all stress-free if the object's proper geometry is Euclidean.

If you then place that object hovering near a big mass, the object's proper geometry cannot stay Euclidean. So the beams cannot all be stress-free, no matter how smoothly you support the structure against gravity.

I will make a self correction which implies we were both right/wrong in a sense. Again, I think the correct analog to a well understood similar case is spinning up a disk in SR. Your position amounts to saying that if you spin up the disk rigidly it will have extra stresses compared to a disc built spinning. This is wrong - there is no memory effect. Instead, and this is where I was wrong, having forgotten some aspects of this problem, it is simply impossible to spin up the disc rigidly - its not even a matter if infinite force - such an operation is simply impossible. That much, I got right. But what this does say is that if you imagine absolutely rigid bonds between atoms, then they must break as soon as you start spinning up the disc, and you are left with disassociated atoms (until they re-form bonds). In a more realistic model, bonds expand or contract, producing additional stresses compared to those acting against rotational “force”. The key point is that there is no such thing as moving a rigid body radially (assuming we are right about the generalization of Herglotz-Noether to GR). It must deform as soon as you start such motion, breaking bonds if you idealize them as perfectly rigid. More plausibly, they just deform. If such deformation becomes too large, bonds break, but this is just tidal shredding by another name. And having deformed, you are left with an object no different from one constructed at the smaller radius.

Also, in my view, this has nothing really to do with spatial geometry. All separation of spacetime into space plus time has no more physical meaning than a simultaneity convention. As I noted above, you measure different geometry for an object depending on what family of instruments you use. The invariant facts must couple to spacetime as a whole, and this is captured the behavior of congruences.

 

[A.T.]

Yes, but are not the tidal gravity forces just those originating from the difference between flat and curved spacetime?

Yes, or from gravity gradient. But if you distribute the external suport forces smoothly throughout the hovering object, you can account for that, and provide each infinitesimal part exactly the proper acceleration it locally needs to hover, thus eliminating any internal stresses steming from supporting the body. Any stresses that remain are from a mismatch beteween the object's stress-free proper geometry and the spatial geometry it hovers in.

 

[Ibix]

Perhaps a slightly different experiment might help. Consider a flock of small rockets. Initially they are at rest far from a black hole and arranged in concentric circles. The smallest circle is of radius R. We can join the the rockets by radial and circumferential light threads, with lengths chosen so that they have no tension but no slack.

Now we move to a black hole of mass M, M<R/2, and arrange our rockets around the hole, programming the rockets to hover wherever we leave them. We can either arrange the rockets so that the circumferential threads are just taut (as in flat spacetime) but the radial threads are under tension, or we can arrange the radial threads to be just taut and the circumferential threads will have slack.

So far I think that's uncontroversial and entirely stated in terms of invariants.

The bit that's causing argument, I think, is why the threads have different stresses in the two cases. If you work in Schwarzschild coordinates, it's because the spatial geometry is different. If you work in Gullstrand-Painleve coordinates, though, the spatial geometry is flat in both cases and the reason for the stresses is that the threads are the wrong length (remember that the threads are the intersection between 2d worldsheets and the spatial planes, and G-P spatial planes are not Schwarzschild spatial planes so the same thread has a different length in the two systems). Or you can take the 4d view that the spacetime/congruence is different in the two cases.

Am I following this thread right? Does that make sense?

 

[PeterDonis]

Every beam/strut has a relaxed proper length

But what does that relaxed proper length depend on?

You seem to be saying that, if I build two beams of the exact same material, one in flat spacetime and one in curved spacetime, they can have different relaxed proper lengths.

I don't think that's right. The relaxed proper length of the beam should depend only on what it's made of, not on the spacetime geometry in which it was built. Two beams of the same material should have the same relaxed proper length. If one is in flat spacetime and one is in curved spacetime, they will "fit" differently into the spacetime geometry around them, but that won't change their relaxed proper length.

The spacetime geometry will make a difference in the distribution of stress inside the two beams, but that's because, in flat spacetime, for example, a beam in free fall can be unstressed everywhere--its actual proper length will be exactly equal to its unstressed proper length. The presence of tidal gravity in the curved spacetime will prevent that beam from being unstressed everywhere even if its center of mass is in free fall; its actual proper length won't be exactly equal to its unstressed proper length.

 

[PAllen]

Perhaps a slightly different experiment might help. Consider a flock of small rockets. Initially they are at rest far from a black hole and arranged in concentric circles. The smallest circle is of radius R. We can join the the rockets by radial and circumferential light threads, with lengths chosen so that they have no tension but no slack.

Now we move to a black hole of mass M, M<R/2, and arrange our rockets around the hole, programming the rockets to hover wherever we leave them. We can either arrange the rockets so that the circumferential threads are just taut (as in flat spacetime) but the radial threads are under tension, or we can arrange the radial threads to be just taut and the circumferential threads will have slack.

So far I think that's uncontroversial and entirely stated in terms of invariants.

Maybe not. I don’t see why you couldn’t arrange the threads to have no slack in the second case. The threads would just have different lengths than the first case. And this gets at the difference between my view ant @A.T. If you imagine moving a sector of this arrangement radially, threads will stretch, break, or develop slack. But you don’t get any choice that keeps them the same length with different stress. That possibility simply doesn’t exist.

 

[PeterDonis]

The threads would just have different lengths than the first case.

I think the intent was to have the same physical threads in the second case--that you would take the structure built in flat spacetime and move it into the curved spacetime, and get it into whatever equilibrium state it can be in in the curved spacetime. The unstressed proper lengths of the threads would be held constant during the process.

 

[Ibix]

I don’t see why you couldn’t arrange the threads to have no slack in the second case.

I think the intent was to have the same physical threads in the second case-

As Peter says, you'd tie each end of a thread to one of a pair of rockets, so the segment of thread between any pair is fixed. Then you can transport the resulting net in a folded state, then place one rocket, then the ones it's attached to and so on.

I agree you could attach the threads adjustably (with a bulldog clip or a lockable spool or something) so you could pay out thread to relax it or take up slack. But that was not my intent.

 

[Tomas Vencl]

Yes, or from gravity gradient. But if you distribute the external suport forces smoothly throughout the hovering object, you can account for that, and provide each infinitesimal part exactly the proper acceleration it locally needs to hover, thus eliminating any internal stresses steming from supporting the body. Any stresses that remain are from a mismatch beteween the object's stress-free proper geometry and the spatial geometry it hovers in.

I do not understand, how we choose the exact position of each infinitensimal part, where it hovers ? How we measure , that it is at right place relative to other parts ? What is the right place ? Thinking about positions with no stress at all, but then there are no remain streses. (Maybe this is the case, when we are talking about rigid measuring rods, for example for measuring radial distance )
Maybe the 1D radially oriented rod is the simplest example how to grasp a problem.

 

[PAllen]

As Peter says, you'd tie each end of a thread to one of a pair of rockets, so the segment of thread between any pair is fixed. Then you can transport the resulting net in a folded state, then place one rocket, then the ones it's attached to and so on.

I agree you could attach the threads adjustably (with a bulldog clip or a lockable spool or something) so you could pay out thread to relax it or take up slack. But that was not my intent.

Ok, that is fine.

However, what is true includes that, in an invariant sense, you can't place the rockets in the new locations at the same mutual distances. This isn't just a choice like a spatial slice. It follows from the congruence analysis. So I still claim there is no sense in which you can talk about bringing the 'low curvature constructed body' to the high curvature region, preserving its configuration, just with extra stresses.

Also, any real material will behave more like balls connected by a network of springs. These springs will adjust to do all of the following:

- compress to balance a force applied from below to hold the body stationary
- stretch and compress in different directions to balance tidal forces
- if moved radially, the springs will change in length, no matter how stiff you make them.

I further claim that after moving, the result is no different than constructing the arrangement locally delta a short settle time if the movement is 'too fast'. Also, that items 2 and 3 in the list are basically both manifestations of tidal gravity.

 

[PAllen]

Having thought about generalizing rigid motion theory from SR to GR, I believe we can all agree on the following for SC spacetime:

- a rigid stationary congruence is possible
- a rigid congruence moving circumferentially about a BH is possible
- a rigid congruence representing rotation about a radial axis is possible

These are all killing motions. Possibly, no other rigid motions are possible.

 

[PAllen]

A further nuance is that I think GR may allow a few cases that are not related to KVFs of the spacetime. Consider a stationary rigid configuration in an FLRW spacetime. There is no timelike KVF. Yet I believe this rigid motion is possible because of homogeneity. It is simply necessary that the congruence lines have position varying inward (for the expanding case) motion relative to the Hubble flow (with one congruence line following the Hubble flow). Similar to size limits in SR for Fermi-Normal flows, there would be a maximum size for such a rigid configuration depending on rate of acceleration of expansion.

 

[PeterDonis]

Consider a stationary rigid configuration in an FLRW spacetime. There is no timelike KVF. Yet I believe this rigid motion is possible because of homogeneity.

Hm. This congruence would be irrotational, but I don't think its orthogonal hypersurfaces would be flat. So it would not meet the requirements of the H-N theorem.

I'm not sure my intuition says that such a congruence is possible, because I don't think the proper acceleration required to keep it rigid would be constant. But I'd have to work through the math to see; my confidence in my intuition for this case is not very high.

 

[Ibix]

So I still claim there is no sense in which you can talk about bringing the 'low curvature constructed body' to the high curvature region, preserving its configuration, just with extra stresses.

Well, we can definitely arrange the rings of rockets so that their circumferences are the same as they were in flat spacetime. And we can find a Euclidean foliation, the G-P one. So in those spatial planes the distances between rings must also match what they were in flat spacetime. So I think there is a sense in which you can talk about having preserved the structure just with different stresses, but it's coordinate dependant.

 

[PAllen]

Hm. This congruence would be irrotational, but I don't think its orthogonal hypersurfaces would be flat. So it would not meet the requirements of the H-N theorem.

I'm not sure my intuition says that such a congruence is possible, because I don't think the proper acceleration required to keep it rigid would be constant. But I'd have to work through the math to see; my confidence in my intuition for this case is not very high.

There is no requirement for constant proper acceleration. Constant position lines in FN coordinates based on a completely arbitrary world (in SR) produce a rigid congruence. As to hypersurface being flat, this is one of those questions not addressed due to no references for the details of HN in curved spacetime. I don’t see why flatness would be required. I think homogeneity is enough here. In each surface, you just have a smaller and smaller (in coordinate terms) representation of the configuration.

 

[PeterDonis]

There is no requirement for constant proper acceleration.

If the proper acceleration of each worldline is not constant, the congruence is not stationary.

 

[PeterDonis]

Constant position lines in FN coordinates based on a completely arbitrary world (in SR) produce a rigid congruence.

Wouldn't this violate the H-N theorem if it were true?

 

[PAllen]

Wouldn't this violate the H-N theorem if it were true?

No. See the end of paragraph 2 in irrotational motion in the current Wikipedia article on born rigid motion. Hate to use this, but the last sentence of the paragraph gives a reference.

 

[PeterDonis]

No. See the end of paragraph 2 in irrotational motion in the current Wikipedia article on born rigid motion. Hate to use this, but the last sentence of the paragraph gives a reference.

Ah, got it, thanks! The key point is that the form given by Moller (1952) explicitly shows that the orthogonal spatial slices are flat. I'd forgotten that that was demonstrated as a theorem for flat spacetime.