Motion in Spacetime (using internal forces), swimming/gliding.

[Sbaraka]

I'm here to discuss motion in spacetime and how it works to hopefully get a better understanding of it.

Specifically "spacetime swimming", and the motion of a "relativistic glider", which is talked about in this article "Surprises from General Relativity: "Swimming" in Spacetime" By Eduardo Guéron.

Swimming

For example, in a curved space, a body can seemingly defy basic physics and “swim” through a vacuum without needing to push on anything or be pushed by anything.

Gliding

Curved spacetime also allows a kind of gliding, in which a body can slow its fall even in a vacuum.

so any information you guys can provide on this topic of how these two forms of motion work would be of great help.
also besides any general information you can provide about it. i also have a few questions.
1. does "swimming" give the object/person a velocity?
from my understanding of it and what I've researched swimming is completely different to motion on earth, and does not have acceleration or velocity involved. it is simply a translation of the centre of mass resulting in change of displacement in relation to other bodies of mass.

2. how does gliding work specifically?
upon research gliding only works in conjunction with utilization of the gravitation pull of lagrange points 4 or 5. where the object gliding enters the orbit of the points L4 or L5 and uses that gravitation pull to slow its descent caused by the gravitation pull of the planet.

3. What effect does rotating have on a body of mass?
gliding works via the spaceship rotating (changing its orientation) which in when in spacetime curvature will cause the centre of mass to move as well? but what sort of motion does this put on the spaceship?

any information you guys can provide would be of great assistance in helping to further my understanding of motion in spacetime. thanks.

 

[JesseM]

For reference, here's a previous thread with some discussion of the subject:

https://www.physicsforums.com/showthread.php?t=326206

 

[Sbaraka]

thanks i forgot to search the forums before posting.
but I'm looking to go into more detail than what is said in that thread. (refer to my questions).

 

[bcrowell]

Here are some sources of information that might be more easily accessible than the Sci Am article:

MTW, p. 1120, ex. 40.8

"Swimming in Spacetime: Motion in Space by Cyclic Changes in Body Shape" Jack Wisdom 2003, Science , 299 , 1865. http://groups.csail.mit.edu/mac/users/wisdom/

"The relativistic glider," Eduardo Gueron and Ricardo A. Mosna, Phys.Rev.D75:081501,2007. http://arxiv.org/abs/gr-qc/0612131

"'Swimming' versus 'swinging' in spacetime", Gueron, Maia, and Matsas, http://arxiv.org/abs/gr-qc/0510054

 

[bcrowell]

As long as we've got a thread going on this subject, I'd like to see if my own understanding of it holds up to inspection. There are three main points that I get from this.

1) This kind of effect occurs in both Newtonian mechanics and GR. There's nothing inherently relativistic about it, but some aspects of it work out differently in the relativistic context. All the examples in the Wisdom paper are nonrelativistic except for the very last one. However, the Newtonian analogue is not an object in a Newtonian gravitational field, it's an object constrained to a curved surface.

2) It can occur in a space of constant curvature, which makes it different from superficially similar examples involving deformable bodies in Newtonian gravity.

3) I think the basic idea is that although momentum is conserved, you can move your c.m. without ever locally violating conservation of momentum. A nonrelativistic example is that a robot living on a sphere stretches out its accordion arms and clasps its hands on the opposite side. Then it detaches one arm from its shoulder and contracts both arms again. Momentum has always been conserved locally, but the robot's c.m. has moved, which shows that we can't have anything like a global conservation law for momentum, if we confine ourselves to observables that are intrinsic to the surface of the sphere. (If you built a model of this, the sphere would recoil, but that isn't an interpretation that you could use if you were really living on this spherical manifold.)

Re the distinction between local and global, if I take equation (10) from Gueron, http://arxiv.org/abs/gr-qc/0510054 , and make some approximations, I find that a static observer sees the swimmer as experiencing a maximum force (assuming it flails its limbs at c) that's on the order of $GMm\ell/r^3$, where M is the mass of the body producing the (Schwarzschild) spacetime curvature at a distance r, and m and $\ell$ are the mass and size of the swimmer. I guess the fact that it scales like $\ell^p$ with p=1 is a measure of how local or global the effect is. It's not obvious to me whether p=1 is really big enough to make the word "local" applicable in the sense that the stress-energy tensor's zero divergence is a "local" conservation law.

[Edit] It occurs to me now that the robot example may not be totally apropos. I'll post below about that.

 

[Jonathan Scott]

I've just had a look at Wisdom's paper.

The part about rotations is fine, and I can also believe the bit about translations in a space which has intrinsic curvature, like getting around on a frictionless sphere.

I don't fully understand his model for the GR case, but I'm not yet convinced. For a start, it seems to me that we need to consider space-time rather than just space and that the closest equivalent of intrinsic curvature is the mass-energy density tensor, which is zero in a vacuum.

I also know that there are various non-obvious techniques which can be used even in Newtonian gravity, such as stabilization in the direction of a tidal gradient, and I suspect that it may well be theoretically possible to achieve other unexpected forms of motion using extended structures to interact with the variations in the gravitational field of a central object, but all of these techniques still strictly obey the standard conservation laws.

 

[bcrowell]

Re my robot example in #5, it occurred to me later that the example also works on a cylinder, which has no intrinsic curvature, so it's really a topological example, not one having to do with curvature (except to the extent that curvature can constrain topology). So I'm not sure that it's even qualitatively a good example of the Wisdom/Gueron idea.

I don't fully understand his model for the GR case, but I'm not yet convinced. For a start, it seems to me that we need to consider space-time rather than just space and that the closest equivalent of intrinsic curvature is the mass-energy density tensor, which is zero in a vacuum.

Well, both the Riemann tensor $R^a_{bcd}$ and the Einstein tensor $G_{ab}$ are measures of the curvature. The Einstein tensor is proportional to the mass-energy tensor, so it has to vanish in a vacuum, but the Riemann tensor doesn't. If you take a look at the Gueron paper, http://arxiv.org/abs/gr-qc/0510054 , he works it out in some detail in the Schwarzschild metric, which is a vacuum solution.

I also know that there are various non-obvious techniques which can be used even in Newtonian gravity, such as stabilization in the direction of a tidal gradient, and I suspect that it may well be theoretically possible to achieve other unexpected forms of motion using extended structures to interact with the variations in the gravitational field of a central object, [...]

My current understanding is that it is really not analogous to those effects. The Newtonian gravity version has to be done in sync with the orbital motion (like pumping on a swing). The relativistic version can be done at high frequencies, and it doesn't require a periodic orbit.

[...]
but all of these techniques still strictly obey the standard conservation laws.

The only standard conservation laws in GR are local conservation laws. There are no standard global conservation laws. Wisdom explicitly uses local conservation laws to calculate the effect. Gueron calculates the effect in GR, which also has a local conservation law built in. The reason the effect can exist is really very closely related to the reason that GR doesn't have global conservation laws. The Sci Am article has a nice explanation of this, via the fact that when you try to compute a center of mass on a spherical surface, the result depends on the order in which you do the addition. This is exactly the same idea as why GR doesn't have global conservation laws: because vectors at distant points can only be related by parallel transport, which is path-dependent.

 

[JesseM]

The only standard conservation laws in GR are local conservation laws. There are no standard global conservation laws.

Energy is conserved globally in a spacetime that is either stationary or asymptotically flat according to this (see the paragraph that begins 'In certain special cases...'), would the same be true of momentum? The claims about "swimming" or "gliding" in spacetime probably don't require the use of a non-stationary and non-asymptotically-flat metric.

 

[bcrowell]

I think I've figured out a satisfactory check on the scaling/local/global issue.

Let's start with an E&M example. Suppose you have an electric current density $J=(ax+bx^2)\hat{x}$. The divergence, evaluated at x=0, equals a, so $a\ne 0$ violates local conservation of charge. The divergence basically means constructing a tiny box with sides of length dx, and comparing the flux on two sides of the box. The b term has no effect on the divergence at x=0, because it's too small to have a big enough effect on the value of J(x=0)-J(x=0+dx). If you have an ambiguity due to the path-dependence of parallel transport, then its effect on this subtraction is of order dx², which, like the b term, is too small to affect the divergence.

In relativity, the analogous conservation law is $\partial_aT^{ab}$=0. The swimming force in the Gueron paper is of order Rdx, where dx is the size of the swimmer and R is the Riemann tensor. (The Riemann tensor falls off like r^-3.) If you make a little box in spacetime with sides of length dx, then the change in momentum between the initial side and the final side due to swimming is (F)(dt), which is on the order of (dx)(dx)=dx². But the ambiguity in parallel transport between a momentum vector at the initial side of the box and a momentum vector at the final side is of order Rdx². Therefore if $R\ne 0$, the swimmer is not violating the local conservation law.

 

[Sbaraka]

loving the discussion here, although all the maths is way over my head, I am slowly researching to help my understanding of what you guys are saying.

however it would be of great help if someone could refer to my 3 questions in the OP, as no one has answered or discussed the motion in those terms.
thanks.