# Swimming in space-time

1. Jul 21, 2009

### Jonathan Scott

"Swimming" in space-time

On the main forums page, there is a link in the "Scientific American" section to an article http://www.scientificamerican.com/article.cfm?id=surprises-from-general-relativity".

I've had a look at the article and I don't even begin to believe it.

Even if you forgive the illustrative device of macroscopically-curved space, the idea that one could wriggle around in a way which escapes conservation laws seems extremely far-fetched. Even his simplified example of "swimming" on an ordinary sphere seems to be in obvious violation of conservation laws.

I checked the article date and it's not April 1.

Does anyone think this could REALLY be correct?

(I'd go so far as to say that if GR really did work like that, I'd count it as evidence of a problem in GR).

Last edited by a moderator: Apr 24, 2017
2. Jul 21, 2009

### tiny-tim

Hi Jonathan!

I think the clue is in the example Guéron gives of swimming on a sphere (http://www.scientificamerican.com/article.cfm?id=surprises-from-general-relativity&page=2") …

I'll change it slightly …

Suppose you have two bodies, of masses m and 2m, stationary at O.

Stage 1: Let them push each other apart, a distance 3r. Since the c.o.m. will still be at O, that means the m mass is now twice as far from O as the 2m mass, and in the opposite direction.

Stage 2: Let the 2m mass split into two equal parts (each therefore of mass m, the same as the other mass), which push each other apart, perpendicular to the original direction, until the three masses form an equilateral triangle.

Stage 3: Join the three masses to C, the centre of the equilateral triangle, and let them pull each other in, obviously coming together at C.

In flat space, C is at a distance r, say, from each vertex of the equilateral triangle, and a distance r.sin30º, = r/2, from each side of the equilateral triangle, and so C and O are the same point, so the whole mass of 3m is back where it started.

But on the surface of a sphere, C is at a distance r from each vertex of the equilateral triangle, and a distance r.sinθ, which is slightly more than r/2, from each side of the equilateral triangle (where θ 60º and 90º are the angles of the obvious smaller triangle), and so C and O are not the same point, so the whole mass of 3m has moved slightly.

And the same applies in any curved space, whether positively or negatively curved …
Basically, in curved space, the "average position" of three bodies is not well-defined, since it depends on the order in which you do the averaging.

In other words, the c.o.m. of three bodies is not well-defined, so there is no conservation law relating to the c.o.m., and so there is no paradox in the c.o.m. moving.

Last edited by a moderator: Apr 24, 2017
3. Jul 21, 2009

### George Jones

Staff Emeritus
4. Jul 21, 2009

### Jonathan Scott

Re: "Swimming" in space-time

I read that but I'm still having difficulty in believing it.

I'm assuming his "swimming on a sphere" case is like literally being a 2D object on a frictionless sphere (with the usual 3D laws of physics) and trying to move around. That can in theory work, in that it's certainly possible to rearrange things. If for example you split a mass at the equator into two parts which are pushed apart until they land up at the poles, then you can pull them back together to any other point on the equator. This doesn't violate any conservation laws because this will apply some force perpendicularly to the surface of the sphere which will shift the sphere itself a tiny bit.

However, I don't see how the curvature of GR could be like applying a force in a perpendicular direction to space, and in any case I find it hard to believe that one could be in a frame of reference where space alone is curved without that being associated with major acceleration or non-zero mass density. I think I'd need some convincing on that too.

I'd agree that in a local bit of curved space in geometry (not GR) such oddities could perhaps be achieved, but in GR the masses affect the space as well as the space affecting the masses and conservation of momentum and energy still hold on a scale large enough to approximate flatness even if there are regions of significant curvature locally.

Still, you're welcome to try to convince me that the "swimming" effect can work in GR.

5. Jul 21, 2009

### tiny-tim

No, Guéron's and my examples treat the sphere as fixed: the masses start in the same position, and finish in the same position, but have moved relative to the sphere, but without violating conservation.
And the force is not perpendicular to the sphere, or to space: it is along the surface of the sphere, or within space.

6. Jul 22, 2009

### Nickelodeon

Re: "Swimming" in space-time

I think he hasn't factored in the modification to the local geodesic caused by the mass of the weights on the end of his arms together with a modification caused by the acceleration and deceleration of these weights which will also modify the geodesic.

I think what will happen as he swims towards his rocket is that his small amount of movement, if any, towards the rocket will move the rocket an equal amount away from him.

7. Jul 22, 2009

### tiny-tim

You mean he can move both himself and the rocket in the same direction, and without even touching the rocket?

8. Jul 22, 2009

### Nickelodeon

Re: "Swimming" in space-time

Strange I know but yes. Taking an extreme example, if you have a satellite moving around the earth in orbit and you give the earth a nudge it will effect the motion of the satellite.

In this case, he slightly moves the earth towards him which in turn moves the rocket

9. Jul 22, 2009

### Stingray

Re: "Swimming" in space-time

Similar effects exist even in Newtonian gravity. I disagree with the author of that article that the relativistic versions are fundamentally different. They do allow for a wider array of phenomena, but I think it's a very similar mechanism.

In either case, there are no conservation laws violated. The point is that an extended object in an inhomogenous gravitational field can "push" or "pull" off of those inhomogeneities. In Newtonian gravity, these are the forces that couple to the quadrupole and higher moments of an extended mass. The same interpretations also arise in GR, although the relevant formalism is too complicated to be taught in most textbooks.

10. Jul 22, 2009

### tiny-tim

Hi Stingray!
No … those inhomogeneities are the difference … see http://www.scientificamerican.com/article.cfm?id=surprises-from-general-relativity&page=3" of Guéron's article …
The Newtonian method works only because Earth's gravitational field is inhomogeneous, but Wisdom's method (in Guéron's article) works even in a completely homogeneous gravitational field.

And the Newtonian method uses the "springiness" of Earth's gravitational field, and must be timed to "harmonise" with it, but Wisdom's method can be timed quite randomly and still work.

Last edited by a moderator: Apr 24, 2017
11. Jul 22, 2009

### fleem

Re: "Swimming" in space-time

But if you can move between two points when they happen to be at the same gravitational potential, then you should be able to move up an infinitesimal gravitational gradient using the same method, and thus up any gravitational potential using the same method.

12. Jul 22, 2009

### Stingray

Re: "Swimming" in space-time

I disagree. In their papers, Wisdom and Gueron both consider motion in Schwarzchild fields. These are inhomogeneous. Furthermore, a "uniform gravitational field" is flat spacetime. I don't think it's controversial to say that there are no swimming effects in that case.

To generalize slightly, no curvature inhomogeneities implies either Minkowski or de Sitter spacetime. In both cases, there are a full set of 10 conservation laws associated with the background Killing fields. This implies that no amount of internal deformation can affect the center-of-mass motion (ignoring beamed radiation or anything like that). The only way out of this is if the body is so large that it fills almost the entire universe. This is the only case where I'd agree with Gueron that centers-of-mass are intrinsically hopeless to define.

It is possible to get motion through internal deformation in (cosmological) spacetimes that are homogeneous only in spatial directions. I actually wrote a paper about this. Intuitively, components of the fluid moving with respect to the cosmological rest frame no longer see spatially homogeneity. They can push and pull off of the "timelike curvature." This is essentially the Newtonian effect with allowances for Lorentz boosts.

I need to think more about the timing argument.