Stability of relativistic double star system

[01030312]

We consider two stars which initially were at rest with each other, and were gravitationally influencing each other. We assume a kind of gravitation force which propagates at light speed. Now we give equal velocity (in relatively opposite direction) to both stars. Since gravity is propagated at 'c', and hence delayed, the direction of force on one star will be at retarded position of other. Clearly line of force will be slightly deviated from center of mass, thus leading to a non-central component in force. One can easily see that this force will in some time lead to exponential separation in both stars, if energy of the system is conserved. Thus system won't be stable.
I think that such a situation will never be achieved due to gravitational radiation. Is it true or is there some way to clear the stability problem?

 

[01030312]

P.S.- 1st paragraph is considered with geometry being a rigid structure, that is, no geometrodynamics. But gravity at light speed is of course there. 2nd paragraph considers geometrodynamics for stability.

 

[bcrowell]

We consider two stars which initially were at rest with each other, and were gravitationally influencing each other. We assume a kind of gravitation force which propagates at light speed. Now we give equal velocity (in relatively opposite direction) to both stars. Since gravity is propagated at 'c', and hence delayed, the direction of force on one star will be at retarded position of other. Clearly line of force will be slightly deviated from center of mass, thus leading to a non-central component in force. One can easily see that this force will in some time lead to exponential separation in both stars, if energy of the system is conserved. Thus system won't be stable.
I think that such a situation will never be achieved due to gravitational radiation. Is it true or is there some way to clear the stability problem?

An argument along these lines can be used to prove the existence of gravitational radiation. See Spacetime Physics by Taylor and Wheeler. I think you're wandering off track starting at "One can easily see..." The effect is in the opposite direction compared to what you thought. It isn't two effects that partially cancel. The effect you're talking about *is* gravitational radiation.

 

[01030312]

"bcrowell"
I completely lost you at "the effect you are talking about is gravitational...". I guess I am talking about a force on either of stars which must have a radial and a tangential component( tangential and approx. in direction of velocity). The tangential force deviates the trajectory from purely circular motion. This can be avoided if the system looses energy by gravitational radiation in geometrodynamic theory. Is this what you are stating?

 

[espen180]

01030312: This is what you had in mind, correct?

[PLAIN]http://img571.imageshack.us/img571/8158/binarystargr.png

The evolution of such a system seems to be that the distance between the bodies are increasing, and their speeds are increasing as well, but I don't know if such a system satisfies energy conservation unless energy is radiated off as gravitational radiation? Would mechanical energy decrease over time otherwise?

 

[bcrowell]

I see. Espen180's drawing makes it more clear to me what 01030312 presumably had in mind.

OK, to connect that to what I was saying, imagine a highly elliptical orbit instead of a circular one. This is essentially the case that Taylor and Wheeler talk about (although they phrase it in terms of Atlas taking two planets and moving them in and out as if he's working out in the gym). As the planets are approaching one another, each one feels a retarded force that is weaker than it "should" be according to Newton, so the amount of positive work done on each is smaller than the Newtonian value. As they recede from one another, the retarded force is smaller than the Newtonian value, so the negative work is greater than Newtonian. The result is that with each cycle, they lose energy. If we assume that energy should be conserved, then the only possible way of resolving the problem is to assume that this energy is radiated away as gravitational waves. So the retarding effect is not a countervailing effect that partially cancels the gravitational radiation, it *is* the effect of the gravitational radiation.

In the circular orbit case, each planet is doing positive work on the other, so by the work-kinetic energy theorem, they're gaining kinetic energy. But remember that when a non-contact force does acts on a particle, $\int F\cdot dr$, where r is the position of the object being acted on, doesn't give the change in the object's energy, it just gives the change in the object's *kinetic* energy. Each planet gains KE. As they gain KE, they remain in circular orbits, and the only way that can happen is if the radius of the orbit decreases. This leads to a loss of PE that is twice as big as the loss in KE, so over all, there is a loss of energy in the system. This is exactly what we observe in, e.g., the Hulse-Taylor system: the period is shortening over time. So again we have loss of energy from the system, the energy loss is accounted for by gravitational radiation, and there is only one effect, not two.

BTW, there is nothing in any of this that is specific to GR. Every argument that we've made in this thread applies equally to electromagnetism. We know how the analysis turns out in E&M, so it should be clear that it turns out the same way in GR.

 

[01030312]

'bcrowell'
Exactly that is gravitation radiation. Thanks for the point of view. And John Wheeler had a real sense of humor.