Capture by Gravitational Radiation in 2-Body System?

[Yazzledore]

Is it possible for enough energy to be dissipated in the form of gravitational radiation in a two-body system to allow for capture? From what I remember, you would need extremely massive bodies passing extremely close to each other: I'd like to know how massive and how close.

It has been a few years since I did any GR and don't feel confident in my ability to do the math anymore (but I think I can still understand it). If it is possible, I was hoping someone had seen a good paper to which they could link me (I couldn't find one).

 

[PeterDonis]

Is it possible for enough energy to be dissipated in the form of gravitational radiation in a two-body system to allow for capture?

I assume you mean a two-body system that does not start out bound. I haven't done the calculation, but I would guess that this is, while not impossible in principle, extremely unlikely. The time scale for gravitational radiation to carry significant energy away from a system is likely to be much longer than the time scale for a pair of unbound objects to swing past each other and escape.

The basic intuition here is that gravitational radiation requires many orbits to carry away significant energy. But for a pair of unbound objects passing each other, you only have one "orbit"--they pass once and then fly apart. It doesn't seem to me that gravitational radiation could do enough during that one brief passage. But, as I said, I haven't done the calculation.

 

[pervect]

Is it possible for enough energy to be dissipated in the form of gravitational radiation in a two-body system to allow for capture? From what I remember, you would need extremely massive bodies passing extremely close to each other: I'd like to know how massive and how close.

It has been a few years since I did any GR and don't feel confident in my ability to do the math anymore (but I think I can still understand it). If it is possible, I was hoping someone had seen a good paper to which they could link me (I couldn't find one).

I don't think you need gravitational radiation to have capture in a 2 body system in GR. Consider a test particle that falls from infinity and passes within the photon sphere of a black hole, for instance. The test particle can't orbit - no orbit exists inside the photon sphere - so it must be captured.

[add]And - a test particle follows a geodesic, it doesn't emit any gravitational radiation.

 

[Vanadium 50]

If the capture-ee comes in on a parabolic trajectory, if the system loses any energy at all the trajectory will become elliptical.

 

[PeterDonis]

I don't think you need gravitational radiation to have capture in a 2 body system in GR.

You don't. But the question I understand the OP to be asking is, given a scenario in which, leaving gravitational radiation out of account, the 2 body system would not result in capture, could the effects of gravitational radiation change that outcome so that there would be capture?

a test particle follows a geodesic, it doesn't emit any gravitational radiation.

Following a geodesic is not a sufficient condition for not emitting gravitational radiation. The binary pulsars whose orbital decays have been used to confirm GR predictions of gravitational radiation are traveling on geodesic orbits.

Being a test particle, in itself, is a sufficient condition for not emitting gravitational radiation, however, because by definition a test particle cannot contribute at all to any system's gravitational properties, including those which cause gravitational radiation to be emitted (i.e., a time varying mass quadrupole moment).

 

[pervect]

Following a geodesic is not a sufficient condition for not emitting gravitational radiation. The binary pulsars whose orbital decays have been used to confirm GR predictions of gravitational radiation are traveling on geodesic orbits.

It's getting a bit off topic, but do you have a reference for this point? I would think that the momentum and energy carried away by the gravitational radiation would cause departures of the non-test particle from a geodesic.

It's easy enough to come up with a test particle orbit that does not decay - if a massive body could follow the same non-decaying orbit and still emit gravitational radiation, I don't see how energy-at-infinity could be conserved.

 

[PeterDonis]

do you have a reference for this point?

I can try to dig one up, but AFAIK the models that generate the predictions for which Hulse and Taylor won the Nobel prize, and similar predictions for other binary pulsars, assume geodesic orbits, so the fact that they match the data so well is strong evidence that the binary pulsars are in fact traveling on geodesic orbits, at least to a very, very good approximation.

I would think that the momentum and energy carried away by the gravitational radiation would cause departures of the non-test particle from a geodesic.

No, they just mean that the geodesics are geodesics of the full spacetime geometry including the effects of gravitational waves, rather than the geodesics of an idealized Schwarzschild geometry without gravitational waves, which are the kinds of orbits we intuitively think about when we think about orbiting objects.

It's easy enough to come up with a test particle orbit that does not decay

Sure, if you ignore the effects of gravitational waves. But if you include those effects, the spacetime geometry is no longer Schwarzschild, so the non-decaying geodesic orbits you are intuitively thinking of no longer exist.

If, OTOH, you just mean that if all we have is one massive body with test particles orbiting it, there are no gravitational waves period and the geometry is just the static Schwarzschild geometry, that I agree with. But it's irrelevant to the case under discussion.

if a massive body could follow the same non-decaying orbit and still emit gravitational radiation

Which it can't. You are right that this would violate energy conservation. But the energy being conserved here is not what you appear to think it is. See below.

I don't see how energy-at-infinity could be conserved.

Energy at infinity is only a conserved quantity along a geodesic orbit if the spacetime is stationary. If gravitational waves are present, the spacetime is not stationary, so there is no such conserved quantity.

The only conserved energy in the case we are discussing, a spacetime with gravitational waves but which is still asymptotically flat, is the ADM energy of the spacetime as a whole. But that's not the same as energy at infinity for an orbiting object. If an object could follow a non-decaying geodesic orbit in a spacetime containing gravitational waves, then the ADM energy would not be conserved (heuristically, because the non-decaying orbit would be making a constant contribution to the ADM energy, while the gravitational waves would be making an increasing contribution--whereas if the orbit is decaying, its contribution to the ADM energy decreases, by the same amount that the gravitational wave contribution increases).