Do Falling Particles in Schwarzschild Solution Radiate?

[Passionflower]

Consider the Schwarzschild solution where millions of test particles move towards the event horizon.

Will the particles falling in radiate? If so, how can they radiate as they do not have any proper acceleration?

 

[bcrowell]

Consider the Schwarzschild solution where millions of test particles move towards the event horizon.

Will the particles falling in radiate?

If the infalling cloud is spherically symmetric, then there is no radiation: http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity)

In the case of a single infalling particle, the radiated power is proportional to the mass of the test particle raised to some power p, where p>1. (I believe p=2 for a small mass infalling toward a large mass.) The whole idea of a test particle is that we're taking the limit where its mass is small, and therefore if the radiated power is proportional to $m^p$, it becomes negligible, in the sense that the square of an infinitesimal number dx is negligible compared to the original number dx. If radiation by a test particle wasn't negligible in this sense, then there would be no way to define geodesics using test particles.

If so, how can they radiate as they do not have any proper acceleration?

If a mass m infalls toward another mass, yes, you will get gravitational radiation. If m is infinitesimally small (the case of a test particle), then there is no radiation. If m is not infinitesimally small, then it radiates, but it also doesn't follow a geodesic, so it's not true that it experiences zero proper acceleration.

 

[Passionflower]

In the case of a single infalling particle, the radiated power is proportional to the mass of the test particle raised to some power p, where p>1. (I believe p=2 for a small mass infalling toward a large mass.)

Agreed, assuming you speak of gravitational radiation here.

If m is not infinitesimally small, then it radiates, but it also doesn't follow a geodesic, so it's not true that it experiences zero proper acceleration.

So in addition to gravitational radiation, when it accelerates is must also radiate EM radiation do you agree?

 

[bcrowell]

Agreed, assuming you speak of gravitational radiation here.


So in addition to gravitational radiation, when it accelerates is must also radiate EM radiation do you agree?

Was your original question intended to be about gravitational radiation, or elecromagnetic radiation? I took it as a question about gravitational radiation, since you didn't state that the particles were charged.

A uniformly charged infalling spherical cloud will not radiate EM radiation. The reason is the same as for gravitational radiation: neither gravitational waves nor EM waves have propagating monopole modes.

Anyway, the issue of EM radiation by free-falling particles is complicated. The classic paper is C. Morette-DeWitt and B.S. DeWitt, "Falling Charges," Physics, 1,3-20 (1964), available at http://www.physics.princeton.edu/~mcdonald/examples/EM/dewitt_physics_1_3_64.pdf This issue has been discussed many times on this forum within the last year, so there's not much point in repeating the discussion.