Spacetime and Geometry · Problem 7.8

Cosmology — Sean Carroll

Physics — General Relativity · gravitational-bremsstrahlung

Problem

The gravitational analog of bremsstrahlung radiation is produced when two masses scatter off each other. Consider what happens when a small mass

m

scatters off a large mass

M

with impact parameter

b

and total energy

E = 0

. Take

M \gg m

and

M/b \ll 1

. The motion of the small mass can be described by Newtonian physics, since

M/b \ll 1

. If the orbit lies in the

(x, y)

plane and if the large mass sits at

(x, y, z) = (0, 0, 0)

, calculate the gravitational wave amplitude for both polarizations at

(x, y, z) = (0, 0, r)

. Since the motion is not periodic, the gravitational waves will be burst-like and composed of many different frequencies. On physical grounds, what do you expect the dominant frequency to be? Estimate the total energy radiated by the system. How does this compare to the peak kinetic energy of the small mass? Hint: The solution for the orbit can be found in Goldstein (2002). The solution is:

r = \frac{2b}{1 + \cos\theta},

t = \sqrt{\frac{2b^3}{M}} \left(\tan\frac{\theta}{2} + \frac{1}{3}\tan^3\frac{\theta}{2}\right).

Time runs from

t = (-\infty, \infty)

. Rather than using the above implicit solution for

\theta(t)

you might want to use

\dot{\theta} = \sqrt{\frac{M}{8b^3}}\,(1 + \cos\theta)^2.

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