PHYS 5770 Grav Theory Spring 2008. Problem Set 6. Due Tue 29 Apr
1. Will you be torn apart when two black holes merge?
This question is posed on behalf of Phil Plait, the Bad Astronomer, whose upcoming (Oc-
tober 2008) book “Death from the Skies!” contains a chapter “Seven ways a black hole can
kill you”. One of the ways, says Phil, is to stand near a pair of merging black holes, and be
torn apart by the tidal forces from the gravitational waves. Is it true?
(a) Quadrupole moment
Consider a pair of masses M1and M2in circular orbit, with position vectors r1and r2relative
to their center of mass. Argue that the quadrupole moment –
Iij of the mass distribution
–
Iij ≡X
masses a
Ma(ra,i ra,j −1
3δij r2
a) (1.1)
is
–
Iij =mr2(ˆriˆrj−1
3δij) (1.2)
where r≡r2−r1is the orbital separation, and mis the reduced mass
m≡M1M2
M, M ≡M1+M2.(1.3)
[Hint: Assume for simplicity that the orbit is described by classical Newtonian mechanics.
After all, our aim is to decide whether we die, and Newton is good enough for that.]
(b) Tensor components
Suppose that the orbital plane is inclined at inclination angle ιto the line-of-sight. Choose
the observer’s locally inertial frame so that the x-axis is the line-of-sight direction from the
center of mass of the binary to the observer, and the y-axis points in the plane of the orbit.
Argue that the orbital separation ris
r=r[(ˆxcos ι−ˆzsin ι) cos ωt + ˆysin ωt] (1.4)
where ωis the orbital frequency (units G= 1)
ω2=M
r3.(1.5)
Deduce that the tensor components of the quadrupole moment are
–
I+≡1
2(–
Iyy −–
Izz ) = 1
4mr2cos2ι−(1 + sin2ι) cos 2ωt,(1.6a)
–
I×≡–
Iyz =−1
2mr2sin ιsin 2ωt . (1.6b)
(c) Tensor perturbation
Deduce the tensor perturbations h+and h×at distance xfrom the orbiting masses from the
quadrupole formula
hij =−1
x¨
–
Iij
tensor
.(1.7)
1
