Lecture Notes for 12/2/08
Benjamin Hall and Hidris Pal
December 4, 2008
0.1
Continuing linear perturbation
As stated in the last lecture, we decompose the metric perturbation hµν as:
hµν = (Φ, Ψ, wi , S ij )
(1)
where S ij is traceless. The vector part wi is decomposed further: wi =
i
w⊥
+ wki where
i
∂i w ⊥
= 0
(2)
k
∂i w k
(3)
ijk
= 0.
i
These conditions imply that w⊥
has 2 degrees of freedom and wki = ∂i ξ has
only one degree of freedom.
S ij can be similarly decomposed: S ij = S⊥ij + SSij + Skij where
∂i S⊥ij = 0
∂i ∂j SSij = 0
l
ijk ∂j ∂l Skk
= 0.
(4)
(5)
(6)
Again, Skij has only 1 d.o.f, SSij has 2, and S⊥ij has the remaining 2. This is
because we can write
1
Skij = (∂i ∂j − ∇2 δij )Θ
2
SSij = ∂(i Ξj) .
1
(7)
(8)
As was stated in previous classes, gravitational waves are spin 2 (under
spatial rotations) traceless tensors; the only part of the perturbation matching this criterion is S ij , specifically S⊥ij , as it is the only spin-2 part of the
metric perturbation.
For the remainder of these notes we will use the traditional notation
hijT T ≡ S⊥ij , where the “TT” stands for “transverse traceless.” If expanded
in spherical harmonics (for example) hijT T has only terms with L ≥ 2, so
it doesn’t include the gravitational monopole (∆m, S = 0, L = 0), the
gravitational dipole (relating to motion against the background, S = 0, L =
1) or the (static) angular momentum (S = 1, L = 1) terms that may be
present from the source. Various choices of gauge can push the gravitational
waves to other parts of the perturbation by reducing spin, but cannot change
L.
0.2
Gravitational Waves
For this section we will consider S⊥ij = hijT T = Aexp[i(ωt − ~k · ~x)]; ki S⊥ij = 0
and also set ~k = (0, 0, k z ). Under these assumptions we can write:
hµν =
h+ hx 0
hx −h+ 0
0
0
0
(9)
The action of a pure h+ or hx polarized gravitational wave on a circular array
of masses is shown in Figure 1.
Note, this formalism only describes the weak-field limit (small sources or
large separation between source and detector). It is totally insufficient to
describe the generation of gravitational waves close to strong sources such
as black holes. We can thus write Gµν (η + h) = 8πTµν for weak sources and
slow motion. We know from conservation that time and spatial derivatives
of T µν can be related:
∂µ T µ0 = 0 −→ ∂0 T 00 = −∂i T i0
∂µ T µi = 0 −→ ∂0 T 0i = −∂j T ji .
(10)
(11)
After much work, we can then write the metric perturbation as:
hij =
2G d2
[Iij (t − r)]
r dt2
2
(12)
where hµν ≡ (h − 12 hρρ )µν . The quantity Iij is the mass quadrupole moment,
defined as
Z
0
Iij (t ) = d3 y y i y j T 00 (t0 , y)
(13)
p
evaluated at t0 = t − (xi − y i )2 . y i is the distance between a portion of the
source and the point where the measurement is being done. With care, this
formulation also works at large distances from even moderate sources.
This gives us the amplitude, but to get the energy-density, we need to consider terms quadratic in the perturbation (2nd order perturbations). Since
the waves are being measured far from the source (at infinity, to be precise),
there are no sources, which means Tµν = 0. This implies that we can write
G(η + h) = G0 + G1 + G2 + · · ·
(14)
where each term is of the indicated order in h. The energy density can be
written as
1
h∂µ hTijT ∂ν hTijT i
(15)
tµν =
32πG
(brackets indicate an average over a suitable region) and the power radiated
to infinity as
G d3
d3
P = − 5 h 3 J ij 3 Jij i
(16)
5c dt
dt
where J ij = I ij − 31 Ikk δ ij .
For 2 newtonian-like (ignoring higher order effects) masses in circular
orbit: define φ = GM
. 1, P = − 128G
M 2 Ω6 R4 . We know from previous
Rc2
5c5
GM 1/2
for Schwartzchild. This implies that P ∼ φ5 =
work that Ω = ( 4R3 )
5 GM
− 2c
(
)5 1. For solar-mass black holes, this is (for a small time) more
5G Rc2
luminous than stars. As the black holes spiral in, the amplitude grows as
1
and the frequency increases (both with modulation due to the structure
r5
of the infalling objects); afterwards, the frequencies and decay periods are
completely characterized by the total mass and total angular momentum of
the new black hole.
The maximum amplitudes expected can be estimated:
Z L0 √
L0 =
dx2
(17)
0
Z L0 p
Z L0
2
L0 + δL =
dx (1 + 2h+ ) ∼
dx(1 + h+ )
(18)
0
0
δL
∴h∼
. 10−21
L
3
(19)