Pre-Phase A Report - Lisa - Nasa

4.4 Data analysis 111
with the phase modulation Φ(t) as given in Eq. (4.69) and the modulated amplitude
T (h+ + h×) T t (cf. Eq. 4.64). To see how the gravitational-wave detector works, recall
that General Relativity predicts that a ray of light connects a set of points by an interval
of zero or
ds 2 =0. (4.72)
For simplicity, let us consider first one arm of the the detector, which we take to lie in
first quadrant of the x-y plane, at an angle α to the x-axis. The above equation then
becomes
0 = ds 2
= gµνdx µ dx ν
= −c 2 dt 2 +(1+Hxx(t, x)) dx 2 +(1+Hyy(t, x)) dy 2 + Hxy(t, x) dxdy
+Hyx(t, x) dydx
= −c 2 dt 2 +[(1+Hxx(s)) cos 2 α +(1+Hyy(s)) sin 2 α
+( Hxy(s)+Hyx(s))sinαcos α] ds 2 , (4.73)
where ds ≡ dx 2 + dy 2 . In the standard deDonder gauge in which we are working,
the freely falling masses at the two ends of the arm maintain fixed coordinate locations
(x, y, z). Thus the light travel time τ between the two ends is determined by
c
0
τ
dt =
L
0
1+Hxx(s)cos 2 α + Hyy(s)sin 2 α + 1
2
Hxy(s)+Hyx(s) sin 2α ds.
(4.74)
Equivalently, we can say the arm length has changed by an amount δL given by (treating
the metric perturbation as approximately constant during the trip, and expanding the
square root):
δL = 1
2 L
Hxx cos 2 α + Hyy sin 2
α + Hxy sin(2α) . (4.75)
LISA is designed to measure the difference in the arm length changes, δ(L1−L2). Actually,
since there are three arms, LISA can measure two independent differences. We shall refer
to the combination L1−L2 as interferometer I, and to the combination (L1+L2−2L3)/ √ 3
as interferometer II. For simplicity, we let L1 make a 15 0 angle to the x-axis, while L2
makes a 75 0 angle (i.e., α = π/12 and 5π/12 for L1 and L2, respectively).
From Eq. (4.75) one easily shows that the strains associated with these particular combinations
are
√
3
hI =
4 (Hxx − Hyy) ,
√
3
hII =
4 (Hxy + Hyx) . (4.76)
Thus hI and hII directly measure the + and ×-polarisation components of a wave travelling
perpendicularly to the plane of the detector. One can therefore think of LISA
Corrected version 2.08 3-3-1999 9:33