Pre-Phase A Report - Lisa - Nasa

1.1 Theory of gravitational radiation 13
field, by the equivalence principle. Detectors work only because they sense the changes
in this acceleration across them. If two parts of a detector are separated by a vector L,
then it responds to a differential acceleration of order
L ·∇(∇Φ) ∼ LΦ/λ 2 . (1.3)
SincewehaveseenthatΦ∼ hc 2 (dropping the indices of hαβ in order to simplify this
order-of-magnitude argument), the differential acceleration is of order Lω 2 h.
If the detector is a solid body, such as the bar detectors described in Section 2.2.1,
the differential acceleration will be resisted by internal elastic stresses, and the resulting
mechanical motion can be complex. Bars are made so that they will “ring” for a long
time after a gravitational wave passes, making detection easier. If the detector consists
of separated masses that respond to the gravitational wave like free particles, then the
situation is easier to analyse. This is the case for interferometers, including LISA.
For two free masses separated by the vector L, the differential acceleration given by
Equation 1.3 leads to an equation for the change in their separation δ L,oforder
d2δL dt2 ∼ Lω2h. Since the time-derivatives on the left-hand-side just bring down factors of ω, we arrive at
the very simple equation δL/L ∼ h. A careful derivation shows that this is exact with a
further factor of 2:
δL 1
= h. (1.4)
L
Here we make contact with the geometrical interpretation of general relativity. The distances
L and δL should be interpreted as proper distances, the actual distances that a
meter-stick would measure at a given time. Then we see that h is indeed a metric, a distance
measure: as a gravitational wave passes, it stretches and shrinks the proper distance
between two free bodies. This equation also explains why interferometric detectors should
be made large: the technical problem is always to measure the small distance change δL,
and for a given wave amplitude h this distance change increases in proportion to L.
Polarization of gravitational waves. We have managed to discover much about gravitational
waves by ignoring all the indices and the full complexity of the field equations,
but this approach eventually reaches its limit. What we cannot discover without indices
is how the differential accelerations depend on the direction to the source of the wave.
Here there are two important results that we simply quote without proof:
2
• Gravitational waves are transverse. Like electromagnetic waves, they act only in
a plane perpendicular to their direction of propagation. This means that the two
separated masses will experience the maximum relative distance change if they are
perpendicular to the direction to the source; if they lie along that direction there
will be no change δL.
• In the transverse plane, gravitational waves are area preserving. This means that
if a wave increases the proper distance between two free masses that lie along a given
direction, it will simultaneously decrease the distance between two free masses lying
along the perpendicular direction in the transverse plane. The consequence of this
is illustrated in the standard polarization diagram, Figure 1.1 .
Corrected version 2.08 3-3-1999 9:33