LISALISA - iucaa

Chapter 1 Scientific Objectives
1.1.2 The nature of gravitational waves in general relativity
Tidal accelerations.
We remarked above that the observable effects of gravity lie in the tidal forces. A gravitational
wave detector would not respond to the acceleration produced by the wave (as given by ∇Φ),
since the whole detector would fall freely in this 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)
Since we have seen that Φ ∼ hc 2 (dropping the indices of h αβ in order to simplify this order-ofmagnitude
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, of order
d 2 δL
dt 2 ∼ Lω 2 h.
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
L = 1 h. (1.4)
2
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:
• 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.
13-9-2000 11:47 12 Corrected version 1.04