Pre-Phase A Report - Lisa - Nasa

10 Chapter 1 Scientific Objectives
measure different energies and different densities (volume is Lorentz-contracted), so
the actual source has to include not only energy but also momentum, and not only
densities but also fluxes. Since pressure is a momentum flux (it transfers momentum
across surfaces), relativistic gravity can be created by mass, momentum, pressure,
and other stresses.
Among the consequences of this that are observable by LISA are gravitational
effects due to spin.
These include the Lense-Thirring effect, which is the gravitational analogue of spinorbit
coupling, and gravitational spin-spin coupling. The first effect causes the
orbital plane of a neutron star around a spinning black hole to rotate in the direction
of the spin; the second causes the orbit of a spinning neutron star to differ
from the orbit of a simple test particle. (This is another example of the failure of
the equivalence principle for a macroscopic “particle”.) Both of these orbital effects
create distinctive features in the waveform of the gravitational waves from the
system.
Gravitational waves themselves are, of course, a consequence of special relativity
applied to gravity. Any change to a source of gravity (e.g. the position of a star)
must change the gravitational field, and this change cannot move outwards faster
than light. Far enough from the source, this change is just a ripple in the gravitational
field. In general relativity, this ripple moves at the speed of light. In principle,
all relativistic gravitation theories must include gravitational waves, although they
could propagate slower than light. Theories will differ in their polarization properties,
described for general relativity below.
Special relativity and the equivalence principle place a strong constraint on the
source of gravitational waves. At least for sources that are not highly relativistic, one
can decompose the source into multipoles, in close analogy to the standard way of
treating electromagnetic radiation. The electromagnetic analogy lets us anticipate
an important result. The monopole moment of the mass distribution is just the
total mass. By the equivalence principle, this is conserved, apart from the energy
radiated in gravitational waves (the part that violates the equivalence principle for
the motion of the source). As for all fields, this energy is quadratic in the amplitude
of the gravitational wave, so it is a second-order effect. To first order, the monopole
moment is constant, so there is no monopole emission of gravitational radiation.
(Conservation of charge leads to the same conclusion in electromagnetism.)
The dipole moment of the mass distribution also creates no radiation: its time
derivative is the total momentum of the source, and this is also conserved in the
same way. (In electromagnetism, the dipole moment obeys no such conservation law,
except for systems where the ratio of charge to mass is the same for all particles.)
It follows that the dominant gravitational radiation from a source comes from the
time-dependent quadrupole moment of the system. Most estimates of expected
wave amplitudes rely on the quadrupole approximation, neglecting higher multipole
moments. This is a good approximation for weakly relativistic systems, but only an
order-of-magnitude estimate for relativistic events, such as the waveform produced
by the final merger of two black holes.
3-3-1999 9:33 Corrected version 2.08