Pre-Phase A Report - Lisa - Nasa

1.1 Theory of gravitational radiation 11
The replacement of Newtonian gravity by general relativity must, of course, still
reproduce the successes of Newtonian theory in appropriate circumstances, such as
when describing the solar system. General relativity has a well-defined Newtonian
limit: when gravitational fields are weak (gravitational potential energy small compared
to rest-mass energy) and motions are slow, then general relativity limits to
Newtonian gravity. This can only happen in a limited region of space, inside and
near to the source of gravity, the near zone. Far enough away, the gravitational
waves emitted by the source must be described by general relativity.
The field equations and gravitational waves. The Einstein field equations are inevitably
complicated. With 10 quantities that can create gravity (energy density, 3 components
of momentum density, and 6 components of stress), there must be 10 unknowns,
and these are represented by the components of the metric tensor in the geometrical language
of general relativity. Moreover, the equations are necessarily nonlinear, since the
energy carried away from a system by gravitational waves must produce a decrease in the
mass and hence of the gravitational attraction of the system.
With such a system, exact solutions for interesting physical situations are rare. It is
remarkable, therefore, that there is a unique solution that describes a black hole (with 2
parameters, for its mass and angular momentum), and that it is exactly known. This is
called the Kerr metric. Establishing its uniqueness was one of the most important results
in general relativity in the last 30 years. The theorem is that any isolated, uncharged
black hole must be described by the Kerr metric, and therefore that any given black hole is
completely specified by giving its mass and spin. This is known as the “no-hair theorem”:
black holes have no “hair”, no extra fuzz to their shape and field that is not determined
by their mass and spin.
If LISA observes neutron stars orbiting massive black holes, the detailed waveform
will measure the multipole moments of the black hole. If they do not
conform to those of Kerr, as determined by the lowest 2 measured moments,
then the no-hair theorem and general relativity itself may be wrong.
There are no exact solutions in general relativity for the 2-body problem, the orbital motion
of two bodies around one another. Considerable effort has therefore been spent over
the last 30 years to develop suitable approximation methods to describe the orbits. By
expanding about the Newtonian limit one obtains the post-Newtonian hierarchy of approximations.
The first post-Newtonian equations account for such things as the perihelion
shift in binary orbits. Higher orders include gravitational spin-orbit (Lense-Thirring) and
spin-spin effects, gravitational radiation reaction, and so on. These approximations give
detailed predictions for the waveforms expected from relativistic systems, such as black
holes spiralling together but still well separated, and neutron stars orbiting near massive
black holes.
When a neutron star gets close to a massive black hole, the post-Newtonian approximation
fails, but one can still get good predictions using linear perturbation theory, in which the
gravitational field of the neutron star is treated as a small perturbation of the field of the
black hole. This technique is well-developed for orbits around non-rotating black holes
(Schwarzschild black holes), and it should be completely understood for orbits around
general black holes within the next 5 years.
Corrected version 2.08 3-3-1999 9:33