Of course, some systems cannot be properly described by any
post-Newtonian approximation because their behavior is
fundamentally controlled by strong gravity. These include the
imploding cores of supernovae, the final merger of two compact
objects, the quasinormal-mode vibrations of neutron stars and
black holes, the structure of rapidly rotating neutron stars, and
so on. Phenomena such as these must be analysed using different
techniques. Chief among these is the full solution of Einstein's
equations via numerical methods. This field of ``numerical
relativity'' is a rapidly growing and maturing branch of
gravitational physics, whose description is beyond the scope of
this article. Another is black hole perturbation theory
(see [93] for a review).
Damour [39] calls this the ``effacement'' of the bodies' internal
structure. It is a consequence of the strong equivalence
principle (SEP), described in Section
3.1.2
.
General relativity satisfies SEP because it contains one and
only one gravitational field, the spacetime metric
. Consider the motion of a body in a binary system, whose size is
small compared to the binary separation. Surround the body by a
region that is large compared to the size of the body, yet small
compared to the separation. Because of the general covariance of
the theory, one can choose a freely-falling coordinate system
which comoves with the body, whose spacetime metric takes the
Minkowski form at its outer boundary (ignoring tidal effects
generated by the companion). There is thus no evidence of the
presence of the companion body, and the structure of the chosen
body can be obtained using the field equations of GR in this
coordinate system. Far from the chosen body, the metric is
characterized by the mass and angular momentum (assuming that one
ignores quadrupole and higher multipole moments of the body) as
measured far from the body using orbiting test particles and
gyroscopes. These asymptotically measured quantities are
oblivious to the body's internal structure. A black hole of mass
m
and a planet of mass
m
would produce identical spacetimes in this outer region.
The geometry of this region surrounding the one body must be
matched to the geometry provided by the companion body.
Einstein's equations provide consistency conditions for this
matching that yield constraints on the motion of the bodies.
These are the equations of motion. As a result the motion of two
planets of mass and angular momentum
,
,
and
is identical to that of two black holes of the same mass and
angular momentum (again, ignoring tidal effects).
This effacement does not occur in an alternative gravitional
theory like scalar-tensor gravity. There, in addition to the
spacetime metric, a scalar field
is generated by the masses of the bodies, and controls the local
value of the gravitational coupling constant (i.e.
G
is a function of
). Now, in the local frame surrounding one of the bodies in our
binary system, while the metric can still be made Minkowskian far
away, the scalar field will take on a value
determined by the companion body. This can affect the value of
G
inside the chosen body, alter its internal structure
(specifically its gravitational binding energy) and hence alter
its mass. Effectively, each mass becomes several functions
of the value of the scalar field at its location, and several
distinct masses come into play, inertial mass, gravitational
mass, ``radiation'' mass, etc. The precise nature of the
functions will depend on the body, specifically on its
gravitational binding energy, and as a result, the motion and
gravitational radiation may depend on the internal structure of
each body. For compact bodies such as neutron stars, and black
holes these internal structure effects could be large; for
example, the gravitational binding energy of a neutron star can
be 40 percent of its total mass. At 1PN order, the leading
manifestation of this effect is the Nordtvedt effect.
This is how the study of orbiting systems containing compact objects provides strong-field tests of general relativity. Even though the strong-field nature of the bodies is effaced in GR, it is not in other theories, thus any result in agreement with the predictions of GR constitutes a kind of ``null'' test of strong-field gravity.
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The Confrontation between General Relativity and
Experiment
Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |