In the solar system, gravity is weak, in the sense that the Newtonian gravitational potential and related
variables () are much smaller than unity everywhere. This is the basis for the
post-Newtonian expansion and for the “parametrized post-Newtonian” framework described in Section 3.2.
“Strong-field” systems are those for which the simple 1PN approximation of the PPN framework is no
longer appropriate. This can occur in a number of situations:
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 analyzed 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 review
(see [165, 24] for reviews). Another is black hole perturbation theory (see [188, 146, 235
] for
reviews).
When dealing with the motion and gravitational wave generation by orbiting bodies, one finds a remarkable simplification within GR. As long as the bodies are sufficiently well-separated that one can ignore tidal interactions and other effects that depend upon the finite extent of the bodies (such as their quadrupole and higher multipole moments), then all aspects of their orbital behavior and gravitational wave generation can be characterized by just two parameters: mass and angular momentum. Whether their internal structure is highly relativistic, as in black holes or neutron stars, or non-relativistic as in the Earth and Sun, only the mass and angular momentum are needed. Furthermore, both quantities are measurable in principle by examining the external gravitational field of the bodies, and make no reference whatsoever to their interiors.
Damour [70] calls this the “effacement” of the bodies’ internal structure. It is a consequence of the 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
and a planet of mass
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.
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
inside the chosen body, alter
its internal structure (specifically its gravitational binding energy) and hence alter its mass.
Effectively, each body can be characterized by several mass functions
, which depend on
the value of the scalar field at its location, and several distinct masses come into play, such as
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 10 - 20 percent of
its total mass. At 1PN order, the leading manifestation of this phenomenon is the Nordtvedt
effect.
This is how the study of orbiting systems containing compact objects provides strong-field tests of GR. 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.
![]() |
http://www.livingreviews.org/lrr-2006-3 |
© Max Planck Society and the author(s)
Problems/comments to |