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4.1 Strong-field systems in general relativity

4.1.1 Defining weak and strong gravity

In the solar system, gravity is weak, in the sense that the Newtonian gravitational potential and related variables (U (x,t) ∼ v2 ∼ pāˆ•ρ ∼ ε) 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 [16524] for reviews). Another is black hole perturbation theory (see [188Jump To The Next Citation Point146235Jump To The Next Citation Point] for reviews).

4.1.2 Compact bodies and the strong equivalence principle

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 [70Jump To The Next Citation Point] 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 gμν. 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 m1, m2, J1, and J2 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 φ 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 φ). 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 φ0 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 body can be characterized by several mass functions mA (φ), 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.

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