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2 The motivation for strong-field tests

Most physical scientists would agree that there is very little need to motivate testing one of the fundamental theories of physics in a regime that experiments have probed only marginally, so far. However, in the particular case of testing the strong-field predictions of general relativity, there exist at least three arguments that provide additional strong support to such an endeavor. First, there is no fundamental reason to choose Einstein’s equations over other alternatives. Second, gravitational tests to date seldom probe strong gravitational fields. Finally, it is known that general relativity breaks down in the strong-field regime. I will now elaborate on each of these arguments.

∙ There is no fundamental reason to choose Einstein’s equations over other alternatives. – All theories of gravity, including Newton’s theory and general relativity, have two distinct ingredients. The first describes how matter moves in the presence of a gravitational field. The second describes how the gravitational field is generated in the presence of matter. For Newtonian dynamics, the first ingredient is Newton’s second law together with the assertion that the gravitational and inertial masses of an object are the same; the second ingredient is Poisson’s equation. For general relativity, the first ingredient arises from the equivalence principle, whereas the second is Einstein’s field equation.

The equivalence principle, in its various formulations, dictates the geometric aspects of the theory [181Jump To The Next Citation Point]: it is impossible to tell the difference between a reference frame at rest and one freely falling in a gravitational field by performing local, non-gravitational (for the Einstein Equivalence Principle) or even gravitational (for the Strong Equivalence Principle) experiments. Moreover, the equivalence principle encompasses the Lorentz symmetry, as well as our belief that there is no preferred frame and position anywhere in the universe. Because of its central importance in any gravity theory, there have been many attempts during the last century at testing the validity of the equivalence principle. These were performed mostly in the weak-field regime and have resulted in upper limits on possible violations of this principle that are as stringent as one part in 1012 [181Jump To The Next Citation Point].

Contrary to the case of the equivalence principle, there are no compelling arguments one can make that lead uniquely to Einstein’s field equation. In fact, Einstein reached the field equation, more or less, by reverse engineering (see the informative discussion in [102Jump To The Next Citation Point117]) and, soon afterwards, Hilbert constructed a Lagrangian action that leads to the same equation. The Einstein–Hilbert action is directly proportional to the Ricci scalar, R,

4 ∫ --- S = -c---- d4x√ − g (R − 2Λ ) , (1 ) 16πG
where g ≡ det|gμν|, gμν is the spacetime metric, c is the speed of light, G is the gravitational constant, and Λ is the cosmological constant. While such a theory is entirely self-consistent at the classical level, it may represent only an approximation that is valid at the scales of curvature that are found in terrestrial, solar, and stellar-system tests.

Indeed, a self-consistent theory of gravity can also be constructed for any other action that obeys the following four simple requirements [102]. It has to: (i) reproduce the Minkowski spacetime in the absence of matter and the cosmological constant, (ii) be constructed from only the Riemann curvature tensor and the metric, (iii) follow the symmetries and conservation laws of the stress-energy tensor of matter, and (iv) reproduce Poisson’s equation in the Newtonian limit. Of all the possibilities that meet these requirements, the field equations that are derived from the Einstein–Hilbert action are the only ones that are also linear in the Riemann tensor. Albeit simple and elegant, a more general classical action of the form [154Jump To The Next Citation Point]

4 ∫ S = --c--- d4x √ −-gf(R ) , (2 ) 16πG
also obeys the same requirements. Indeed, the action (2View Equation) results in a field equation that allows for the Minkowski solution in the absence of matter, is constructed only from the Riemann tensor, obeys the usual symmetries and conservation laws [178Jump To The Next Citation Point], and can be made to produce negligible corrections at the small curvatures probed by weak-field gravitational experiments. On the other hand, the predictions of the theory may be significantly different at the strong curvatures probed by gravitational tests involving compact objects.

The single, rank-2 tensor field gμν (i.e., the metric) of the Einstein–Hilbert action may also not be adequate to describe completely the gravitational force (although, if additional fields are introduced, then the strong equivalence principle is violated, with important implications for the frame and time-dependence of gravitational experiments). In fact, a variant of such theories with an additional scalar field, the Brans–Dicke theory [22], has been the most widely used alternative to general relativity to be tested against experiments. Today, scalar-tensor theories are among the prime candidates for explaining the acceleration of the universe at late times (the “dark energy” [121Jump To The Next Citation Point]). Depending on the coupling between the metric, the scalar field, and matter, the relative contribution of such additional fields may become significant only at the high curvatures found in the early universe or in the vicinity of compact objects.

Although the above discussion has considered only the classical action of the gravitational field in a phenomenological manner, it is also important to note that corrections to the Einstein–Hilbert action occur naturally in quantum gravity theories and in string theory. For example, if we choose to interpret the metric gμν as a quantum field, we can take Equation (1View Equation) as a quantum field-theoretic action defined at an ultraviolet scale (such as the Planck scale), and proceed to perform quantum-mechanical calculations in the usual way [50Jump To The Next Citation Point]. However, radiative corrections will induce an infinite series of counter-terms as we flow to lower energies and such counter-terms will not be reabsorbed into the original Lagrangian by adjusting its bare parameters. Instead, such terms will appear as new, higher-derivative correction terms in the Einstein–Hilbert action (1View Equation).

Finally, it is worth emphasizing that the previous discussion focuses on Lagrangian gravity in a four-dimensional spacetime. In the context of string theory, general relativity emerges only as a leading approximation. String theory also predicts an infinite set of non-linear terms in the scalar curvature, all suppressed by powers of the Planck scale. Moreover, the low-energy effective action of string theory contains additional scalar (dilatonic) and vector gravitational fields [67]. Motivated by ideas of string theory, brane-world gravity [90515253] also provides a theory that is consistent with all current tests of gravity.

All the above strongly support the notion that the field equation that arises from the Einstein–Hilbert action may be appropriate only at the scales that have been probed by current gravitational tests. But how deep have we looked?

∙ Gravitational tests to date seldom probe strong gravitational fields. – All historical tests of general relativity have been performed in our solar system. The strongest gravitational field they can, therefore, probe is that at the surface of the Sun, which corresponds to a gravitational redshift of

z⊙ ≃ GM--⊙-≃ 2 × 10− 6 , (3 ) R⊙c2
and to a spacetime curvature of
GM --3-⊙2-≃ 4 × 10− 28 cm −2 . (4 ) R ⊙c
Coincidentally, the gravitational fields that have been probed in tests using double neutron stars are of the same magnitude, since the masses and separation of the two neutron stars in the systems under consideration are comparable to the mass and radius of the Sun, respectively1. These are substantially weaker fields than those found in the vicinities of neutron stars and stellar-mass black holes, which correspond to a redshift of ∼ 1 and a spacetime curvature of ≃ 2 × 10–13 cm–2.

It is also instructive to compare the degree to which current tests verify the predictions of general relativity to the increase in the strength of the gravitational field going from the solar system to the vicinity of a compact object. Current constraints on the deviation of the PPN parameters from the general relativistic predictions are of order ≃ 10–5 [181]. It is conceivable, therefore, that deviations consistent with these constraints can grow and become of order unity when the redshift of the gravitational field probed is increased by six orders of magnitude and the spacetime curvature by fifteen!

Is it possible, however, that general relativity still accurately describes phenomena that occur in the strong gravitational fields found in the vicinity of stellar-mass black holes and neutron stars?

∙ General relativity breaks down in the strong-field regime. – Our current understanding of the physical world leaves very little doubt that the theory of general relativity itself breaks down at the limit of very strong gravitational fields. Considering the theory simply as a classical, geometric description of the spacetime leads to predictions of infinite matter densities and curvatures in two different settings. Integrating the Oppenheimer–Snyder equations, which describe the collapse of a cloud of dust [112], forward in time leads to the formation of a black hole with a singularity at its center. Integrating the Friedmann equation, which describes the evolution of a uniform and isotropic universe, backward in time always results in a singularity at the beginning of time, the Big Bang. Clearly, the outcome in both of these settings is unphysical.

It is widely believed that quantum gravity prohibits these unphysical situations that occur at the limit of infinitely strong gravitational fields. Even though none of the observable astrophysical objects offer the possibility of testing gravity at the Planck scale, they will nevertheless allow the placing of constraints on deviations from general relativity that are as large as ∼ 10 orders of magnitude more stringent compared to all other current tests. This is the best result we can expect in the near future to come out of the detection of gravitational waves and the observation of the innermost regions of neutron stars and black holes with NASA’s Beyond Einstein missions. If the history of the recent detection of a minute, yet non-zero, cosmological constant is any measure of our inability to predict even the order of magnitude of gravitational effects that we have not directly probed, then we might be up for a pleasant surprise!

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