Modern observations of black holes and neutron stars in the galaxy provide ample opportunity for testing the predictions of general relativity in the strong-field regime, as discussed in the previous section. In several cases, astrophysical complications make such studies strongly dependent on model assumptions. This will be remedied in the near future, with the anticipated advances in the observational techniques and in the theoretical modeling of the various astrophysical phenomena. A second difficult hurdle, however, in performing quantitative tests of gravity with compact objects will be the lack of a parametric extension to general relativity, i.e., the equivalent of the PPN formalism, that is suitable for calculations in the strong-field regime.

In the past, bona fide tests of strong-field general relativity have been performed using particular parametric extensions to the Einstein–Hilbert action. This appears, a priori, to be a reasonable approach for a number of reasons. First, deriving the parametric field equations from a Lagrangian action ensures that fundamental symmetries and conservation laws are obeyed. Second, the parametric Lagrangian action can be used over the entire range of field strengths available to an observer and, therefore, even tests of general relativity in the weak-field limit (i.e., with the PPN formalism) can be translated into constraints on the parameters of the action. This is often important when strong-field tests lead to degenerate constraints between different parameters. Finally, phenomenological Lagrangian extensions can be motivated by ideas of quantum gravity and string theory and, potentially, help constrain the fundamental scales of such theories. There are, however, several issues that need to be settled before any such parametric extension of the Einstein–Hilbert action can become a useful theoretical framework for strong-field gravity tests (see also [154] and references therein).

First, gravity is highly non-linear and strong-field phenomena often show a non-perturbative dependence on small changes to the theory. I will illustrate this with scalar-tensor theories that result from adding a minimally coupled scalar field to the Ricci curvature in the action. Such fields have been studied for more than 40 years in the form of Brans–Dicke gravity [180] and have been recently invoked as alternatives to a cosmological constant for modeling the acceleration of the universe [121]. In the context of compact-object astrophysics, constraints on the relative contribution of scalar fields coupled in different ways to the metric have been obtained from observations of the orbital decay of double neutron stars [182, 39] and compact X-ray binaries [182, 125]. More recently, similar constraints on scalar extensions to general relativity have been placed using the observation of redshifted lines from an X-ray burster [43] and of quasi-periodic oscillations observed in accreting neutron stars [44]. The oscillatory modes of neutron stars in such theories and the prospect of constraining them using gravitational-wave signatures have also been studied [152, 153].

The general form of the Lagrangian of a scalar-tensor theory is given, in the appropriate frame, by the Bergmann–Wagoner action (see [180] for details)

where and are two arbitrary functions, and is the action for the matter field . In the strong-field regime, the potential term in the action (22) is typically negligible and is set to zero. On the other hand, the functional form of the coupling function can be parameterized to measure deviations from general relativity.Damour and Esposito-Farèse [39] considered a second-order parametric form

with a background cosmological value for the scalar field and and the two parameters of the theory to be constrained by observations. The linear term, parameterized by , can be best constrained with weak-field tests. On the other hand, constraining significantly the non-linear term, parameterized by , requires strong-field phenomena, such as those found around neutron stars. Indeed, the two main PPN parameters for such a scalar-tensor theory are The deviation of the PPN parameters from the general relativistic values is of second order in and of third order in the product . As a result, a very good limit on the parameter renders the parameter practically unconstrainable by weak-field tests.The study of Damour and Esposito-Farèse [39] revealed one of the main reasons that necessitate careful theoretical studies of possible extensions of general relativity that are suitable for strong-field tests. The order of a term added to the Lagrangian action of the gravitational field is not necessarily a good estimate of the expected magnitude of the observable effects introduced by this additional term. For example, because of the non-linear coupling between the scalar field and matter introduced by the coupling function (23), the deviation from general relativistic predictions is not perturbative. For values of less than about –6, it becomes energetically favorable for neutron stars to become “scalarized”, with properties that differ significantly from their general relativistic counterparts [39]. Such non-perturbative effects make quantitative tests of strong-field gravity possible even when the astrophysical complications are only marginally understood.

A similar situation, albeit in the opposite regime, arises in an extended gravity theory in which a term proportional to the inverse of the Ricci scalar curvature, , is added to the Einstein–Hilbert action in order to explain the accelerated expansion of the universe [30]. Although one would expect that such an addition can only affect gravitational fields that are extremely weak, it turns out that it also alters to zeroth order the post-Newtonian parameter and can, therefore, be excluded by simple solar-system tests [33].

Second, Lagrangian extensions of general relativity often suffer from serious problems with instabilities. This issue can be understood by considering a Lagrangian action that includes terms of second order in the Ricci scalar, i.e., , as well as the terms of similar order that can be constructed with the Ricci and Riemann tensors. For the sake of the argument, I will consider here the parametric Lagrangian

with , , and the three parameters of the theory. Such terms arise naturally as high-order corrections in quantum gravity and string theory and their relative importance increases with the curvature of the metric [50, 27]. They have also been invoked as alternatives to the inflation paradigm for the early expansion of the universe [158]. The predictions for astrophysical objects of extended-gravity theories that incorporate high-order terms have been reported only for a few limited cases in the literature. The dependence of the stellar properties on terms in the action has been studied by Parker and Simon [119], who simply derived the generalized Tolmann–Oppenheimer–Volkoff equation without solving it, and by Barraco and Hamity [9], who attempted to solve the problem using a perturbation analysis (unfortunately, this last study suffers from a large number of errors).This second-order gravity theory has a number of unappealing properties (see discussion in [149, 150]). Classically, a high-order gravity theory requires more than two boundary conditions, which is a fact that appears to be incompatible with all other physical theories. Quantum mechanically, second-order gravity theories lead to unstable vacuum solutions. Both these phenomena could be artifacts of the possibility that the action (25) may arise as a low-energy expansion of a non-local Lagrangian that is fundamentally of second order [149, 150]. Phenomenologically speaking, these problems can be overcome by requiring the field equations to be of second order, when extremizing the action. This procedure leads to a generalized, high-order gravity theory that remains consistent with classical expectations and is stable quantum-mechanically (according to the procedure outlined in [149, 150]), but requires a different than usual derivation of the field equations [45].

Even if we neglect these issues, the terms proportional to and lead to field equations with solutions that suffer from the Ostrogradski instability [184]. And even if these terms are dropped and only actions that are generic functions of the Ricci scalar alone are considered, then the resulting solutions for the expansion of the universe [49] and for spherically symmetric stars [144], can be violently unstable, depending on the sign of the second-order term.

A potential resolution to several of these problems in theories with high-order terms in the action appears to be offered by the Palatini formalism. In this approach, the field equations are derived by extremizing the action under variations in the metric and the connection, which is considered as an independent field [155]. For the simple Einstein–Hilbert action, both approaches are equivalent and give rise to the equations of general relativity; when the action has non-linear terms in , the two approaches diverge. Unfortunately, the Palatini formalism leads to equations that cannot handle, in general, the transition across the surface layer of a star to the matter-free space outside it, and is, therefore, not a viable alternative [6].

Finally, it is crucial that we identify the astrophysical phenomena that can be used in testing particular aspects of strong-field gravity. For example, in the case of the classical tests of general relativity, it is easy to show that the deflection of light during a solar eclipse and the Shapiro time delay depend on one (and the same) component of the metric of the Sun (i.e., on the PPN parameter ). Therefore, they do not provide independent tests of general relativity (as long as we accept the validity of the equivalence principle). On the other hand, the perihelion precession of Mercury and the gravitational redshift depend on the other component of the metric (i.e., on the PPN parameter ) and, therefore, provide complementary tests of the theory. Understanding such degeneracies is an important component of performing tests of gravity theories.

In the case of strong gravitational fields, this issue can be illustrated again by studying the high-order Lagrangian action (25) in the metric formalism (see also [27]). In principle, as the strength of the gravitational field increases, the terms that are of second-order in the Ricci scalar become more important and, therefore, affect the observable properties of neutron stars and black holes. However, because of the Gauss–Bonnet identity,

variations, with respect to the metric, of the term proportional to in Equation (25), can be expressed as variations of the terms proportional to and . Therefore, for all non-quantum gravity tests, the predictions of the theory described by the Lagrangian action (25) are identical to those of the Lagrangian As a result, astrophysical tests that do not invoke quantum-gravity effects can only constrain a particular combination of the parameters, i.e., and . It is only through phenomena related to quantum gravity, such as the evaporation of black holes, that the parameter may be constrained.When the spacetime is isotropic and homogeneous, as in the case of tests using the cosmic evolution of the scale factor, an additional identity is satisfied, i.e.,

This implies that, for cosmological tests, the predictions of the theory described by the Lagrangian action (25) are identical to those of the Lagrangian As a result, such cosmological tests of gravity can only constrain a particular combination of the parameters, i.e., .The parameters and can be independently constrained using observations of spacetimes that are strongly curved but not isotropic and homogeneous, such as those found in the vicinities of black holes and neutron stars. Measuring the properties of neutron stars, such as their radii, maximum masses and maximum spins, which require the solution of the field equations in the presence of matter, will provide independent constraints on the combination of parameters and . However, one can show that in the absence of matter, the external spacetime of a black hole, as given by the solution to Einstein’s field equation, is also one (but not necessarily the only) solution of the parametric field equation that arises from the Lagrangian action (25). As a result, tests that involve black holes will probably be inadequate in distinguishing between the particular theory described by Equation (25) and general relativity [130].

This is, in fact, a general problem of using astrophysical observations of black holes to test general relativity in the strong-field regime. The Kerr solution is not unique to general relativity [130]. For example, there is strong analytical [165, 13, 71] and numerical evidence [142] that, in Brans–Dicke scalar-tensor gravity theories, the end product of the collapse of a stellar configuration is a black hole described by the same Kerr solution as in Einstein’s theory. The same appears to be true in several other theories generated by adding additional degrees of freedom to Einstein’s gravity; the only vacuum solutions that are astrophysically relevant are those described by the Kerr metric [130]. Until a counter-example is discovered, studies of the strong gravitational fields found in the vicinities of black holes can be performed only within phenomenological frameworks, such as those involving multipole expansions of the Schwarzschild and Kerr metrics [138, 35, 66].

To date, it has only been possible to test quantitatively the predictions of general relativity in the strong-field regime using observations of neutron stars, as I will discuss in the following section. In all cases, the general relativistic predictions were contrasted to those of scalar-tensor gravity, with Einstein’s theory passing all the tests with flying colors.

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