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 [181]:
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 [181
].
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 [102, 117]) 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,
,
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 [154]
The single, rank-2 tensor field (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” [121
]). 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
as a quantum field, we can take Equation (1
) 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 [50
]. 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 (1
).
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 [90, 51, 52, 53] 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
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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