One of the biggest puzzles of modern physics is the present-day accelerated expansion of the Universe. The acceleration is usually attributed to the presence of a mysterious dark energy, a yet unknown substance of the Universe. Although in the framework of conventional General Relativity (GR), a cosmological constant can be added to mimic dark energy, the fine tuning required to adjust its value makes this explanation unsatisfactory. We can then ask whether a modification of General Relativity can naturally accommodate the present-day cosmic acceleration.
Massive Gravity is a quite natural modification of GR, which modifies its large-distance behavior, thus being able, in principle, to solve the dark energy problem. Similar to, e.g., a scalar field that can be massless or massive, massive gravity can be viewed as a massive version of GR. However, giving mass to the graviton is not as straightforward as in the scalar case, since this procedure requires the introduction of a second (fiducial) metric. This modification changes the theory much more drastically than in the case of a simple scalar field. In particular, extra degrees of freedom emerge that can drastically alter the phenomenology and the viability of the theory. A sister theory to massive gravity is its bi-gravity extension, in which the fiducial metric is also given dynamics via the introduction of an Einstein-Hilbert term in the action for this metric.
The proposal of a theoretically viable family of massive gravity has lead in the past few years to a considerable amount of work in order to understand whether such a modification of gravity can lead to a physically sound theory, able to reproduce the highly-constrained weak-field regime of gravity. Hopefully, we will also start to grasp the surface of strong-field gravity in the next decade with the advent of the next generation of telescopes and gravitational-wave detectors. As such, black holes, fully nonlinear solutions of almost any theory of gravity, are an excellent testing ground to look for possible deviations from GR. In GR, black holes are fascinating objects, possessing remarkable properties. In massive gravity and its extensions, black holes generically have a more complicated structure that we hope may be exploited to hunt for the presence of modifications of gravity.
In our recent CQG paper, we describe the state of the art of this new and rapidly developing field of research – black holes in massive gravity and its bi-gravity extension. The classes of black hole solutions are richer than in GR and there is a simple reason for that: the Birkhoff theorem does not apply. However, despite the fact that massive gravity is more complicated than GR, analytic solutions can be constructed. Spherically symmetric black holes in these theories can be divided into two classes: bidiagonal and non-bidiagonal ones, depending on whether the metrics are aligned or not. These classes of exact solutions feature solutions of GR, since they are sought in such a way that the mass term is either trivial or effectively gives a cosmological constant. Similarly, charged and rotating black holes can be constructed in massive gravity. All these exact solutions can be thought as hairless black holes belonging to the Kerr family. On the other hand, numerical solutions of black holes with “massive graviton” hair have been explicitly constructed, which do not correspond to any black hole solutions in GR, and are thus a clear evidence that deviations from GR are indeed possible.
With such a plethora of solutions, the first question that one should ask is whether black holes in massive gravity are stable or not. This question, which we have actively worked on in the past two years, requires detailed analysis and the answer depends on the solution. One of the standard procedures is to use perturbation theory, which not only helps to understand the stability of the solutions but can also give hints on the existence of new solutions. Only recently, a few steps in this long-term program have been made, already yielding very interesting results. Perhaps the most striking result is that the simplest solution of massive bi-gravity, namely the bi-Schwarzschild solution (with two metrics equal to each other, so that the mass term is zero on the solution) is unstable. The reason is that there is an extra propagating mode, the helicity-0 component of the massive graviton, which is pure gauge in GR. Notably, it turns out that the perturbative equation of motion for the graviton in this background is equivalent to the equation for a Kaluza-Klein mode of a black string – a four-dimensional Schwarzschild black hole extended into a flat higher-dimensional spacetime. This mode was known to suffer from the so-called Gregory-Laflamme instability, and thus the bidiagonal Schwarzschild black hole must also be generically unstable against spherically symmetric perturbations! Outstanding questions that remain open are the end-state of this instability, and not less importantly, what is the final outcome of gravitational collapse in these theories.
Another type of instabilities were also shown to occur for a particular family of Kerr solutions. This instability, known as “superradiant instability”, is related to the process of superradiance, in which low-frequency bosonic waves are amplified by extracting energy and angular momentum from the spinning black hole. Massive bosonic fields, which naturally emerge in massive gravity in the form of a massive spin-2 field, are able to confine superradiant modes and thus repeatedly extract angular momentum from the black hole. The black hole spin-down continues until the instability becomes ineffective against spin-up effects, such as accretion of gas. A generic prediction of this scenario is that massive black holes should have a maximum spin below the extremal Kerr bound. Because we do observe fastly-spinning black holes one can put upper limits on the graviton mass of the order 10-23 eV, comparable to the best limits coming from other models.
Our paper succinctly describes all these exciting recent developments and ends with a list of open problems that we believe, aside from being interesting theoretical issues, might end-up being decisive to decide whether theories with massive gravitons are physically viable theories of gravity.
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