We obtain an infinite number of exact static, Ricci-flat spherically symmetric vacuum solutions for a class of f(R) theories of gravity. We analytically derive two exact vacuum black-hole solutions for the same class of f(R) theories. The two black-hole solutions have the event-horizon at the same point; however, their asymptotic features are different. Our results suggest that no-hair theorem may not hold for generic modified gravity theories. We discuss the implications of our work to distinguish modified gravity theories from general relativity in gravitational wave detections.
Over the last two decades, motivations for modified gravity have emerged from both theoretical and observational levels. f(R) and Chern-Simons gravity have received more attention as they are the simplest generalization. However, f(R) and Chern-Simons gravity contain only an additional scalar (spin-0) degree of freedom and, as a result, do not include other modes of modified theories of gravity. In contrast, quadratic gravity (also referred to as Stelle gravity) is the most general second-order modification to 4-D general relativity and contains a massive spin-2 mode that is not present in f(R) and Chern-Simons gravity. Using two different physical settings—the gravitational wave energy-flux measured by the detectors and the backreaction of the emitted gravitational radiation on the spacetime of the remnant black hole—we demonstrate that massive spin-2 mode carries more energy than the spin-0 mode. Our analysis shows that the effects are pronounced for intermediate-mass black holes, which are prime targets for LISA.
This work tests the no-hair conjecture in f (R) gravity models. No-hair conjecture asserts that all black holes in General Relativity coupled to any matter must be Kerr-Newman type. However, the conjecture fails in some cases with non-linear matter sources. Here, we address this by explicitly constructing multiple slow-rotating black hole solutions, up to second order in rotational parameter, for a class of f (R) models. We analytically show that two vacuum solutions satisfy the field equations up to the second order in the rotational parameter. The uniqueness of our result stems from the fact that these are obtained directly from metric formalism without conformal transformation. We discuss the kinematical properties of these black hole solutions and compare them with slow-rotating Kerr. Specifically, we show that the circular orbits for the black holes in f (R) are smaller than that of Kerr. This implies that the inner-most stable circular orbit for black holes in f (R) is smaller than Kerr's; hence, the shadow radius might also be smaller. Finally, we discuss the implications of our results for future observations.
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