Rational orbits around 4$\mathcal{D}$ Einstein–Lovelock black holes

Abstract

The orbital dynamics in the strong gravitational field might present unique features of quantum gravity and high-dimensional theory. In this paper, a timelike particle’s periodic orbits around the 4-dimensional Einstein–Lovelock (4$\mathcal{D}-$EL) black holes are investigated by employing a classification of the zoom–whirl structure with a rational number $q$. It is found that these rational (periodic) orbits around 4$\mathcal{D}-$EL black holes have a higher or lower energy than the ones around Schwarzschild black holes during the inspiral stage of detecting the gravitational waves. It depends strongly on the sign of a coupling constant in the theory. These results might provide hints for distinguishing 4$\mathcal{D}-$EL black holes from Schwarzschild ones by using the orbital dynamical approach in the strong gravitational field.

Introduction

Black holes are perfect laboratories in the strong gravitational field of the Universe, allowing one to probe the nature of gravitation deeply, as well as to get an insight into an unified theory beyond the Einstein’s theory of general relativity (GR). During the last five years, the great leap made in astronomy for the first time is to detect gravitational waves from black hole binaries [1], [2], [3], [4], [5], [6] and to image the supermassive black hole shadow in the center of galaxy M87 [7], [8], [9], [10], [11], [12]. Meanwhile, these direct observations of black holes also leave the window for testing alternative theories in the extreme environments open [13], [14], [15], [16], [17], [18], [19], [20], [21]. It is interesting to note that some influences of alternative theories on the strong gravitational field can present unique features and be well studied by using the orbital dynamical approach for a test particle’s motion around a black hole.

It is well-known that an extreme mass ratio inspiral is formed from a stellar-mass black hole closely whirling around one supermassive black hole, which can be approximatively regarded as a timelike particle orbiting around the supermassive black hole. During the inspiral stage for detecting the gravitational wave, the timelike particle’s path should be located in bound orbits around the black hole. For the timelike particle, these bound orbits mainly depend on two crucial regions: one is the marginally bound orbits (MBO); the other is the innermost stable circular orbits (ISCO). Bound orbits for test particles’ motions around black holes have been extensively studied in their geodesic motions [22], [23], [24], [25], [26], [27], [28], long-term effects on the radial velocities [29], [30], dynamical effects [31], their energy level diagrams [32], gravitational redshifts [33], “no-hair” theorems [34] and these black holes’ evolutions [35], [36], [37].

As one kind of bound orbits, periodic orbits play an important role in our understandings of celestial bodies’ motions in the Solar System, their long-term stabilities and dynamical evolutions, and even the adiabatic inspiral stage for detecting the gravitational wave. At the inspiral regime, for example, a series of periodic orbits consist of one continuing transitional orbit and play an important part in the detection of the gravitational wave [36]. In view of the fact, a classification of the zoom–whirl structure has been proposed in Ref. [38] with a rational number $q$, which is described by three integers $(z,w,\nu )$. $z$ and $w$ respectively count the numbers of zooms and whirls for a timelike particle’s orbits around black holes. $\nu$ denotes the vertex number, which indicates geometrically a different structure of periodic orbits with the same whirl and zoom numbers. Given one angular momentum and energy of the particle’s motion in bound orbits, a rational number $q$ corresponds to a closed periodic orbit. The classification of the zoom–whirl structure with $q$ for representing periodic orbits has been widely investigated in Kerr black holes [38], Reissner–Nordström black holes [39] and others black holes [40], [41], [42], [43], [44].

Although GR has been tested and proved to be quite effective to describe some observations in the Solar System and beyond with unprecedented accuracy (see Refs. [45], [46], [47], [48], [49], [50], for reviews), it ineffectively describes others phenomena such as cosmic inflation, the problems of spacetime singularities, unification theories, etc. As one competitive candidate of unification theories, string theory are proposed with spacetime more than four dimensions. Especially, in the low-energy limit of string theory, higher curvature corrections induced considerable interest to Einstein–Lovelock theories and Einstein–Gauss–Bonnet. Based on dimensional regularization procedure [51], a modified theory that caught a lot of attention recently is the so-called the 4-dimensional Einstein–Gauss–Bonnet gravity, which could be treated as a reduced 4-dimensional theory of high-dimensional string theory and satisfies Lovelock’s theorem [52]. Although the authors claim that this gravity bypasses the Lovelock’s theorem and avoids Ostrogradsky instability, $D\to 4$ limit of higher-dimensional Einstein–Gauss–Bonnet gravity is currently under the debate [51] and whether the theory is well defined is constantly attracting criticism [51], [53], [54], [55], [56], [57]. In Ref. [52], it is argued that a physical observer could never reach this curvature singularity given the repulsive effect of gravity at short distances. However, considering the geodesic equations, this claim has been refuted by Ref. [58]. The authors in Ref. [58] explicitly showed that the infalling particle starts at rest will reach the singularity with zero velocity as attractive and repulsive effects compensate each other along the trajectory of the particle.

Since Einstein–Lovelock theories include an infinite series of corrections to the corresponding Einstein term with an increasing power of the curvature, a general covariant modified theory called the 4-dimensional Einstein–Lovelock (4$\mathcal{D}-$EL) gravity has been recently developed [51], [53]. By performing the dimensional regularization of the Einstein–Lovelock equations, the authors in Ref. [51], [53] formulate and propose the most general static four-dimensional metric in this theory with diffeomorphism invariance. The 4$\mathcal{D}-$EL gravity not only covers the solution of 4-dimensional Einstein–Gauss–Bonnet gravity with the quadratic in curvature (e.g., $\tilde{m}=2$) by the covariance and general method, but also contains the most general static 4$\mathcal{D}$ black-hole solution allowing for a $\Lambda$-term (either positive or negative) and the electric charge $Q$. In addition, it is shown that if one is limited by positive values of coupling constants in front of Lovelock terms, the eikonal instability works as an effective cut-off for higher order terms in the 4$\mathcal{D}-$EL gravity [51], [53]. And basic observable quantities such as quasinormal modes, frequencies at the innermost stable circular orbit and radius of the black-hole shadow change almost indistinguishably when the Lovelock corrections of higher than the fourth order in curvature are included (see Ref. [51], [53] for details). Some fascinating properties in 4$\mathcal{D}-$EL black holes have been attentively studied in thermodynamics [54], [59], [60], the gravitational collapse [56], [57], geodesic and orbital dynamics [61], [62], gravitational lensing [63], [64], (in)stability [55], [65], whereas the classification of the zoom–whirl structure and rational orbits in 4$\mathcal{D}-$EL black holes is still missing in the literature.

In the light of some unique features of the classification, in the present work, we primarily concentrate on investigating periodic orbits of a timelike particle around 4$\mathcal{D}-$EL black holes. In what follows, $G=c=1$ and the metric signature is $(-,+,+,+)$. The paper is organized in the following way. In Section 2, we study the effective potential and bound orbits for the timelike particle around 4$\mathcal{D}-$EL black holes. By considering the classification of the zoom–whirl structure [38] and bound orbits, Section 3 will be devoted to investigate periodic orbits and rational numbers in 4$\mathcal{D}-$EL black holes. In Section 4, conclusions and discussion will be represented.