We study the motion of particles in the background of a scalar–tensor theory of gravity in which the scalar field is kinetically coupled to the Einstein tensor, and we present the null geodesic structure for asymptotically flat, AdS, and dS Horndeski black holes, studying the effect of the cosmological constant on the orbits. Also, we consider three classical tests of gravity in the solar system, namely the bending of the light, the gravitational redshift, and the Shapiro time delay, in order to constraint the coupling parameters of the scalar field to gravity. Calculating the Lyapunov exponent, we explore the stability of these geodesics for various values of the cosmological constant.

Chile

D. Carvajal

P. González

Marco Olivares

Greece

Eleftherios Papantonopoulos

Yerko Vásquez

Study of null geodesics and their stability in Horndeski black holes

We consider scalar field perturbations in the background of black holes immersed in perfect fluid dark matter (PFDM). We find, by using the sixth-order Wentzel–Kramers–Brillouin (WKB) approximation, that the longest-lived modes are those with a higher angular number for a scalar field mass smaller than a critical value, known as the anomalous decay rate of the quasinormal modes, while beyond this critical value the opposite behavior is observed. Moreover, we show that it is possible to recover the real part of the quasinormal frequencies (QNFs), the imaginary part of the QNFs, and the critical scalar field mass of the Schwarzschild background for different values of the PFDM intensity parameter k. For values of k smaller than these values, the abovementioned quantities are greater than the Schwarzschild background. However, beyond these values of k, these quantities are smaller than the Schwarzschild background.

Ramón Bécar

Massive scalar field perturbations of black holes surrounded by dark matter

In this work we consider the propagation of massive scalar fields in the background of Weyl black holes, and we study the effect of the scalar field mass in the spectrum of the quasinormal frequencies (QNFs) via the Wentzel–Kramers–Brillouin (WKB) method and the pseudo-spectral Chebyshev method. The spectrum of QNFs is described by two families of modes: the photon sphere and the de Sitter modes. Essentially, we show via the WKB method that the photon sphere modes exhibit an anomalous behaviour of the decay rate of the QNFs; that is, the longest-lived modes are the ones with higher angular numbers, and there is a critical value of the scalar field mass beyond which the anomalous behaviour is inverted. We also analyse the effect of the scalar field mass on each family of modes and on their dominance, and we give an estimated value of the scalar field mass where the interchange in the dominant family occurs.

Felipe Moncada

Massive scalar field perturbations in Weyl black holes

The time like structure of the five-dimensional Schwarzschild and Reissner–Nordström anti-de Sitter black holes is studied in detail. Different kinds of motion are allowed and studied by using an adequate effective potential. Then, by solving the corresponding equations of motion, several trajectories and orbits are described in terms of Weierstraß elliptic functions and elementary functions for neutral particles.

J. Villanueva

Time like geodesics for five-dimensional Schwarzschild and Reissner–Nordström anti-de Sitter black holes

We consider the propagation of a charged massive scalar field in the background of a four-dimensional Ernst black hole and study its stability analyzing the quasinormal modes (QNMs), which are calculated using the semi-analytical Wentzel–Kramers–Brillouin method and numerically using the continued fraction method. We mainly find that for a scalar field mass less than a critical mass, the decay rate of the QNMs decreases when the harmonic angular number  $$\ell $$  <math> <mi>ℓ</mi> </math>   increases; and for a scalar field mass greater than the critical mass, the behavior is inverted, i.e., the longest-lived modes are always the ones with the lowest angular number recovering the standard behavior. Also, we find a critical value of the external magnetic field, as well as a critical value of the scalar field charge that exhibits the same behavior with respect to the angular harmonic numbers. In addition, we show that the spacetime allows stable quasibound states, and we observe a splitting of the spectrum due to the Zeeman effect. Finally, we show that the unstable null geodesic in the equatorial plane is connected with the QNMs when the azimuthal quantum number satisfies  $$m= \pm \ell $$  <math> <mrow> <mi>m</mi> <mo>=</mo> <mo>±</mo> <mi>ℓ</mi> </mrow> </math>   in the eikonal limit.

Quasinormal modes of a charged scalar field in Ernst black holes

We study neutral massless scalar field perturbations around an extreme dilaton black hole in 2 + 1 dimensions: the wave equations of the massless scalar field is shown to be exactly solvable in terms of Whittaker functions. Thus, the quasinormal modes are computed exactly and shown to be purely imaginary: we show the existence of stable and unstable modes. Interestingly, the quasinormal modes do not depend on the black holes parameters and the fundamental mode is always unstable and depends only on the parameters of the test field. Also, we determine the quasinormal frequencies via the improved asymptotic iteration method which shows a good agreement with the analytical results.

United States

Sharmanthie Fernando

Extreme dilaton black holes in 2 + 1 dimensions: quasinormal modes

The anomalous decay rate of the quasinormal modes occurs when the longest-lived modes are the ones with higher angular number. Such behaviour has been recently studied in different static spacetimes, for uncharged scalar and fermionic perturbations, being observed in both cases. In this work we consider the propagation of charged massive scalar fields in the background of Reissner-Nordström-de Sitter black holes and we mainly study the effect of the scalar field charge in the spectrum of quasinormal frequencies, as well as, its effect on the anomalous decay rate. Mainly, we show that the anomalous behaviour is present for massive charged scalar fields as well, and a critical value of scalar field mass exists, beyond which the behaviour is inverted. However, there is also a critical value of the parameter qQ of the charge of the scalar field and of the charge of the black hole, which increases when the cosmological constant increases, and beyond the critical value the anomalous behaviour of the decay rate could be avoided for the fundamental mode.

Joel Saavedra

Quasinormal modes for massive charged scalar fields in Reissner-Nordström dS black holes: anomalous decay rate

We study the propagation of scalar fields in the background of an asymptotically de Sitter black hole solution in f(R) gravity. The aim of this work is to analyze in modified theories of gravity the existence of an anomalous decay rate of the quasinormal modes (QNMs) of a massive scalar field which was recently reported in Schwarzschild black hole backgrounds, in which the longest-lived modes are the ones with higher angular number, for a scalar field mass smaller than a critical value, while that beyond this value the behavior is inverted. We study the QNMs for various overtone numbers and they depend on a parameter  $$\beta $$  <math> <mi>β</mi> </math>   which appears in the metric and characterizes the f(R) gravity. For small  $$\beta $$  <math> <mi>β</mi> </math>  , i.e. small deviations from the Schwarzschild–dS black hole the anomalous behavior in the QNMs is present for the photon sphere modes, and the critical value of the mass of the scalar field depends on the parameter  $$\beta $$  <math> <mi>β</mi> </math>   while for large  $$\beta $$  <math> <mi>β</mi> </math>  , i.e. large deviations, the anomalous behavior and the critical mass does not appear. Also, the critical mass of the scalar field increases when the overtone number increases until the f(R) gravity parameter  $$\beta $$  <math> <mi>β</mi> </math>   approaches the near extremal limit at which the critical mass of the scalar field does not depend anymore on the overtone number. The imaginary part of the quasinormal frequencies is always negative leading to a stable propagation of the scalar fields in this background.

Almendra Aragón

Quasinormal modes and their anomalous behavior for black holes in f(R) gravity

We consider a three-dimensional rotating AdS black hole, which is a solution of Hořava gravity in the low-energy limit that corresponds to a Lorentz-violating version of the BTZ black hole, and we analyze the effect of the breaking of Lorentz invariance on the possibility that the black hole can act as a particle accelerator by analyzing the energy in the center-of-mass (CM) frame of two colliding particles in the vicinity of its horizons. We find that the critical angular momentum of particles increases when the Hořava parameter  $$\xi $$  <math> <mi>ξ</mi> </math>   increases and when the aether parameter b increases. Also, the particles can collide on the inner horizon with arbitrarily high CM energy if one of the particles has a critical angular momentum, possible for the BSW process. Here it is essential that, while for the extremal BTZ black hole the particles with critical angular momentum only can exist on the degenerate horizon, for the Lorentz-violating version of the BTZ black hole the particle with critical angular momentum can exist in a region away from the degenerate horizon. It is worth mentioning that the results exposed in this manuscript can be applied for the covariant version of Hořava gravity, where the covariant definition of the center-of-mass energy is well defined.

Collision of particles near a three-dimensional rotating Hořava AdS black hole

The study of the motion of photons around massive bodies is one of the most useful tools to find the geodesic structure associated with said gravitational source. In the present work, different possible paths projected in an invariant hyperplane are investigated, considering a five-dimensional Reissner–Nordström anti-de Sitter black hole. Also, we study some observational tests, such as the bending of light and the Shapiro time delay effect. Mainly, we found that the motion of photons follows the hippopede of a Proclus geodesic, which is a new type of trajectory of the second kind, the Limaçon of Pascal being their analog geodesic in four-dimensional Reissner–Nordström anti-de Sitter black hole.