Advances in High Energy Physics

Volume 2018, Article ID 5427158, 8 pages

https://doi.org/10.1155/2018/5427158

## Particle Motion around Charged Black Holes in Generalized Dilaton-Axion Gravity

^{1}Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India^{2}Department of Physics, Gheorghe Asachi Technical University, 700050 Iasi, Romania^{3}Department of Civil Engineering, University of Thessaly, 383 34 Volos, Greece^{4}Department of Mathematics, Nagar College, P.O. Nagar, Dist. Mursidabad, West Bengal, India

Correspondence should be addressed to Farook Rahaman; ni.aacui.setaicossa@namahar

Received 14 May 2018; Accepted 8 August 2018; Published 16 September 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 Susmita Sarkar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The behaviour of massive and massless test particles around asymptotically flat and spherically symmetric, charged black holes in the context of generalized dilaton-axion gravity in four dimensions is studied. All the possible motions are investigated by calculating and plotting the corresponding effective potential for the massless and massive particles as well. Further, the motion of massive (charged or uncharged) test particles in the gravitational field of charged black holes in generalized dilaton-axion gravity for the cases of static and nonstatic equilibrium is investigated by applying the Hamilton-Jacobi approach.

#### 1. Introduction

Recently, scientists have focused their attention on the black hole solutions in various alternative theories of gravity, particularly theories of gravitation with background scalar and pseudoscalar fields. In the low energy effective action, usually string theory based-models are comprised of two massless scalar fields, the dilaton, and the axion (see, e.g., [1]). Sur, Das, and SenGupta [2] employed the dilaton and axion fields coupled to the electromagnetic field in a more generalized coupling with Einstein and Maxwell theory in four dimensions in the low energy action. Exploiting this new idea, they have found asymptotically flat and nonflat dilaton-axion black hole solutions. The vacuum expectation values of the various moduli of compactification are responsible for these couplings. These black hole solutions have been studied extensively in the literature; e.g., their thermodynamics has been investigated [3], thin-shell wormholes have been constructed from charged black holes in generalized dilaton-axion gravity [4], the energy of charged black holes in generalized dilaton-axion gravity has been calculated [5], the statistical entropy of a charged dilaton-axion black hole has been examined [6], and the superradiant instability of a dilaton-axion black hole under scalar perturbation has been investigated [7]. Among various properties of such black hole solutions, a subject of great interest is the study of the behaviour of a test particle in the gravitational field of such black holes.

In this paper, we study the behaviour of the time-like and null geodesics in the gravitational field of a charged black hole in generalized dilaton-axion gravity. The solution under study describes an asymptotically flat black hole and the motions of both massless and massive particles are analyzed. The effective potentials are calculated and plotted for various parameters in the cases of circular and radial geodesics. The motion of a charged test particle in the gravitational field of a charged black hole in generalized dilaton-axion gravity is also investigated using the Hamilton-Jacobi approach.

The present paper has the following structure: in Section 2 the charged black hole metric in generalized dilaton-axion gravity is presented. Section 3 focuses on the geodesic equation in the cases of massless particle motion () and massive particle motion (). In Section 4 the effective potential is studied in both cases of the massless and the massive particle. Section 5 is devoted to the study of the motion of a test particle in static equilibrium as well as in nonstatic equilibrium. For the latter case, a chargeless () and a charged test particle are considered. Finally, in Section 6, the results obtained in this paper are discussed.

#### 2. Charged Black Hole Metric in Generalized Dilation-Axion Gravity

Recently, Sur, Das, and SenGupta [2] have discovered a new black hole solution for the Einstein-Maxwell scalar field system inspired by low energy string theory. In fact, they have considered a generalized action in which two scalar fields are minimally coupled to the Einstein-Hilbert-Maxwell field in four dimensions (in the Einstein frame, see, e.g., [8, 9]) having the formwhere , R is the curvature scalar, describes the Maxwell field strength and , are two massless scalar/pseudoscalar fields depending only on the radial coordinate which are coupled to the Maxwell field through the functions and . Here, acquires a nonminimal kinetic term of the form due to its interaction with (, can be identified with the scalar dilaton field and the pseudoscalar axion field, respectively), while is the Hodge-dual Maxwell field strength.

Indeed, with the action described by (1), a much wider class of black hole solutions has been found, whereby two types of metrics, asymptotically flat and asymptotically nonflat, for the black hole solutions have been obtained.

For our study we use the asymptotically flat solution to analyze the behaviour of massive and massless test particles around a spherically symmetric, charged black hole in generalized dilaton-axion gravity. The asymptotically flat metric considered is given bywithand

In (3) and (4), according to [2], in order to have nontrivial and fields, the exponent is a dimensionless constant strictly greater than 0 and strictly less than 1. The other various parameters are given as follows:andwhere is the mass of the black hole and . The parameters and determine the inner and outer event horizons, respectively. Also, for , there is a curvature singularity and the parameters obey the condition .

#### 3. Geodesic Equation

The geodesic equation for the metric (2) describing the motion in the plane is as follows [10]:where is known as the Lagrangian having the values 0 for a massless particle and −1 for a massive particle and , are constants identified as the energy per unit mass and the angular momentum, respectively.

Now we proceed to discuss the motion of the massless and the massive particle for the radial geodesic.

The radial geodesic equation () isUsing (13), (14) becomesThen, by inserting from (3), (15) reads

##### 3.1. Massless Particle Motion ()

For the motion of a massless particle the Lagrangian vanishes. In this case the equation for the radial geodesic (16) becomesAfter integrating we getAgain from (14) we obtain for from which we have a relationshipIn Figure 1 (left) is plotted with respect to the radial coordinate and in Figure 1 (right) the proper time () is plotted with respect to radial coordinate for a massless particle.