Abstract
Motivated by black hole solutions with matter fields outside their horizon, we study the effect of these matter fields on the motion of massless and massive particles. We consider as background a four-dimensional asymptotically AdS black hole with scalar hair. The geodesics are studied numerically and we discuss the differences in the motion of particles between the four-dimensional asymptotically AdS black holes with scalar hair and their no-hair limit, that is, Schwarzschild AdS black holes. Mainly, we found that there are bounded orbits like planetary orbits in this background. However, the periods associated to circular orbits are modified by the presence of the scalar hair. Besides, we found that some classical tests such as perihelion precession, deflection of light, and gravitational time delay have the standard value of general relativity plus a correction term coming from the cosmological constant and the scalar hair. Finally, we found a specific value of the parameter associated to the scalar hair, in order to explain the discrepancy between the theory and the observations, for the perihelion precession of Mercury and light deflection.
Highlights
In this work, motivated by black hole solutions with matter fields outside their horizon, we study their effect in the motion of massless and massive particles in the background of a four-dimensional asymptotically AdS black hole with scalar hair [20]
We study classical tests such as perihelion precession, deflection of light and gravitational time delay in order to determine the contribution that arises from the scalar hair
These solutions asymptotically give the Schwarzschild anti-de Sitter solution, and they are characterized by a scalar field with a logarithmic behavior, being regular everywhere outside the event horizon and null at spatial infinity, and by a self-interacting potential, which tends to the cosmological constant at spatial infinity
Summary
The hairy black hole that we consider is solution of the Einstein–Hilbert action with a negative cosmological constant and a neutral scalar field minimally coupled to the curvature having a self-interacting potential V (φ) [20]. = −6l−2/κ, l being the length of the AdS space and κ = 8π G N , with G N the Newton constant This potential has a global maximum at φ = 0. In Eq (6), dσk is the metric of the spatial 2-section, which can have positive, negative or zero curvature, and the coordinates are defined in the ranges 0 < r < ∞, −∞ < t < ∞, 0 ≤ θ < π, 0 ≤ φ < 2π. In order to show that these parameters yield a hairy black hole solution we plot in Fig. 1 the behavior of the metric function
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