Abstract
Abstract We study the null geodesics in a static circularly symmetric (SCS) black hole spacetime, which is a solution in the $(2+1)$D massive gravity proposed by Bergshoeff, Hohm, and Townsend (BHT massive gravity). We obtain analytic solutions for the null geodesic equation in the SCS black hole background and find the explicit form of deflection angles. We see that, for various values of the impact parameter, the deflection angle can be positive, negative, or even zero in this black hole spacetime. The negative deflection angle indicates the repulsive behavior of the gravity that comes from the gravitational hair parameter that is the most characteristic quantity of the BHT massive gravity.
Highlights
It is well known that in (2 + 1) dimensions, the Riemann tensor can be expressed in terms of the Ricci tensor and the Ricci scalar since the Weyl tensor is identically zero
In this paper we have studied the null geodesics in the static circularly symmetric black hole in the BHT massive gravity
We obtained analytic solutions for the geodesic equation for massless particles and found that the gravity behaves repulsively for the null geodesics with the parameters corresponding to Eq (4.7)
Summary
It is well known that in (2 + 1) dimensions, the Riemann tensor can be expressed in terms of the Ricci tensor and the Ricci scalar since the Weyl tensor is identically zero. We show that the deflection angle in the SCS black hole spacetime can be negative This fact indicates the repulsive behavior of the gravity. It is known that the Lifshitz black hole [16, 17], which is another type of black hole spacetime in the BHT massive gravity, does not have the gravitational hair parameter and the geodesics in its black hole background do not show such a repulsive behavior of the gravity. The last section is devoted to the conclusion and the discussion
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