By Dias, O J C
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Extra info for Black Hole Solutions and Pair Creation of Black Holes in Three, Four and Higher Dimensional Spacetimes
Sample text
Moreover, the case ω > −1 yields gravities whose black holes have a structure and properties similar to the BTZ black hole, but with a feature that might be useful: the ω > −1 black holes have dynamical degrees of freedom, which implies for example that the origin has a curvature singularity and that gravitational waves can propagate in the spacetime. The neutral black holes of this Brans-Dicke theory were found and analyzed by S´ a, Kleber and Lemos [69, 70]. The pure electric charged black holes have been analysed by Dias and Lemos [71], and the pure magnetic solutions have been discussed by Dias and Lemos [72].
We work at the level of the Newtonian limit, but the full general relativity analysis yields the same result. Consider a static distribution of matter with an energy-momentum tensor given by T µν = diag(ρ, p1 , p2 , p3 ), where ρ = T 00 is the mass density of the system and pi = Ti are the pressures along the three directions. The Newtonian limit of Einstein equations for this distribution is given by ∇2 Φ = 4πG(ρ − T ii ), where Φ is the gravitational potential. For non-relativistic matter, pi ≪ ρ, and we recover the classical Poisson equation, ∇2 Φ = 4πGρ.
The dashed line represents a null ray that is sent from the vicinity of the event horizon of one of the black holes towards the other black hole. (a) In the case of the C-metric this ray can never reach the other black. Thus, there is no gravitational interaction between the black holes. (b) In the Israel-Khan solution the null ray can reach the second black hole, and so they attract each other gravitationally. In this solution, the two black holes are connected by a strut that exerts an outward pressure which cancels the inward gravitational attraction, and so the distance between the two black holes remains fixed.