Fivedimensional black holes with lensspace horizon topology black

Five-dimensional black holes with lens-space horizon topology (black lenses) Phys. Rev. D 78 (2008) 064062 Yu Chen and Edward Teo Department of Physics, National University of Singapore

Outline n Review of 4 D and 5 D black hole solutions n n n 4 D black hole solutions 5 D black hole and black ring solutions 5 D black lens solutions n n n Review of lens-space Static black lens Rotating black lens

Review of 4 D black hole solutions n n We are interested in the vacuum solutions of Einstein’s field equation. Uniqueness theorem: In 4 D asymptotically flat space-time, a black hole is uniquely determined by its mass M, angular momentum J and charge Q, and the only allowed topology of horizon is a sphere S 2. In vacuum case, Q=0, it coincides with the Kerr black hole. n Kerr black hole: rotating black hole, whose line element takes the following form (with mass M=m and angular momentum J=ma) By setting a=0, we recover the Schwarzschild black hole.

Only allowed horizon topology in 4 D is S 2 At physical infinity we recover a Minkowski space-time (meaning asymptotically flat): The horizon of the Schwarzschild black hole is located at r=2 m. For constant time slice, it has an induced metric Obviously the horizon has a topology S 2. We can do similar analysis for the horizon of Kerr black hole. The topology is also a S 2. And the uniqueness theorem asserts that it is the only allowed topology, so black holes with topology S 1×S 1 do not exist in 4 D asymptotically flat space-time. θ

Review of 5 D black hole solutions n n n Higher dimensional black holes have attracted a lot of attention towards unifying gravity with other forces in recent years, and production of these black holes is predicted in certain theories. But a complete classification of these black holes is far from known. Recent uniqueness considerations on 5 D asymptotically flat stationary black holes with two axial symmetries have restricted their horizon topology to three possibilities: either a sphere S 3, a ring S 1×S 2, or a lens-space L(p, q). 5 D Myers-Perry black hole: S 3 horizon topology, rotating along two independent axes in two orthogonal planes (with mass M=m and angular momentum J 1=ma 1, J 2=ma 2).

5 D black holes and black rings n n Emparan-Reall black ring/Pomeransky-Senkov black ring: S 1×S 2 horizon topology. The striking thing is that the black ring can take the same mass and angular momenta as the Myers-Perry black hole in certain cases. This indicates a discrete nonuniqueness of the black holes in 5 D asymptotically flat space-time. Myers-Perry BH Emparan-Reall BR in Does a black hole with lens-space L(p, q) horizon topology exist in 5 D? in

5 D black lens Review of lens-space n A lens-space L(p, q) is a quotient space of 3 -sphere S 3. More precisely, n A 3 -sphere S 3 can be defined to be the set n We define the cyclic group Zp={0, 1, 2…p-1} which acts on S 3 freely by n Then the lens-space is defined as L(p, q)=S 3/Zp. Some special cases of the lens space L(p, q): L(1, q)= S 3, L(2, 1)= RP 3, L(0, 1)=S 1×S 2 (a degenerate limit)

Static black lens n The local metric for a static black lens was previously found by Ford et al in ar. Xiv: 0708. 3823 and by Lu et al in ar. Xiv: 0804. 1152. But they never made a black lens interpretation. n In a new form (known as C-metric form), the solution reads n What is the horizon topology of this space-time?

Horizon topology of the static black lens n The induced metric on the horizon is homeomorphic to n But identifications must be made through n We see that if and have periods 2π, the horizon is a S 3, but the above identifications form a cyclic group Zn. To see this more clearly, define a map n Hence the horizon topology is a lens space L(n, 1).

Rotating black lens n n n It can be shown a conical singularity is present in the static black lens spacetime to prevent it from collapsing due to the self-gravitation. Can we eliminate it by making the black lens rotate such that the centrifugal force balances the selfgravitation (like in the black ring case)? We have constructed a rotating black lens in asymptotically flat space-time using the inverse scattering method (ISM). But unfortunately it turns out that the rotation alone cannot balance the self-gravitation. The conical singularity is still present. Some properties of the rotating black lens A) asymptotically flat B) L(n, 1) horizon topology C) possesses an angular momentum D) a conical singularity is needed to balance the self-gravitation

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