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Five-dimensional black holes with lens-space horizon topology (black lenses)

Five-dimensional black holes with lens-space horizon topology (black lenses). Yu Chen and Edward Teo Department of Physics, National University of Singapore. Phys. Rev. D 78 (2008) 064062. Outline. Review of 4D and 5D black hole solutions 4D black hole solutions

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Five-dimensional black holes with lens-space horizon topology (black lenses)

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

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

  3. Review of 4D black hole solutions • We are interested in the vacuum solutions of Einstein’s field equation. • Uniqueness theorem: In 4D 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 S2. In vacuum case, Q=0, it coincides with the Kerr black hole. • 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.

  4. Only allowed horizon topology in 4D is S2 At physical infinity we recover a Minkowski space-time (meaning asymptotically flat): The horizon of the Schwarzschild black hole is located at r=2m. For constant time slice, it has an induced metric θ Obviously the horizon has a topology S2. We can do similar analysis for the horizon of Kerr black hole. The topology is also a S2. And the uniqueness theorem asserts that it is the only allowed topology, so black holes with topology S1×S1 do not exist in 4D asymptotically flat space-time.

  5. Review of 5D black hole solutions • 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 5D asymptotically flat stationary black holes with two axial symmetries have restricted their horizon topology to three possibilities: either a sphere S3, a ring S1×S2, or a lens-space L(p, q). • 5D Myers-Perry black hole: S3 horizon topology, rotating along two independent axes in two orthogonal planes (with mass M=m and angular momentum J1=ma1, J2=ma2).

  6. 5D black holes and black rings • Emparan-Reall black ring/Pomeransky-Senkov black ring: S1×S2 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 non-uniqueness of the black holes in 5D asymptotically flat space-time. Myers-Perry BH Emparan-Reall BR in in Does a black hole with lens-space L(p, q) horizon topology exist in 5D?

  7. 5D black lensReview of lens-space • A lens-space L(p, q) is a quotient space of 3-sphere S3. More precisely, • A 3-sphere S3 can be defined to be the set • We define the cyclic group Zp={0,1,2…p-1} which acts on S3 freely by • Then the lens-space is defined as L(p, q)=S3/Zp. • Some special cases of the lens space L(p, q): • L(1, q)= S3, • L(2, 1)= RP3, • L(0, 1)=S1×S2 (a degenerate limit)

  8. Static black lens • The local metric for a static black lens was previously found by Ford et al in arXiv: 0708.3823 and by Lu et al in arXiv: 0804.1152. But they never made a black lens interpretation. • In a new form (known as C-metric form), the solution reads • What is the horizon topology of this space-time?

  9. Horizon topology of the static black lens • The induced metric on the horizon is homeomorphic to • But identifications must be made through • We see that if and have periods 2π, the horizon is a S3, but the above identifications form a cyclic group Zn. To see this more clearly, define a map • Hence the horizon topology is a lens space L(n, 1).

  10. Rotating black lens • It can be shown a conical singularity is present in the static black lens space-time 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 self-gravitation (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

  11. Thank you!

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