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 5D black hole and black ring solutions

5D black lens solutions Review of lens-space Static black lens Rotating black lens

Review of 4D black hole solutions We are interested in the vacuum solutions of

Einstein’s field equation. 02

1 uvuv RgR

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.

By setting a=0, we recover the Schwarzschild black hole.

Kerr black hole: rotating black hole, whose line element takes the following form (with mass M=m and angular momentum J=ma)

Only allowed horizon topology in 4D is S2

θ

2S

The horizon of the Schwarzschild black hole is located at r=2m. For constant time slice, it has an induced metric

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.

At physical infinity we recover a Minkowski space-time (meaning asymptotically flat):

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).

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.

3S 4Rin

Myers-Perry BH

2S

1S12 SS in 4R

Emparan-Reall BR

Does a black hole with lens-space L(p, q) horizon topology exist in 5D?

5D black lensReview of lens-space

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)

A 3-sphere S3 can be defined to be the set

A lens-space L(p, q) is a quotient space of 3-sphere S3. More precisely,

We define the cyclic group Zp={0,1,2…p-1} which acts on S3 freely by

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.

What is the horizon topology of this space-time?

In a new form (known as C-metric form), the solution reads

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).

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

Thank you!