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

2. Outline Review of 4D and 5D 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. in
Myers-Perry BH in
Emparan-Reall BR
Does a black hole with lens-space L(p, q) horizon topology exist in 5D?

7. 5D black lens Review 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: and by Lu et al in arXiv: 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

12. Thank you!