Stability of five-dimensional Myers-Perry black holes with equal angular momenta

Kyoto University, JapanKeiju Murata & Jiro Soda

IntroductionIn string theory

large extra dimension scenario (ADD model, brane world, …)

It is important to study the higher dimensional black holes.

Our spacetime should be higher dimension.4D spacetime + compactified space

evidence for higher dimensional spacetime

Mini black holes may be produced at LHC.

It need not be planck scale.

The compactified space can be mm scale.

The fundamental Planck mass can be TeV scale.

variety of higher dimensional black holes

They can have same masses and angular momenta.

Ｍ , J Ｍ , JMyers-Perry BH black ring

There is no uniqueness theorem in higher dimensional spacetime.

What kind of black holes are formed in colliders?

To answer this problem, we need the stability analysis of higher dimensional black holes.

We focus on the stability analysis of Myers-Perry black holes.

(Emparan & Reall, 2006)

What is the stability analysis?

We consider the perturbation of the background metric.

If the grows exponentially, the background spacetime is unstable.

If the oscillates , the background spacetime is stable.

In general, the perturbation equation is given by PDE.

However, in some cases, the perturbtion equation can be separated and becomes ODE.

substituteperturbation equation

ex) Schwarzschild BH

stability analysis of rotating black holes

Killing vectors of general D-dimensional Myers-Perry BH are

(n+1) Killing vectors

n+1 < D-1 (for D >= 4)

In general, the symmetry is not enough to separate the perturbation equation.

However, in some cases, the symmetry is enhanced and the perturbation equation of the Myers-Perry spacetime becomes separable.

We constructed the formalism for stability analysis of 5-dimensional Myers-Perry BH with equal angular momenta

and

We gave the strong evidence for stability of this BH.

Myers-Perry black holes with equal angular momenta

(KM & J.Soda, Prog.Theor.Phys.120:561-579,2008 [arXiv:0803.1371 ])

The spaceime symmetry is enhanced and we can separate the perturbation equations.

5D Myers-Perry BH with equal angular momenta

PDEs of

The PDEs can be reduced to ODEs by the mode decomposition.

stability analysis

Symmetry

SU(2)Killing

are invariant forms of SU(2).

part has a rotational symmetry.

that is SU(2) symmetry

U(1) symmetry

The symmetry of this spacetime is

We decompose by using this symmetry and obtain ODEs.

mode function

, and are simultaneously diagonalizable.

Wigner functions

eigenfunction

vector Wigner function

We can also construct vector and tensor eigen functions.

Making use of these mode functions, we can separate the tensor field as

Substituting this into ,

we can get ODEs labeled by (J, K, M).

In general, the resultant ODEs are coupled each other and cannot reduce to single ODE.

However, special modes are decoupled and reduce to single ODE.

We studied the stability of these mode.

tensor Wigner function

Master equationsmaster equation for J=M=K=0 mode symmetric mode

tortoise coordianate

master variable

gauge conditions

We obtain Schrodinger type master equation.

master equation

From bottom to top, a/a_max = 0.25, 0.5, 0.75, 0.99

r / r_h

V_0

The potential is positive definite.

This mode is stable.

J=M=0, K=±1

∀J, ∀M, K=±(J+2)

(J=0, M=0, K=0,±1) and (J, M, K=±(J+2)) modes reduce to single ODE.

These equations are not Schrodinger type equations.

We solved the equations numerically and found that there is not mode with Im(ω) > 0.

These mode are stable.

SummaryWe have studied the stability of 5-dimensional Myers-Perry black holes with equal angular momenta.

The perturbation equations can be separate and reduce to ODEs.

Some modes are decoupled and reduce to single ODEs.

We got the master equations for these modes and showed the stability.

evidence for the stability of this BH.

We found that these modes are stable.evidence for stability of this spacetime

4-dimensional Schwarzschild BH

Killing vectors of this spacetime are

time translation symmetry

spherical symmetry

We define operators

are simultaneously diagonalizable.

Eingen functions are and

We also find

Modes with different do not couple each other in perturbation equation.

separable

example