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arXiv:hep-th/9204046v1 15 Apr 1992

TIFR/TH/92-20

hepth@xxx/9204046

April, 1992

ROTATING CHARGED BLACK HOLE SOLUTION

IN HETEROTIC STRING THEORY

Ashoke Sen

Tata Institute of Fundamental Research

Homi Bhabha Road, Bombay 400005, India

e-mail address: sen@tifrvax.bitnet

Abstract

We construct a solution of the classical equations of motion arising in the low energy

effective field theory for heterotic string theory. This solution describes a black hole in four

dimensions carrying mass M, charge Q and angular momentum J. The extremal limit of

the solution is discussed.

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It has been realized recently that the low energy effective field theory describing string

theory contains black hole (or, more generally, black p-brane) solutions which can have

properties which are qualitatively different from those that appear in ordinary Einstein

gravity[1]. Most of these solutions are characterized by one or more charges associated

with Yang-Mills fields or the antisymmetric tensor gauge field, and a non-trivial dilaton

field. In the absence of any charge, the solution reduces to the ordinary Schwarzschild

solution.

Rotating charge neutral black hole solutions can also be constructed in string theory, and

are identical to the Kerr solution[2] of ordinary Einstein gravity with the dilaton taking a

constant value. Recently, rotating charged black hole solutions in these theories have been

analyzed[3] in the limit of small angular momentum. In fact, in ref.[3] the authors consider

a more general class of theories than those which arise as the low energy effective action

in string theory, by allowing dilaton couplings to the Maxwell field of the type which is

not necessarily the one induced in string theory. They, however, consider only the dilaton-

graviton system, and do not consider more general action that also includes antisymmetric

tensor gauge field.

In this paper we shall construct an exact classical solution in the low energy effective

field theory describing heterotic string theory, which describes a black hole carrying finite

amount of charge and angular momentum. Our solution, however, differs from that of

ref.[3] even in the limit of small angular momentum since it involves the antisymmetric

tensor field in a non-trivial way. In fact, since a rotating charged black hole also carries a

magnetic dipole moment, the antisymmetric tensor field background is induced automati-

cally if we take into account the coupling of the antisymmetric tensor gauge field strength

to the Chern-Simons three form constructed from the gauge fields.

The method that we shall be using for obtaining the solution is the twisting procedure[4]-

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[8] that generates inequivalent classical solutions starting from a given classical solution

of string theory. In particular, in ref.[7] it was shown how one can generate charged

blackhole solutions starting from a charge neutral solution. With the help of the same

transformations, we shall generate the rotating charged black hole solution by starting

from a rotating black hole solution carrying no charge, i.e. the Kerr solution[2].

We shall first summarize the results of ref.[7] applied to the particular problem at hand.

We begin with the string theory effective action in four dimensions:

S = −

?

d4x√−det Ge−Φ(−R +

1

12HµνρHµνρ− Gµν∂µΦ∂νΦ +1

8FµνFµν) (1)

Here Gµνis the metric, R is the scalar curvature, Fµν= ∂µAν−∂νAµis the field strength

corresponding to the Maxwell field Aµ, Φ is the dilaton field, and,

Hµνρ= ∂µBνρ+ cyclic permutations − (Ω3(A))µνρ

(2)

where Bµνis the antisymmetric tensor gauge field, and,

(Ω3(A))µνρ=1

4(AµFνρ+ cyclic permutations) (3)

is the gauge Chern-Simons term. There are several points we need to mention at this

stage. They are the followings.

1. We are considering a theory where 6 of the 10 dimensions have been compactified (say,

to a Calabi-Yau manifold). The massless fields arising from compactification have not

been included in the effective action.

2. We have included only a U(1) component of the full set of non-abelian gauge fields

present in the theory. This will suffice for our purpose, since we shall look for solutions

carrying U(1) charge only.

3. The metric Gµν used here is the metric that arises naturally in the σ-model, and is

related to the Einstein metric G(E)

µν through the relation:

G(E)

µν= e−ΦGµν

(4)

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After this field redefinition (together with a rescaling Φ → 2Φ, Aµ→ 2√2Aµ) one can

recover the action of ref.[3], except for the HµνρHµνρterm appearing in eq.(1).

4. We have truncated the action to contain only those terms that contain two or less number

of derivatives. Thus, for example, the Lorentz Chern Simons term has not been included

in the definition of Hµνρ, since the corresponding terms in the action will contain more

than two derivatives.

We shall look for solutions that are independent of the time coordinate t. In the following

analysis we shall use matrix notation to describe the various fields. In this notation, Gµν

and Bµν will be treated as 4 × 4 matrices, and Aµwill be treated as a four dimensional

column vector, with the fourth row and/or column corresponding to the time coordinate

t. Let us now define the matrices K, η and M as,

Kµν= −Bµν− Gµν−1

4AµAν

(5)

ηµν= Diag(1,1,1,−1)(6)

and

M =

(KT− η)G−1(K − η)

(KT+ η)G−1(K − η)

−ATG−1(K − η)

(KT− η)G−1(K + η)

(KT+ η)G−1(K + η)

−ATG−1(K + η)

−(KT− η)G−1A

−(KT+ η)G−1A

ATG−1A

(7)

Here T denotes transposition of a matrix. Eq.(7) defines a 9× 9 matrix M. The result of

ref.[7] then says that if {Gµν,Bµν,Φ,Aµ} describes a time independent solution of the clas-

sical equations of motion derived from the action given in eq.(1), then {G′

also describes a solution of the same equations of motion, if the primed variables are related

µν,B′

µν,Φ′,A′

µ}

to the unprimed ones through the relation,

M′= ΩMΩT,Φ′− lndetG′= Φ − lndetG (8)

where,

Ω =

I7

coshα

sinhα

sinhα

coshα

(9)

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Here I7denotes a 7 × 7 identity matrix, and α is an arbitrary number. Eqs.(8) uniquely

determine all the primed fields in terms of the unprimed ones.

We now apply this transformation to the charge neutral rotating black hole solution.

This is given by the standard Kerr solution[2]

ds2= −ρ2+ a2cos2θ − 2mρ

ρ2+ a2cos2θ

sin2θ

ρ2+ a2cos2θ{(ρ2+ a2)(ρ2+ a2cos2θ) + 2mρa2sin2θ}dφ2−

dt2+

ρ2+ a2cos2θ

ρ2+ a2− 2mρdρ2+ (ρ2+ a2cos2θ)dθ2

+

4mρasin2θ

ρ2+ a2cos2θdtdφ

Φ =0,Bµν= 0,Aµ= 0

(10)

The transformed solution is given by,

ds′2= −(ρ2+ a2cos2θ − 2mρ)(ρ2+ a2cos2θ)

(ρ2+ a2cos2θ + 2mρsinh2 α

ρ2+ a2cos2θ

ρ2+ a2− 2mρdρ2+ (ρ2+ a2cos2θ)dθ2

+ {(ρ2+ a2)(ρ2+ a2cos2θ) + 2mρa2sin2θ + 4mρ(ρ2+ a2)sinh2α

(ρ2+ a2cos2θ)sin2θ

(ρ2+ a2cos2θ + 2mρsinh2 α

−4mρacosh2 α

(ρ2+ a2cos2θ + 2mρsinh2 α

2)2

dt2

+

2+ 4m2ρ2sinh4α

2}

×

2)2dφ2

2(ρ2+ a2cos2θ)sin2θ

2)2

dtdφ

(11)

Φ′= −lnρ2+ a2cos2θ + 2mρsinh2 α

ρ2+ a2cos2θ

2

(12)

A′

φ= −

2mρasinhαsin2θ

ρ2+ a2cos2θ + 2mρsinh2 α

2

(13)

A′

t=

2mρsinhα

ρ2+ a2cos2θ + 2mρsinh2 α

2

(14)

B′

tφ=

2mρasinh2 α

ρ2+ a2cos2θ + 2mρsinh2 α

2sin2θ

2

(15)

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The other components of A′

µand B′

µνvanish. The Einstein metric ds′2

E≡ e−Φ′ds′2is given

by,

ds′2

E= −

ρ2+ a2cos2θ − 2mρ

ρ2+ a2cos2θ + 2mρsinh2 α

2

dt2+ρ2+ a2cos2θ + 2mρsinh2 α

ρ2+ a2− 2mρ

4mρacosh2 α

ρ2+ a2cos2θ + 2mρsinh2 α

2

dρ2

+ (ρ2+ a2cos2θ + 2mρsinh2α

2)dθ2−

2sin2θ

2

2+ 4m2ρ2sinh4α

dtdφ

+ {(ρ2+ a2)(ρ2+ a2cos2θ) + 2mρa2sin2θ + 4mρ(ρ2+ a2)sinh2α

sin2θ

ρ2+ a2cos2θ + 2mρsinh2 α

2

2}

×

dφ2

(16)

This metric describes a black hole solution with mass M, charge Q, angular momentum

J, and magnetic dipole moment µ given by,

M =m

2(1+coshα),Q =

m

√2sinhα,J =ma

2(1+coshα),µ =

1

√2masinhα (17)

so that the g-factor[3] is given by,

g ≡2µM

QJ

= 2 (18)

We shall now analyze various properties of this solution, and also discuss its extremal

limit. For this purpose, it will be more convenient to express m, a and α in terms of the

independent physical parameters M, J and Q by inverting the relations given in eq.(17).

We get,

m = M −Q2

2M,

sinhα =

2√2QM

2M2− Q2,a =

J

M

(19)

The coordinate singularities (horizon) occur on the surfaces

ρ2− 2mρ + a2= 0 (20)

which gives,

ρ = m ±

?

m2− a2= M −Q2

2M±

?

(M −Q2

2M)2−J2

M2≡ ρ±

H

(21)

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The area of the outer event horizon with the metric given in eq.(16) is given by,

A = 8πM(M −Q2

2M+

?

(M −Q2

2M)2−J2

M2) (22)

From eq.(21) we see that the horizon disappears unless,

|J| ≤ M2−Q2

2

(23)

Thus the extremal limit of the black hole corresponds to |J| → M −

A → 8π|J|. Hence the event horizon remains to be of finite size in this limit, as is expected

from the general arguments of ref.[3]. Note the amusing result that in the extremal limit

Q2

2M. In this limit,

the area of the event horizon depends only on the angular momentum J. Surprisingly,

this result is identical to the corresponding result for the rotating charged black hole in a

different model discussed in ref.[9].

The angular velocity Ω at the horizon is determined by demanding that the Killing

vector

∂

∂t+ Ω∂

∂φis null at the horizon[2][3]. In other words,

Gtt+ 2GtφΩ + GφφΩ2= 0(24)

This gives,

Ω =

J

2M2

1

M −

Q2

2M+

?

(M −

Q2

2M)2−

J2

M2

(25)

As we approach the extremal limit, Ω →

vanishes, as can be directly seen from eq.(25). It is interesting to note that in the extremal

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2Msign(J) as long as |J| ?= 0. If J = 0, then Ω

limit, |Ω| depends only on the mass of the black hole.

Finally, the surface gravity κ (or the Hawking temperature TH= κ/2π) is calculated at

the pole as,

κ = lim

ρ→ρ+

H

√gρρ∂ρ

√−gtt|θ=0=

?(2M2− Q2)2− 4J2

2M(2M2− Q2+?(2M2− Q2)2− 4J2)

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(26)

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Thus in the extremal limit κ → 0 if J ?= 0. On the other hand, if J = 0, then κ =

agreement with the results of refs.[1].

1

4M, in

To summarize, in this paper we have constructed a rotating charged black hole solution

in four dimensional heterotic string theory and studied its various properties. The extremal

limit of the solution was also discussed, and, for J ?= 0, was found to have features that

are qualitatively similar to the extremal rotating black hole rather than extremal charged

black hole, as was conjectured in ref.[3].

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