*f*, and find that both the extremal and near-extremal black holes cannot be overspun.

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- Chinese Physics C
- Vol. 44, Issue 1, (2020)

Abstract

1. Introduction

It is widely believed that spacetime singularities are formed at the end of a gravitational collapse. At the singularities, all physical laws break down. To avoid the destruction caused by the singularities, Penrose proposed the weak cosmic censorship conjecture (WCCC) in 1969 [

The Gedanken experiment proposed by Wald is the first attempt to test the validity of WCCC [

The validity of WCCC was examined from the aspect of fields in Refs. [

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In this paper, we review the thermodynamics and WCCC in BTZ black holes using the scattering of a scalar field. The BTZ black holes are generic in the sense that they constitute an open set in the space of solutions of the Einstein equation. The change of the energy and angular momentum of the black hole during a time interval relies on the fluxes of energy and angular momentum of the ingoing wave function. To get the wave function, we introduce a tortoise coordinate and solve the second-order differential equation at the event horizon. Here, the time interval is infinitesimal. For the non-extremal BTZ black hole, the increase of the event horizon ensures that WCCC is valid. The first law of thermodynamics is recovered by the scattering. For the near-extremal and extremal BTZ black holes, Eq. (21) is divergent at the event horizons. Therefore, we need to resort to other methods to test the validity of WCCC. Its validity is tested by evaluating the minimum value of the function *f*. It is found that the horizons do not disappear and the singularities are always hidden behind them.

The paper is organized as follows. The BTZ black hole solution is given and its thermodynamics are discussed in the next section. In Section 3, the first law of thermodynamics for the non-extremal BTZ black hole is recovered by the scattering of a scalar field. In Section 4, the validity of WCCC in the extremal and near-extremal BTZ black holes is tested using the minimum of the function *f*. Section 5 is devoted to our discussion and conclusion.

2. BTZ black holes

The BTZ metric is given by [

where

It describes a local three-dimensional rotating AdS spacetime. The parameter
*M* and *J* are the ADM mass and angular momentum. They determine the asymptotic behavior of the spacetime. The event (inner) horizons are located at

When

The entropy, Hawking temperature, angular velocity and ADM mass are respectively

Here, the expression for the entropy used in [

We show in the next section that the first law of thermodynamics is recovered by the scattering of a scalar field. The variations are caused by the interaction between the scalar field and the black hole. Due to this interaction, the energy and angular momentum are transferred, and are evaluated by the energy flux and angular momentum flux. Therefore, we first write the action and calculate the energy-momentum tensor.

3. Thermodynamics of non-extremal BTZ black holes

The action of the minimally coupled complex scalar field in the BTZ spacetime is

where

To evaluate the energy-momentum tensor, we need to know the wave equation

Due to the existence of the Killing vectors

where
*j* denote the energy and angular momentum. Inserting the contravariant components of the BTZ metric and the separation of variables (9) into the Klein-Gordon equation (8), yields a second-order differential equation. To solve this equation, we introduce the tortoise coordinate [

where
*T* is the Hawking temperature. The second-order differential equation then becomes

Since we are interested in the scattering near the event horizon, the equation needs to be solved at the horizon. Let

In the above equation,

where the solutions with

The interaction between the field and the black hole transfers the energy and angular momentum. Since we are discussing the changes in the horizon after the black hole absorbs the energy and angular momentum, we focus our attention on the ingoing wave equation.

The two Killing vectors
*E* and the angular momentum *L*. When the fluxes of energy and angular momentum flow into the event horizon and are absorbed by the black hole, the energy and angular momentum of the black hole change. The energy flux and the angular momentum flux are respectively

Combining the fluxes with the energy-momentum tensor and the ingoing wave equation yields

In this derivation,
*J*. Therefore, the increase of the energy and angular momentum during a time interval

which may be negative or positive, depending on the sign of

In the following discussion, the time interval is assumed to be infinitesimal, and the variations of the energy and angular momentum are also infinitesimal. The scattering changes the function *f* and the horizon radius

where

To derive

When

which shows that the entropy does not decrease with the scattering of the field. This result supports the second law of thermodynamics, and is a simple consequence of the fact that the system satisfies the null energy condition. From Eqs. (17) and (21), we get

Therefore, the first law of thermodynamics in the non-extremal BTZ black hole is recovered by the scattering of the scalar field.

In the thought experiments, it is usually preferred to study systems which are inferred to be close to the critical condition. In the next section, we investigate this case, namely the near-extremal and extremal BTZ black holes. For these black holes, Eq. (21) is divergent at the event horizons. Thus, the above method cannot be applied to the extremal and near-extremal BTZ black holes, and we need to resort to other methods to test WCCC.

4. WCCC in near-extremal and extremal BTZ black holes

WCCC in the near-extremal and extremal BTZ black holes has been tested, and it was found that the near-extremal BTZ black hole has the possibility to be overspun [*f* in the final state. Due to the interaction between the black hole and the field, the energy and angular momentum of the black hole change, and the value of the function *f* changes. In the metric, there are two roots (corresponding to the inner and event horizons) for

The time interval is assumed to be infinitesimal, and the transferred energy and angular momentum in the scattering are also infinitesimal. The minimum value of *f* is expressed as
*M* and *J*. Thus we get

where

and

For the extremal BTZ black hole, the event and inner horizons are coincident and the temperature is zero. Thus, the term
*f*, namely

This result shows that the extremal black hole is also extremal after scattering with a new mass and angular momentum. Therefore, the extremal BTZ black hole cannot be overspun. This is in full accordance with the result obtained by Rocha and Cardoso in [

For the near-extremal BTZ black hole, we have

For convenience of discussion, we rewrite Eq. (23) as a function of

The maximum, located at

where

5. Discussion and conclusion

In this paper, we investigated the thermodynamics and WCCC in the BTZ black holes using the scattering of a scalar field. The variations of the energy and angular momentum of the black holes in an infinitesimal time interval were calculated. The first law of thermodynamics in the non-extremal BTZ black hole is recovered by the scattering. The increase of the horizon radius ensures that the singularity is hidden behind the event horizon of the non-extremal black hole. For the near-extremal and extremal BTZ black holes, since Eq. (21) is divergent, we tested WCCC by evaluating the minimum of the function *f* in the final state. We found that these black holes maintain respectively their near-extremity and extremity. This result is in full accordance with that obtained by Wald and Jorse [

In a recent work, D
*n* is the azimuthal wave number. When there is no superradiation, the frequency is in the range

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Deyou Chen. Weak cosmic censorship conjecture in BTZ black holes with scalar fields *[J]. Chinese Physics C, 2020, 44(1):

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