Non-linear collisional Penrose process – How large energy can a black hole release? –

Non-linear collisional Penrose process
– How large energy can a black hole release? –

   Ken-ichi Nakao    Hirotada Okawa    Kei-ichi Maeda
Abstract

Energy extraction from a rotating or charged black hole is one of fascinating issues in general relativity. The collisional Penrose process is one of such extraction mechanisms and has been reconsidered intensively since Bañados, Silk and West pointed out the physical importance of very high energy collisions around a maximally rotating black hole. In order to get results analytically, the test particle approximation has been adopted so far. Successive works based on this approximation scheme have not yet revealed the upper bound on the efficiency of the energy extraction because of lack of the back reaction. In the Reissner-Nordström spacetime, by fully taking into account the self-gravity of the shells, we find that there is an upper bound on the extracted energy, which is consistent with the area law of a black hole. We also show one particular scenario in which the almost maximum energy extraction is achieved even without the Bañados-Silk-West collision.

Non-linear collisional Penrose process

[.5em] – How large energy can a black hole release? –

Kei-ichi Maeda

                 \@hangfrom Subject Index E01, E31

1 Introduction

Energy extraction from a black hole is one of the interesting and important issues not only in general relativity but also in astrophysics (engines of -ray bursts, energy sources of jets from AGN, origins of ultra-high-energy cosmic rays, etc.). In 1969 [1], Penrose pointed out that it is possible to extract the rotational energy of a Kerr black hole, which is a stationary and axi-symmetric rotating black hole, through the decay of a particle falling from the infinity to create two particles in the ergo-region, in the case that one is bounded with negative energy, whereas the other escapes to infinity with positive energy. Successive works revealed that this mechanism does not work quite efficiently in the astrophysical situation [2, 3]. A bit modified version of the Penrose process called the collisional Penrose process, in which two particles collide with each other in the ergo-region instead of a single particle decay, was first noticed by Piran, Shaham and Katz [4], but its efficiency as modest as the original process was reported.

Recently, the collisional Penrose process again attracts people since Bañados, Silk and West (BSW) showed that there is no upper bound on the center-of-mass energy of two particles colliding with each other almost at the event horizon of an extremal Kerr black hole [5]. This fact does not necessarily mean the unbounded energy extraction from the black hole, as the particle escaping to the infinity wastes its energy to run up the deep gravitational potential. Nevertheless, some works consistently show that fine-tuned parameters of the particles result in the energy output about 14 times larger than the input energy [6, 7, 8, 9, 10, 11]. It is worthwhile to notice that the same conclusion is derived even though the deformation of the event horizon caused by energetic particles swallowed by the black hole is taken into account in accordance with the hoop conjecture [12]. More efficient extraction mechanism of the energy from a black hole, which has been named the super-Penrose process, was suggested [13, 14], but there is still an argument [10].

In order to know how large energy a black hole can really release through the Penrose process, one should fully take into account the nonlinearity of the Einstein equations. It is much complicated and not so easy to treat a Kerr black hole with the gravitational backreaction by the particles. Here it is worthwhile to notice that the similar phenomenon to the BSW collision [15] and the collisional Penrose process  [16, 17] can occur in the case of the Reissner-Nordström black hole which is a spherically symmetric charged black hole. The BSW collision can also occur between two infinitesimally thin charged dust shell concentric to the Reissnr–Nordström black hole although non-linear effects are taken into account through Israel’s formalism [18]. In this paper, we shall study the collisional Penrose process in the similar situation to that studied in Ref. [18] and analyze the energy extraction efficiency.

This paper is organized as follows. In Section 2, we explain our setup and derive the equations of motion for a spherically symmetric infinitesimally thin charged shell in accordance with Israel’s formalism. Also in this section, we derive the formulation to get the conditions of two thin shells concentric with each other just after a collision with the mass transfer by imposing the 4-momentum conservation. We estimate the maximum extraction from the central black hole by analytic means in Section 3. Section 4 is devoted to concluding our analyses.

We adopt the abstract index notation: small Latin indices indicate a type of a tensor, whereas Greek indices denote components of a tensor with respect to the coordinate basis vector [19]. We also adopt the signature of the metric and the convention of the Riemann tensor used in Ref. [19]. The geometrized unit is adopted.

2 Setup and basic equations

2.1 Setup

We consider two spherically symmetric shells concentric with each other. Each shell is infinitesimally thin and generates a timelike hypersurface through its motion. We will often refer this hypersurface as a shell. These shells will collide with each other, and divide the spacetime into four regions (see Fig.1). Before the collision, we call these shells Shell 1 and Shell 2, respectively. After the collision, the shell which faces on a region together with Shell 2 is called Shell 3, and the other shell is called Shell 4. The region whose boundary is formed by Shell 1 and Shell 4 is called Region 1, while the region between Shell 1 and Shell 2 is called Region 2. Similarly, the region whose boundary is formed by Shell 2 and Shell 3 is called Region 3, and the region between Shell 3 and Shell 4 is called Region 4. For notational convenience, Region 1 is often called Region 5. Hereafter, we use capital Latin indices, , and , to specify a shell or a region: runs from to , takes the values and , which represents the shells before the collision, and and , labeling the shells after the collision.

Figure 1: A schematic diagram of the collision between two spherical shells. The vertical direction is temporal, while the horizontal direction is radial.

2.2 Equations of motion for shells

Let be a unit outward space-like vector normal to Shell , and define the projection operator as . Each shell is characterized by the surface stress-energy tensor which is given by

where is a Gaussian normal coordinate ( on the shell).

The extrinsic curvature of a timelike hypersurface generated by the motion of Shell is defined by

where is the covariant derivative.

The Einstein equations lead to the jump condition for the extrinsic curvatures and the conservation law for [20]:

(1)
(2)

and

(3)

where the quantity with the subscript is defined in the region to which the unit normal points, whereas that with the subscript is evaluated on another side.

Now let us turn the spherically symmetric case. We assume that the line element in Region is given in the form

(4)

where is not specified in this section so that the results obtained here is applicable to various cases: a vacuum spacetime, one with the Maxwell field, one with a cosmological constant, and so on.

The components of the 4-velocity of Shell are expressed as

(5)

where an over dot represents a derivative with respect to the proper time naturally defined on the shell. Here note that the time coordinate, , is not continuous across the shell, although the circumferential radius, , the azimuthal angle, , and the polar angle, , are everywhere continuous. Hence two different time coordinates are assigned to each shell, and there are two kinds of time components of the 4-velocity. Using these components of the 4-velocity, we obtain the components of the unit vector normal to Shell as

(6)

The surface-stress-energy tensor of the spherical shell takes the following form

where is the surface energy density, corresponds to the tangential pressure, and is the 2-sphere metric with the radius . Then the conservation law (3) leads to

(7)

where

(8)

is the proper mass of Shell . In the case of , we often call Shell a dust shell and Eq. (7) implies that is constant. On the other hand, in the case of non-vanishing , depends on the proper time, if the shell is moving. We assume the reasonable energy conditions, so that .

Now, we assume the outward normal which is directed from region to region , whereas the direction of is from region to region . This assumption implies, together with Eq. (6), that the circumferential radius is increasing across the shell (shell ) from region (region ) to region (region ). Then, the junction condition (1) leads to

(9)

and

(10)

where

(11)
(12)

As shown later, corresponds to the specific Misner-Sharp (MS) energy [21] (MS energies per unit mass) of Shell .

From the normalization of 4-velocity and Eqs. (9) and (10), we obtain

(13)
(14)

and

(15)
(16)

2.3 Momentum conservation

In order to determine the motions of the shells after the collision, we impose the “momentum conservation” at the collision event;

(17)

where is the conserved total 4-momentum of two shells (see Appendix A). Using this conservation law (17), in what follows, we will show how and are determined when and are fixed. The 4-velocities and contain the information carried by two shells after collision, as we will show the details later.

For this purpose, we write down in the linear combination form of and , and describe the components of with respect to the coordinate basis in Region 3. This is because the components of and with respect to the coordinate basis in Region 3 are given as the initial data before the collision. We also write down in the linear combination form of and and describe the components of with respect to the coordinate basis in Region 1 by the similar reason.

In general, scattering problems are extremely simplified in the center of mass frame. Hence, we define the dyad basis corresponding to the center of mass frame as

(18)
(19)

where

Here, we write the 4-velocities of Shell 3 and Shell 4 in the form

(20)
(21)

The dyad components of the momentum conservation (17) lead to

(22)
(23)

From Eqs. (22) and (23), we have

(24)
(25)

By subtracting each side of Eq. (25) from that of Eq. (24), we have

and hence

(26)

By the similar manipulation, we also have

(27)

Since we consider the situation of Fig. 1, is positive, whereas is negative;

(28)

If we assume the proper masses and of the shells after the collision, and are determined by Eqs. (20) and (21) with the coefficients given by Eqs. (26)–(28). By using Eqs. (18) and (19), we write down in the form of the linear combination of and .

In order to write down the components of , we first replace and in and by the linear combinations of and . For notational simplicity, we introduce

(29)
(30)

We have

(31)
(32)

From the normalizations of and and the above equations, we have

(33)

Since and are desicribed by the linear combinations of and by using Eqs. (31) and (32), can also be described by the linear combinations of and through Eq. (20). We also perform the similar manipulation for by using

(34)
(35)

As a result, we have

(36)
(37)

where

(38)
(39)
(40)
(41)

2.4 Components with respect to the coordinate basis

From Eqs. (36) and (37), we obtain the components of with respect to the coordinate basis in Region 3 and those of with respect to the coordinate basis in Region 1, writing down the components of and with respect to the coordinate basis in Region 3 as

(42)
(43)

and the components of and with respect to the coordinate basis in Region 1 as

(44)
(45)

where for notational simplicity, we introduce

(46)
(47)

When we write and as

(48)

and

(49)

where

(50)
(51)

we find the relation between and . Note that , which correspond to the specific Killing energies for test particles, may describe the energies of the shells but they are not conserved because of self-gravity effects of the shells.

Hereafter, all components are evaluated at , i.e., at the collision event. By using Eqs. (43)–(48), Eq. (36) leads to

(52)
(53)

By using Eqs. (43)–(48), Eq. (37) leads to

(54)
(55)

The components of , , and with respect to the coordinate basis in Region 2 are given by

(56)
(57)
(58)

Using these components, Eqs. (29) and (30) lead to

(59)
(60)

Once we know the initial conditions of shells just before the collision ( and at the collision event) and the masses of shells just after the collision, , we can obtain and by Eqs. (26)–(28), and by Eqs. (59) and (60), and then by Eqs. (52)–(41); Note that the information about is equivalent to and , whereas that about is equivalent to and . By the definition of , the value of the metric function at the collision event is given by

(61)

We will use Eq. (61) for deriving the mass parameter of Region 4 in the next section.

3 Maximum energy extraction by the collision of charged shells

Here we consider the situation in which the collision of two spherical shells occurs around a Reissner-Nordström black hole. Each shell is assumed to be concentric to the black hole which is located in Region 1. Then, we study the maximum energy extraction from the black hole through the collisional Penrose process by two shells: Shell 4 falls into the black hole, whereas Shell 3 goes away to the infinity with the energy larger than the total energy carried initially by Shell 1 and Shell 2. In the test-shell limit , the present system recovers the situations studied in Refs. [16, 17].

The metric function of Region is given by

where and are the mass and charge parameters, respectively. The gauge one-form in the region is given by

The charge of Shell is denoted by . Gauss’s law leads to

or equivalently

The above equations lead to the conservation of total charge through the collision:

(62)

Equation (61) leads to

In the case of the spherically symmetric system, almost all of the quasi-local energies proposed until now agree with the so-called Misner-Sharp energy [21]. In the present case, the Misner-Sharp energy within the sphere with the circumferential radius is given by

(63)

Hence, the Misner-Sharp energy carried by Shell is given by

If Shell has a non-vanishing charge, depends on the radius due to the electric interaction. Then the energies of Shell 1, Shell 2 and Shell 3 found by the observers at infinity, are given by

(64)
(65)
(66)

where

Before proceeding to the non-linear analysis, it is intriguing to consider the case in which the test-shell approximation is applicable. In this case, is regarded as the charge parameter of the fixed background spacetime, whereas is the charge of Shell 3 going away to the infinity. Here, we assume , and it should be noted that as long as the charge conservation (62) holds, Shell 3 can have arbitrary large charge fixing under the test-shell approximation. Thus, if very large amount of charge is transferred from Shell 4 to Shell 3 by the collision so that and then the extracted energy can be much larger than the initial total energy of the shells , the large amount of energy is extracted from the black-hole spacetime [see Eq. (66)]. There is no upper bound on the efficiency of the energy extraction, which is defined by

(67)

This is the case pointed out by Zaslavskii [16]. If we take into account the non-linearity of the shell contribution, it is not trivial whether there exist an upper bound on the efficiency or not, although the extracted energy is finite since holds.

3.1 Upper bound on the extracted energy

Now we evaluate the upper bound on by using the fact that the Misner-Sharp energy has a non-decreasing nature with respect to , i.e., in the direction of just on Shell  [22]. From Eq. (63), the following inequality should hold;

(68)

From Eq. (68), we have

If the central black hole is extremal, i.e., , and vanishes, we have

Then the collision at the horizon radius in Region 1 gives the largest upper bound:

(69)

This is also easily understood from the view of the irreducible mass of the initial black hole. Since the initial black hole is described by an extremal Reissner-Nordström solution with the mass , the irreducible mass is given by . The rest energy is one from electromagnetic contribution and can be extracted by some mechanism. As a result, The extracted energy is bounded by , which is Eq. (69).

Inequality (69) leads to

(70)

Since Shell 4 will be absorbed into the black hole, the black hole eventually becomes charge-neutral. Then, the area of its event horizon is larger than which is equal to the initial value of the extremal BH. This result is consistent to the area law of the event horizon.

It should be noted that the largest upper bound on is achieved by the collision on the event horizon. This fact seems to imply that the BSW type collision is a necessary condition for the large efficiency in contrast to the test particle case. However, we will see in the following example that it is not necessarily the case.

3.2 An example of almost maximum energy extraction

In this subsection, we focus on the case of and . By this restriction, the expressions of the energy-momentum transfer through the collision become so simple that we obtain analytically an example of almost maximum energy extraction. The same system has been studied by Ida and one of the present authors [23], although they has not focused on the collisional Penrose process.

We have

(71)
(72)
(73)
(74)

Substituting Eqs. (71)–(74) into Eqs. (52)–(55) and by using Eqs. (A2)–(A5), we have

(75)
(76)
(77)
(78)

It is worthwhile to notice that and hold because of . Note also that holds by its definition, and hence and hold.

We again assume that the black hole is initially extremal and finally charge-neutral as a result of the absorption of Shell 4 by the black hole:

(79)

Furthermore, we assume

(80)

Since we assume the collision takes place near the horizon, we write the circumferential radius at the collision event, , in the form of

(81)

with .

Hereafter, a character with a tilde denotes a quantity normalized by the initial mass of the black hole, , i.e., and . Since Shell 1 and Shell 2 approach the black hole from infinity, should be larger than or equal to unity. We focus on the case that is of order unity.

Together with Eqs. (79)–(81), Eq. (75) leads to

(82)

The assumptions (79)–(81) lead to

(83)
(84)

Since the collision should occur outside the black hole, we have from Eqs. (83) and (84) the following constraints

(85)
(86)

Equation (85) implies that should be at most of order , and then Eq. (86) implies that both of and should also be at most of order . Hereafter we assume

We consider the situation in which holds at the collision event. Then, is approximately estimated at

(87)
(88)

Therefore, the asymptotic energy of Shell 3 is given by