Particle Acceleration in Kerr-(anti-) de Sitter Black Hole Backgrounds

Particle Acceleration in Kerr-(anti-) de Sitter Black Hole Backgrounds

Yang Li    Jie Yang    Yun-Liang Li    Shao-Wen Wei    Yu-Xiao Liu Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China
Abstract

Banãdos, Silk and West (BSW) proved that Kerr black holes could act as particle accelerators with arbitrarily high center-of-mass energy, if the two conditions are satisfied: (1) These black holes are extremal; (2) one of the colliding particles has critical angular momentum. In this paper, we extend the research to the cases of Kerr-(anti-) de Sitter black holes and find that the cosmological constant has an important effect on the result. In order for the case of Kerr-anti-de Sitter black holes (with negative cosmological constant) to get arbitrary high center-of-mass energy, we need an additional condition besides the above two for Kerr ones. While, for the case of general Kerr-de Sitter black holes (with positive cosmological constant), the collision of two particles can take place on the outer horizon of the black holes and the center-of-mass energy of collision can blow up arbitrarily if the above second condition is satisfied. Hence, non-extremal Kerr-de Sitter black holes could also act as particle accelerators with arbitrarily high center-of-mass energy.

pacs:
97.60.Lf, 04.70.-s
thanks: E-mail: liyang09@lzu.cnthanks: E-mail: yangjiev@lzu.edu.cn, Corresponding authorthanks: E-mail: liyl09@lzu.cnthanks: E-mail: weishw@lzu.edu.cnthanks: E-mail: liuyx@lzu.edu.cn

I Introduction

Two years ago, Banãdos, Silk and West reported a process (BSW process) that two particles may collide on the horizon of the extremal Kerr black hole with the arbitrarily high center-of-mass (CM) energy Banados1 (). Although it has been pointed out in Refs. Berti (); Jacobson () that there are astrophysical limitations preventing a Kerr black hole to be extremal, and the gravitational radiation and backreaction effects should be counted in this process, similar processes have been found in other kinds of black holes or naked singularities and the BSW process of the Kerr black hole has been studied more deeply Jacobson (); Grib1 (); Lake (); Grib2 (); Wei1 (); Grib3 (); Zaslavskii1 (); Wei2 (); Mao (); Harada (); Grib4 (); Banados2 (); Patil (). On the other hand, a general analysis of this BSW process has been done for rotating black holes Zaslavskii2 () and for most general black holes Zaslavskii3 (); WuPRD (). Some efforts have also been made to draw some implications concerning the effects of gravity generated by colliding particles in Kimura ().

In this paper, we investigate the BSW process of the Kerr-(anti-) de Sitter black hole, and our goal is to see the effect of the cosmological constant on the BSW process. There are good reasons to believe that our results can be reduced to the ones of BSW given in Banados1 () as the cosmological constant turns to zero. Besides, because the Kerr-(anti-) de Sitter black hole does not have a simple horizon structure as the previous studied black holes, we have to use a different method to study the BSW process.

This paper is organized as follows. In Sec. II, we study the horizon structure of Kerr-(anti-) de Sitter black holes. In Sec. III, we calculate the CM energy of the particle collision on the horizon of the black holes, and derive the critical angular momentum to make the CM energy to blow up. In Sec. IV, we find the BSW process requirements for the black hole and the colliding particle from the geodesic motion of the colliding particle. The conclusion is given in the last section.

Ii Extremal Kerr-(anti-) de Sitter black holes

In this section, we would like to study the extremal the Kerr-(anti-) de Sitter black holes. First, the vacuum metric of the Kerr-anti-de Sitter (Kerr-AdS) black holes in Boyer-Lindquist coordinate system with units is given by

(1)

where

(2)
(3)
(4)
(5)

is related to the mass of the black hole, is related to the black hole’s spin angular momentum per mass by , and is related to the cosmological constant by . And for the Kerr-de Sitter (Kerr-dS) black hole, the form of the vacuum metric will remain the same, but in and should be replaced by .

The horizon is given by . We can make a comparison of coefficients in for the Kerr-AdS case, where () denotes the zeros of Hackmann (). From this comparison we can conclude that for the Kerr-AdS black hole, there are two separated positive horizons at most, and is positive outside the outer horizon of the Kerr-AdS black hole. In the same way, we can conclude for the Kerr-dS black hole that there are three separated positive horizons at most, and is negative outside the outer horizon of the black hole. For the Kerr-AdS (Kerr-dS) black hole, when two horizons of the black hole coincide, the black hole is extremal.

If consider the extremal Kerr-AdS black hole, we have to make a comparison of coefficients in with being real Hackmann (). From this comparison we can get

(6)
(7)

In these equations, is positive and related to the coincided horizon of the extremal Kerr-AdS black hole. Then can be solved as

(8)

Analogously we can get these equations for the Kerr-dS case:

(9)
(10)

where is positive and related to the coincided horizon of the extremal Kerr-dS black hole. Also can be solved as

(11)
(12)

These two solutions are both the coincided horizons of the extremal Kerr-dS black hole. By computing at and , we can see that and are related to the inner and outer coincided horizons, respectively.

We can also solve and from Eqs. (9) and (10):

(13)
(14)

from which we can see that and has a range of . So for an extremal Kerr-dS black hole, there are upper limits for the extremal horizon and the angular momentum of the black hole. While for a Kerr-AdS black hole, also has an upper limit . When reaches , the metric will be singular, and when exceeds , the Kerr-AdS black hole will be unstable due to the superradiance Hawking ().

Iii The CM Energy of the collision on the horizon of the Kerr-(anti-) de Sitter black hole

To investigate the CM energy of the collision on the horizon of the Kerr-(anti-) de Sitter Black Hole, we have to derive the 4-velocity of the colliding particle. And we only study the particle motion on the equatorial plane (, ).
The generalized momenta can be given as

(15)

where the dot denotes the derivative with respect to the affine parameter and . Thus, in equatorial motion, generalized momenta and are turned out to be

(16)
(17)

and are constants of motion. In fact, and correspond to the test particle’s energy per unit mass and the angular momentum parallel to the symmetry axis per unit mass , respectively. And in the following discussion we will just regard these two constants of motion as and Hackmann ().

The affine parameter can be related to the proper time by , where is given by the normalization condition with for timelike geodesics and for null geodesics. For a timelike geodesic, the affine parameter can be identified with the proper time, and thus from Eqs. (16) and (17), we can solve 4-velocity components and :

(18)
(19)

For the remained component of the equatorial motion, we can obtain it from the Hamilton-Jacobi equation of the timelike geodesic

(20)

with the ansatz

(21)

where is the function of . Inserting the ansatz into (20), and with the help of the metric (1), we get

(22)
(23)

On the other hand, we have

(24)

Thus, we get the square of the radial 4-velocity component

(25)

Here we have obtained all nonzero 4-velocity components for the equatorial motion geodesic. Next we would like to study the CM energy of the two-particle collision in the backgrounds of Kerr-(anti-) de Sitter black holes. We assume that the two particles have the angular mentum per unit mass and energy per unit mass , respectively. For simplicity, the particles in consideration have the same rest mass . The expression of the CM energy of this two-particle collision is given by Banados1 ()

(26)

where are the 4-velocity vectors of the two particles (). With the help of Eqs. (18, 19, 25), we obtain the CM energy

(27)

where

(28)

For simplicity, we can rescale the CM energy as . We would like to see when the particles collide on the horizon. So we have to make at Eq. (27). The denominator of is zero, and the numerator of it is

(29)
(30)

When , the numerator will be zero and the value of on the horizon will be undetermined; but when , the numerator will be negative finite value and on the horizon will be negative infinity. So it should have , and for the CM energy on the horizon, we have to compute the limiting value of equation (27) as , where is related to the horizon of the black hole.

We can make in Eq. (27). Then we expand Eq. (27) at , which is the horizon under the consideration. When , the remaining term in the expansion of (27) is the zero-order term. In fact, the zero-order term is of the lowest order in the expansion of (27). So the limiting value of as is given by the zero-order term of (27) as

(31)

where

(32)

When , will be

(33)

So we can see that when and , the CM energy on the horizon will blow up. We will call the angular momentum per unit mass that make the critical angular momentum , and is given as

(34)

We can also prove that when and , the CM energy will not blow up. So if we need the CM energy to be arbitrarily high, one of the colliding particles must have the critical angular momentum and the other particle must not have the critical angular momentum.

We can see that the critical angular momentum depends on the horizon , and when we consider different horizons of the black hole, the critical angular momentums correspond to the horizons will be different. This result can reduce to the critical angular momentum for the case of the Kerr black hole when the cosmological constant becomes zero.

In order to get arbitrarily high CM energy on the horizon of the Kerr-AdS(dS) black hole, the colliding particle with the critical angular momentum must be able to reach the outer horizon of the black hole. We will study this part in next section.

Iv The radial motion of the particle with the critical angular momentum near the outer horizon of the black hole.

In this section, we will study the conditions under which the particle with the critical angular momentum can reach the outer horizon of the black hole. In order for a particle to reach the horizon of the black hole, the square of the radial component of the 4-velocity in Eq. (25) has to be positive in the neighborhood outside of the black hole’s horizon.

We denote as . Obviously, when the particle has the critical angular momentum, on the horizon of the Kerr-AdS or Kerr-dS black hole. So if the particle with the critical angular momentum can reach the horizon of the black hole, the derivative of with respect to must be positive at the horizon , i.e.,

(35)

Before doing the computation, we would like to make a parameter replacement

(36)

for the Kerr-AdS black hole, and

(37)

for the Kerr-dS black hole. After this parameter replacement, will be the horizon of the black hole (). So we can start to discuss the black hole’s horizon and let be identified with . Thus we will only write in the following discussion. For the Kerr-dS black hole, must not exceed to avoid a negative . After this parameter replacement, computing at for the particle with the critical angular momentum will give

(38)

for the Kerr-AdS black hole, and

(39)

for the Kerr-dS black hole, where

(40)
(41)
(42)

Notice that because , whether is positive only depends on the sign of or . Both and only depend on the parameters of the black hole. Next, we will discuss the cases of the Kerr-AdS and Kerr-dS black holes respectively.

iv.1 The Kerr-AdS case

In Kerr-AdS black hole case, by solving , we get

(43)

We draw the shape of in (43) with in Fig. 1, in which every point is related to a combination of the black hole horizon and the black hole spin , and the point on the line means that the corresponding is zero. Because is a continuous function of and , the different regions in Fig. 1 separated by the line () relate to different signs of . So we call is the boundary case. We can verify that above the line, and make ; and below the line, and make .

Figure 1: The shape of in (43) with . The point in region 2 will make positive, which means the particle with critical angular momentum can reach the horizon .

When , the particle with critical angular momentum can reach the horizon . But we have to make sure that is the outer horizon of the black hole. Notice that Eq. (43) is just the same as Eq. (8), which means in the boundary case of is the extremal horizon of the Kerr-AdS black hole. This means when the point is on the line in Fig. 1, the black hole is extremal. But when is off the line, is the black hole extremal or not? To answer that, we must make the parameter replacement (36) in the extremal horizon equations (6, 7), and solve them for . Eq. (43) must be one of the solutions, and this solution is related to the extremal horizon. Recall that the Kerr-AdS black hole can have two positive horizons at most. So if the black hole is extremal, these two horizons must coincide and the black hole will only have one horizon, namely . Thus Eq. (43) is the only solution. This means that only if is on the line in Fig. 1, the Kerr-AdS black hole can be extremal.

So the line in Fig. 1 can also be regarded as the boundary case of the horizon situation of the black hole. This is because when we pick a point off the line in Fig. 1, if we find is the inside horizon of the black hole, cannot turn into the outer horizon by crossing the other horizon or the number of the black hole horizons cannot change unless the point cross the line. In this case, when the point is above the line, is the outer horizon and the black hole has two horizons; when is on the line, is the extremal horizon and the black hole has only one extremal horizon; when is below the line, is the inner horizon and the black hole has two horizons. This means that if we want the particle with critical angular momentum to reach the outer horizon of the Kerr-AdS black hole, the only chance is the black hole is extremal. Thus, we must choose these points on the line in Fig. 1. But in this boundary case, and we must calculate :

(44)

where

(45)

and

(46)

In above calculation we have already used Eq. (43). If , the particle with the critical angular momentum can reach the only horizon of the extremal Kerr-AdS black hole. It can be proved that if ,

(47)
(48)

And if , we can prove that must be negative. Notice that the upper limit of black hole spin in Eq. (47) is still below .

Now we summarize the result for the case of Kerr-AdS black hole and give a comparison to the case of Kerr black hole. We find that, for a non-extremal Kerr-AdS black hole, the particle with the critical angular momentum cannot reach the outer horizon of the black hole, which is the same with the case of Kerr black hole. However, for an extremal Kerr-AdS black hole, if the additional conditions (47) and (48) are satisfied, the particle with the critical angular momentum can reach the outer horizon of the black hole. While, for an extremal Kerr black hole, this process can always happen.

iv.2 The Kerr-dS case

Analogously to the Kerr-AdS case, we solve the boundary case and get

(49)
(50)

We draw these two boundary lines in Fig. 2 with . We can see that these two boundary lines join together at . So actually there is only one boundary line in Fig. 2. Like the Kerr-AdS case, we verify that inside the boundary line, makes ; and outside the boundary line, makes .

Figure 2: The shapes of in (49) (the below line) and in (50) (the above line) with . In region 2, can make , which means that when is in region 2, the particle with the critical angular momentum can reach the black hole horizon . In the contrast, when is in region 1, the particle with the critical angular momentum cannot reach the black hole horizon .

When , we still have to make sure that is the outer horizon of the black hole. Notice that Eqs. (49) and (50) are the same as Eqs. (11) and (12). This means that the boundary line in Fig. 2 is also the boundary line of the horizon situation of the black hole. But unlike the Kerr-AdS case, the Kerr-dS black hole can have three positive horizons at most and this means there are other boundary lines of the horizon situation of the black hole. To find them, we make the parameter replacement (37) in Eqs. (9) and (10), and solve it for . Obviously, Eqs. (49) and (50) are two solutions. And there are two other solutions which relate to the situations that the black hole is extremal but is not the extremal horizon. We draw all this boundary lines of the horizon situation of the black hole in Fig. 3 with .

Figure 3: Four boundary lines and the two axes separate the plane into four different regions. The horizontal line on the top in the figure refers to the upper limit of .

To see the effect of the boundary lines, we draw a vertical line crossing all the boundary lines in Fig. 3 and we let moving along this line to see the change of the horizon situation of the black hole. On this vertical line, we choose one point in each different region (denoted by ), and for each point, we draw the horizon situation of the black hole in Fig. 4.

Figure 4: The different types of the horizon situation of the Kerr-dS black hole. The coordinates in this figure have no meanings. are related to the points in Fig. 3.

Comparing Figs. 2, 3 and 4, we can find that when is in region 1, region 2 or on line 1 in Fig. 3, the particle with critical angular momentum can reach the outer horizon of the Kerr-dS black hole. When is on line 2, is the outer extremal horizon and . By computing , it can be proved that is positive. So When is on line 2, the particle with critical angular momentum can reach the outer horizon of the Kerr-dS black hole.

As a summary, we find as long as is the outer horizon of the Kerr-dS black hole, the particle with critical angular momentum can always reach the horizon . This means the particle with critical angular momentum can always reach the outer horizon of the Kerr-dS black hole without constraints coming from the geodesic motion of the particle. This is very different from the case of Kerr and Kerr-AdS black holes.

iv.3 From the Kerr-AdS case to the Kerr-dS case

Here we analyze in detail that how the Kerr-AdS (Kerr-dS) case changes into the Kerr case when the cosmological constant changes from negative to positive.

When the cosmological constant turns from the negative to zero, Eq. (43) becomes

(51)

We denote . We draw in Fig. 5. So we can see can serve as a boundary line. Thus, the particle with critical angular momentum can reach Kerr black hole’s outer horizon only if the Kerr black hole is extremal.

Figure 5: The shape of for the case of the extremal Kerr black hole. This figure is similar to Fig. 1.

When the cosmological constant turns from positive to zero, the right part of Eq. (49) also changes into and the right part of Eq. (50) becomes positive infinite.

Recalling Figs. 1, 2 and 5, we find that the curved boundary line in Fig. 1 will become the straight boundary line in Fig. 5 when the cosmological constant turns from negative to zero. And the straight boundary line in Fig. 5 will bend back in Fig. 2. So it can surround the region 1 in Fig. 2 completely when the cosmological constant turns from zero to the positive. In this way, the region 1 in Fig. 2 where can be bounded and outside the boundary, and can still be the outer horizon. This is why particles with critical angular momentum can reach the outer horizon of the Kerr-dS black hole without requiring the black hole to be extremal.

In fact, can be rewritten as

(52)

and when ,

(53)

For the Kerr-AdS or the Kerr black hole, on the outer non-extremal horizon, ; and on the extremal horizon, and . So as , cannot be positive on the outer non-extremal horizon. Thus the outer horizon has to be extremal, and as must be positive on the extremal horizon, from Eq. (53), we can see that the parameters of the black hole and the particle must be confined. In fact, we can still get (47) and (48) in this way.

For the Kerr-dS black hole, on the outer non-extremal horizon, ; and on the outer extremal horizon, and . So from Eqs. (52) and (53), we can see that on the outer non-extremal horizon, is positive; and on the outer extremal horizon, and . Thus, the Kerr-dS black hole needs not to be extremal and there is no additional condition needed.

From above analysis, we know why the Kerr-AdS and Kerr cases are similar and why the Kerr-dS case is so different.

V Conclusion

In this work, we have analyzed the possibility that Kerr-(anti-) de Sitter black holes could act as particle accelerators. We find that the result is different from the case of Kerr black holes because of the non-vanishing cosmological constant in the background spacetime. In order for two particles to collide on outer horizon of the Kerr, Kerr-AdS, or Kerr-dS black holes and to reach arbitrary high CM energy, one and only one of the colliding particles should have a critical angular momentum. Besides, for the case of the Kerr black hole, it has to be extremal. For the Kerr-AdS one, it has to be extremal, and an additional condition should be satisfied. However, for the case of the Kerr-dS black hole, it does not need to be extremal and no additional condition need to be satisfied. Hence, non-extremal Kerr-de Sitter black holes could also act as particle accelerators with arbitrarily high CM energy, which is very different from the cases of the Kerr and Kerr-AdS black holes. By analyzing how the Kerr-AdS (Kerr-dS) case changes into the Kerr case when the cosmological constant vanishes, we have seen exactly why the Kerr-dS case is so different.

Acknowledgements

Y. Li is grateful to Dr. Pujian Mao for the valuable discussion. This work was supported by the National Natural Science Foundation of China (No. 11075065), the Doctoral Program Foundation of Institutions of Higher Education of China (No. 20070730055), the Fundamental Research Funds for the Central Universities (lzujbky-2010-171) and the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (LZULL200907).

References

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