Center of mass energy of colliding electrically neutral particles and super-Penrose process

TL;DR

This paper classifies collision scenarios of neutral particles near rotating spacetimes, identifying conditions under which unbounded energy outcomes and super-Penrose processes can occur without frame transformations.

Contribution

It provides a unified classification of collision scenarios leading to super-Penrose processes in rotating spacetimes, based on exact dynamic characteristics.

Findings

Five nonequivalent collision scenarios identified.

Relation established between lapse function behavior and super-Penrose process.

Unified framework without explicit frame transformations.

Abstract

We consider collisions of electrically neutral particles in the background of axially symmetric rotating space-times. For reactions of the kind 1+2$% \rightarrow $3+4 we give full classification of possible scenarios based on exact expressions for dynamic characteristics of particles 3 and 4 depending on particles 1 and 2. There are five nonequivalent scenarios valid for different types of space-time (black hole, naked singularity, etc.). We are mainly interested in the situations when particle collisions can give unbounded outcome for (i) the energy $E_{c.m.}$ in the center of mass frame and (ii) Killing energy $E$ at infinity. If (i) is fulfilled, this may or may not lead to (ii). If (ii) holds, this is called the super-Penrose process (SPP). For equatorial particle motion, we establish close relation between the type of behavior of $E_{c.m.}$ depending on the lapse function $% N $ in the point of collision and the possibility of the SPP. This unites separate previous observations in literature in a unified picture as a whole. In doing so, no explicit transformations to the center of mass frame and back are needed, so all consideration is carried out in the original frame. As a result,we suggest classification of sub-classes of scenarios that are able (or unable) to give rise to the SPP particle collision, super-Penrose process, center of mass frame.

Equations

$$d{s}^2=-N^{2}d{t}^2+g_{\varphi }(d\varphi -\omega dt{)}^2+\frac{d{r}^2}{A}+g_{\varphi }d{\theta }^2\text{,}$$

$$m\dot{t}=\frac{X}{N^2}\text{,}$$

$$m\frac{N}{\sqrt{A}}\dot{r}=P_r=\sigma P\text{,}$$

$$m\dot{\varphi }=\frac{L}{g_{\varphi }}+\frac{\omega X}{N^2}\text{, }$$

$$X=E-\omega L\text{,}$$

$$P=\sqrt{X^2-N^2{\tilde{m}}^2},$$

$${\tilde{m}}^2=m^2+\frac{L^2}{g_{\varphi }}\text{.}$$

$$X\ge 0\text{.}$$

$$E_0=E_1+E_2=E_3+E_4\text{,}$$

$$L_0=L_1+L_2=L_3+L_4\text{,}$$

$$X_0=X_1+X_2=X_3+X_4\text{.}$$

$$-P_0={\sigma }_2{P}_2+{\sigma }_1{P}_1.$$

$$-P_0={\sigma }_3{P}_3-P_4.$$

$$m_0^2=E_{c.m.}^2,$$

$$E_{c.m.}^2=-(p_{1\mu }+p_{2\mu })(p_1^{\mu }+p_2^{\mu })\text{,}$$

$$E_{c.m.}^2=m_1^2+m_2^2+2m_1{m}_2\gamma ,$$

$$\gamma =-u_{1\mu }{u}_2^{\mu }\text{.}$$

$$\gamma m_1{m}_2={\left(\frac{X_1{X}_2-{\sigma }_1{\sigma }_2{P}_1{P}_2}{N^2}-\frac{L_1{L}_2}{g_{\varphi }}\right)}_c\text{.}$$

$${\left(X_3\right)}_c=\frac{1}{2{\tilde{m}}_0^2}(X_0{\Delta }_{+}+P_0\sqrt{D}\delta {)}_c\text{,}$$

$${\left(X_4\right)}_c=\frac{1}{2{\tilde{m}}_0^2}(X_0{\Delta }_{-}-P_0\sqrt{D}\delta {)}_c\text{,}$$

$${\Delta }_{\pm }={\tilde{m}}_0^2\pm ({\tilde{m}}_3^2-{\tilde{m}}_4^2).$$

$${\Delta }_{\pm }>0.$$

$$D={\Delta }_{+}^2-4{\tilde{m}}_0^2{\tilde{m}}_3^2={\Delta }_{-}^2-4{\tilde{m}}_0^2{\tilde{m}}_4^2.$$

$$D\ge 0.$$

$${\tilde{m}}_0\ge {\tilde{m}}_3+{\tilde{m}}_4\text{.}$$

$$P_3=\frac{\left|P_0{\Delta }_{+}+\delta X_0\sqrt{D}\right|}{2{\tilde{m}}_0^2},$$

$$P_4=\frac{\left|P_0{\Delta }_{-}-\delta X_0\sqrt{D}\right|}{2{\tilde{m}}_0^2},$$

$$H_3\equiv \frac{X_0^{2}D-P_0^2{\Delta }_{+}^2}{{\tilde{m}}_0^2}={\Delta }_{+}^2{N}^2-4{\tilde{m}}_3^2{X}_0^2$$

$${\epsilon }_i=sgn{H}_i\text{, }$$

$$1(+,+,+,-),2(+,+,-,-),3(-,-,-,+),4(-,+,-,+),5(-,-,-,-),6(-,-,+,-).$$

Full text

particle collision, super-Penrose process, centre of mass frame

I Introduction

In recent years, a possibility of black holes to serve as supercolliders attracts attention after findings by Bañados, Silk and West (the BSW effect, after the names of its authors) ban who showed that under some conditions, collision between two particles near the extremal Kerr metric leads to the unbounded energy $E_{c.m.}$ in the centre of mass (CM) frame. As a result, interest to previous findings on high-energy collisions near black holes pir1 - pir3 also revived. This effect was extended to rather generic black holes including nonextremal ones. Meanwhile, especially interesting is a question about the energy $E$ measured by an observer at infinity since it would give a chance to register debris of corresponding collision in the laboratory, at least in principle. If $E$ can be unbounded, this is called the super-Penrose process (SPP). However, detailed investigations showed that rotating neutral black holes are not pertinent for this purpose in the BSW scenario, so in spite of unbounded $E_{c.m.}$, the energy $E$ gained in the BSW process, remains quite modest j - z . Another type of scenario in which a fine-tuned outgoing particle experiences head-on collision with an incoming one increases the efficiency of the process shnit but, anyway, it remains bounded cons - frac .

There have been attempts to extend the results of shnit to the case when both particles are generic (so-called usual), not fine-tuned. Numerically, it was observed in card that such a collision near the horizon formally leads to the SPP. This was found independently in mod analytically for rather generic black holes. The escape probability for such processes was found analytically for the Kerr black hole esc . There is a problem, however, that  such a usual particle with a finite energy cannot emerge from a black hole frac although this is possible for white holes gpw . Examination of more involved scenarios of multiple scattering showed that this is also impossible for black holes, provided all characteristics of initial ingoing articles are finite pir15 , epl . Thus, one is led to reject black holes as potential supercolliders of particles coming to infinity (although near the horizon this is quite possible).

From the other hand, there are scenarios of collisions in the background of naked singularities and wormholes headon - owh when the SPP does occur. Thus there is a series of particular observations for some kinds of physical objects with a positive result in this sense.

The question arises, what is the  underlying reason, why the SPP is possible in the aforementioned case but is forbidden for black holes? In the present work we consider rotating neutral space-times and suggest explanation. We argue that so different behavior follows unambiguously from one feature - the dependence of $E_{c.m.}$ on the lapse function $N$ in the point of collision. Thus we relate key properties of the effect under consideration in the region of strong gravity (say, near the horizon of a black hole) and at infinity. At the same time, we present simple and exact formulas for collisions in the equatorial plane that can be useful in a quite general context for other purposes as well. This enabled us to classify all relevant scenarios of collisions in the equatorial plane including those pertinent to the SPP. Thus we give a unified picture as a whole from which previous observations about (im)possibility of the SPP follow as particular cases.

We would like to stress that in the present paper we consider collisions of electrically neutral particles only. Experience shows that for the electrically charge case rn the results can be very different from the neutral one described above.

We use the geometric system of units in which fundamental constants $G=c=1$.

II Basic formulas

Let us consider the metric

[TABLE]

where the coefficients do not depend on $t$ and $\phi$. (To simplify formulas, we use notation $g_{\phi}$ for the component of the metric tensor $g_{\phi\phi}$). We suppose that the equatorial plane is a plane of symmetry and are interested in the motion within this plane only. On the horizon $r=r_{+}$ we have $N=0$. Equations of motion read

[TABLE]

[TABLE]

[TABLE]

where dot denotes differentiation with respect to the proper time $\tau$,

[TABLE]

$E$ being the conserved energy, L\,\conserved angular momentum, $\sigma=\pm 1$ depending on the direction of motion.

[TABLE]

[TABLE]

The forward-in-time condition $\dot{t}>0$ entails

[TABLE]

III Collision vs decay

Our main goal consists in the analysis of particle collisions. Particle 1 and 2 collide to produce particles 3 and 4. To simplify formulas, we conditionally represent collision of particles 1 and 2 as creation of some effective particle 0 that decays immediately to particles 3 and 4. Hereafter, we use subscript $i$ ($i=1,2,3,4)$ to enumerate characteristics of particles ($E_{i}$, $L_{i}$, $m_{i}$, $P_{i}$, etc.). Then, the energy and angular momentum are given by the conservation laws:

[TABLE]

[TABLE]

As a consequence, we have also the equality

[TABLE]

It is assumed that $X_{0}$ is some finite quantity. Then, as (8) is valid for each particle, $X_{3}$ and $X_{4}$ are also bounded.

We also assume the initial radial momentum to be negative (particle 0 moves towards a black hole), so particles 3 and 4 cannot have $P_{r}>0$ both. For definiteness, we assume that it is particle 4 that moves inside always, so $\sigma_{4}=-1$. The conservation of the radial momentum in the reaction $1+2\rightarrow 0$ reads

[TABLE]

In a similar way, for decay $0\rightarrow 3+4$ we have

[TABLE]

Taking the square of (12), one can check that

[TABLE]

where $E_{c.m.}$ is the energy in the CM frame,

[TABLE]

$p_{i}^{\mu}=m_{i}u_{i}^{\mu}$, where $u_{i}^{\mu}$ is the four-velocity of particle i. It follows from (15) that

[TABLE]

the Lorentz factor of relative motion

[TABLE]

In terms of characteristics of particles 1 and 2, one obtains from (2) - (4) that

[TABLE]

Hereafter, subscript ”c” indicates the point of collision. The similar formulas hold if we calculate $E_{c.m.}$ between particles 3 and 4. Then, $E_{c.m.}$ is the same due to the conservation laws.

IV Explicit solutions of equations

One can solve eq. (13) exactly, taking into account (6). Then, one finds

[TABLE]

[TABLE]

where $\delta=1$ or $\delta=-1$.

[TABLE]

The positivity of $X_{3.4}$ entails

[TABLE]

[TABLE]

It is necessary that

[TABLE]

Then, after simple algebraic manipulations, (21) - (24) lead to

[TABLE]

In the above formulas, all quantities related to particles 1 and 2 (hence, those of effective particle 0 as well) are fixed. We also assume that masses $m_{3,4}$ are fixed for any given process. Meanwhile, one of two angular momentum (say, $L_{3}$) remains a free parameter.

Then, by substitution into (6), one finds

[TABLE]

[TABLE]

where we omitted subscript ”c” for shortness. It is easy to check that $X_{i}^{2}-P_{i}^{2}=\tilde{m}_{i}^{2}N^{2}$ as it should be. The difference

[TABLE]

shows which term dominates the numerator in (26). In a similar way, $H_{4}$ is defined, with $\Delta_{+}$ replaced with $\Delta_{-}$ and $\tilde{m}_{3}$ replaced with $\tilde{m}_{4}$. Correspondingly, it is convenient to introduce the factors $\varepsilon_{i}$ according to

[TABLE]

$i=3,4$.

Then, we can characterize the process by the set $(\sigma_{3},\varepsilon_{3},\varepsilon_{4},\delta)$. Direct inspection by substitution of (26), (27) in (13) shows that the list of possible scenarios includes six cases:

[TABLE]

Cases 3 and 5 are equivalent. They differ by particle labels 3 and 4 only. Thus actual number of different scenarios is five. In turning points, $H_{i}=0$ and the difference between $\varepsilon_{i}=+1$ and $\varepsilon_{i}=-1$ becomes irrelevant.

Below, we examine three typical cases depending on the behavior of $E_{c.m.}=m_{0}$ and the region where collision occurs. Our main goal is to establish, when the SPP is possible. In all cases we assume that all characteristics ($m,E,L$) of initial particles 1 and 2 are finite.

To classify particles, in what follows we use the standard terminology but in somewhat extended manner. Namely, we call a particle usual if $\lim_{N_{c}\rightarrow 0}X\neq 0$ and critical if $\lim_{N_{c}\rightarrow 0}X=0$. For black holes, this coincides with the standard definition ban , prd . But it includes also the case of naked singularities and wormholes if collision occurs in the point with small but nonzero $N_{c}>\left(N_{c}\right)_{\min}$ when $\left(N_{c}\right)_{\min}\rightarrow 0$   and $N_{c}\rightarrow 0$ owh .

V Energy in CM frame is finite

Let us consider collisions with finite $m_{0}$. It was observed in inf for the overspinned Kerr metric that if $m_{0}$ is finite, no SPP occurs. This was generalized and explained in terms of the Wald identities wald in waldcol . In this sense, the absence of the SPP is clear in advance. However, in this Section we reconsider this issue again to demonstrate how our approach works. Collisions under discussion can be realized in two typical situations: (i) in some intermediate point with $N_{c}=O(1),$ (ii) for two usual particles near the horizon, where $N\rightarrow 0$ ban , prd . Both $X_{3}$ and $X_{4}$ are finite and nonzero. As $X_{3}$ should be finite, the property $E_{3}\rightarrow\infty$ requires $L_{3}\rightarrow\infty$ as well to compensate big $E_{3}$. In doing so, $L_{4}=L_{0}-L_{3}\rightarrow-\infty$. As a result, both $\tilde{m}_{3}~{}$and $\tilde{m}_{4}\rightarrow\infty$ according to (7). As $\tilde{m}_{0}$ remains finite by assumption, we see that eq. (25) is violated and so is condition (24). As a result, the scenario under discussion is impossible, so there is no SPP in this case, as expected.

VI Square of Energy in CM frame growing as $N_{c}^{-2}$

Now, we consider collisions with $N_{c}\rightarrow 0$ and

[TABLE]

This can be realized if two particles experience head-on collision, so $\sigma_{1}\sigma_{2}=-1$ in (18). There are different objects and scenarios with these features. Particle 1 can bounce back from the potential barrier existing in the case of naked singularity and collide with an ingoing particle 2 headon , inf , waldcol . Then, considering reflection from the potential barrier and making transformation between the original stationary frame, LNRF (locally nonrotating frame) and the CM one, we obtain for generic rotating axially symmetric over-spinned space-time that $E_{3}$ can be unbounded, so the SPP does occur - see Sec. 3.5 of waldcol . In another version, particles 1 and 2 can come from the opposite mouths of a wormhole kras , owh .

Now, we are in position to show that the dependence (31) itself leads to the SPP, independently of the details of the scenario. In doing so, our approach is simpler than in waldcol since we are using one frame only.

It follows from (16), (18) that

[TABLE]

Then, we can again take the limit $L_{3}\rightarrow\infty$. But now, there is some additional restriction to guarantee the finiteness of $X_{3}$, $X_{4}$ and the condition that, according to (6),

[TABLE]

We consider at first the case when the centrifugal contribution to  $P_{3,4}$ (6) due to $L_{3,4}^{2}$ has the same order as $X_{3,4}^{2}$. To this end, we assume that $L_{3}$ is adjusted to the position of collision according to

[TABLE]

where $l$ is some constant.

From (31), we have

[TABLE]

since $L_{0}$ entering (7) is supposed to be  finite. Meanwhile,

[TABLE]

where

[TABLE]

[TABLE]

We must require

[TABLE]

so $l^{2}\leq\frac{\alpha}{4}\left(g_{\phi}\right)_{c}$. This agrees with (25). Then, we get from (19), (20)

[TABLE]

Here, for $N_{c}\rightarrow 0$,

[TABLE]

according to (6) and (32). The procedure is self-consistent, so we can achieve $E_{3}\rightarrow\infty$, if we take $N_{c}$ sufficiently small, with $L_{3}$ given by (34). This hints at the SPP, provided the escape condition is satisfied (see below).

To identify a type of scenario, we must evaluate (28):

[TABLE]

We want to realize scenario 1 from list (30) to have particle 3 escaping directly to infinity. Otherwise, this usual particle would fall in a black hole. Then, we must require, in addition to (39), also

[TABLE]

It is seen from (11) and (32) that

[TABLE]

so the the expression inside the equare root in (41) is indeed non-negative, as it should be, and condition (43) is more tight that (39).

This is not the end of story since the desired conditions near the point of collision do not guarantee escaping. For escape to occur, we must require (33) in any point, not only in the point of collision. Bearing in mind (5), we can write (33) as

[TABLE]

For large $L_{3}>0$, it is seen from (7) that

[TABLE]

Then, this condition  translates to

[TABLE]

The quantity $\omega_{+}$ has a simple physical meaning. As is known, it defines the maximum angular velocity of a particle in the ergoregion.

We want to have inequality $Y_{c}=\left(X_{3}\right)_{c}-\frac{l}{\sqrt{\left(g_{\phi}\right)_{c}}}=\left(X_{3}\right)_{c}-b>0$. Taking into account (38), (40) we can write

[TABLE]

If (39) is satisfied, it is seen, with (44) taken into account, that the left hand side of (48) is positive. Then, it is easy to check that (48) is valid without any additional assumptions.

Thus we checked the validity of (47) in the point of collision for the scenario under discussion. We want it to hold for any $r>r_{c}$ as well. If $\frac{d\omega_{+}}{dr}<0$, then $\frac{dY}{dr}>0$ that is sufficient for the validity of (47) everywhere. For example, in the Kerr and Kerr-Newman background $\frac{d\omega_{+}}{dr}<0$ for extremal black holes and slightly overspinning metrics - e.g., see Fig. 1 and discussion in waldcol and Fig. 2 in dc .

In general, even if there exists a small interval where $\frac{d\omega_{+}}{dr}>0$, $\omega_{c}-\omega_{+}\approx-BN_{c}$ with the constant $B>0$ and $N_{c}\ll 1$, this only adds some inessential restriction on $l$ but does not spoil the SPP. Indeed, in this interval we should require, according to the positivity of (47), that

[TABLE]

where $N_{\max\text{ }}$is the maximum value of $N$ within this interval. Then, (34) entails that

[TABLE]

Thus, under rather weak assumptions that are satisfied in the Kerr and Kerr-Newman cases, the SPP does exist. This includes also a wormhole obtained by gluing two such metrics owh .

Now, we can generalize (34) and consider

[TABLE]

If $s>1$, this violates condition (25) and, hence, (24) as well. Therefore, this case should be rejected. Case $s=1$ is already considered above. Let $s<1$. Then, $\Delta_{\pm}\approx\tilde{m}_{0}^{2}$, in (28) $H_{3.4}>0,$ so $\varepsilon_{3}=\varepsilon_{4}=+1$, and only scenario 1 can be realized. We have from (19)

[TABLE]

Neglecting in $P_{0}$ terms of the order $N^{2}$, we have

[TABLE]

This expression is finite. It is easy to check that in the point of collision (47) is satisfied. As $\sigma_{3}=+1$ for scenario 1, particle 3 moves after collision in the outward direction. If the assumption about monotonic decrease of $\omega_{+}(r)$ (see above) is fulfilled, we are faced with the SPP. Thus holds for any $0<s\leq 1$.

VII Square of Energy in CM frame growing as $N_{c}^{-1}$

In this section, we consider another asymptotic behavior,

[TABLE]

It is typical of the BSW ban and Schnittman shnit processes that occur near the horizons of extremal black holes when critical particle 1 collides with usual particle 2. Our goal is to trace, how it happens that the SPP is impossible. The details of the concrete energy bounds can be found in j - max .

Using the Taylor expansion near the horizon

[TABLE]

one can infer j that $X_{1}=B_{1}N_{c}L_{1}+O(N^{2})$, and

[TABLE]

where

[TABLE]

We know already that for finite $L_{3}$ the SPP is impossible. We would like to examine, what happens if, from the very beginning, we adjust a value of $L_{3}$ to the point of collision in such a way that $L_{3}$ is formally divergent in the limit $N_{c}\rightarrow 0$. One of simplest choices is

[TABLE]

Then, it follows from (19), (20) that

[TABLE]

[TABLE]

notation $b$ from (37) is used, $E_{3}=\omega_{c}L_{3}+\left(X_{3}\right)_{c}$, so

[TABLE]

Formally, we have large positive $E_{3}$ and large negative $E_{4}$. Now, $\Delta_{\pm}=O(N_{c}^{-1})$, so in (28) the negative term dominates and $\varepsilon_{3}=\varepsilon_{4}=-1$. It is seen that scenario 3 or 5 can be realized.

Thus we have two usual particles that move towards a black hole and fall in it since there are no turning points. In principle, this case (not discussed in literature before) can be of some interest for more involved scenarios that include consideration of events inside a black hole. However, for our purposes, the case in question is completely useless since nothing comes out to infinity and the SPP is impossible.

Instead of (58), one can take

[TABLE]

with an arbitrary positive $\rho$. If $\rho>\frac{1}{2}$, condition (25) is also violated. Thus this case cannot be realized. Case $\rho=\frac{1}{2}$ is already considered above. Let now $\,0<\rho<\frac{1}{2}$. Then, repeating consideration step by step, one can check that $\Delta_{\pm}\approx\tilde{m}_{0}^{2}\approx\frac{\beta}{N_{c}}$, $\varepsilon_{3}=\varepsilon_{4}=-1$, so only $\sigma_{3}=-1$ is possible as is seen from (30), we deal with scenario 3 or 5. In doing so, there is no turning point, $P_{3}\approx X_{0}\frac{\tilde{m}_{3}^{2}}{\tilde{m}_{0}^{2}}\approx X_{0}\frac{b^{2}}{\beta}N^{1-2p},$ particle 3 approaches the horizon and falls in a black hole. Therefore, the SPP is impossible similarly to the case $\rho=\frac{1}{2}$.

To conclude this Section, we will discuss briefly also collisions near nonextremal black holes. Although the BSW effect was found and remains more popular just for extremal black holes, in somewhat modified form it exists also for nonextremal ones. This was shown in gp for the Kerr metric and generalized to arbitrary axially symmetric stationary black holes in prd . In contrast to the extremal case, for nonextremal one the critical particle cannot reach the horizon and at first it would seem that this fact makes the BSW effect impossible. However, it turned out that if one takes a near-critical particle instead of the critical, with deviation from the critical relation between the energy and angular momentum having the order $N$ (see eq. 18 of gp and eq. 18 in prd , where $\varepsilon$ is implied to have the order $N$ with the restriction (18) given in prd ), then the effect does take place with (54) being valid. Then, the consideration of this Section applies directly with the same conclusion that the SPP is absent.

VIII Discussion and conclusions

Thus we developed a scheme that enables us to classify all possible scenarios of collisions in the equatorial plane. We found only 5 nonequivalent types of such scenarios. The approach works both for finite and divergent energy in the CM frame and is applicable to diverse types of space-times. Information about possibility or impossibility of the SPP is encoded in the rate with which the energy in the CM frame diverges when $N_{c}\rightarrow 0$.

This allowed us to explain and unify previous observations. In doing so, physical nature of the metric reveals itself indirectly, so qualitatively different space-times and scenarios can give the same result. For example, the dependence $m_{0}^{2}\sim N_{c}^{-2}$ can arise either in the two-step scenario in the background of a naked singularity for slightly overspinned metric or in collisions of two usual particles one of which comes from the white hole horizon directly. The result is the same, in both cases the SPP is possible. The results are summarized in Table 1.

[TABLE]

Table 1. Possibility/impossibility of the super-Penrose process in different cases.

We would like to stress that our conclusions are derived in a simple and straightforward manner, directly in the original frame, we did not need to pass into the centre of mass frame and back in contrast to inf , waldcol . We believe that the corresponding approach can be useful also in other problems connected with high energy particle collisions in the region of strong gravity. It is of interest to generalize it to the charged case whose properties are expected to be very different from the neutral one.

Acknowledgements.

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.