New scenarios of high-energy particle collisions near wormholes
aa r X i v : . [ g r- q c ] N ov New scenarios of high-energy particle collisions near wormholes
O. B. Zaslavskii
Department of Physics and Technology,Kharkov V.N. Karazin National University,4 Svoboda Square, Kharkov 61022, Ukraine andInstitute of Mathematics and Mechanics, Kazan Federal University,18 Kremlyovskaya St., Kazan 420008, Russia ∗ We suggest two new scenarios of high-energy particle collisions in the backgroundof a wormhole. In scenario 1 the novelty consists in that the effect does not requiretwo particles coming from different mouths. Instead, all such scenarios of high energycollisions develop, when an experimenter sends particles towards a wormhole fromthe same side of the throat. For static wormholes, this approach leads to indefinitelylarge energy in the center of mass. For rotating wormholes, it makes possible thesuper-Penrose process (unbounded energies measured at infinity). In scenario 2, oneof colliding particles oscillates near the wormhole throat from the very beginning. Inthis sense, scenario 2 is intermediate between the standard one and scenario 1 sincethe particle under discussion does not come from infinity at all.
PACS numbers: 04.70.Bw, 97.60.Lf
I. INTRODUCTION
During last decade, a lot of efforts was devoted to description of high-energy collisionsin the region of the strong gravitation field. This was stimulated by the observation aboutpossibility to obtain an indefinitely large energy E c.m. in the centre of mass frame of twocolliding particles [1] (see also earlier works [2] - [4]). These findings were made for thecase of rotating black holes. Meanwhile, later, similar results were obtained for anotherstrongly gravitating objects. Thus, the unbounded energies E c.m. were found for processes ∗ Electronic address: [email protected] near naked singularities and wormholes. In the present article, it is the latter case which weare interested in.One should distinguish between two kinds of energy. The first one is E c.m. that canbe measured by an observer who is present just in the point of collision. The second oneis the Killing energy E measured at infinity in the asymptotically flat space-time. In thepresent work, we will discuss both of them. It is essential that if E c.m. is finite, E is finite aswell. This was shown in [5] for the Kerr metric and in [6] for a more general case. (Thereis a very special case [7] when the parameters of the metric themselves diverge, but wewill not discuss it further.) Therefore, the necessary (although not sufficient) condition forobtaining unbounded E consists in the consideration of processes with unbounded E c.m. . Inwhat follows, we will use the term super-Penrose process (SPP), if E is unbounded.For the first time, collisions with unbounded E c.m. near wormholes were considered in[8] for a particular type of wormholes, so-called Teo wormholes [9]. They are necessarilyrotating, the corresponding space-time does not have an asymptotically flat region. In thenext work, it was shown that collisions near such wormholes can also produce unbounded E [10]. Later, it was noticed [11] that high-energy collisions can be realized even for staticwormholes (for example, if two Schwarzschild-like wormholes are glued by means of the ”cutand past” technique, see e.g. Sec. 15.2.1 of [12]). In the Krasnikov’s scenario [11], unbounded E c.m. do occur but unbounded E are forbidden since this would require the presence of theergosphere where E <
0. Meanwhile, such a region is absent for the Schwarzschild-likemetric. In our previous paper it was shown that the SPP is possible for rather generalrotating wormholes [13].It is worth stressing that there exists nontrivial dependence between the behavior of E c.m. ( N ), where N is the lapse function, and the existence or nonexistence of the SPP. Thisrelation was established in [14], where general classification was constructed. In doing so, E c.m. ( N ) itself is determined by the relative sign of radial momenta of colliding particles. Itis head-on collision that leads to the existence of the SPP. In this sense, it is of interest todescribe possible ways, how to realize head-on collisions. It is this point that we make theaccent on. As far as wormholes are concerned, in previous works it was assumed that twoparticles come from opposite mouths and meet near the throat (head-on collision). Thusthe corresponding experiment had ”mixed” nature involving observers from different sidesof Universe.In the present work, we suggest two completely new, alternative scenarios. We show thatan indefinitely large energies E c.m. and E can occur even if both particles are sent from thesame side of the throat. However, this requires two-step process. Also, we exploit the factthat in wormhole space-times there exist bound states (impossible for black holes) when aparticle can oscillate between two turning points [15].One reservation is in order. All scenarios connected with using wormholes for obtainingunbounded E c.m. share the same feature. Namely, the lapse function near the throat shouldbe small. This leads to an indefinite growth of the curvature invariants (say, the Kretschmannscalar K ) there. Meanwhile, one can reconcile large E c.m. and K remaining below thePlanckian scale by choosing the parameters of the system accordingly [16].Below, we consider two types of scenarios in which high energy phenomena reveal them-selves in (i) indefinite growth of E c.m. (with E remaining modest), (ii) in the SPP. To thisend, we consider separately (i) collision in static spherically symmetric wormholes and (ii)in rotating axially symmetric ones. In the first case, only E c.m. can be unbounded, in thesecond one we explain why also E can be made as large as one likes.The paper is organized as follows. In Sec. II we give basic formulas for the sphericallysymmetric case including the metric, equations of motion and the energy in the center ofmass of two colliding particles. In Sec. III we describe a scenario in which one of twoparticles reflects from the potential barrier, so that an ingoing particle converts into theoutgoing one. In Sec. IV we show that, choosing in the point of collision the metric functionsmall enough, we can achieve indefinitely large E c.m. . In Sec. V we describe another scenarioin which one of colliding particles does not come from infinity but oscillates near the throatbetween turning points. In Sec. VI we give the general metric and equations of particlemotion in the case of rotating wormholes. In Sec. VII we describe a general scheme ofparticle collisions in such a background. In Sec. VIII we analyze possible output of particleswith ultrahigh energy. In Sec. IX we discuss the role of trajectories with negative energyplayed in the high energy processes under consideration. In particular, we discuss how theycan be used in an alternative scenario of collision. In Sec. X we summarize the results andoutline some perspectives.We use the geometric system of units in which fundamental constants G = c = 1. II. SPHERICALLY SYMMETRIC CASE: BASIC FORMULAS
Let us consider the spherically symmetric metric ds = − f dt + dρ f + r ( ρ ) dω , dω = dθ + sin θdφ , (1)where we used a so-called quasiglobal coordinate ρ (see, e.g. Sec. 3.3.2 of [17]). Motion offree particles occurs in the plane which we choose to be the equatorial one θ = π . Equationsof motion read m ˙ t = Ef , (2) m ˙ ρ = σP , (3) m ˙ φ = Lr , (4)where dot denotes differentiation with respect to the proper time τ , E being the conservedenergy, L conserved angular momentum, σ = ± P = p E − f ˜ m , (5)˜ m = m + L r . (6)The forward-in-time condition ˙ t > E > E c.m. = − ( m u µ + m u µ )( m u µ + m u µ ) = m + m + 2 m m γ . (7)Here, u µ is the four-velocity, subscript label particles, γ = − u µ u µ is the Lorentz factor ofrelative motion. Using (2) - (4) one obtains m m γ = E E − σ σ P P f − L L r . (8)In what follows we consider the manifold to be a wormhole. For simplicity, we assume thatthe function r ( ρ ) has one minimum at ρ = ρ , so r ≥ r ≡ r ( ρ ). (9)In this section, we restrict ourselves by pure radial motion L = 0 since this simplified casecaptures the main features of the phenomenon under discussion. Then,˙ ρ = σp , (10) p = p ε − f , (11)where ε = Em , γ = ε ε − σ σ p p f . (12) III. SCENARIO OF COLLISION 1: TWO PARTICLES COME FROM THESAME MOUTH
Let us consider the following scenario. Particle 1 has the energy E > m and starts itsmotion, say, from the right infinity. In some point it decays to two particles 2 and 3. Weassume that particle 2 has the energy E < m , whereas particle 3 has E > m , σ = − r ,where p = 0, its position is given by f ( r ) = ε . (13)We assume that f is a monotonic function of r in each half-space, so there is one value of r but there are two turning points in terms of ρ in which r ( ρ ) = r . It is also clear that f attains its minimum f at point ρ , f = f ( r ( ρ )).Particle 2 oscillates between both turning points. Let it collide in point ρ with one moreparticle 4 having (for simplicity) the same mass that comes from infinity, ε > σ = − σ = +1. From (12), we have γ = ε ε + p ( ρ ) p ( ρ ) f . (14) IV. UNBOUNDED E c.m.
Now, we consider configurations with small f ≪ ε < ε . Then, p ( ρ ) ≈ ε , p ( ρ ) ≈ ε , γ ≈ ε ε f . (15)When f → γ grows unbounded, and so does E c.m. We would like to remind a reader that there are few scenarios of high energy particlecollisions in which unbounded E c.m. is obtained in head-on collisions. The key point of suchscenarios is to obtain somehow a particle that moves in the opposite direction (with respectto another particle that falls from infinity) and arrange collision in the point where the lapsefunction is very small. This can be realized (i) near white holes [18], (ii) in the backgroundof a naked singularity [19], (iii) in the background of a wormhole. In case (ii) there is a two-step scenario in which a particle bounces back from an indefinitely high potential barrierand meets a new particle coming from infinity. In case (iii), there are two options. One ofthem (iii-a) consists in that two particles comes from opposite mouths [11]. Meanwhile, inour scenario (iii-b) all particles participating in the process, start in our universe.Thus in our scenario we can probe the other side of a wormhole starting the experimenton our side of it and remaining only there. V. SCENARIO OF COLLISION 2: INTERMEDIATE CASE
In this section, we describe one more scenario. Let us remind a reader that the keyingredient for obtaining unbounded E c.m. is a head-on collision of two particles near thethroat, under an additional condition that the metric coefficient f is small enough in thecorresponding point. Thus we have two alternatives: (i) both particles come from oppositemouths [8], [11], (ii) particles come from the same mouth (see above). Meanwhile, there isalso one more possibility based on the property of wormholes having no analog in the blackhole case. It was shown in [15] that there exist states such that a particle performs boundedmotion between two turning points. Choosing an appropriate phase when particle 2 moves,say, from the left to the right, while particle 1 comes from the right infinity, for small f weobtain the result similar to (15) with one difference: now ε is to be replaced by ε .To make presentation self-closed, we write down the metric in the same form as in [15]: ds = − dt ( g ( r ) + λ ) + dr g ( r ) + r dω . (16)Here, for simplicity, g = 1 − r + r , r ≥ r + , λ is a constant, r + has the meaning of the throatradius. If λ < ε < λ , a trajectory oscillates between two turning points. Let collisionoccur in the phase when both particles move in opposite directions.Repeating our calculations step by step, we obtain for collision of particles 1 and 2 inpoint r , moving in opposite directions radially, the expression γ = ε ε + p ( r ) p ( r ) g ( r ) + λ , (17) p ( r ) = q ε − ( g + λ ) (18)instead of (14).Choosing r = r + , we have γ = 2 ε ε λ . (19)If λ is sufficiently small, γ can be made as big as one likes.Such a scenario can be thought of as an intermediate case between the aforementionedscenarios in the sense that particle 2 comes neither from the left infinity nor from the rightone. It was present near the throat because of initial conditions. And, this scenario 2 hasadvantage as compared to scenario 1 in that we should not arrange two-step process. It issufficient to arrange one-step collision. VI. ROTATING WORMHOLES
Now, we consider a more general metric that takes into account the effect of rotation: ds = − N dt + g φ ( dφ − ωdt ) + dρ A + g φ dθ , (20)where the coefficients do not depend on t and φ , ω >
0. (To simplify formulas, we usenotation g φ for the component of the metric tensor g φφ ). We suppose that the equatorialplane is a plane of symmetry and are interested in the motion within this plane only. Insteadof (2) - (4), equations of motion read now m ˙ t = XN , (21) m N √ A ˙ ρ = P r = σP , (22) m ˙ φ = Lg φ + ωXN , (23)where X = E − ωL , (24) P = √ X − N ˜ m , (25)˜ m = m + L g φ . (26)The forward-in-time condition gives us X ≥
0. (27)We assume that our metric has a wormhole character. This means that g φ has a minimumin some point ρ . For simplicity we assume that N has also minimum in this point, N ( ρ ) = 0and N ( ρ ) ≪ VII. COLLISIONS NEAR THROAT OF ROTATING WORMHOLE: SCENARIO1
Again, we consider the two-step scenario. Our aim is to elucidate, whether or not theenergy extraction from a wormhole is possible and whether or not it can be unbounded. Ingeneral, energy gain in this context is nothing else than the Penrose process [20]. Let usrepeat that, if it is formally (in the test particle approximation) unbounded, it is called thesuper-Penrose process (SPP). If the Penrose process is realized in the scenario that involvescollision, it is called the collisional Penrose process (for black holes, this process is reviewedin [21]). On the first stage, particle 1 decays to particles 2 and 3. Particle 3 escapes to theleft infinity while particle 2 moves to the right. Both E > E >
0. On the secondstage, particle 4 comes from infinity and collides with particle 2 near the throat. This ishead-collision like in the static case. As a result, particles 5 and 6 are created. We assumethat E < E >
0, particle 6 escapes to the right infinity.Here, there are two essential differences now as compared to the static case. First, weassumed that the ergoregion does exist that makes it possible to have
E <
0. Such optionwas forbidden in the limit ω → E + E = E + E , (28) L + L = L + L . (29)It follows from (28) and (29) that X tot ≡ X + X = X + X . (30)As, by assumption, E < E > E > E . Eq. (27) with ω > L < E large positive, so E should be large negative. Formally, E → −∞ , E → + ∞ . Meanwhile, as all energies and angular momenta of particles on the1st stage are supposed to be finite, the quantities X and X are finite as well. Taking intoaccount that X > X > L → −∞ , L → + ∞ . Thus divergences in the right hand sides of (28) and (29) should compensateeach other. It is seen from (24) that E = X ( ρ ) + ω ( ρ ) L . Then, for finite X ( ρ ) , ω ( ρ )and L → + ∞ , the energy E → + ∞ as well. This realizes the super Penrose process, whenthe energy E detected by an observer at infinity is as large as one likes. VIII. OUTPUT OF COLLISION
In Section VII, we outlined the desired features of the process, but the question remained,whether or not it can be realized. In principle, further analysis is required that, apartfrom the conservation of the energy and angular momentum, takes into account also theconservation of the radial momentum. This is the most subtle and crucial point. Happily,there is no need in carrying out such analysis here since we reduced the problem to the onethat has been already investigated in [13] and generalized in [14]. Namely, the followingstatement was proved there.Let (i) two particles collide in the point where N ≪ E c.m. ( N ) for small N has the same form E c.m. ( N ) ∼ N − and this gives rise to unbounded E - see line 3 in Table 1 on page 6 in [14]. Independently of origination of head-on collision nearthe throat with very small N , once it occurred, it leads to unbounded energies at infinity E .0 IX. TRAJECTORIES WITH NEGATIVE ENERGIES AND SCENARIO 2
The key role in the scenario under discussion, as well as in any Penrose process, is playedby the states with negative energy. Strange as it may seem, only quite recently the propertiesof such trajectories were elucidated and described in [22] for the Kerr metric. Later on, theywere generalized in [23]. It turned out that corresponding geodesics cannot stay foreverin the region external with respect to the horizon. The complete curve inevitably crossesthe horizon. Correspondingly, a particle with
E <
E <
E <
E < → . If the lapse function in the point of collision (say, exactly in the throat) issmall enough, we again obtain indefinitely large E , provided L is big and negative. Theessential difference between this scenario and the one described above in Sec. VII consistsin that there is no need in a two-step process. X. CONCLUSIONS
One of the methods of obtaining the super-Penrose process consists in arranging the head-on collision in the point with a small value of the lapse function. To this end, a particlethat was ingoing converts into an outgoing due to reflection from the potential barrier withsubsequent collision with another particle coming from infinity. This is realized in the metricwith naked singularities [5], [6], [19] where the potential barrier has indefinitely big height.1Meanwhile, in the present work we considered wormholes, the potential barrier being finite.In the present work, we suggested two new scenarios. In scenario 1, both particles aresent from infinity from the same side of a wormhole. It turned out that two main featuresare inherent to this scenario. For pure static wormholes, it warrants unbounded E c.m. If awormhole is rotating, it also leads to unbounded E , i.e. the super-Penrose process. A sep-arate question arises, how a remote observer who registers high-energy particles at infinity,can distinguish between a naked singularity and a wormhole.In scenario 2, particle 1 comes from infinity while particle 2 oscillates between turningpoints from the very beginning. It can be considered as an intermediate scenario betweena standard one (when both particles come from different mouths) and scenario 1 outlinedabove. In doing so, particle 2 does not come from infinity at all.It turns out that trajectories with finite motion near the wormhole throat can play adouble role. First, they can serve as initial conditions in collisions leading to unbounded E c.m. Second, after collisions, one of product of reaction can sit on such a trajectory. Thus,either motion along a trajectory under discussion can be specified as some initial condition ora particle can appear there as a result of a previous collision. Anyway, one cannot determinethe origin of such a trajectory without additional assumptions.All discussion was carried out in the test particle approximation. As long as the energyvalue does not exceed the parameters of the metric, this looks quite reasonable. Say, inthe case of Kerr-like wormholes with the parameters M and a , one can obtain the energy1 ≪ Em ≪ a, M. Especially interesting is to make attempt of finding self-consistent solutionswith the backreaction taken into account but this problem is beyond of our task.
Acknowledgments
The work is performed according to the Russian Government Program of CompetitiveGrowth of Kazan Federal University. [1] M. Ba˜nados, J. Silk and S.M. West, Kerr Black Holes as Particle Accelerators to ArbitrarilyHigh Energy, Phys. Rev. Lett. (2009) 111102 [arXiv:0909.0169]. [2] T. Piran, J. Katz, and J. Shaham, High efficiency of the Penrose mechanism for particlecollision, Astrophys. J. , L107 (1975).[3] T. Piran and J. Shaham, Production of gamma-ray bursts near rapidly rotating accretingblack holes, Astrophys. J. , 268 (1977).[4] T. Piran and J. Shanam, Upper Bounds on Collisional Penrose processes near rotating blackhole horizons, Phys. Rev. D , 1615 (1977)[5] Patil, M., Harada, T., Nakao, K., Joshi, P.S., Kimura, M.: Infinite efficiency of collisionalPenrose process: can over-spinning Kerr geometry be the source of ultra-high-energy cosmicrays and neutrinos? Phys. Rev. D , 104015 (2016). [arXiv:1510.08205].[6] I. V. Tanatarov, O. B. Zaslavskii, Collisional super-Penrose process and Wald inequalities,Gen Relativ Gravit
119 (2017), [arXiv:1611.05912].[7] O. B. Zaslavskii, Rapidly rotating spacetimes and collisional super-Penrose process, Gen.Relat. and Gravitation (2016) 67 [ arXiv:1511.00844].[8] Tsukamoto, N., Bambi, C.: High energy collision of two particles in wormhole spacetimes,Phys. Rev. D , 084013 (2015). [arXiv:1411.5778].[9] E. Teo, Rotating traversable wormholes, Phys. Rev. D , 024014 (1998)[arXiv:gr-qc/9803098].[10] N. Tsukamoto and C. Bambi, Collisional Penrose Process in Rotating Wormhole Spacetime,Phys. Rev. D , 104040 (2015) [arXiv:1503.06386].[11] S. Krasnikov, Schwarzschild-Like Wormholes as Accelerators, Phys. Rev. D , 104048 (2018)[arXiv:1807.00890].[12] M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP Press, New York, 1995).[13] O. B. Zaslavskii, Super-Penrose process and rotating wormholes, Phys. Rev. D , 104030(2018), [arXiv:1807.11033].[14] O. B. Zaslavskii, Center of mass energy of colliding electrically neutral particles and super-Penrose process, Phys. Rev. D , 024050 (2019) [arXiv:1904.04874].[15] T. Damour and S. N. Solodukhin, Wormholes as black hole foils. Phys. Rev. D , 024016(2007) [arXiv:0704.2667].[16] S. V. Sushkov and O. B. Zaslavskii, Horizon closeness bounds for static black hole mimickers.Phys. Rev. D , 067502 (2009) [arXiv:0903.1510].[17] K. A. Bronnikov and S. G. Rubin, Black holes, cosmology and extra dimensions (World Scientific, 2013).[18] A. Grib and Yu. V. Pavlov, Are black holes totally black? Grav. Cosmol. , 13 (2015),[arXiv:1410.5736].[19] M. Patil and P. Joshi, Kerr naked singularities as particle accelerators, Class. Quantum Grav. (2011) 235012, [arXiv:1103.1082].[20] R. Penrose, Riv. Nuovo Cimento (1969) 252; reprinted in Gen. Relat. Gravit. (2002)1141.[21] J. D. Schnittman, The collisional Penrose process, Gen Relativ Gravit (2018) 77,[arXiv:1910.02800].[22] A. A. Grib, Yu. V. Pavlov and V. D. Vertogradov, Geodesics with negative energy in theergosphere of rotating black holes, Mod. Phys. Letters A
29 (2014) 1450110, [arXiv:1304.7360].[23] O. B. Zaslavskii, On geodesics with negative energies in the ergoregions of dirty black holes,Mod. Phys. Lett. A30