Particle collisions near static spherically symmetric black holes
PParticle collisions near static spherically symmetric black holes
Eva Hackmann, ∗ Hemwati Nandan,
2, 3, † and Pankaj Sheoran ‡ ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany. Department of Physics, Gurukul Kangri Vishwavidyalaya, Haridwar-249 407, India. Center for Space Research, North-West University, Mafikeng 2745, South Africa. Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo,Edificio C-3, 58040 Morelia, Michoac´an, M´exico. (Dated: July 10, 2020)It has been shown by Ba˜nados, Silk and West (BSW) that the center of mass energy ( E cm ) of test particlesstarting from rest at infinity and colliding near the horizon of a Schwarzschild black hole is always finite. Inthis short note, we extent the BSW scenario and study two particles with different energies colliding near thehorizon of a static spherically symmetric black hole. Interestingly, we find that even for the static sphericallysymmetric (i.e., Schwarzschild like) black holes it is possible to obtain an arbitrarily high E cm from the two testparticles colliding near but outside of the horizon of a black hole, if one fine-tunes the parameters of geodesicmotion. I. INTRODUCTION
In 2009, Ba˜ndos, Silk and West (BSW) first showed [1]that an extremal axially symmetric rotating black hole (BH)can act as a particle accelerator: an extremely high amount ofcenter of mass energy ( E cm ) can be produced from the col-lision of two test particles starting from rest at infinity. Thisis in contrast to static spherical symmetric BHs where E cm isalways finite. They pointed out that the extremal rotating BHsmay therefore be used as an important probe of high energyscale physics. Since then collisions of geodesic (and charged)particles near the horizon of legion of BHs [2–15] has beenanalysed. The study of BSW effect is not limited to geodesicparticles only but extended to spinning and accelerated parti-cles as well [16–18]. However, all these different cases of col-lisions of particles (i.e., geodesic, charged, spinning, and ac-celerated) in the vicinity of a BH are accompanied by some se-rious limitations. For the case of geodesic particles, extremelyhigh E cm is obtained if the collision occurs near the horizonof an extremal rotating BH and the energy E and the angularmomentum L of one of the colliding particles meet a criticalcondition [1]. On the other hand, when the collision of spin-ning particles is considered far away from the event horizonof a BH, the main difficulty is the unavoidable occurrence ofa superluminal region (i.e. the region where the four velocityof the spinning particle changes from timelike to spacelike)[17]. Finally, one can obtain the arbitrarily high E cm even forthe static spherically symmetric case, but it requires that oneof the colliding particles must be accelerated, which in turnrequires the existence of some external source (i.e. which actas an engine to accelerate the particle) [18]. With these limita-tions on the BSW effect, it is natural to investigate whether aBH in its simplest form (i.e. static spherically symmetric BH)can act as an accelerator for colliding geodesic particles.In this paper, we study a collision of two particles near thehorizon of a static spherically symmetric BH. We assume that ∗ [email protected] † [email protected] ‡ [email protected] one of the colliding particles originates near the horizon ofthe BH and posses very small radial motion with energy suchthat E = (cid:15) (cid:28) . On the contrary, the second particle isgeneric, for instance falling from rest at infinity with energy E as shown in Fig. 1. This is a new situation for the collidingparticles that has not been studied yet, and some novel featuresof particle acceleration process in the vicinity of a BH arereported. BH 𝐸 FIG. 1. Schematic view of two geodesic particles with different en-ergies ( E (cid:54) = E ) colliding near the horizon of static and sphericallysymmetric BH. The paper is organized as follows. In Sec. II, we brieflydescribe the metric of the static spherically BH, the equationof motion of the geodesic particles, and the expression for thecenter of mass energy E cm . In Sec. III, we explain how astatic spherically symmetric BH can act as a particle accelera-tor for two geodesic particles colliding in its vicinity. Finally,in Sec. IV, we summarize our main results with a concludingremark. Throughout this paper, we choose the ( − , + , + , +) signature for the metric tensor, Greek indices run from to ,and both the fundamental constants G and c are set equal tounity. a r X i v : . [ g r- q c ] J u l II. COLLISION OF GEODESIC PARTICLES
A static and spherically symmetric spacetime is given by, g = − f ( r ) dt + dr f ( r ) + r ( dθ + sin θdφ ) . (1)We assume that we have a BH solution with an event horizonlocated at r = r H . Dependent on the choice of the metricfunction f ( r ) we may have additional horizons, for instancea cosmological or a Cauchy horizon. The two Killing vectors u t = − E and u φ = L thus give rise to two constants ofmotion as below, − E = g tt ˙ t , (2) L = g φφ ˙ φ , (3)where a dot denotes a derivative with respect to the propertime of the particle. Due to spherical symmetry, without lossof generality, we entirely restrict to the plane θ = π/ . Fromthe normalization condition g µν ˙ x µ ˙ x ν = − , one can thenobtain the following equation of geodesic motion, ˙ r = E − f ( r ) (cid:18) L r (cid:19) =: R ( r ) . (4)Let us now consider two particles of masses m and m moving on geodesics in a BH spacetime. Their energy in thecenter of mass frame is given by, E = − g µν ( p µ + p µ )( p ν + p ν ) (5) = m + m − g µν p µ p ν , (6)where p µi denotes the four momentum of the two particles. Ifwe assume m = m = m , one can easily obtain, E m = 1 − L L r + E E − (cid:112) R ( r ) R ( r ) f ( r ) , (7)where the indices refer to the two different particles. Re-gardless of the apparent singularity at the horizon, where f ( r H ) = 0 , for generic particles the center of mass energyremains finite as the nominator also vanishes at r = r H . Thelimit is given by, lim r → r H E m = 1 − L L r H + E E (cid:18) L r H (cid:19) + E E (cid:18) L r H (cid:19) . (8) In the original paper by BSW, it was assumed from the begin-ning that the two particles start from rest at infinity, implying E = E = 1 . In this case, we see that the center of massenergy is always finite. Later, it was pointed out by Zaslavskiithat for charged particles in a Reissner-Nordstr¨om spacetime,or geodesic particles in a stationary (but non-static) and ax-ially symmetric spacetime the parameters can be fine-tunedsuch that the expression Eq. (8) becomes arbitrarily large. Forthe Schwarzschild spacetime, Grib et al [19] show that, in thelimit r → r H considered above, if the point of collision coin-cides with the radial turning point ˙ r = 0 of one of the particlesthan the energy may also grow without bound. We generalisehere the argument from [19], allowing for general static spher-ically symmetric spacetimes, and removing the condition thatthe point of collision coincides with the radial turning point. III. ARBITRARILY LARGE CENTER OF MASS ENERGY
We immediately notice from the limit Eq. (8) that the centerof mass energy diverges if one of the particles has zero energy.However, as evident from eq. (4), this means that geodesicmotion is impossible. Instead, we may consider a particle witha very low energy. Assume that particle 1 has E = (cid:15) (cid:28) and particle 2 is generic. Then from Eq. (8), one can observe, lim r → r H E m = E (cid:15) (cid:18) L r H (cid:19) + O ( (cid:15) ) . (9)This result was basically derived in [19], although we did notexplicitly assume that ˙ r = 0 at the horizon. However, asmentioned above, geodesic motion with E (cid:28) is only pos-sible if ˙ r = 0 very close to the horizon. Note that if bothparticles have E (cid:28) , the center of mass energy becomesagain finite.More generally, we can also obtain arbitrarily large centerof mass energy for collisions close to the horizon, but outsideof it. Let r c be the point of collision with r c − r H (cid:28) .We rewrite the metric function f as f ( r ) = ( r − r H ) ˜ f ( r ) , inparticular ˜ f ( r ) = 1 /r for Schwarzschild spacetime. Let uschoose for, say particle 1, E = C √ r c − r H . Here C is apositive constant, taken such that geodesic motion is possiblein the region ( r H , r c ) . We then find the following expressionfrom Eq. (7) at the point of collision r c , E m = 1 − L L r c + E C √ r c − r H ( r c − r H ) ˜ f ( r c ) − r c − r H ) ˜ f ( r c ) ×× (cid:20) ( r c − r H ) (cid:18) C − ˜ f ( r c ) (cid:18) L r c (cid:19)(cid:19) (cid:18) E − ( r c − r H ) ˜ f ( r c ) (cid:18) L r c (cid:19)(cid:19)(cid:21) (10) = 1 − L L r c + E C √ r c − r H ˜ f ( r c ) − E (cid:114) C − ˜ f ( r c ) (cid:16) L r c (cid:17) √ r c − r H ˜ f ( r c ) + O ( √ r c − r H ) . (11)Therefore, for small but nonvanishing r c − r H , we can ob-tain arbitrarily large center of mass energy provided we canchoose the constant C such that C − (cid:115) C − ˜ f ( r c ) (cid:18) L r c (cid:19) > . (12)For instance, we could choose C = (cid:115) f ( r c )( r c + L ) r c . (13)Then, at the point of collision we find from Eq. (4) that ˙ r | r = r c = ( r c − r H ) > as well.A caveat of both calculations explained above is that theparticle 1 is assumed to have a very small radial motion ve-locity although it is very close to the horizon. This is a veryspecial situation, which will usually not be encountered, andprobably will only arise if the particle originated from a fore-going collision. IV. SUMMARY AND CONCLUSION
In this short paper, we delved into the BSW effect and stud-ied the collision of two geodesic particles near the horizonof a static spherically symmetric BH, generalising a scenarioshortly discussed in [19]. It is assumed that one of the collid-ing particles (say particle 1) originates near the horizon of theBH and posses a very small radial velocity and energy E (cid:28) (despite the fact that it is very close to r H ). On the other hand,the second colliding particle with energy E is considered tobe generic, for instance the usual particle coming from rest atinfinity. It is shown that when the collision of the above men-tioned geodesic particles takes place near the event horizon ofa static spherically symmetric BH, it is still possible to havean arbitrarily high E cm . This is in contrast to the case whenthe colliding particles needs to have a spin as pointed out in [16, 17], a charge as considered in [20], or to the case whenone needs an external engine (i.e. source) to accelerate one ofthe colliding particles in order to achieve arbitrarily high E cm for static spherically symmetric BHs as pointed out in [18].It is noteworthy to mention that even a static sphericallysymmetric BH can also act as a particle accelerator forgeodesic particles and one do not need special types of parti-cles (i.e. spinning or accelerated) in order to obtain arbitrarilyhigh E cm . However, the scenario we discussed here is veryspecial and is only possible if the particle 1 is originated nearto horizon of BH by some process, maybe through a forgoingcollision of particles.In response to an earlier version of this paper, Zaslavskii[21] analysed the physical relevance of the scenario under dis-cussion here. He found that the for the Schwarzschild space-time it is unphysical, requiring infinite forces to place thecritical particle near the horizon. However, for the extremalReissner-Nordstr¨om spacetime the critical (neutral) particlecould be produced by a foregoing collision of charged par-ticles. For other static and spherically symmetric spacetimesthe question of physical relevance remains open. ACKNOWLEDGMENTS
HN would like to thank Science and Engineering ResearchBoard (SERB), New Delhi, India for financial support throughgrant no. EMR/2017/000339. HN is also thankful to IUCAA,Pune, India (where a part of the work was completed) for sup-port in form of academic visits under its Associateship pro-gramme. PS would like to thank
Programa de Desarrollo Pro-fesional Docente (PRODEP) of the
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