Finite escape fraction for ultrahigh energy collisions around Kerr naked singularity
aa r X i v : . [ g r- q c ] M a y Finite escape fraction for ultrahigh energy collisions around Kerr naked singularity
Mandar Patil ∗ and Pankaj S. Joshi † Tata Institute of Fundamental ResearchHomi Bhabha Road, Mumbai 400005, India
We investigate here the issue of observability of high energy collisions around Kerr naked singu-larity and show that the results are in contrast with the Kerr black hole case. We had shown earlierthat it would be possible to have ultrahigh energy collisions between the particles close to the loca-tion r = M around the Kerr naked singularity if the Kerr spin parameter transcends the unity by aninfinitesimally small amount a → + . The collision is between initially ingoing particle that turnsback as an outgoing particle due to angular momentum barrier, with another ingoing particle. Weassume that two massless particles are produced in such a collision and their angular distribution isisotropic in the center of mass frame. We calculate the escape fraction for the massless particles toreach infinity. We show that escape fraction is finite and approximately equal to half for ultrahighenergy collisions. Therefore the particles produced in high energy collisions would escape to infinityproviding signature of the nature of basic interactions at those energies. This result is in contractwith the extremal Kerr black hole case where almost all particles produced in high energy collisionsare absorbed by the black hole, thus rendering the collisions unobservable. PACS numbers: 04.20.Dw, 04.70.-s, 04.70.Bw
I. INTRODUCTION
Banados, Silk and West demonstrated an intriguingpossibility of collisions of particles with arbitrarily largecenter of mass energy around extremal Kerr black hole[1]. The collisions were between two ingoing particlesthat start from rest at infinity and fall freely towards theblack hole. The collision takes place at a location thatis close to the event horizon. It turns out that in orderfor collision to be energetic the angular momentum ofone of the particles must be finetuned to a specific value[3]. This particle approaches the event horizon asymptot-ically. Thus the proper time required for the collision isinfinite. A similar process was also studied in the contextof many other known examples of black holes by differentauthors. While analyzing the possible implications of thehigh energy collision process for the distant observers atinfinity the escape fraction was subsequently calculated.Escape fraction tells us the fraction of particles createdin the high energy collisions that manage to escape toinfinity. It turns out that since the collision takes placeat a location close to the event horizon most of the par-ticles produced are absorbed by the black hole. Thus theescape fraction in the case of a black hole turns out to bevanishingly small [2, 4]. Thus although ultrahigh energycollisions would take place around extremal Kerr blackholes, they would be inconsequential from the perspec-tive of observational implications and in such a case thephysics at high energies would remain unprobed.We studied earlier the process of high energy collisionsin the context of Kerr naked singularity [6]. We showedthat it would be possible to have ultrahigh energy parti- ∗ Electronic address: [email protected] † Electronic address: [email protected] cle collisions around Kerr naked singularity provided theKerr spin parameter is larger than but arbitrarily closeto unity. Collisions take place at a location which isaway from the singularity. The collision is between ingo-ing and outgoing particles. Outgoing particles naturallyarise as initially ingoing particles are reflected back dueto the potential barrier offered by the angular momen-tum of the particle in the equatorial plane and due tothe repulsive nature of gravity along the axis of symme-try. We also studied the similar process in the context ofReissner-Nordstr¨om naked singularity when charge pa-rameter transcends the mass parameter by infinitesimalamount [7]. In this case, exploiting the spherical symme-try we could replace the particles by thin spherical shellsand take into account the effect of self-gravity of collidingparticles on the center of mass energy of collisions. Weshowed that in a reasonable astrophysical scenario theupper bound on the center of mass energy of collisionbetween two proton like particles was much larger thanthe Planck scale. Whereas a similar calculation in thecontext of black holes showed the upper bound on cen-ter of mass energy of collision to be much smaller thenthe Planck energy. We also showed that neither blackholes nor naked singularities are necessary for high en-ergy collisions to occur. We demonstrated this point inthe context of the Bardeen spacetime. It turns out thatultrahigh energy collisions can take place in the space-times that do not contain black hole [8].In this paper we address the issue of observability ofhigh energy collisions in the absence of an event horizon.We specifically work in the context of Kerr naked singu-lar spacetime. We calculate the escape fraction for theparticles produced in the high energy collisions aroundKerr naked singularity. Following [2] we make variousassumptions. The collision considered is between twoidentical massive particles starting from rest at infinity.The collision, as stated before, is between ingoing andoutgoing particle. We assume that the particles move onthe equatorial plane. Collision products are taken to betwo massless particles. In the center of mass frame col-liding particles move in in the opposite directions withsame speed. In this frame two massless particles pro-duced move in the opposite directions with equal mag-nitude of momenta. We assume that the two masslessparticles also travel in the equatorial plane. The dis-tribution of the direction of motion of massless particleswith respect to the direction of colliding particles is takento isotropic. We focus attention on only one of the par-ticles as we intend to compute the order of magnitude ofescape fraction.We show that the escape fraction for the collision prod-ucts to escape will be finite and would be approximatelyhalf. This implies that around half of the particles pro-duced eventually hit the singularity and disappear whileremaining half manage to escape away to infinity. Thussignificant fraction of collision products escape to infinityand would carry the information about nature of basicinteractions at extremely large energy scales. This is incontrast with the black hole case where high energy col-lisions that take place in the vicinity of event horizonwould be unobservable.
II. KERR GEOMETRY
In this section we briefly describe the Kerr naked sin-gularity geometry. We work in the units where the grav-itational radius, speed of light and mass of the collidingparticles is set to unity r g = GMc = c = µ = 1.The Kerr metric when restricted to the equatorialplane, in Boyer-Lindquist coordinates is given by, ds = − (cid:18) − r (cid:19) dt − ra Σ dtdφ + (cid:18) r ∆ (cid:19) dr + r dθ + (cid:18) r + a + 2 a r (cid:19) dφ (1)where ∆ = r + a − r and a is the angular momen-tum parameter. When a >
1, event horizon is absentin the spacetime. However, there is a singularity locatedat r = 0 , θ = π , which turns out to be visible for theasymptotic observer. Thus the Kerr solution representsa naked singularity.We now analyze the geodesic motion in the spacetime.Let U = (cid:0) U t , U r , U θ , U φ (cid:1) be the velocity of the particle.We assume for simplicity that particle follows geodesicmotion in the equatorial plane θ = π so that U θ = 0.The Kerr spacetime admits Killing vectors ∂ t , ∂ φ andthere are corresponding conserved quantities E = − U.∂ t , L = U.∂ φ associated with the geodesic motion which areinterpreted as the conserved energy and orbital angularmomentum per unit mass. Using these relations, U t , U φ can be written down and U r can be obtained using thenormalization condition for velocity U.U = − m . Here r FIG. 1: The variation of center of mass energy of collisionbetween the ingoing and outgoing particles with angular mo-menta L = 0 . , L = 1 . r = 1 in aspacetime containing naked singularity with spin parameterclose to unity, with a − ǫ = 10 − . m = µ = 1 for the massive particle and m = 0 for themassless particle. The velocity components are given by, U t = 1∆ (cid:20)(cid:18) r + a + 2 a r (cid:19) E − ar L (cid:21) U φ = 1∆ (cid:20)(cid:18) − r (cid:19) L + 2 ar E (cid:21) , U θ = 0 (2) U r = u s E − m + 2 m r − ( L − a ( E − m )) r + 2 ( L − aE ) r where u = ± E = 1. III. ULTRAHIGH ENERGY COLLISIONS
In this section we briefly describe the process of ultra-high energy particle collisions in the Kerr naked singular-ity geometry with the spin parameter transcending unityby an extremely small amount a − ǫ = 1 + [6].We consider collision between ingoing and outgoingparticles at r = 1. We assume that both the particlesstart from infinity at rest. Thus the conserved energyper unit mass for each of the particles is E = 1. Oneof the initially ingoing particles must turn back as anoutgoing particle at the radial location r <
1. The angu-lar momentum per unit mass of this particle must lie inthe range 2 (cid:0) − √ a (cid:1) ≤ L < (cid:0) − √ a − (cid:1) . Thelower limit on its value corresponds to the angular mo-mentum it must have for it to turn back at all, whereasthe upper limit comes from the restriction that it mustturn back at the radial location r <
1. The second par-ticle is an ingoing particle at the collision location. Therequirement that it should not turn back before it reaches r = 1 puts the following restriction on its angular mo-mentum L < (cid:0) − √ a − (cid:1) [6].The center of mass energy of collision between the par-ticles with velocities U , U is given by E c.m. = 2 m (1 − g µν U µ U ν ) (3)Approximate expression for the center of mass energy ofcollision in the situation that we described is given by E c.m. ≃ T T ǫ (4)where T , T are factors that can be written down interms of the energy and angular momenta of the par-ticles and they are of O (1). Clearly the center of massenergy of collision is large in the limit ǫ = a − → r = 1 is shown in Fig 1 for Kerr geometry with ǫ = 10 − . IV. ESCAPE FRACTION FOR COLLISIONPRODUCTS
We now compute the escape fraction for the collisionproducts. We assume that two identical massive particleseach with mass µ (= 1) participate in a collision and pro-duce two massless particles. In the center of mass framecolliding particles travel in the opposite direction withequal magnitude of three-momenta. This is also the casewith massless particles produced in the collision. The dis-tribution of massless particles in the center of mass frameis assumed to be isotropic for simplicity and we focus onlyon one of the particles since we intend to calculate theorder of magnitude of escape fraction. We assume thatthe massless particles travel in the equatorial plane. Wefirst calculate the conditions under which massless parti-cle will be able to escape to infinity. We then move overto the center of mass frame. The conditions for particlesto escape can be translated to the escape cones in thecenter of mass frame within which particles must travelif it were to escape to infinity. Sum of angles of thesecones divided by 2 π will be the estimate of the escapefraction. A. Escape conditions for massless particles
We now derive the conditions for massless particles toescape to infinity. Null geodesics are characterized bythe ratio of conserved energy and angular momentum LE , rather than by both conserved energy and angularmomentum as in the case of timelike geodesics. This canbe seen from (2) by dividing it throughout by E andchoosing a new affine parameter ˜ λ = Eλ where λ is theold affine parameter. Equivalently, one can choose E = 1.The angular momentum required for the particle toturn back at the radial coordinate r can be obtained from L2 H r L L2 H r L L1 H r L L2maxrmax1 2 3 4 5 6 r - - - H r L FIG. 2: The variation of angular momentum b ( r ) = L ( r ) , L ( r ) required for the massless particle to turn backfrom radius r . The lower branch of L ( r ) admits a maxi-mum L = L max ≈ − r = r max ≈ a ≈ (2) and is given by b ( r ) = L ( r ) = 1 r − (cid:16) − a + p r + a r − r (cid:17) = L ( r ) = 1 r − (cid:16) − a − p r + a r − r (cid:17) (5)Two branches of b ( r ) namely L ( r ) , L ( r ) are plotted inFig.2. The lower branch of L ( r ) admits a maximum L max ≃ − r = r max ≃ r could either be ingoing or outgoing corre-sponding to u = ±
1. The particle will escape to infinityif r < r max , u = +1 , L ,max < L < L ( r ) (6) r < r max , u = − , a < L < L ( r ) r > r max , u = +1 , L < L < L ( r ) r > r max , u = − , a < L < L (0) r > r max , u = − , L ( r ) < L < L ,max The first condition in (6) indicates that if the collisiontakes place at radial location r and if the massless parti-cle produced in the collision is to be radially ingoing thenit would escape to infinity if its angular momentum liesin the range L ,max < L < L ( r ). This is evident fromFig.2. Other four conditions can be interpreted similarly.These conditions can be translated into the escape frac-tion as we demonstrate below. B. Center of mass frame
We now consider the procedure to move over to thecenter of mass frame. We first introduce locally non-rotating frame [5]. Given a vector in Boyer-Lindquistcoordinate system V µ , its components in the locally non-rotating frame V ˜ µ are given by V ˜ µ = e ˜ µν V ν (7)where e ˜ µν = r g tφ − g tt g φφ g φφ g tφ √ g φφ √ g rr √ g θθ
00 0 0 √ g φφ in the above the g µν being the Kerr metric. We thenmake a rotation in the ˜ r − ˜ φ plane in the locally non-rotating frame so that the net three velocity of two col-liding particles is oriented along the new radial direction.We further make a boost along new radial direction sothat all the components of total three velocity now van-ish. This is the center of mass frame.The transformation to center of mass frame from lo-cally nonrotating frame can be given as V ˆ µ = Λ ˆ µ ˜ ν V ˜ ν (8)where the transformation matrix is composed of rotationand boostΛ = Λ boost Λ rot = γ − βγ cos α − βγ sin α − βγ γ cos α γ sin α − sin α α (9)The boost parameter β and rotation parameter α aregiven in terms of components of total velocity U tot = U + U in the locally non-rotating frame as β = q U ˜ r tot + U ˜ φ tot U ˜ ttot α = arccos U ˜ rtot q U ˜ r tot + U ˜ φ (10)Given the energy and angular momentum parametersof the colliding particles one can write down the veloc-ity components in the Boyer-Lindquist coordinate systemand also in the locally non-rotating orthogonal tetrad.The rotation and boost parameters can be written down once components of the colliding particles are known inthe non-rotating frame. Thus we can write down trans-formation laws that will allow us to write down compo-nents of any vector in the center of mass frame given itscomponents in the Boyer-Lindquist coordinate system. C. Escape Fraction
We assume that the massless particles produced in thecollisions travel in the equatorial plane and their distribu-tion is isotropic in the center of mass energy frame. Par-ticles would escape to infinity if (6) are satisfied. Theseconditions can be translated into the escape cones in thecenter of mass frame such that if the direction of motion(in other words three-velocity) lies in the cone, particleescapes to infinity. Directions that define edges of theescape cones would correspond to the extreme angularmomentum values in (6) given the radial location of col-lision and whether particle is ingoing or outgoing. Giventhe extreme momentum values one can write down thevelocity components of the massless particle in the Boyer-Lindquist coordinate system. Then one can make a tran-sition to the center of mass frame and write down thethree-velocities as U i = U ˆ i = Λ ˆ i ˜ µ e ˜ µν U ν . (11)Angle between two extreme three-velocities will be theescape angle. When escape angle is divided by 2 π we getthe escape fraction, EF ( r, a ) = Θ( r − r max ( a )) A ( r, a ) + Θ( r max ( a ) − r ) A ( r, a )2 π Here Θ is the step function. The first term makes a con-tribution if collision takes place outside r max ( a ), whereasthe second term contributes if collision takes place inside r max ( a ). Since we are dealing with the case where a → r = 1. In this situation r max ( a ) ≃ A . A ( r, a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) arccos (cid:20) U ( r, a, L = L ( r ) , u = 1) .U ( r, a, L = L ( r ) , u = 1) | U ( r, a, L = L ( r ) , u = 1) || U ( r, a, L = L ( r ) , u = 1) | (cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) arccos (cid:20) U ( r, a, L = a, u = − .U ( r, a, L = L (0) , u = − | U ( r, a, L = a, u = − || U ( r, a, L = L , u = − | (cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) arccos (cid:20) U ( r, a, L = L ( r ) , u = − .U ( r, a, L = L max , u = − | U ( r, a, L = L ( r ) , u = − || U ( r, a, L = L max , u = − | (cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r (a)
200 400 600 800 1000Ecm0.49960.49970.49980.49990.5000E.F. (b)
FIG. 3: Fig 3a shows the variation of escape fraction with radius. Highly energetic collisions take place around r = 1. Here ǫ = a − − . The collision is between ingoing and outgoing particles with angular momenta L = 0 . , L = 1 .
4. The escapefraction is more or less constant and takes a value
E.F. ≈ .
5, unlike the extremal Kerr black hole case where it decreases asone approaches the horizon r = 1. Fig 3b depicts the slight increase in the escape fraction with the center of mass energy ofcollision, unlike the Kerr black hole case where there is a sharp fall in the escape fraction with the center of mass energy ofcollision. A ( r, a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) arccos (cid:20) U ( r, a, L = L ( r ) , u = 1) .U ( r, a, L = L max , u = 1) | U ( r, a, L = L ( r ) , u = 1) || U ( r, a, L = L max , u = 1) | (cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) arccos (cid:20) U ( r, a, L = L ( r ) , u = − .U ( r, a, L = a, u = − | U ( r, a, L = L ( r ) , u = − || U ( r, a, L = a, u = − | (cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) We numerically calculate the escape fraction for highenergy collisions that take place around r = 1. The spinparameter is slightly larger than unity ǫ = a − → − .Collision is between ingoing and outgoing particles withangular momenta L = 0 . , L = 1 .
4. The escape frac-tion as a function of radial coordinate is plotted in Fig3a. Escape fraction in this region around radial coordi-nate r = 1 takes a value which is approximately half.Variation of the escape fraction with the center of massenergy of collision is depicted in Fig 3b. Escape fractionincreases slowly with the center of mass energy. This be-havior is in contrast with the extremal Kerr black holecase where there is a sharp decrement in the escape frac-tion with center of mass energy for the collision as weapproach the horizon r = 1 [2]. Thus the flux of the par-ticles produced in the high energy collisions around theKerr naked singularity will be large as approximately halfof the particles manage to escape to infinity while remain-ing half of the particles eventually hit the naked singular-ity. The high energy collisions, unlike the black hole case,would be observable and in principle would shed light onthe nature of basic interactions at very large energies. V. CONCLUSIONS AND DISCUSSION