Collision of high-energy closed strings: Formation of a ring-like apparent horizon
aa r X i v : . [ g r- q c ] O c t Alberta-Thy-04-07
Collision of high-energy closed strings:Formation of a ring-like apparent horizon
Hirotaka Yoshino
Department of Physics, University of Alberta,Edmonton, Alberta, Canada T6G 2G7
Tetsuya Shiromizu
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan (Dated: June 30, 2007)
Abstract
We study collisions of two high-energy closed strings in the framework of D -dimensional generalrelativity. The model of a high-energy closed string is introduced as a pp -wave generated by aring-shaped source with the radius R . At the instant of the collision, the positions of two stringsare assumed to coincide precisely. In this setup, we study the formation of two kinds of apparenthorizons (AHs): the AH of topology S D − (the black hole AH) and the AH of topology S × S D − (the black ring AH). These two AHs are solved numerically and the conditions for the formationof the two AHs are clarified in terms of the ring radius R . Specifically, we demonstrate that theblack ring AH forms for sufficiently large R . The effects of an impact parameter and the relativeorientation of incoming strings in more general cases are briefly discussed. PACS numbers: 04.70.-s, 04.50.+h, 04.70.Bw . INTRODUCTION The trans-Planckian collisions of particles attract renewed interests motivated by thescenarios of large extra dimensions [1, 2]. In these scenarios, the gravity becomes higher di-mensional at microscopic scale and the Planck energy could be as low as TeV. This indicatesthe possibility of the observation of quantum gravity phenomena at near-future accelerators.There are two ways to approach this subject. One is the string theory, in which one canstudy the regime where the string length is important but gravity is not so strong (see[3, 4, 5] for recent studies). Since the regime of strong gravity cannot be investigated by thestring theory, the theory of general relativity is often used as the alternative approach. Inthis approach, the intermediate state is expected to be a black hole. See [6, 7, 8] for outlinesof the expected phenomena of tiny black holes at accelerators.To study the process of the black hole production in general relativity, the apparenthorizon (AH) is a very useful tool, since formation of an AH implies the existence of theevent horizon outside of it. In [9], the grazing collision of high-energy particles was studiedusing the Aichelburg-Sexl particle model [10], and the analytic formula for the AH wasobtained in the four-dimensional case. The numerical code for solving the AHs of thissystem in the higher-dimensional cases was developed by one of us and Nambu [11], and theresults of [11] were further improved by adopting a better slice [12].Since the Aichelburg-Sexl particle is a simplified model of a particle, several efforts havebeen made in order to take into account the potentially important effects. The effect ofelectric charge was studied in [13], of which results were used to improve the estimate ofthe black hole production rate at accelerators [14]. In [15, 16], the validity of the generalrelativistic model was examined and the importance of wavepacket effects was pointed out.In a recent paper, the effects of the particle spin and of the wavepacket were studied in [17]using the gyraton model [18, 19]. See also [20, 21] for trials to incorporate the effects fromthe string theory.In this paper, we study the collisions of high-energy extended objects in the frameworkof general relativity. Specifically, we consider the system of two “closed strings” whosegravitational field is of the pp -wave type. Figure 1 shows the configuration of the system thatwe study in this paper. The two gravitational shocks collide in a D -dimensional spacetime.Each shock field is generated by a ring-shaped source whose radius is R . The positions of2 R Shock 1Shock 2
FIG. 1: The system that we study in this paper. Two shock gravitational fields generated byhigh-energy closed strings with the radius R collide in a D -dimensional spacetime. The positionsof the two strings coincide at the instant of the collision. the two strings exactly coincide at the instant of the collision. In this setup, formation ofAHs is investigated.This study has two meanings. First, it is expected that if the length of strings is taken intoaccount, the black hole formation in high-energy collisions becomes more difficult comparedto the cases of pointlike particles (e.g., [5, 20, 22, 23]). By considering the collisions ofextended objects, we would be able to incorporate some of effects of the strings on the blackhole formation. Such an approach was already done in [20]. In that paper, the collisions ofuniform disks were studied. The reason for this setup comes from the expectation that thestrings behave as quasi-homogeneous beams [24]. However, since how high-energy stringscouple to gravity is still an open problem, it is meaningful to study the collisions of high-energy shocks with one-dimensional sources. Furthermore, this work will clarify how theresults in [20] depend on the energy density distributions of incoming objects.Next, we can provide a possible scenario of producing a black ring as a result of thegravitational collapse. The black ring [25] is the solution of the asymptotically flat five-dimensional spacetime with the event horizon of topology S × S . Although the black ringsolution has been found only in the five-dimensional case, there are strong indications forthe existence of ( D ≥ S × S D − . Unfortunately the black rings are expected to be unstable (see,e.g., Sec. I of [27]). However, since they would be able to exist as intermediate states ofgravitational collapses, it is of interest to provide scenarios of producing AHs of topology3 × S D − (say, black ring AHs). The possibility of producing black rings in collisions oftwo high-energy particles was discussed in [28]. In [27], it was clarified that multi-particlesystems can make the black ring AHs. In this paper, we demonstrate that collisions ofclosed strings also lead to the formation of the black ring AHs. We study both the blackring AH and the AH of topology S D − (say, black hole AH), and clarify the conditions forthe formation of the two AHs in terms of the ring radius R .This paper is organized as follows. In the next section, we introduce the model of ahigh-energy closed string and set up the collision. In Sec. III, we show the AH equationand the boundary conditions, and demonstrate existence of the black ring AH for large R .The numerical methods for solving the two AHs are also explained. Then the numericalresults are shown in Sec. IV. Section V is devoted to discussions on more general collisionsand implications for trans-Planckian collisions in the scenarios of large extra dimensions. InAppendix A, formulas necessary for calculating the metrics of high-energy closed strings arepresented. II. SYSTEM SETUP
In this section, we introduce the model of a high-energy closed string and set up thecollision.
A. Model of a high-energy closed string
In order to introduce the model of a high-energy closed string in a D -dimensional space-time, we assume the following metric of the pp -wave form: ds = − d ¯ ud ¯ v + D − X i =1 d ¯ x i + Φ(¯ x i ) δ (¯ u ) d ¯ u . (1)The spacetime is flat except at ¯ u = 0. There exists a gravitational shock wave at ¯ u = 0,which propagates at the speed of light. The properties of the shock gravitational field ischaracterized by the function Φ(¯ x i ), which we call the shock potential hereafter. The nonzerocomponent of the energy-momentum tensor of this spacetime has the form T ¯ u ¯ u = ˆ ρ (¯ x i ) δ (¯ u ) . (2)4ere, ˆ ρ (¯ x i ) indicates the energy density in the shock. The Einstein equation is¯ ∇ Φ = − πG ˆ ρ (¯ x i ) (3)with the D -dimensional gravitational constant G and the flat space Laplacian ¯ ∇ in thecoordinates ¯ x i . The Aichelburg-Sexl particle [10] has the shock potentialΦ = − Gp log ¯ r, ( D = 4);16 πGp ( D − D − ¯ r D − , ( D ≥ . (4)Here, ¯ r := pP i ¯ x i and Ω D − is the area of the ( D − ρ = pδ D − (¯ x i ), the Aichelburg-Sexl particle representsa point particle with the energy p .We would like to generalize the Aichelburg-Sexl particle to high-energy closed strings.Let us introduce coordinates (¯ x, ¯ y, ¯ z i ) by¯ x = ¯ x , ¯ y = ¯ x , ¯ z i = ¯ x i +2 , (5)with i = 1 , ..., D −
4, and put the energy source on the (¯ x, ¯ y )-plane in the shape of a ringwith the radius R . If we introduce coordinates ( ¯ W , ¯ φ ) by¯ x = ¯ W cos ¯ φ, ¯ y = ¯ W sin ¯ φ, (6)the energy density we assume is ˆ ρ = p δ ( ¯ W − R )2 π ¯ W δ D − (¯ z i ) . (7)For D ≥
5, the shock potential for this energy density is given byΦ = 16 πGp ( D − D − Z π dζ / π (cid:2) (¯ x − R cos ζ ) + (¯ y − R sin ζ ) + ¯ Z (cid:3) ( D − / (8)with ¯ Z := qX ¯ z i . (9)Hereafter, we adopt r := (cid:18) πGp Ω D − (cid:19) / ( D − (10)5s the unit of the length. For D = 5–11, the value of r is related to the gravitational radius r h (2 p ) of the system as r ≃ . r h (2 p ). In order to calculate the black hole AH, it is usefulto introduce the spherical-polar coordinates (¯ r, ¯ θ, ¯ φ ):¯ x = ¯ r sin ¯ θ cos ¯ φ, ¯ y = ¯ r sin ¯ θ sin ¯ φ, ¯ Z = ¯ r cos ¯ θ. (11)In these coordinates, the shock potential is written asΦ = I D π ( D − , (12) I D := Z π dζ ( a − b cos ζ ) ( D − / , (13)with a = ¯ r + R , b = 2 R ¯ r sin ¯ θ. (14)The results of the integration of I D for D = 5–11 are summarized in the appendix. Theyare given in terms of elementary functions for even D while in terms of the complete ellipticintegrals for odd D .The above formula holds only for D ≥
5. For four dimensions, the shock potential isΦ = − r, (¯ r ≥ R ); − R, (¯ r ≤ R ) , (15)which has the same form as that of the Aichelburg-Sexl particle outside of the ring. There-fore, in the case D = 4, we can easily see what happens in the head-on collision of closedstrings using the results of [9]. The black hole AH forms if and only if R <
1. We alsofind that the black ring AH does not form for any values of R . This is consistent with thetheorem for the topology of AHs in four dimensions [29]. Hereafter we will consider higherdimensional cases with D ≥ D = 5 coincides with theone obtained by taking the lightlike limit of the boosted black ring solution [31]. As wementioned in Sec. I, it is expected that the black ring solutions exist also for D ≥
6. Hence6e conjecture that the lightlike boosts of such black ring solutions will give the above metricsafter discovery of exact solutions for D ≥ ρ, ¯ ξ, ¯ φ ) by¯ x = ( R + ¯ ρ cos ¯ ξ ) cos ¯ φ, ¯ y = ( R + ¯ ρ cos ¯ ξ ) sin ¯ φ, ¯ Z = ¯ ρ sin ¯ ξ. (16)These coordinates are also useful for studying the black ring AH. In these coordinates, thering is located at ¯ ρ = 0. In the regime ¯ ρ/R ≪
1, the shock potential behaves asΦ ≃ − πR log (cid:16) ¯ ρ R (cid:17) , ( D = 5); α D R ¯ ρ D − , ( D ≥ , (17)with α D := Γ(( D − / √ π Γ( D/ − . (18)These formulas will be used later to show the existence of the black ring AH in the case R ≫ B. Continuous and smooth coordinates
Because the delta function in the metric (1) indicates the discontinuity of the coordinates(¯ u, ¯ v, ¯ x i ) at ¯ u = 0, it is necessary to introduce the continuous and smooth coordinates( u, v, x i ) in order to set up the collision. Such a coordinate transformation is given in [9] as¯ u = u, (19)¯ v = v + Φ θ ( u ) + 14 uθ ( u )( ∇ Φ) , (20)¯ x i = x i + 12 u ∇ i Φ( x ) θ ( u ) . (21)Here, θ ( u ) denotes the Heaviside step function. In these coordinates, the line v, x i = const . is the null geodesic and u is its affine parameter. The metric is written as ds = − dudv + H ik H jk dx i dx j , (22) H ij = δ ij + 12 ∇ i ∇ j Φ( x ) uθ ( u ) . (23)7 I v II IIIIV shock1 shock2
FIG. 2: Schematic structure of the spacetime for the system of two high-energy strings. Themetric in the regions I, II, and III can be written down. We study the AHs on the slice indicatedby a dotted line.
C. Setup of the collision
Now we set up the collision of two high-energy closed strings. Figure 2 shows theschematic picture of the spacetime structure. The spacetime is divided into regions I:( u ≤ , v ≤ u ≥ , v ≤ u ≤ , v ≥ u ≥ , v ≥ ds = − dudv + h H (1) ik H (1) jk + H (2) ik H (2) jk − δ ij i dx i dx j (24) H (1) ij = δ ij + 12 ∇ i ∇ j Φ( x ) uθ ( u ) , (25) H (2) ij = δ ij + 12 ∇ i ∇ j Φ( x ) vθ ( v ) . (26)Since we use the same formula of the shock potential for two incoming strings, their positionsexactly coincide at the instant of the collision.Region IV is the interaction region and its spacetime structure is not known. However,it is still possible to confirm the black hole formation by studying an AH on some slicein regions I, II, and III. In the next section, we explain how to find the AHs on the slice u ≤ v and v ≤ u (the dotted line in Fig. 2).8 source x yz i BH R C source x yz BR i R FIG. 3: Schematic pictures of the boundary C BH for the black hole AH (left) and the boundary C BR for the black ring AH (right). III. FINDING APPARENT HORIZONS
In this section, we show the equations and the boundary conditions for the black holeAH and the black ring AH on the slice u ≤ v and v ≤ u . These AH equations canbe solved analytically in the two cases R = 0 and R ≫
1. Then, the numerical methods forsolving the two AHs are presented.
A. The equation and the boundary conditions
The AH is defined as a ( D − S : v = − Ψ ( x )in v ≤ u and S : u = − Ψ ( x ) in u ≤ v . Here, the surfaces S , are connected on acommon boundary C in u = v = 0. The equation and the boundary conditions for Ψ , werederived in [9] (see also [20, 21]). Here we just comment on the results. The AH equationis found as ∇ ( Ψ , − Φ , ) = 0, by imposing the expansion of the null geodesic congruenceof each surface to be zero. The boundary conditions are Ψ , = 0 and ∇ Ψ · ∇ Ψ = 4 on C , which come from the continuity of the surface and the null tangent vectors, respectively.There are two boundary conditions because the boundary C itself is an unknown surface tobe solved.As the two incoming strings are identical in our setup, we can simply put Ψ := Ψ = Ψ .In numerical calculations, it is convenient to introduce a function h := Ψ − Φ. In terms of9 , the AH equation and the boundary conditions become ∇ h = 0 , within C, (27) h = Φ and [ ∇ (Φ − h )] = 4 , on C. (28)These equation and boundary conditions can be applied to both the black hole AH andthe black ring AH. However, we have to choose the boundary C appropriately in each case.Figure 3 shows schematic shapes of the boundaries C BH and C BR for the black hole AH andthe black ring AH, respectively. In the case of the black hole AH, we require the boundary C BH to have the topology S D − so that the AH can have the topology S D − . On the otherhand, the boundary C BR of the black ring AH is assumed to have the topology S × S D − in order that the AH can have the topology S × S D − . B. The cases R = 0 and R ≫ Let us consider the black hole AH in the case R = 0. In this case, the system is reducedto the head-on collision of Aichelburg-Sexl particles. Using the spherical-polar coordinates( r, θ, φ ) introduced in Eq. (11) (here bars are omitted since we are now working in thecontinuous coordinates), the solutions of h ( r, θ ) and C BH are given by h ( r, θ ) = D − and r = 1, respectively.Next we consider the limit R ≫
1. In this case, we can show the existence of the blackring AH as follows. In the neighborhood of the ring, the shock potential Φ is approximatedas Eq. (17). Then, the solutions of h ( r, ξ ) and C BR are given by h ( r, ξ ) = Φ( ρ h ) and ρ = ρ h ,respectively, where ρ h = (cid:18) β D R (cid:19) / ( D − . (29)Here, β D := D − α D . Because ρ h becomes small as R is increased, the black ring AH islocated near the ring for large R . It is reminded that the above approximation is valid onlyfor R ≫ R = 0, we can numerically solve itfor R > R . Similarly, since the solution of the black ringAH is known for R ≫
1, the numerical calculation can be done by starting at sufficientlylarge R and gradually decreasing the value of R . In the following, we briefly explain thenumerical methods for solving the black hole AH and the black ring AH, one by one.10 . Numerical method for black hole AHs In order to solve the black hole AH, the spherical-polar coordinates ( r, θ, φ ) are useful.In these coordinates, the AH equation is h ,rr + h ,θθ r + D − r h ,r + cot θ − ( D −
5) tan θr h ,θ = 0 . (30)Let the boundary C BH be given by r = g ( θ ). We further perform a coordinate transformation˜ r := r/g ( θ ) , (31)for numerical convenience. In the coordinates (˜ r, θ ), the AH equation becomes (cid:18) g ′ g (cid:19) h , ˜ r ˜ r − g ′ g h , ˜ rθ ˜ r + h θθ ˜ r + (cid:20) D − − (cid:18) g ′′ g − g ′ g (cid:19)(cid:21) h , ˜ r ˜ r + cot θ − ( D −
5) tan θ ˜ r (cid:18) h ,θ − ˜ r g ′ g h , ˜ r (cid:19) = 0 . (32)Then, we can write down the finite difference equations with the second order accuracy.At the coordinate singularity ˜ r = 0, we cannot use the above equation. Instead, weused the ( W, Z ) coordinates introduced in Eqs. (6) and (9). In these coordinates, the AHequation is 2 h ,W W + ( D − h ,ZZ = 0 (33)at W = Z = 0 (i.e., ˜ r = 0). We wrote down the finite difference equation of Eq. (33),which can be rewritten as the equation to determine the value of h at ˜ r = 0 in the originalcoordinates (˜ r, θ ).In order to solve this problem, we used the code developed in [11], which makes both h (˜ r, θ ) and g ( θ ) simultaneously converge to the real solutions. We used the grid numbers(50 × × D = 5–11. D. Numerical method for black ring AHs
In order to solve the black ring AH, we use the coordinates ( ρ, ξ, φ ) introduced in Eq. (16).In these coordinates, the AH equation is h ,ρρ + h ,ξξ ρ + (cid:18) D − ρ + cos ξR + ρ cos ξ (cid:19) h ,ρ + 1 ρ (cid:20) ( D −
5) cot ξ − ρ sin ξR + ρ cos ξ (cid:21) h ,ξ = 0 . (34)11imilar to the case of the black hole AH, we give the boundary C BR by ρ = f ( ξ ). Applyinga coordinate transformation ˜ ρ := ρ/f ( ξ ) , (35)the AH equation becomes (cid:18) f ′ f (cid:19) h , ˜ ρ ˜ ρ − f ′ f h , ˜ ρξ ˜ ρ + h ξξ ˜ ρ + (cid:20) ρ (cid:18) D − − f ′′ f + 2 f ′ f (cid:19) + cos ξR/f + ˜ ρ cos ξ (cid:21) h , ˜ ρ + 1˜ ρ (cid:20) ( D −
5) cot ξ − ˜ ρ sin ξR/f + ˜ ρ cos ξ (cid:21) (cid:18) h ,ξ − ˜ ρ f ′ f h , ˜ ρ (cid:19) = 0 . (36)Then the finite difference equations with the second order accuracy can be written down.Similar to the case of the black hole AH, a careful treatment is required at ˜ ρ = 0. Weused the coordinates ( X, Y ) := ( ρ cos ξ, ρ sin ξ ), in which the AH equation becomes h ,XX + ( D − h ,Y Y + h ,X R = 0 (37)at X = Y = 0 (i.e., ˜ ρ = 0). Since there is no symmetry with respect to the Y axis inthis case, the discretization becomes complicated. It is still possible to prepare the finitedifference equation with the second order accuracy and rewrite it in terms of the originalcoordinates ( ˜ ρ, ξ ).We solved h ( ˜ ρ, ξ ) and f ( ξ ) by rewriting the code of [11] using the grid numbers (50 × × D = 5–11. IV. NUMERICAL RESULTS
In this section, we show the numerical results for the black hole AH and the black ringAH. The amounts of energy trapped by the produced AHs are also evaluated. Then webriefly discuss the interpretation of the obtained results.
A. Black hole AHs
Figure 4 shows shapes of the boundary C BH of the black hole AH on the ( x, z i )-plane for D = 5–8. The y -axis is suppressed. As the ring radius R is increased, the black hole AHbecomes oblate. There is some critical value R (BH)max such that the solution of the black hole12 - - x r - - z i r D = - - - x r - - z i r D = - - - x r - - z i r D = - - - x r - - z i r D = FIG. 4: Shapes of the boundary C BH (the solid lines) for the black hole AH on the ( x, z i )-planefor D = 5–8. The boundary C BH becomes oblate as R is increased. The values of the ring radius R/r are 0 . ◦ ), 0 . • ), 0 . ♦ ), 0 . (cid:7) , shown only for D = 6–8), and R (BH)max /r ( ⋆ ). For R = R (BH)max ,the black ring AH (the dashed line) exists inside of the black hole AH. R r r min r D = D = FIG. 5: The relation between the ring radius R and the minimum radius of the C BH , i.e. r min := g (0), for D = 5–11. The value of dr min /dR becomes −∞ at R = R (BH)max . AH cannot be found for
R > R (BH)max . At R = R (BH)max , the black ring AH also exists inside ofthe black hole AH. For large D , their shapes agree well except at small θ .The function g ( θ ) that specifies the location of C BH takes a minimum value at θ = 0.13 - - x r - - z i r D = - - - x r - - z i r D = - - - x r - - z i r D = - - - x r - - z i r D = FIG. 6: Shapes of the boundary C BR (the solid lines) for the black ring AH on the ( x, z i )-planefor D = 5–8. The values of the ring radius R/r are 2 . ◦ ), 2 . • ), 1 . ♦ ), 1 . (cid:7) ), and R (BR)min /r ( ⋆ ). For R = R (BR)min , the black hole AH (the dashed line) exists outside of the black ring AH. In Fig. 5, we plot this “minimum radius” r min := g (0) as a function of R for each D . Theminimum radius r min is a monotonically decreasing function of R , and its gradient becomes −∞ at R = R (BH)max . The values of R (BH)max are somewhat smaller than r , as summarized inTable I. 14 .8 1 1.2 1.4 1.6 1.8 2 R r r in r D = D = FIG. 7: The relation between the ring radius R and the inner radius of C BR , i.e. r in := R − f ( π ),for D = 5–11. The value of dr in /dR becomes ∞ at R = R (BR)min . B. Black ring AHs
Figure 6 shows shapes of the boundary C BR of the black ring AH on the ( x, z i )-plane for D = 5–8. Since the boundary C BR has the topology S × S D − , it has two characteristicscales: the radius of the S circle and the radius of the S D − sphere. For large R , the radiusof the S D − sphere is approximately equal to ρ h defined in Eq. (29), and becomes large as R is decreased. The black ring AH disappears at some critical value R = R (BR)min . At R = R (BR)min ,the black ring AH is surrounded by the black hole AH.Let us look at the behavior of the radius of the S circle. We define the “inner radius” of C BR by r in := R − f ( π ), which represents the radius of the inner side of C BR (i.e., ξ = π ).In Fig. 7, we plot r in as a function of R for each D . The inner radius r in is a monotonicallyincreasing function of R , and its gradient diverges at R = R (BR)min . For D ≥
7, the value of r in is very small at R = R (BR)min and thus the inner side of the C BR almost touches as shownin Fig. 6. The values of R (BR)min are summarized in Table I. R (BR)min is slightly less than R (BH)max .This indicates that at least one of the two AHs forms in this system for arbitrary values of R . 15 .25 0.5 0.75 1 1.25 1.5 1.75 2 R r M AH p D = BH BR
R r M AH p D = BH BR0.25 0.5 0.75 1 1.25 1.5 1.75 2
R r M AH p D = BH BR 0.25 0.5 0.75 1 1.25 1.5 1.75 2
R r M AH p D = BH BR
FIG. 8: The relation between the ring radius R and the trapped energy M AH / p by the black holeAH (solid lines) and the black ring AH (dashed lines) for D = 5–8. C. Trapped energy
Since the event horizon is located outside of the AH, the area of the AH A D − gives thelower bound on the area of the produced black hole. Therefore the following quantity M AH := ( D − D − πG (cid:18) A D − Ω D − (cid:19) ( D − / ( D − (38)provides the lower bound on the mass of the produced black hole (or the black ring) andindicates the amount of the energy trapped by the AH.Figure 8 shows the values of M AH of the black hole AH and the black ring AH as functionsof R . The value of M AH of the black hole AH decreases as R is increased, but the blackhole AH traps more than 50% of the system energy for all 0 ≤ R ≤ R (BH)max . In the range R (BR)min ≤ R ≤ R (BH)max , M AH has two values since both the black hole AH and the black ring AHexist. The two values of M AH are different, but the difference is very small for D ≥
6. Thevalue of M AH of the black ring AH becomes small as R is increased. For large R , M AH / p becomes proportional to R − ( D − D − D − since A D − asymptotes to 4 π Ω D − Rρ D − h .16 ABLE I: The values of R (BH)max /r , R (BR)min /r and β / ( D − D for D = 5 , ...,
11. The black hole AHand the black ring AH exist for R ≤ R (BH)max and R ≥ R (BR)min , respectively. In all dimensions, thetransition from a black hole AH to a black ring AH occurs in a range β / ( D − D < R/r < D R (BH)max /r .
796 0 .
836 0 .
867 0 .
889 0 .
905 0 .
917 0 . R (BR)min /r .
774 0 .
832 0 .
865 0 .
887 0 .
903 0 .
915 0 . β / ( D − D .
564 0 .
630 0 .
679 0 .
715 0 .
744 0 .
767 0 . D. Interpretation
Let us briefly discuss the physical interpretation of the obtained results. As the valueof R is increased, the “transition” from the black hole AH to the black ring AH occurs at R (BR)min ≤ R ≤ R (BH)max . One of the plausible interpretations for the values of the transitionradius is as follows. For R &
1, the formation of the black hole AH cannot be expected sincethe incoming strings have characteristic scales larger than the gravitational radius of thesystem. On the other hand, if R is less than the value of ρ h defined in Eq. (29), the blackring AH cannot be expected to form because the radius of the S circle is smaller than theradius of the S D − sphere of C BR . Since R = ρ h is equivalent to R = β / ( D − D , the transitionfrom a black hole AH to a black ring AH is expected to occur at β / ( D − D < R <
1. Thevalues of β / ( D − D are summarized in Table I. We can confirm that both R (BR)min and R (BH)max are larger than β / ( D − D and smaller than 1 for all D .We also point out the similarity between our study and that of [26]. In that paper, severalmomentarily static initial data sets in five-dimensional spacetimes were investigated. Oneof the studied system is ring-shaped matter distribution, for which the black hole AH andthe black ring AH form for small and large values of the ring radius, respectively. Theirinterpretation is that for large ring radius, the gravitational field in the transverse directionof the ring is four dimensional, and thus the black ring AH can form. We consider thatthe same argument holds for our system, since the behavior of the shock potential in thetransverse direction of the D -dimensional string is similar to that of the ( D − RR FIG. 9: The example of positions of two closed strings at the instant of the head-on collisionin five-dimensional case. The direction of the motion is suppressed. Because the shock is threedimensional, there is a nonzero angle θ between the two planes on which the strings exist. V. DISCUSSION
In this paper, we studied the collision of high-energy closed strings. The positions of twoclosed strings were assumed to coincide at the instant of the collision. In this setup, wefound that the black hole AH forms for small ring radius R and the black ring AH forms forlarge ring radius R . Therefore, the collision of high-energy closed strings with large radiuswill lead to the formation of the black ring.Because the setup of the collision in this paper is quite limited, let us discuss whathappens in a more general setup. For simplicity, let us assume that the radii of two stringsbe the same. In general collisions, we have to take account of the impact parameter b .Here, b is defined as the distance of two centers of incoming strings when two shocks collide.Furthermore, even in the head-on collision b = 0, the orientations of two strings do notcoincide at the instant of the collision. Figure 9 depicts the example in the five-dimensionalcase. Because the shock is three dimensional, the two planes on which two strings exist crosseach other at angles θ . Similarly, in the D -dimensional case, the relative orientation of twostrings will be specified by angles θ , ..., θ D − .Let us first discuss the black hole AH. In the head-on collision b = 0, it is expected thatthe main factor for the AH formation is the ring radius, because the discussion in Sec. IVD18olds for any angles θ i . Namely, the maximum ring radius R max for the formation of theblack hole AH would be about r and would weakly depend on θ i . As the impact parameter b is increased, R max will become small and eventually go to zero. This happens at b ≃ r ,since in the Aichelburg-Sexl particle case ( R = 0), the maximal impact parameter for the AHformation is b max ≃ r as demonstrated in [11, 12]. Therefore, we expect that the conditionfor the black hole formation will be given by b . r − R . This is the same criterion as theone discussed in [20].Next we discuss the black ring AH. The condition for the formation of the black ring AHwill strongly depend on the angles θ i even in the head-on collision cases b = 0, since thedistance between the two strings should be less than ρ h . Namely, in the five-dimensionalcase, the angle θ should satisfy | θ | . ∆ θ := ρ h /R ∼ R − . Similarly, in the D -dimensionalcase, all the angles θ i should satisfy | θ i | . ∆ θ := ρ h /R ∼ R − ( D − / ( D − . Therefore, even inthe head-on collision cases, the black ring formation is quite difficult. If we cannot controlthe orientations of the incoming strings, the black ring will form only with the probability ∼ ∆ θ D − ∼ R − ( D − . Furthermore, the impact parameter should be smaller than the theradius of the S D − sphere of C BR , i.e. b . ρ h ∼ R − / ( D − , which is much smaller than r .Finally we discuss some implication for the trans-Planckian collisions of fundamentalstrings in the scenarios of large extra dimensions. Let us consider the collisions of closedstrings with length λ s = 2 πR and assume that our system can describe these processes.Under this assumption, we find that R . r h (2 p ) is necessary for the black hole formation.For R & r h (2 p ), the black ring formation would be expected instead, with the probabilitydiscussed above. However, we point out here that an additional condition is required for theproduction of a classical black ring. Because the black ring can be a classical object onlywhen the radius of the S D − sphere of C BH is larger than the Planck length l p , the systemshould satisfy ρ h & l p . This is rewritten as Rl p . (cid:16) r h (2 p ) l p (cid:17) D − . Therefore, the parameterrange of R for the black ring formation is restricted also from above.Since we studied the collision of closed strings and the closed strings do not stand forgauge particles, our results cannot be directly applied for the phenomena in accelerators.We are planning to generalize our result to the collision of high-energy open strings.19 cknowledgments HY thanks the Killam Trust for financial support. The work of TS was supported byGrant-in-Aid for Scientific Research from Ministry of Education, Science, Sports and Cultureof Japan (No. 13135208, No. 14102004, No. 17740136 and No. 17340075), the Japan-U.K.and Japan-France Research Cooperative Program.
APPENDIX A: CALCULATION OF THE SHOCK POTENTIAL
In this section, we show the formulas of I D defined in Eq. (13). We begin with even D values. For D = 6, I = 2 π √ a − b . (A1)By taking derivatives of this formula with respect to a , we obtain I = 2 πa ( a − b ) / ; (A2) I = π a + b ( a − b ) / . (A3)Next we show the cases of odd D values. In these cases, the integrals are expressed interms of the complete elliptic integrals of the first and second kinds: K ( k ) := Z π/ dθ p − k sin θ ; (A4) E ( k ) := Z π/ dθ p − k sin θ. (A5) I is found to be I = 4 √ a + b K ( k ) (A6)with k := p b/ ( a + b ). By taking derivatives of this formula with respect to a , we obtain I = 4( a − b ) √ a + b E ( k ); (A7) I = 43( a − b ) ( a + b ) / [4 aE ( k ) − ( a − b ) K ( k )] ; (A8) I = 415( a − b ) ( a + b ) / (cid:2) (23 a + 9 b ) E ( k ) − a ( a − b ) K ( k ) (cid:3) . (A9)20
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