The dark side of fuzzball geometries
Massimo Bianchi, Dario Consoli, Alfredo Grillo, Jose F. Morales
PPrepared for submission to JHEP
PREPRINT
The dark side of fuzzball geometries
M. Bianchi, a,b
D. Consoli, a,b
A. Grillo, a J.F. Morales b a Dipartimento di Fisica, Universit`a di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133Roma, Italy b I.N.F.N. Sezione di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Black holes absorb any particle impinging with an impact parameter below acritical value. We show that 2- and 3-charge fuzzball geometries exhibit a similar trappingbehaviour for a selected choice of the impact parameter of incoming massless particles. Thissuggests that the blackness property of black holes arises as a collective effect whereby eachmicro-state absorbs a specific channel.
Keywords:
Black holes, fuzzballs, D-branes, micro-states a r X i v : . [ h e p - t h ] J u l ontents ϑ = 0 geodesics 145.3 ϑ = π/ The interest in objects colloquially known as black holes (BH’s) has been revived not onlyby their role in the generation of the first gravitational wave signal detected by the LIGO-Virgo collaboration [1] but also by the possibility that primordial BH’s may account for a(small) fraction of the dark matter in the universe [2] and rotating BH’s and similar objectsmay accelerate cosmic rays thanks to Penrose mechanism [3, 4].In String Theory it is natural to describe BH’s as ensembles of micro-states representedby smooth, horizonless geometries without closed time-like curves, the so called “fuzzballs”[5–11]. The counting of micro-states for extremal 3- and 4-charged black hole states in fiveand four dimensions has proven to be very successful [12–16], while the identification ofthe corresponding geometries in the supergravity regime has revealed to be much harder[17–37]. To go one step further one can probe fuzzball geometries with particles, waves andstrings and test the proposal at the dynamical level [38]. Elaborating on our recent work on– 1 –-charge systems [38], our present focus will be on the geodetic motion of massless particleson a class of 3-charge micro-state geometries introduced in [39]. This should capture therelevant physics not only for large impact parameters where the eikonal approximationof the scattering process is valid even quantitatively [40–51], but also for small impactparameters whereby the particles get trapped or absorbed, at least at a qualitative level.We leave the analysis of waves and strings or other classes of smooth geometries (such asJMaRT [52]) to the future.The picture that emerges from our analysis is that the blackness property of blackholes arises as a collective/statistical effect where each micro-state absorbs a specific chan-nel. More interestingly, this universal property of fuzzball geometries suggests the possibleexistence of more exotic distributions of micro-state geometries looking effectively as grav-itational filters obscuring only a band in the light spectrum of distant sources, or morebizarre black looking objects such as rings, spherical shells, etc. A more detailed analysisshould take into account radiation damping, i.e. the energy lost in gravitational waveemission by an accelerated particle. Contrary to the case of an accelerated charged par-ticle, we expect gravitational brems-strahlung to be anyway negligible for a vast range ofkinematical parameters.The paper is organised as follows. In section 2 we introduce the class of micro-stategeometries we will consider, discuss the general behaviour of massless geodesics in thesebackgrounds and summarise our results. In particular we will introduce the notions of turning points and critical geodesics , characterising geodesics that either bounce back toinfinity or get trapped spinning around the gravitational source, respectively. In Section 3-5 we analyse the behavior of massless geodesics in the case of 3-charge black holes, 2-chargeand 3-charge fuzzballs respectively. The analysis of 2-charge fuzzballs is performed in fullgenerality, while the analysis of the 3-charge case is restricted to geodesic motion along orperpendicular to the plane of the string profile characterising the fuzzball. The latter caselacks spherical symmetry and exhibits an intricate non-completely separable dynamics. Asimple solution in this class is presented in some detail. Section 6 contains our conclusionsand outlook.
In this section we introduce the fuzzball geometries we will be interested in and summariseour results. We write down the general form of the metric, the Lagrangian governingthe dynamics of massless neutral particles and the geodesic equations. We then identifythe conjugate momenta and the Hamiltonian and describe how to take advantage of theisometries when present. We also discuss the classification of the geodesics when the systemis integrable. – 2 – .1 The 3-charge fuzzball metrics
We will consider 3-charge BPS micro-state geometries belonging to the general class con-structed in [39]. The ten-dimensional metric can be written in the form ds = √ Z Z Z ds + (cid:114) Z Z ds T . (2.1)where ds T is the metric on a T torus (or a K3 surface, in fact) while the 6-dimensionalmetric ds describes a 5-dimensional space-time times a compact circle of radius R y . Thismanifold can be parametrized with coordinates { t, (cid:126)X, y } or alternatively by introducingthe null coordinates u = t − y √ and v = t + y √ and the oblate spheroidal coordinate system X + i X = (cid:112) ρ + a sin ϑ e i ϕ , X + i X = ρ cos ϑ e i ψ . (2.2)By doing so one obtains ds = g mn dx m dx n = − dv + β m dx m ) ( du + γ m dx m ) + Z ds . (2.3)where ds is the flat metric of R ds = (cid:0) ρ + a cos ϑ (cid:1) (cid:18) dρ ρ + a + dϑ (cid:19) + (cid:0) ρ + a (cid:1) sin ϑ dϕ + ρ cos ϑ dψ . (2.4)The functions Z , Z , Z , β m , γ m depend on the coordinates (cid:126)x of R and on v , their explicitexpression is as follows Z = 1 + L ρ + a c ϑ + ε R ∆ n s ϑ cos 2 φL ( ρ + a c ϑ ) Z = 1 + L ρ + a c ϑ Z = 2 ε R ∆ n s ϑ cos φ ( ρ + a c ϑ ) Z = Z Z − Z β ϕ = a R s ϑ ρ + a c ϑ β ψ = − a R c ϑ ρ + a c ϑ γ ϕ = α β ϕ − n ε R L ∆ n cos 2 φ s ϑ γ ψ = − α β ψ γ v = F n γ ϑ = − ε R L ∆ n sin 2 φ s ϑ c ϑ γ ρ = − ε R L ∆ n ρ sin 2 φ s ϑ (2.5)with s ϑ = sin ϑ, c ϑ = cos ϑ and φ = ϕ + nvR R = R y √ F n = − ε a (cid:20) − (cid:18) ρ ρ + a (cid:19) n (cid:21) ∆ n = a ρ + a (cid:18) ρ ρ + a (cid:19) n α = 1 − F n − n ε L ∆ n cos 2 φ s ϑ (2.6) In the notation of this reference, we focus on solutions with k = 1, m = 0 and n an arbitrary positiveinteger. For the class of solutions we are interested in the components β ρ , β ϑ , β u , β v and γ u are identically zero. – 3 –egularity of the metric near ρ = 0, ϑ = π/ a = L L R − ε , ε = ε (cid:18) a nL (cid:19) . (2.7)The conserved charges and the angular momenta J and ˜ J are given by Q = L , Q = L , Q P = ε n , J = ˜ J = Ra √ (cid:54) = 0 . (2.8)or equivalently J ϕ = J + ˜ J = √ Ra , J ψ = J − ˜ J = 0 (2.9)We will study the scattering of massless neutral particles in the following special cases ofthe family of BPS metrics introduced above: • a → • ε = n = 0 in the 3-charge metric. • ϑ = 0 and ϑ = π/ We are interested in null geodesics in the 6-dimensional geometry that solve the Euler-Lagrange equations derived from the Lagrangian L = g mn ˙ x m ˙ x n , (2.10)with g mn the six-dimensional metric, and dots denoting derivatives with respect to anaffine parameter τ . Null geodesics are specified by solutions x m ( τ ) of the Euler-Lagrangeequations satisfying L = 0. Equivalently one can introduce the Hamiltonian H = P m ˙ x m − L = g mn P m P n (2.11)expressed in terms of the conjugate momenta P m = ∂ L ∂ ˙ x m = g mn ˙ x n . (2.12)It will prove useful to keep in mind that2 P u P v = E − P y ≥ , (2.13)where E and P y are the momenta conjugate to t and y , respectively. In the Hamiltonianformulation, geodesics are described by the velocities˙ x m = ∂ H ∂P m (2.14) The never vanishing factor √ Z Z Z in front of the 6-dimensional metric (2.3) can be absorbed in aredefinition of the affine parameter τ , and neglected when dealing with 6-dimensional geodesics. – 4 –ith P m a solution of the system of equations H = g mn P m P n = 0 (2.16)˙ P m = − ∂ H ∂x m (2.17)The metric is independent of the variables u and ψ , so the momenta P u and P ψ will alwaysbe conserved. The Hamiltonian can be written in the compact form H = − P u (cid:98) P v + 12 Z (cid:34) ( ρ + a ) (cid:98) P ρ ρ + a c ϑ + (cid:98) P ϑ ρ + a c ϑ + (cid:98) P ϕ ( ρ + a ) s ϑ + (cid:98) P ψ ρ c ϑ (cid:35) (2.18)in terms of the shifted momenta (cid:98) P m = P m − β m ( P v − γ v P u ) − γ m P u (2.19)The velocities become˙ ρ = ( ρ + a ) (cid:98) P ρ Z ( ρ + a c ϑ ) , ˙ ϑ = (cid:98) P ϑ Z ( ρ + a c ϑ )˙ ϕ = (cid:98) P ϕ Z ( ρ + a ) s ϑ , ˙ ψ = (cid:98) P ψ Z ρ c ϑ (2.20)with more involved formulae for ˙ u and ˙ v . The Hamiltonian constraint H = 0 can be solvedby taking (cid:98) P ρ = ± (cid:18) ρ + a c ϑ ρ + a (cid:19) (cid:34) Z P u (cid:98) P v − (cid:98) P ϑ ρ + a c ϑ − (cid:98) P ϕ ( ρ + a ) s ϑ − (cid:98) P ψ ρ c ϑ (cid:35) (2.21)with minus and plus signs for the branches along which the particle approaches or leavesthe gravitational target, respectively. We notice that according to (2.20) (cid:98) P ρ determines theradial velocity of the particle. Starting from infinity, ρ ( τ ) monotonously decreases untilit reaches a point ρ ∗ where (cid:98) P ρ vanishes and flips sign. This is said to be an inversion (or turning ) point. Since ρ is a monotonous function along this branch it can be used inprinciple to parametrize the evolution time, expressing all remaining coordinates x m ( ρ ) asa function of ρ instead of the affine parameter τ . In practice, this is possible only whenthe system is integrable. Examples of integrable geodesics occur for BH’s with or withoutangular momenta, 2-charge circular fuzzballs and geodesics along the plane orthogonal tothe string profile in the 3-charge system. The most difficult and interesting case (motionalong the plane of the profile in the 3-charge case) eludes this simplistic analysis and willbe addressed in section 5.3. We notice that the equations of motion imply˙ H = g mn P m (cid:18) ˙ P n + ∂ H ∂x n (cid:19) = 0 (2.15)so, one of the equations of motion, let us say the one for ρ can be replaced by H = 0. – 5 – igure 1 : Geodesics in the black hole and fuzzball geometries for different values of theimpact parameter b .When the system is integrable and all the variables can be explicitly expressed in termsof ρ , the time (measured by an observer at infinity) required by a geodesic to reach theinversion (or turning) point ρ ∗ starting from a point ρ is given by ∆ t = (cid:90) ρ ∗ ρ dρ (cid:18) dtdτ (cid:19) ρ + a c ϑ ( ρ ) ρ + a Z ( ρ ) (cid:98) P ρ ( ρ ) (2.23)This integral may or may not diverge. Focusing for simplicity on geodesics with zerointernal momenta ( P y = 0) and denoting by K the total angular momentum of the incoming The derivative of the time coordinate t w.r.t. the affine parameter τ is given by dtdτ = − (1 − γ v ) P u + (cid:98) P v √ − √ Z (cid:20) γ ϑ (cid:98) P ϑ + ( ρ + a ) γ ρ (cid:98) P ρ + β ψ (1 − γ v )+ γ ψ ρ cos ϑ (cid:98) P ψ + β ϕ (1 − γ v ) − γ ϕ ( ρ + a ) sin ϑ (cid:98) P ϕ (cid:21) (2.22) – 6 –article the impact parameter is given by b = K/E . We can distinguish three distinctscenari depending on the value of b (see figure 1): • Scattering processes: They occur where either the geodesics encounter a turning point ρ ∗ >
0, i.e. a single zero of (cid:98) P ρ ( ρ ) or when (cid:98) P ρ ( ρ ) is positive everywhere and the timeto reach ρ = 0 is finite. This includes all geodesics on black hole geometries withlarge enough impact parameter and generic geodesics in fuzzball geometries. • Critical falling: They occur when geodesics encounter a critical point ρ ∗ defined as adouble zero of (cid:98) P ρ ( ρ ). In this case, the time to reach ρ ∗ is infinite and the particleasymptotically approaches ρ ∗ without ever reaching it. This class of geodesics existsfor specific choices of the impact parameter, both for black holes and fuzzballs. • Absorption processes: They occur for black hole geometries when geodesics find noturning point before the black hole horizon. In this case (cid:98) P ρ ( ρ ) is positive everywhereand the time to reach the horizon is infinite. In this section we consider massless geodesics in the 3-charge five-dimensional black holegeometry with and without angular momenta.
The non-rotating 3-charge black hole metric is obtained by taking a = n = 0 in (2.1) and(2.3). The Z -functions and one-forms reduce to Z = 1 + L ρ , Z = 1 + L ρ , Z = Z Z γ m dx m = F dv = − L p ρ dv , β m = 0 (3.1)For this choice the oblate radius ρ coincides with the spherical radius r everywhere andthe solution is spherically symmetric. The solution corresponds to a non-rotating five-dimensional black hole with a horizon at ρ = 0 [53, 54]The ‘dressed’ D1-brane charge Q , D5-brane charge Q and Kaluza-Klein momentum Q P are given by Q = L , Q = L , Q P = L p . (3.2)The massless geodesic equation H = 0 can be written in the separable form2 ρ Z H = (cid:2) − ρ Z P u ( P v − F P u ) + ρ P ρ (cid:3) + (cid:34) P ϑ + P ϕ s ϑ + P ψ c ϑ (cid:35) = 0 (3.3)where the two brackets account for ρ and ϑ dependent terms, respectively. The formerequation can be solved by imposing that the combinations inside the brackets be constant,i.e. K = P ϑ + P ϕ s ϑ + P ψ c ϑ = 2 ρ Z P u ( P v − F P u ) − ρ P ρ (3.4)– 7 –he right hand side equation can be solved for P ρ P ρ = − K ρ + 2 P u ( ρ + L )( ρ + L ) ρ (cid:32) P v + L p P u ρ (cid:33) (3.5)We notice that for K < P u L p + 2 P u P v ( L + L ) (3.6)the function P ρ is positive everywhere, so the geodesics extend down to the horizon at ρ = 0. The flight time down to the horizon diverges∆ t ≈ − L L L p (cid:90) ρ dρρ (3.7)as expected for a black hole geometry. The analysis of geodesics in more general black hole backgrounds, extremal or not, with orwithout charges and angular momenta, follows mutatis mutandis the same steps as beforeand the existence of a critical value for the total angular momentum of the incomingparticles can be always displayed. In this section, we illustrate this universal feature byconsidering scattering from a three equal charge supersymmetric black hole with non-trivialangular momentum in five dimensions. The metric of this black hole reads [55] ds S = − (cid:16) − µr (cid:17) (cid:18) dt − µω sin ϑr − µ dϕ − µω cos ϑr − µ dψ (cid:19) ++ (cid:16) − µr (cid:17) − dr + r (cid:0) dϑ + sin ϑ dϕ + cos ϑ dψ (cid:1) (3.8)where µ is the mass parameter and ω accounts for the angular velocity. For concreteness,we focus on geodesics at constant ϑ , let us say ϑ = 0 . Consistently, we set ˙ ϑ = ˙ ϕ = 0, i.e. P ϑ = P ϕ = 0. The corresponding Hamiltonian reduces to H = g mn P m P n = − (cid:16) − µr (cid:17) − E + 12 (cid:16) − µr (cid:17) P + 12 r (cid:18) J − µωEr − µ (cid:19) (3.9)with − E = g tn ˙ x n = − (cid:16) − µr (cid:17) (cid:18) ˙ t − µωr − µ ˙ ψ (cid:19) J = g ψn ˙ x n = µωr − µ (cid:18) ˙ t − µωr − µ ˙ ψ (cid:19) + r ˙ ψP = g rn ˙ x n = (cid:16) − µr (cid:17) − ˙ r . (3.10)The momenta E and J are conserved while P is determined by solving the null condition H = 0 leading to (in the incoming branch) P ( r ) = − r ( r − µ ) (cid:104) E r − (cid:2) J ( r − µ ) − µωE (cid:3) (cid:105) (3.11) The analysis for ϑ = π/ ϕ ↔ ψ – 8 – = = =- - - ω - - - b c Figure 2 : Critical impact parameter b c vs the BH angular velocity ω , both in units of √ µ . For every value of ω we find two different critical parameters, corresponding to theintersections with the solid line.We notice that, if ω < µ , the polynomial inside the brackets is positive for large r andnegative for r = √ µ and therefore it vanishes for some r ∗ > √ µ . For this choice, theparticle either bounces back or gets trapped inside a critical trajectory before it reachesthe horizon at r = √ µ . The trapping behaviour occurs if J = J c such that a point r ∗ existswhere P ( r ∗ ) = P (cid:48) ( r ∗ ) = 0. Parametrising the angular momentum by means of the impactparameter b = J/E , the two equations are solved by taking r ∗ = (cid:12)(cid:12)(cid:12) b c (cid:12)(cid:12)(cid:12) (3.12)with b c a solution of the cubic equation4 b c − µ ( b c + ω ) = 0 (3.13)The solutions are b c = − √ µ sin (cid:32)
13 arctan ω (cid:112) µ − ω + 2 π C (cid:33) , C = − , , C = 0 leads to a zero r ∗ < √ µ inside the horizon, so it should bediscarded. The remaining two roots lead to critical geodesics of the black hole geometry. In this section we consider massless geodesics along 2-charge fuzzball geometries obtainedby setting ε = ε = n = 0 in the three-charge fuzzball solution. The general 2-charge geometry is specified by a profile function (cid:126)F ( v ) with values on R ×T .Here we choose a circular profile (cid:126)F ( v ) in R (cid:126)F ( v ) = a (cid:18) cos 2 πvλ , sin 2 πvλ , , (cid:19) (4.1)– 9 –or which one has Z = 1 + L λ (cid:90) λ (cid:12)(cid:12)(cid:12) ˙ (cid:126)F ( v ) (cid:12)(cid:12)(cid:12) dv (cid:12)(cid:12)(cid:12) (cid:126)X − ˙ (cid:126)F ( v ) (cid:12)(cid:12)(cid:12) = 1 + L ρ + a c ϑ Z = 1 + L λ (cid:90) λ dv (cid:12)(cid:12)(cid:12) (cid:126)X − ˙ (cid:126)F ( v ) (cid:12)(cid:12)(cid:12) = 1 + L ρ + a c ϑ (4.2)and Z = Z Z . Moreover the 1-forms β and γ are given by [56] β = β m dx m = a Rρ + a c ϑ (cid:0) s ϑ dϕ − c ϑ dψ (cid:1) ,γ = γ m dx m = a Rρ + a c ϑ (cid:0) s ϑ dϕ + c ϑ dψ (cid:1) (4.3)with R = R y / √ R y being the radius of S along the y -direction. The geometry has nohorizon for a = L L R (4.4) The Hamiltonian depends only on ϑ and ρ , so the momenta P u , P v , P ψ and P ϕ are allconserved. The Hamiltonian can be separated [57–60] according to2 Z ( ρ + a c ϑ ) H = λ ρ ( ρ, P ρ ) + λ ϑ ( ϑ, P ϑ ) (4.5)with λ ϑ ( ϑ, P ϑ ) = P ϑ + P ψ cos ϑ + P ϑ sin ϑ + 2 a sin ϑ P u P v (4.6) λ ρ ( ρ, P ρ ) = ( ρ + a ) P ρ + a (cid:101) P ψ ρ − a (cid:101) P ϕ ρ + a − ρ + a + L + L ) P u P v (4.7)and (cid:101) P ψ = P ψ + R ( P v − P u ) , (cid:101) P ϕ = P ϕ + R ( P v + P u ) (4.8)Equation H = 0 can be solved by taking λ ϑ = − λ ρ = K (4.9)with K a constant, that can be interpreted as the total angular momentum. Equivalentlyone has P ϑ ( ϑ ) = K − P ψ c ϑ − P ϕ s ϑ − P u P v a s ϑ P ρ ( ρ ) = − a (cid:101) P ψ ρ ( ρ + a ) + a (cid:101) P ϕ ( ρ + a ) + 2 (cid:0) ρ + L + L + a (cid:1) P u P v − K ρ + a (4.10)– 10 –xpressing the velocities in terms of the momenta˙ ϑ = P ϑ ( ϑ ) Z ( ρ + a c ϑ ) , ˙ ρ = ρ + a ρ + a c ϑ P ρ ( ρ ) Z one finds the separable geodesic equation dϑP ϑ ( ϑ ) = dρP ρ ( ρ )( ρ + a ) (4.11)that implicitly determines ϑ ( ρ ) in terms of elliptic integrals. Finally, ϕ ( ρ ) and ψ ( ρ ) followfrom dψ = ρ P ψ + a c ϑ (cid:101) P ψ P ρ ( ρ ) ρ ( ρ + a ) c ϑ dρ , dϕ = ( ρ + a ) P ϕ − a s ϑ (cid:101) P ϕ P ρ ( ρ ) ρ ( ρ + a ) s ϑ dρ (4.12)after integration over ρ . It is convenient to write P ρ ( ρ ) = P ( ρ ) ρ ( ρ + a ) (4.13)and set ρ = x so that P ( x ) = A x + B x + C x + D (4.14)with A = 2 P u P v B = 2 P u P v (2 a + L + L ) − K C = a (cid:104) (cid:101) P ϕ − (cid:101) P ψ + 2 P u P v ( a + L + L ) − K (cid:105) D = − a (cid:101) P ψ (4.15)Since A >
D <
0, the polynomial P ( x ) is positive for large x and negative forsmall x . Therefore it has at least a zero x ∗ (the largest one) for positive x = ρ . Thisis in contrast with the behaviour observed for the black hole geometry, where P ρ ( ρ ) wasshown to be positive everywhere for small enough angular momenta K . We conclude thatmassless probes in the fuzzball metric escape from the gravitational background, even forlow values of the angular momentum K . An exception occurs when the angular momentumis tuned such that x ∗ is a double zero of P ( x ) , i.e. P ( x ∗ ) = P (cid:48) ( x ∗ ) = 0 (4.16)For this choice, the integral (2.23) diverges and the surface ρ ∗ = √ x ∗ looks like a horizonfor the massless geodesics. Indeed, for a critical value of K such that the two largest rootsof P ( x ) collide, the particle winds around the target forever, asymptotically approachingthe ‘circular’ orbit with radius ρ ∗ . Such geodesics will be referred to as critical geodesics . Inthe remaining of this section we will display some explicit choices of kinematics exhibitingsuch trapping behaviour. – 11 –irst, we notice that the conditions A >
D <
0, together with the requirementthat the largest root is double and positive, imply that all three roots are positive and
A, C > , B, D < x ∗ and D one finds x ∗ = 13 A (cid:16) − B + (cid:112) B − AC (cid:17) D = 227 A ( B − AC ) / − B A (2 B − AC ) (4.18)Solutions compatible with (4.17) exist if4 AC ≥ B ≥ AC (4.19)The two extreme cases where the inequalities are saturated are easy to solve in analyticform: • Case I: B = 3 AC . For this choice all three roots collide and D = BC A . From (4.15)one finds (cid:101) P ϕ = (cid:2) K + 2( a − L − L ) P u P v (cid:3) a P u P v (cid:101) P ψ = (cid:2) K − a + L + L ) P u P v (cid:3) a P u P v (4.20)and ρ ∗ = K P u P v − (2 a + L + L ) > K . • Case II: B = 4 AC . For this choice one finds D = 0, (cid:101) P ψ = 0 (cid:101) P ϕ = (cid:2) K − P u P v ( L + L ) (cid:3) a P u P v (4.22)and ρ ∗ = K P u P v − (2 a + L + L ) > To illustrate the trapping behaviour of fuzzballs, let us consider the critical geodesics alongthe plane ϑ = π/
2, for the choice L = L = a , P u = P v , P ψ = 0 (4.24)– 12 – c / a2 4 6 8 10 b / a1234 ( Δ t - Δ t Free )/ a Figure 3 : Time delay between massless particles moving in a 2-charge fuzzball geometryand flat space-time as a function of the adimensionalised impact parameter b/a .For this choice the velocity ˙ y of the particle along the compact circle can be set to zeroalong the full trajectory. The critical geodesics fall into case II above. Introducing theimpact parameter b = P ϕ E = P ϕ √ P u (4.25)and using (5.35), (4.15), (4.10) one finds P ( ρ ) = 2 P u ρ (cid:2) ρ + (3 a − b ) ρ + (3 a − b ) a (cid:3) (4.26)with largest zero ρ ∗ = b − a + (cid:112) ( b − a ) ( b + 3 a )2 (4.27)The turning point exists for b ≤ − a or b ≥ a/
2; when b = 3 a/ b = − a a limit cycleexists at ρ = 0 and ρ = √ a respectively. For values of b in-between P ρ has no zeroes,the probe reaches ρ = 0 in a finite, possibly large, amount of time, surpasses it and getsscattered back at infinity. The time to reach ρ ∗ is given by∆ t = (cid:90) ρ ∗ ρ dρ ρ + 3 a ρ + (3 a − b ) a ρ + a √ P u ρ (cid:112) P ( ρ ) (4.28)In (Fig. 3) we display the difference between the total flight time in the fuzzball geom-etry and in flat space-time as a function of b for a fixed large ρ . As expected, the closer aparticle’s impact parameter approaches the critical one, the longer the time it will spendorbiting around the fuzzball. It is also clear that even though for b < b c the particle willeventually be scattered, it spends a considerable amount of time in the proximity of thefuzzball. In this section we consider scattering on 3-charge fuzzball geometries.– 13 – .1 The Hamiltonian and momenta
Momenta and velocities in the 3-charge geometry are related by P u = − ( ˙ v + β m ˙ x m ) (cid:98) P v = − ( ˙ u + γ m ˙ x m ) (cid:98) P ρ = Z ( ρ + a c ϑ ) ρ + a ˙ ρ (cid:98) P ϑ = Z ( ρ + a c ϑ ) ˙ ϑ (cid:98) P ψ = Z ρ c ϑ ˙ ψ (cid:98) P ϕ = Z ( ρ + a ) s ϑ ˙ ϕ (5.1)The important difference with respect to the 2-charge case is that now β m , γ m and Z , andtherefore the Hamiltonian, explicitly depend on the combination φ = ϕ + nvR and therefore P v and P ϕ are no longer conserved separately but only their combination P ν = P v − nR P ϕ is. Indeed, the equations of motion become˙ P u = ˙ P ν = ˙ P ψ = H = 0˙ P ϑ = − ∂ H ∂ϑ ˙ P ϕ = − ∂ H ∂ϕ = − Rn ∂ H ∂v = Rn ˙ P v (5.2)We observe that the Hamiltonian H is a rational function of cos ϑ and therefore ∂ H ∂ϑ ∼ cos ϑ sin ϑ (5.3)This implies that P ϑ is conserved for ϑ = 0 , π/
2. Moreover at ϑ = 0 , π/ (cid:98) P ϑ = P ϑ and therefore constant P ϑ implies constant ˙ ϑ . We conclude that geodesics starting at ϑ = 0 , π/ ϑ velocity, ˙ ϑ = 0 keep ϑ constant along the whole trajectory.In the following we restrict ourselves on geodesics along these two planes. ϑ = 0 geodesics Let us start by choosing n = 1 and considering the geodesics in the plane ϑ = 0, orthogonalto the circular profile. The functions and forms defining the metric assume the followingexpression Z = 0 β = − a Rρ + a dψγ = a Rρ + a (1 − F ) dψ + F dv F = − ε ρ + a ) Z = Z Z = (cid:18) L ρ + a (cid:19) (cid:18) L ρ + a (cid:19) (5.4)Taking (cid:98) P ϑ = P ϑ = 0 and P ϕ = (cid:98) P ϕ = 0, the Hamiltonian becomes H = − P u (cid:98) P v + 12 Z (cid:32) (cid:98) P ρ + (cid:98) P ψ ρ (cid:33) (5.5)– 14 –ith (cid:98) P v = P v + ε ρ + a ) P u , (cid:98) P ρ = P ρ (cid:98) P ψ = P ψ − a Rρ + a ( P u − P v ) . Recall that P u , P v , P ψ are conserved quantities. Plugging this into (2.21) one finds P ρ = ± (cid:34) Z P u (cid:98) P v − (cid:98) P ψ ρ (cid:35) = ± P ( ρ ) ρ ( ρ + a ) (5.6)with, setting ρ = x as above, P ( x ) = P u x ( x + a + L )( x + a + L ) (cid:2) P v ( x + a ) + ε P u (cid:3) − ( x + a ) (cid:2) P ψ ( x + a ) − a R ( P u − P v ) (cid:3) (5.7)We notice that the polynomial P ( x ) is positive for x → ∞ and negative for x → x axis. Again we denote x ∗ the largestpositive zero. If x ∗ is simple then it is a turning point and the particle gets deflected inthe gravitational background. On the other hand for a critical choice of P ψ for which x ∗ is a double zero the particle gets trapped in the gravitational background, asymptoticallyapproaching ρ ∗ = √ x ∗ .As an illustration of this critical behavior, let us consider a particle with no internalKaluza-Klein momentum P v = P u and L = L = ε / L ≥ a . (5.8)For this choice the polynomial P ( x ) takes the simple form P ( x ) = 2 P u x ( x + a + L ) − ( x + a ) P ψ (5.9)Solving the critical conditions P ( x ) = P (cid:48) ( x ) = 0 one finds a double zero at x ∗ = L − a + L (cid:112) L − a (5.10)for the critical choice of angular momentum P ψ = √ P u L (cid:34) L a − L a (cid:18) − a L (cid:19) / (cid:35) (5.11)In other words, scattering massless particles off the fuzzball geometry, one finds that thecomponents with P ψ satisfying (5.11) are missing in the out-going spectrum, and thefuzzball geometry behaves effectively as a black object for the selected “channel”.– 15 – .3 ϑ = π/ geodesics In this plane, the Hamiltonian, explicitly depends on the combination φ = ϕ + nvR , soit is convenient to introduce the canonically related variables φ , ν (and their conjugatemomenta) ϕ = φ − nvR , P ϕ = P φ v = ν , P v = P ν + nR P φ (5.12)In terms of these variables the equations of motion become˙ P u = ˙ P ν = ˙ P ψ = H = 0˙ P ϑ = − ∂ H ∂ϑ ˙ P φ = − ∂ H ∂φ (5.13)For motion in the plane of the string profile, the metric is given by (2.1) and (2.3) with Z = 1 + L ρ + ε R ∆ n cos 2 φL ρ Z = 1 + L ρ Z = 2 ε R ∆ n cos φρ Z = Z Z − Z β ϕ = a Rρ β ψ = 0 γ ρ = − ε R ρL ∆ n sin 2 φ γ ψ = γ ϑ = 0 γ v = F n (5.14) γ ϕ = a R (1 − F n ) ρ − ε R L ρ ∆ n cos 2 φ ( ρ + a )Taking (cid:98) P ϑ = P ϑ = 0, (cid:98) P ψ = P ψ = 0, the Hamiltonian reads H = − P u (cid:98) P v + ( ρ + a ) (cid:98) P ρ Z ρ + (cid:98) P ϕ Z ( ρ + a ) (5.15)where the hatted conjugate momenta have the form (cid:98) P v = P ν + nR P φ + F n P u (cid:98) P ρ = P ρ + ε RP u ∆ n sin 2 φ ρL (5.16) (cid:98) P ϕ = P φ − a ρ ( n P φ + RP ν + RP u ) + 2 a R P u ρ (cid:20) F n + ε ∆ n ( ρ + a ) cos 2 φ a L (cid:21) with P u and P ν conserved quantities. – 16 –et us focus on the truly dynamical variables ρ and φ . Their velocities are given by ˙ ρ = ρ + a Z ρ (cid:98) P ρ ˙ φ = (cid:98) P ϕ ( ρ − na ) Z ρ ( ρ + a ) − nP u R (5.18)Choosing φ as independent variable, the equations of motion can be written in the form dρdφ = (cid:98) P ρ R ( ρ + a ) (cid:98) P ϕ R ( ρ − na ) − P u Z ρ ( ρ + a ) dP φ dφ = − φ ∂∂φ (cid:34) ( ρ + a ) (cid:98) P ρ Z ρ + (cid:98) P ϕ Z ( ρ + a ) (cid:35) (5.19)and (cid:98) P ρ = ρ ( ρ + a ) (cid:104) Z P u (cid:98) P v ( ρ + a ) − (cid:98) P ϕ (cid:105) (5.20)We are interested in solutions of the geodesic equations (5.19) characterised by trajectoriestrapped in the gravitational background. As before, we expect that for specific values ofthe incoming angular momentum P φ , there exists geodesics ending on trapping trajectoriesbut now both the asymptotic trajectory and the angular momentum will in general varywith φ . Due to the complexity of the three-charge problem along the ϑ = π/ ρ = 0,˙ φ = w . For concreteness we take L = L = L = a , (5.21)According to (5.18), a constant angular velocity can be found by taking ρ = na ⇒ ˙ φ = − nP u R (5.22)while ˙ ρ = 0 requires (cid:98) P ρ = 0 (5.23) The evolution of ν , as well as of the other coordinates, follows from the one of ρ and φ . In particular˙ ν = − P u − a RZ ρ ( ρ + a ) (cid:98) P ϕ (5.17) We choose L = L = a only for illustrative purposes of the general case where the three quantities areof the same order. We notice that this symmetric choice is far different from the standard choice where a is taken much smaller than the D1 and D5 charges, i.e. a << L , . – 17 –r equivalently 2 Z P u (cid:98) P v ( ρ + a ) − (cid:98) P ϕ = 0 (5.24)We notice that at the critical point ρ = na , Z is constant and (cid:98) P ϕ reduces to (cid:98) P ϕ = Rn (cid:20) F n P u − P u − P ν + ε ( n + 1)2 a P u ∆ n cos 2 φ (cid:21) (5.25)Equation (5.24) can therefore be easily solved for P φ P φ = Rn (cid:34) (cid:98) P ϕ P u Z a ( n + 1) − P ν − F n P u (cid:35) (5.26)The two equations of motion (5.19) are satisfied for ρ = √ n a and P φ given by (5.26),quite remarkably this provides an exact solution for the non separable system. It would beinteresting to find a solution interpolating between infinity and these closed trajectories. Finally, we consider massless geodesics in the near horizon geometry. As shown in [61],massless geodesics in this region are described by a separable dynamics that can be in-tegrated in an analytic form. The crucial difference with the case of asymptotically flatsolutions is that in the near the horizon, φ -oscillating terms are missing leading to solutionscarrying no v-dependence. Here we display some simple examples of trapped geodesics inthis region. The geodesics in this region can be viewed as the continuation of trajectoriesstarting from infinity with initial conditions chosen such that no return or critical pointsare found before the particle reaches distances much smaller than L and L .The near horizon geometry is defined by taking L , >> ρ + a (5.27)For this choice important simplifications take place. First, the regularity conditions (5.28)reduce to ε = ε , L L R = 2 a + ε (5.28)with L , L , R taken to be large with fixed ratio L L /R .The functions entering in the six-dimensional metric reduce to Z = ∆ n s ϑ (cid:0) a R − L L (cid:1) + L L (cid:0) a c ϑ + ρ (cid:1) β = a Rρ + a c ϑ (cid:2) s ϑ dϕ − c ϑ dψ (cid:3) γ = a R (1 − F n ) ρ + a c ϑ (cid:2) s ϑ dϕ + c ϑ dψ (cid:3) + F n dv (5.29)with F n = − ε a (cid:20) − (cid:18) ρ ρ + a (cid:19) n (cid:21) ∆ n = a ρ + a (cid:18) ρ ρ + a (cid:19) n (5.30)– 18 –he Hamiltonian depends only on ϑ and ρ , so the momenta P u , P v , P ψ and P ϕ are allconserved. The Hamiltonian can be separated according to2 Z ( ρ + a c ϑ ) H = λ ρ ( ρ, P ρ ) + λ ϑ ( ϑ, P ϑ ) (5.31)with λ ϑ ( ϑ, P ϑ ) = P ϑ + P ψ cos ϑ + P ϕ sin ϑ (5.32) λ ρ ( ρ, P ρ ) = (cid:0) a + ρ (cid:1) P ρ + 2 R P u (cid:0) a F n + (cid:15) (cid:1) ( F n P u − P v ) a + ρ + a ( P ψ + RP v − RP u ) ρ − a ( P ϕ + RP v + RP u − F n RP u ) a + ρ (5.33)The equation H = 0 can be solved by taking λ ϑ = − λ ρ = K (5.34)with K a constant, that can be interpreted as the total angular momentum. Solving thesecond equation for P ρ ( ρ ) one finds P ρ ( ρ ) = P n +1 ( ρ ) ρ ( ρ + a ) (5.35)with P n +1 ( x ) a polynomial of order 2 n + 1. Turning points are associated to zeros of thepolynomial P n +1 ( x ) and critical geodesics to choices of angular momenta such that thetwo largest zeros of P n +1 ( x ) collide.For the sake of simplicity we will discuss only the n = 1 null geodesics, the order 3polynomial reduces to P ( x ) = Ax + Bx + Cx + D (5.36)where the list of coefficients reads A = − K B = a (cid:20) ε P u P v R a + [ P ϕ + R ( P u + P v )] − [ P ψ − R ( P u − P v )] − K (cid:21) C = a (cid:34)(cid:18) ε P u Ra + [ P ϕ + R ( P u + P v )] (cid:19) − P ψ − R ( P u − P v ))] − K (cid:35) D = − a [ P ψ − ( P u − P v ) R ] In order to illustrate the behaviour of the geodesics in this context, as before we choosethe conserved quantities such that D = 0, i.e. P ψ = R ( P u − P v ). A further simplificationoccurs by choosing P ϕ = − R ( P u + P v ) and P v = P u , leading to P ( x ) = − x (cid:2) K x + 2 (cid:0) a K − ε P u R (cid:1) x + (cid:0) a K − ε P u R (cid:1)(cid:3) (5.37)by requiring two coincident roots one finds the relations ρ crit = (cid:113) a − ε , K = ε P u R a − ε (5.38)This shows that critical geodesics exist if a > ε i.e. aR > L L / √ Conclusions and outlook
Relying on a class of micro-state geometries for 3-charge systems in D = 5 constructed in[39], we have further tested the fuzzball proposal by studying massless geodesics in thesebackgrounds. In particular we have shown that 2- and 3-charge fuzzball geometries tendto trap massless neutral particles for a specific choice of their impact parameter. This isat variant with classical BH’s that trap all particles impinging with an impact parameterbelow a certain critical value of the order of the horizon radius. This suggests that theblackness property of black holes arises as a collective effect whereby each micro-stateabsorbs a specific channel.The analysis has been performed in various steps. First we have reviewed the generalform of the metric and written down the geodesic equations for massless neutral probes inboth the Lagrangian and Hamiltonian forms. Then we focused on the cases of (singular)non-rotating BPS black-holes with 3-charge, on micro-states for 2-charge systems with acircular profile and finally on the 3-charge case.We have (implicitly) integrated the geodesic equations for the 2-charge case for genericinitial values of the angle ϑ and of the integration constant K (playing the role of totalangular momentum), thus generalising our previous results for ϑ = 0 (plane orthogonal tothe circular profile) and ϑ = π/ ϑ = 0 (since they leadto separable equations of the same form as in the 2-charge case, previously analysed) andwritten down the equations for ϑ = π/
2, that lead to a non-separable system. A simplesolution of this intricate system has been found.We also considered massless geodesics on asymptotically AdS 3-charge geometries ofthe type studied in [61] . These geometries, unlike their extension to asymptotically flatspace, are characterized by a separable dynamics and massless geodesics can therefore beintegrated in an analytic form. We presented explicit examples of trapped geodesics thatcan be viewed as the end points of the trajectories of massless particles infalling frominfinity without encountering turning or critical points before reaching distances muchsmaller than L and L .In this paper we restricted our attention to the study of scattering of classical point-likemassless neutral probes. It would be interesting to extend this analysis to more generalprobes like massive, possibly charged, particles, waves and strings where tidal effects suchas those studied in [62] can be relevant.Other classes of smooth (non-supersymmetric) geometries (such as JMaRT [52]) leadto interesting effects [4] due to the presence of an ergo-region of finite extent withouthorizons or singularities. In [63], the authors studied the properties of geodesics in theclosely related setup of five and six dimensional supersymmetric fuzzball geometries. Inparticular they used the presence of stably trapped geodesics to argue for the existence ofa non-linear instability even for these BPS microstate geometries. These trapped geodesicsmay be related to the circular orbits considered in section 5.3.1 of the present paper. It We thank the referee for drawing our attention on this work. – 20 –ould be interesting to study linear perturbations and (quasi-)normal modes that maysignal a potential instability of the microstate solutions.Finally, the analysis in [64] has some overlap with section 3 of the present paper, wherefor completeness and comparison with the original results of our analysis we discussed nullgeodesics in rotating and non-rotating singular black-holes in five dimensions.
Acknowledgements
We acknowledge fruitful discussions with Andrea Addazi, Pascal Anastasopoulos, Guil-laume Bossard, Ramy Brustein, Paolo Di Vecchia, Maurizio Firrotta, Francesco Fucito,Stefano Giusto, Elias Kiritsis, Antonino Marcian`o, Lorenzo Pieri, Gabriele Rizzo, RodolfoRusso, Raffaele Savelli, Masaki Shigemori, Gabriele Veneziano, and Natale Zinnato. Partof the work was carried on while D. C. was visiting Erwin Schr¨odingier Institute in Vi-enna and while and M. B. and J. F. M. were visiting Galileo Galilei Institute in Florence.We would like to thank both Institutes for the kind hospitality. We thank the MIUR-PRIN contract 2015MP2CX4002 “Non-perturbative aspects of gauge theories and strings” for partial support. – 21 – eferences [1]
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