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Rotating Black Holes @Colliders Seong Chan Park Korea Institute for Advanced Study (KIAS) 6th RESCEU International Symposium: “Frontiers in Astroparticle Physics and Cosmology” Nov. 6 , 2003 S. Park ,H. S. Song J. Korea Phys. Soc. 43: 30-33 (2003) D. Ida, K. Oda, S. Park Phys. Rev. D67:064025 (2003) Q. Does higher energy lead us to smaller distance? Uncertainty relation: to see smaller distance, we need higher energy. We cannot see the smaller distance than 1/Mp even in trans-Planckian domain : Black Hole production (t’ Hooft 1982) Seong Chan Park, 6th RESCEU, 2003 2 Distance VS Energy L CG We don’t know much QG about this area: 1/Mp String/M ??!? E Mp Seong Chan Park, 6th RESCEU, 2003 3 Review: One minute for Black Hole **Classical: Strong Gravity -nothing goes out (Ve>c) **Quantum: Hawking Radiation -thermal radiation: T=Surface Gravity -Anything goes out! Test? No, Not yet…. :only astronomical indirect observations have been found. Seong Chan Park, 6th RESCEU, 2003 4 Planck Energy **Planck Energy: Energy scale where gravity becomes strong. Well measured only for: r>0.2 mm (Hoyle et. al. PRL03’) Enormous extrapolation assumed!! : from 1mm to 10-34 cm No example such a big extrapolation works in history. Seong Chan Park, 6th RESCEU, 2003 5 New understanding of Planck energy: TeV gravity scenarios **Brane world (1996-) Talks by Langlois, Maartens, Tanaka, Soda, Khoury, Frolov, Ichiki in this symposium Horawa-Witten(1996) ADD(1998) RS(1999) : What did we learn? Newton’s constant is just a derived quantity which is depending on geometry of higher dimensional spacetime. Mp*=TeV possible! Seong Chan Park, 6th RESCEU, 2003 6 Black Holes at Collider? Any colliders with ECM>> MPl have chances to produce BHs. In TeV gravity models, we have Big chances in near future experiments. LHC Ecm=14 TeV, VLHC Ecm=100 TeV or SLHC.. UHECR E>> TeV Such BHs are typically Rh << RCompact Higher Dimensional Rh >> RCompton Semi-Classical Seong Chan Park, 6th RESCEU, 2003 7 Fate of BHs at collider 1. Balding Phase • Dynamical production phase • BH loses its “hair”. 2. Spin Down Phase • BH loses its mass and angular momentum. 3. Schwartzschild Phase • Angular momentum is small. • BH loses its mass. 4. Planck Phase BH radiates mainly • Truly QG, highly unpredictable on the brane. • A few quanta would be emitted. Temperature gets higher and higher. Seong Chan Park, 6th RESCEU, 2003 8 Rotating BH formation parton M/2 Rh b M/2 parton •Condition for BH production (Hoop Conjecture) : BH forms if and only if the energy is compacted into a region whose impact parameter is less than the horizon size of the BH. Rh(M,J) > b/2 •BH must be produced with finite Angular Momentum. : Angular momentum conservation J= M b/2 Don’t forget : we need higher dimensional generalization of Kerr Black Hole Solution! Seong Chan Park, 6th RESCEU, 2003 9 Metric for rotating black hole: D=4+n Myers-Perry 1986 ds 2 = g (4) (r, ϑ ) + r 2 cos 2 ϑ dΩ2 n vanishes on the brane g (4 ) (r, ϑ ) = ∆ − a 2 sin 2 ϑ (∆ − r 2 − a 2 )asin 2 ϑ − 0 0 Σ Σ [(r 2 + a 2 )2 − ∆a 2 sin 2 ϑ ]sin 2 ϑ * 0 0 Σ ∆ 0 0 0 Σ 0 0 0 Σ Σ = r 2 + a 2 cos ϑ 2 µ a2 Seong Chan Park, 6th RESCEU, ∆ = r 1− n+1 + 2 2003 r r 10 ::continued:: Rotating BHs at colliders When we neglect balding phase, phase There is maximum b allowed for b < 2rh ( M ,J ) BH formation. gives condition for −1 BH formation. n + 2 2 n+1 bmax = 21+ rS (M ) rS ( M ) 2 rh = (1+ a* )1/(n+1) 2 2.9 (n = 1) rescaled angular momentum 4.5 (n = 2) a= (n+2)J/2M J max ~ for M / M p = 5 M 11.5 (n = 7) (typical LHC energy) How good is this estimation? Seong Chan Park, 6th RESCEU, 2003 11 Our formula nicely fits numerical result with full GR n 1 2 3 4 5 6 7 RNY 1.056 1.158 1.228 1.276 1.314 1.344 1.368 Rours 1.110 1.170 1.218 1.262 1.300 1.334 1.364 A few % for n =1 and ~1% for n >1. bmax R(n) = rS cf) Numerical result utilizes the Aichelburg-Sexl solution b (Eardley, Giddings 02) Yoshino, Nambu 02 Setup: two particles (SBHs) with • boost , t • mass 0, z • energy: fixed. Closed trapped surface forms There are two direct when b < bmax. consequences. 12 1. Form factor becomes larger in higher dimensions. n 1 2 3 4 5 6 7 FNY 1.084 1.341 1.515 1.642 1.741 1.819 1.883 Four 1.231 1.368 1.486 1.592 1.690 1.780 1.863 σ n + 2 −2 F= 2 2 n+1 πRS σ = πbmax = 41+ 2 πr 2 2 S This growing pattern can be understood when one take angular momentum into account. Seong Chan Park, 6th RESCEU, 2003 13 2. Most BHs are produced with large angular momenta Initial angular momentum: J = bM / 2 dσ 8 πJ / M 2 (J < J max ) = db dJ 0 (J > J max ) dσ = 2πbdb (J max = bmax M /2) Differential cross section increases linearly with J. Seong Chan Park, 6th RESCEU, 2003 14 Production of Black Ring Topological no-go Q. D=5 BR production? theorem in D=4. A. Not so probable. horizon = S2 See the maximum J in D=5: Simple generalization S2+n in D=4+n(??) Black Ring solution in D=5 (Emparan and Reall PRL 2002) horizon = S2 * S1 theorem broken in D>4!! Q. How about D>5 ? We do not have the explicit Gravitational attraction VS BR sol. In D>5, yet!! Centrifugal force Naïve estimation about the minimum value of angular momentum might be helpful. ::SEE NEXT PAGE:: 15 ::Continue from previous page:: Combine above relations gives the minimum angular momentum for rotating Black D=4+n, Black Ring Ring: Horizon = Compare with Maximum value of Angular Momentum: Reduced Schwarzshild D 6 7 8 9 10 11 radius Min/ 0.43 0.31 0.20 0.11 0.06 0.03 Max This means that there L>R could be large Centrifugal force probability to produce > Gravitational attraction BRs in d>5. 16 Hawking Radiations from rotating BH BH radiates mainly into the brane fields via Hawking radiation. radiation Greybody factors for Brane fields determine the evolution of BH. d M 1 Γs,l,m ω − = ∑ gs ∫ dω eω −mΩ m 1m dt J 2 π s ,l,m Greybody factor= absorption cross-section of Black Hole. can be obtained by solving the brane field equations. equations actually g.o. Conventionally people used g.o. approximation: Γ ∝ω 2 approx. is not so nice. Seong Chan Park, 6th RESCEU, 2003 17 Derivation of master equation: Newman-Penrose formalism We set null tetrad as follows: Σ = r 2 + a 2 cos ϑ ϕ Σ r nµ = δµ − asin 2ϑδµ − δµ t ∆ µ a2 ∆ t 1 r ∆ = r 2 1− n+1 + 2 n'µ = 2Σ (δµ − asin 2 ϑδµ )+ δµ ϕ 2 r r i sin ϑ r − ia cos ϑ ϑ mµ = 2(r + ia cos ϑ ) [aδµ − (r 2 + a 2 )δµ ]− t ϕ 2 δµ m'µ = mµ (4 )µν ∇ µ∂ν φ = 0 Scalar: g Spinor: Vector: 18 Master equation for Brane fields −iω t + imϕ Decomposition: φ, ψ0 , φ0 ~ R(r)S(ϑ )e Σ = r 2 + a 2 cos ϑ angular part: 2 µ a2 ∆ = r 1− n+1 + 2 1 d dS r r sin ϑ sin ϑ dϑ dϑ + [(s − aω cos ϑ )2 − (s cot ϑ + m csc ϑ )2 − s(s −1) + A]S = 0 radial part: −sd s +1 dR as 4-dim. Can be treated in a standard manner. Same K = (r 2 + a 2 )ω − ma ∆ ∆ dr dr K 2 ∆,r K + + s 4iωr − i + (∆,rr − 2)+ 2maω − (aω ) − A R = 0 2 ∆ ∆ D=4 case was done by S. A. Teukolsky 1972-1973. Separable for any spin and in any dimensions!! Seong Chan Park, 6th RESCEU, 2003 19 Solving the wave equation D=5 rotating Black Hole D=4 case was done by D. N. Page in 1976 NH limit: FF limit: Matching here! Overlapping region Seong Chan Park, 6th RESCEU, 2003 20 Greybody factors for Rotating Black Hole (D=5) Greybody factors in low frequency expansion. 21 Power Spectrums S=0 dE rh dtdω S=1/2 rhω S=1 22 Angular dependence of emission: a=1.5 S=0 dE rh dt dω d cos ϑ S=1 S=1/2 Seong Chan Park, 6th RESCEU, 2003 23 Summary Lessons from New Paradigm of high energy scattering: High energy does not leads us to smaller distance. Possible TeV gravity scenarios in Brane world. We have shown that Production of rotating Black Holes. Black holes are mostly produced with large angular momentum. Form factor of BH production becomes larger in higher dimensions. Production of Black Rings. Promising in D>5.(need for explicit solution) Evaporation of rotating BH on the brane. We derived the master equation for rotating Black Holes in D>4. Analytic expression of greybody factors for D=5 rotating black hole. Decay signals are significantly affected by the angular momentum of BHs. We are working on to find out greybody factors for D>5 (numerically), to find out greybody factors for (bulk) graviton emission. Seong Chan Park, 6th RESCEU, 2003 24

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