High-Energy, , Cosmology and Cosmology and Strings Strings High-Energy (IHP, Paris, 11-15 December December 2006) 2006) (IHP, Paris, 11-15 Transplanckian String Collisions String Collisions Transplanckian and the information paradox and the information paradox Gabriele Gabriele Veneziano Veneziano (CERN-PH/TH & Collège de France) (CERN-PH/TH & Collège de France)
Has string theory solved the Has string theory solved the information paradox? information paradox? •BH-entropy and counting BH-entropy and counting of states of states agree agree for for extremal BHs extremal BHs • (Strominger-Vafa Strominger-Vafa, ..) , ..) ( •Spectra from quasi-extremal Spectra from quasi-extremal BH BH decay follow Hawking decay follow Hawking iff iff • one traces over over initial initial brane brane configuration (= configuration (= density matrix density matrix) ) one traces Questions (see see e.g. D. Amati, hep-th/0612061): e.g. D. Amati, hep-th/0612061): Questions ( 1. What happens if one if one starts from starts from a pure state? a pure state? Fails at Fails at 1. What happens weak coupling, , may may work at strong coupling work at strong coupling. . weak coupling 2. Are Are there there corrections to a pure thermal corrections to a pure thermal spectrum spectrum? ? 2. 3. How How does this extend does this extend to more to more conventional conventional (Kerr) (Kerr) BHs BHs? ? 3.
Outline Outline The string-black hole correspondence curve The string-black hole correspondence curve 1. 1. Transplanckian string collisions: why and how. string collisions: why and how. Transplanckian 2. 2. 2.1 MGO 2.1 MGO vs vs ACV approach to the problem ACV approach to the problem 2.2 Three scales/regimes in trans-planckian trans-planckian string collisions string collisions 2.2 Three scales/regimes in I) b > R, I) b > R, l l s Easy Easy s II) R > b, II) R > b, l l s Hard Hard s III) III) l l s s > R, > R, b b Easy again? Easy again? 2.3 Approaching gravitational collapse from region III 2.3 Approaching gravitational collapse from region III 2.4 A unitary S-matrix with precocious 2.4 A unitary S-matrix with precocious black-hole-like black-hole-like behaviour behaviour Conclusions Conclusions 3. 3.
The string-black hole The string-black hole correspondence curve correspondence curve
String vs Black-Hole entropy String vs Black-Hole entropy h = c = numerical factors =1 h = c = numerical factors =1 M s , l l s = string mass, length scales M s , s = string mass, length scales Tree-level string entropy Tree-level string entropy ( FV, BM ( ) Counting states ( 70) ) FV, BM (‘ ‘69), HW ( 69), HW (‘ ‘70) Counting states S st = M/M s = L/l L/l s S st = M/M s = s = No. of string bits in the total string length = No. of string bits in the total string length NB: no coupling, no G appears! NB: no coupling, no G appears!
Black-Hole entropy Black-Hole entropy S BH = M R S = (R S /L P ) 2 ~ M 2 S BH = M R S = (R S /L P ) 2 ~ M 2 (GM = R S , 1/T BH = dS/dM dS/dM = R = R S /h) (GM = R S , 1/T BH = S /h) to be contrasted with previous to be contrasted with previous S st = M/M s = L/l s S st = M/M s = L/l s *************************** *************************** S st /S BH > 1 @ small M, S st /S BH < 1 @ large M S st /S BH > 1 @ small M, S st /S BH < 1 @ large M Where do the two entropies meet? Obviously at Where do the two entropies meet? Obviously at R S = l l s i.e. at T T BH = M s ! R S = s i.e. at BH = M s ! “string holes string holes” ” = states satisfying this entropy = states satisfying this entropy “ matching condition matching condition
Using string unification @ the string scale, Using string unification @ the string scale, entropy matching occurs for entropy matching occurs for and the common value of S st and S BH is simply and the common value of S st and S BH is simply In string theory g s is actually a field, the dilaton dilaton. Its . Its In string theory g 2 is actually a field, the s2 value is free in perturbation theory value is free in perturbation theory Consider the (M, g s ) plane Consider the (M, g 2 ) plane s2
The correspondence curve The correspondence curve (critical collapse?) (critical collapse?) M/M s M/M Much more difficult Much more difficult s to establish except to establish except for extremal extremal case case for R S > l s R S > l s Black Holes (= Strings? ) Black Holes (= Strings? ) R S = l s , R S = l s , “ “string hole string hole” ” curve curve R S < l s R S < l many properties match here many properties match here s Strings ≠ ≠ BH BH Strings g s g 2 s2 Safe conclusion since these strings are larger larger than R than R S Safe conclusion since these strings are S
M M strong gravity strong gravity effects effects weak gravity weak gravity effects effects g s g 2 s2 g 0s g 2 0s2 S ~ M S ~ M Collapse @ fixed M. Gravitational binding can increase (log of) Collapse @ fixed M. Gravitational binding can increase (log of) density of states from linear to quadratic from linear to quadratic in the in the physical physical mass. mass. density of states
Turning string entropy into BH entropy Turning string entropy into BH entropy S S Black hole Black hole String (naïve) String (naïve) g s s-2 -2 g g s s-2 -2 M M = g M s = M M sh M = s = sh M M
Evaporation at fixed g s or how to turn a BH Evaporation at fixed g s or how to turn a BH into a string ( (Bowick Bowick, , Smolin Smolin,.. 1987) ,.. 1987) into a string M/M s M/M s trajectory of evaporating BH Black Holes Black Holes R S = l s string-holes Strings Strings g s g 2 s2 Is singularity at the end of evaporation avoided thanks to l l s ? Is singularity at the end of evaporation avoided thanks to s ?
String S-matrix at E >> M P String S-matrix at E >> M P Super-planckian-energy collisions of light particles collisions of light particles Super-planckian-energy within superstring theory. Why care? within superstring theory. Why care? Theoretical Motivations Theoretical Motivations I) As a gedanken experiment gedanken experiment I) As a To reproduce reproduce GR expectations GR expectations at at large distances large distances To To probe how ST modifies GR modifies GR at at short distances short distances To probe how ST II) Information paradox Information paradox II)
“Phenomenological Phenomenological” ” Motivations Motivations “ Signatures of string/quantum gravity @ Signatures of string/quantum gravity @ colliders colliders: : In KK models with large extra dimensions; In KK models with large extra dimensions; In In brane-world brane-world scenarios; in general: scenarios; in general: If we can lower the true QG scale down to the If we can lower the true QG scale down to the TeV TeV NB. Future colliders colliders at best at best marginal marginal for producing for producing BHs BHs! ! NB. Future
Two complementary approaches (> 1987): (> 1987): Two complementary approaches A) Gross & Mende + Mende & Ooguri Ooguri (1987-1990) (1987-1990) A) Gross & Mende + Mende & B) ‘ ‘t-Hooft t-Hooft; ; Muzinich Muzinich & Soldate; ACV (>1987); & Soldate; ACV (>1987); B) Verlinde & & Verlinde Verlinde; ; Kabat Kabat & Ortiz; FPVV; & Ortiz; FPVV;… … Verlinde de Haro; Arcioni Arcioni; ; ‘ ‘t-Hooft t-Hooft; ; … … ( (‘ ‘90s- 90s-’ ’05) 05) de Haro; The two approaches are are very different very different. . Yet they Yet they The two approaches agree incredibly well in in the the ( (small small) ) region region of of agree incredibly well phase space where both can be justified space where both can be justified phase I will limit myself will limit myself to to describing describing B) B) and and, in , in I particular, , the work the work of ACV ( of ACV (the only the only one, one, particular besides A) A) that considers the problem within that considers the problem within besides string theory theory) ) string
Gross-Mende-Ooguri (GMO) (GMO) Gross-Mende-Ooguri Calculation (GM, 1987- (GM, 1987-’ ’88) of 88) of elastic elastic string string Calculation scattering at very high energy and fixed scattering scattering at very high energy and fixed scattering angle θ (h+1 = number number of of exchanged exchanged gravitons): gravitons): angle θ (h+1 = The amplitude amplitude is exponentially suppressed is exponentially suppressed but but the the The suppression decreases decreases as as we increase the number we increase the number of of suppression exchanged gravitons. gravitons. A A resummation was performed resummation was performed exchanged by Mende and Ooguri and Ooguri ( (see below see below) ) by Mende
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