Black Hole Formation in CFT

Transcription

1 Black Hole Formation in CFT Tom Hartman Cornell University 20 Years Later: The Many Faces of AdS/CFT PCTS November 2017

2 = Leinweber 2003 SS 1. How does gravity emerge? 2. In what theories? 3. Locality 4.?

3 = Leinweber 2003 SS 1. How does gravity emerge? 2. In what theories? 3. Locality 4.?

4 = Leinweber 2003 SS 1. How does gravity emerge? 2. In what theories? 3. Locality 4.?

5 2d CFT Gravity sector controlled by Virasoro Effectively, no higher derivative corrections But has an information paradox, firewalls, etc.

6 2d CFT Gravity sector controlled by Virasoro Effectively, no higher derivative corrections But has an information paradox, firewalls, etc. Large-c + Sparse spectrum > 3d gravity c = central charge

7 2d CFT Gravity sector controlled by Virasoro Effectively, no higher derivative corrections But has an information paradox, firewalls, etc. Large-c + Sparse spectrum > 3d gravity c = central charge Higher dimensions Must also explain why Einstein gravity But many lessons from 2d carry 1/N

8 2d CFT Gravity sector controlled by Virasoro Effectively, no higher derivative corrections But has an information paradox, firewalls, etc. Large-c + Sparse spectrum > 3d gravity c = central charge Higher dimensions Must also explain why Einstein gravity But many lessons from 2d carry 1/N This talk An example in 3d gravity - Black hole formation with T. Anous, A. Rovai, J. Sonner

9 A Tale of Two Sums Gravity = Z ho 1 O 2 i DgD exp ( S Einsten [g] S Matter [ ]) CFT = primaries

10 A Tale of Two Sums Gravity = Z ho 1 O 2 i DgD exp ( S Einsten [g] S Matter [ ]) CFT = primaries Under certain conditions, the CFT sum over primaries can be recast as a sum over channels and the sum over channels is the bulk path integral.

11 Example 1: Large-N correlators ho(x 1 )O(x 2 )O(x 3 )O(x 4 )i = h12ih34i + h13ih24i + h14ih23i

12 Example 1: Large-N correlators ho(x 1 )O(x 2 )O(x 3 )O(x 4 )i = h12ih34i + h13ih24i + h14ih23i In a general CFT, the identity dominates in the OPE limits. In a holographic CFT, the full answer at infinite-n is G (identity exchange) channels

13 Example 1: Large-N correlators ho(x 1 )O(x 2 )O(x 3 )O(x 4 )i = h12ih34i + h13ih24i + h14ih23i In a general CFT, the identity dominates in the OPE limits. In a holographic CFT, the full answer at infinite-n is G (identity exchange) channels 1/N corrections: Extended regime of validity of OPE Ex: Conformal Collider ==> Einstein gravity 3-point functions Afkhami-Jeddi, TH, Kundu, Tajdini 16 Kulaxizi, Parnachev, Zhiboedov 17 Costa, Hansen, Penedones 17

14 Example 2: thermal partition function in 2d In any 2d CFT, Z cft ( )= E e E

15 Example 2: thermal partition function in 2d In any 2d CFT, Z cft ( )= E e E Z cft (!1) exp c 12

16 Example 2: thermal partition function in 2d In any 2d CFT, Z cft ( )= E e E Z cft ( Z cft (!1) exp! 0) exp 4 2 c 12 c 12 Cardy formula Cardy 86 Strominger 97

17 In a general CFT, this only tells you about the limits T -> 0, T -> infinity But if we assume c >> 1 and a sparse spectrum ( ). e 2 ( <c/12) then can prove Cardy applies at all temperatures: TH, Keller, Stoica 14 Z( )= E e E e c e 2 c Extended Cardy all temperatures!

18 In a general CFT, this only tells you about the limits T -> 0, T -> infinity But if we assume c >> 1 and a sparse spectrum ( ). e 2 ( <c/12) then can prove Cardy applies at all temperatures: TH, Keller, Stoica 14 Z( )= E e E e c e 2 c Extended Cardy all temperatures! vacuum + vacuum in another channel

19 In a general CFT, this only tells you about the limits T -> 0, T -> infinity But if we assume c >> 1 and a sparse spectrum ( ). e 2 ( <c/12) then can prove Cardy applies at all temperatures: TH, Keller, Stoica 14 Z( )= E e E e c e 2 c 12 Extended all temperatures! vacuum Thermal AdS saddle + vacuum in another channel

20 In a general CFT, this only tells you about the limits T -> 0, T -> infinity But if we assume c >> 1 and a sparse spectrum ( ). e 2 ( <c/12) then can prove Cardy applies at all temperatures: TH, Keller, Stoica 14 Z( )= E e E e c e 2 c 12 Extended all temperatures! vacuum Thermal AdS saddle Gravitational Saddles > Sum over images Dijkgraaf et al; Maloney-Witten + vacuum in another channel BTZ saddle

21 Example 3: 3d Black hole formation Anous, TH, Rovai, Sonner (z k ) (z 2 ) (z 3 )

22 Example 3: 3d Black hole formation Anous, TH, Rovai, Sonner O 1 (z k ) (z 2 ) (z 3 ) O 2

23 Example 3: 3d Black hole formation Anous, TH, Rovai, Sonner Results CFT calculations match gravity calculations (z 2 ) (z 3 ) O 1 (z k ) Thermalization from first principles + Signs of information loss Conformal block sum reorganizes into a sum over bulk worldlines O 2

24 Example 3: 3d Black hole formation Vaidya geometry - Null shell ds 2 = F (r, v)dv 2 +2dvdr + r 2 d' 2 + massive probe (travels on geodesic) O 1 (z k ) (z 2 ) (z 3 ) O 2

25 CFT calculation: G = hv O(x 1 )O(x 2 ) V i in the state V i = ny (z n ) 0i k=1 dust operator Limit: c!1 n!1 with total energy E/c held fixed, and 1 O c

26 G = hv O(x 1 )O(x 2 ) V i in radial quantization: O 1 O 2 Each dot =

27 O 1 i,j,k,... O 2

28 1. Large-c vacuum dominance i,j,k,... F 2 e cf(h i,h j,h k,... ) e cf(0) Includes exchange of identity+ descendants: 1,T,@T,T 2, ) 2, 2. Large-c block from monodromy method 3. In the smooth-shell limit n=, becomes analytically tractable

29 But in which channel? O 1 O 1 O 2 O 2

30 But in which channel? O 1 O 1 O 2 O 2 Minimal answer consistent with crossing symmetry is G (identity exchange) channels cf. extended Cardy

31 Autocorrelation function (equal x): hv O(t 1, 0)O(t 2, 0) V i channels e cf vac max e cf vac apple 1 T cos t 1 2 sinh( Tt 2) 2 sin t 1 2 cosh( Tt 2) 2 O =geodesic on gravity side: Hubeny et al; Balasubramanian et al; Zioga

32 Autocorrelation function (equal x): hv O(t 1, 0)O(t 2, 0) V i channels e cf vac max e cf vac apple 1 T cos t 1 2 sinh( Tt 2) 2 sin t 1 2 cosh( Tt 2) 2 O =geodesic on gravity side: Hubeny et al; Balasubramanian et al; Zioga Thermalization from (nearly) first principles

33 Autocorrelation function (equal x): hv O(t 1, 0)O(t 2, 0) V i channels e cf vac max e cf vac apple 1 T cos t 1 2 sinh( Tt 2) 2 sin t 1 2 cosh( Tt 2) 2 O =geodesic on gravity side: Hubeny et al; Balasubramanian et al; Zioga Thermalization from (nearly) first principles Signs of information loss: G e 2 Ot/ This approximation violates unitarity in the same sense as the bulk geodesic, or Hawking s calculation Maldacena 01 cf. Fitzpatrick, Kaplan, Walters 14

34 The max over channels is identical, term-by-term, to the calculation of the bulk geodesic length. O 1 (z 1 ) (z k ) (z 2 ) (z 3 ) Choice of channel in CFT < > Choice of crossing point in bulk O 2

35 Gravity = Z A Tale of Two Sums DgD exp ( S Einsten [g] S Matter [ ]) CFT = primaries

36 Gravity = Z A Tale of Two Sums DgD exp ( S Einsten [g] S Matter [ ]) e worldlines CFT = primaries O Length

37 Gravity = Z A Tale of Two Sums DgD exp ( S Einsten [g] S Matter [ ]) e worldlines CFT = primaries O Length channels e cf vac

38 Gravity = Z A Tale of Two Sums DgD exp ( S Einsten [g] S Matter [ ]) e worldlines CFT = primaries O Length identical term-by-term channels e cf vac

39 In Euclidean, sum is always dominated by the largest term, channels e cf vac so we cannot really probe the sum. e cf min vac

40 In Euclidean, sum is always dominated by the largest term, channels e cf vac so we cannot really probe the sum. e cf min vac But in Lorentzian kinematics, the sum can have phases. In this case, no single channel dominates we must actually do the sum channels e cf vac e cf vac CFT Saddle Complex crossing point in bulk < > complexified OPE channel

41 Conclusion Applications? Inside the horizon Dynamical black hole entropy Far-from-equilibrium dynamics Eigenstate thermalization Questions 1. How does gravity emerge? 2. In what theories? 3. Locality 4.?