Yun Soo Myung Inje University

Transcription

1 On the Lifshitz black holes Yun Soo Myung Inje University in collaboration with T. Moon

2 Contents 1. Introduction. Transition between Lifshitz black holes and other configurations 3. Quasinormal modes and frequencies 4. Absorption cross section 5. Discussions

3 Why we have to study Lifshitz black holes? The AdS/CFT correspondence (relativistic holography) :AdS Gravity theory (string theory) strongly coupled systems (QCD: quark-gluon plasma, CMP: holographic superconductor) [AdS_5/CFT_4, AdS_4/CFT_3, AdS_3/CFT_ ]

4 AdS/CFT correspondence AdS gravity Boundary gauge theory Bulk field Graviton Maxwell field Scalar and fermion fields AdS black hole Black hole quasinormal modes (perturbation theory) Hawking-Page transition gauge invariant operator stress-energy tensor global current scalar and fermionic operators thermal quantum field theory poles of retarded correlators in momentum space (linear response theory) confinement/deconfinement transition

5 Isotropic scaling anisotropic scaling The Lif/CFT correspondence (non-relativistic holography) :Lif Gravity theory (string theory) strongly coupled systems (CMP?) How to study Lifshitz black holes? Using the tools for studying AdS black holes 1) Thermodynamics analysis (phase transitions) ) Quasinormal modes and stability analysis 3) Absorption cross-section, we wish to point out the difference between Lifshitz and AdS black holes.

6 How to form a black hole in the sky? supernova explosion

7 Before supernova explosion: onion structure

8 Supernova explosion caused by gravitational collapse but not temperature

9 Exceptional case: super-massive 9 black hole with M 610 M Jet (of high energy particles) width and distance from the super-massive black hole in giant elliptical galaxy M87 [Nature 477, 185(011)] We do not know how to form this super-massive black hole.

10 . Transition between Lifshitz black holes and other configurations Formation of a black hole through thermal phase transition 1) For k=1 spherical horizon, Hawking-Page phase transition (first-order ) : thermal AdS unstable small BH stable large BH in Commun. Math. Phys. 87,577(1983) ) Hawking-Page transition confinement/deconfinement transition on the CFT side in hep-th/ (witten) 3) For k=0 Ricci-flat horizon, AdS soliton large BH was suggested in hep-th/ (Surya-)

11 First-order phase transition as Hawking-Page phase transition T<T0 T=T0 T=Tc AdS soliton TAdS T>Tc on-shell free energy Hawking-Page transition in Schwarzshild-AdS BH stable large BH unstable small BH phase transition in AdS soliton-black hole

12 3) For k=-1 hyperbolic horizon, MTZ TBH was suggested to be second-order in hep-th/ ) For k=0 Ricci-flat horizon, Lifshitz brane black brane was suggested in (Taylor) 5) For k=1 spherical horizon, Lifshitz soliton Lifshitz black hole was suggested in (Gonzalez-)

13 Relevant thermal quantities For the black hole phase transitions, we need to know heat capacity(c ) and free energy(f ). C local thermal stability F global thermal stability If C >0 for any size of black hole r+, one important quantity is F

14 r T H Important parameters for phase transitions order parameter ( black hole horizon size) ( r ) order parameter (on-shell temperature) T control parameter (off-shell temperature) F F on off ( r ) increasing (decreasing) black hole by absorption of radiations (Hawking radiation) ( r, T ) growth of black hole via non-equilibrium process

15 Necessary condition for phase transition A phase transition may occur if two configurations have the same asymptote in the gravitational system.

16 .1 Lifshitz black hole (LBH) and Lifshitz soliton(ls) I BHT d x g R R R R 16 G m BTZ black hole ( z 1) =, m 13 1 Lifshitz black hole ( z 3) =, m 3 z r M dr 3D dt Metric element ds 1 r Horizon radius r M, r M z r M Anisotropic scaling: t t,, r, M rd,

17 Penrose diagrams Light-like curvature singularity Light-like curvature singularity BTZ black hole Lifshitz black hole

18 T H M S r, vs Thermal AdS 3 TH 0 BH 1 r r 1 vs Thermal AdS G 1/8 3 M 8G 8G r 4 r vs Thermal AdS 1/8 3 SBH 0 G G on 1 r on 1 F ( r ) M TH SBH vs Thermal AdS3 F 8G 8G off 0 T T First-law of thermodyanmics is satisfied H off 1 r r F ( r, T ) M T SBH T vs Thermal AdS3 F 8G G df dr Thermodynamic quantities for BTZ BH (Thermal AdS as a ground state) off F on

19 T z H M S z BH r z z1 Thermal quantities for LBH (z =3) [Method Euclidean action approach ( )], 1 r 1 r 7 r 4G 4 r 4 r G1/ G z1 z1 z1 z3 not equal to M G1/ ADT z1 z1 on z z 3 r 3 r z ( ) H BH G1/ 4G F r M T S F off z df dr ( r off z z1 z1 z G1/ z 1 r r 1 r, T ) M TSBH T 4 rt 4G G z 0 T T First-law of thermodyanmics is satisfied H

20 Thermodynamic quantities for LS as a ground state T z H 0, M S F F z BH on z off z 0, 3 3, G1/ 4G 3 3, G1/ 4G F on z. because of the absence of horizon.

21 Different asymptotes for LBH and LS Lifshitz soliton: dr ds r dt r d r Lifshitz black hole: z LS dr ds r dt r d r z LBH Phase transition is not allowed

22 Transition from TAdS to BTZ black hole (PLB638,515) OK T<Tc T=Tc T>Tc F r F F r c on on on BTZ ( ) BTZ TAdS 1 F r T F F r r T T off off off BTZ (, ) BTZ TAdS 1 4 1

23 Transition from Lifshitz soliton to Lifshitz black hole is not allowed [different asymptotes] T<Tc T=Tc T>Tc on on on 3 4 FLif ( r) FLBH FTL r 1 off off off 1 4 FLif ( r, T ) FLBH FTL ( r 3) 4 rt 1 Tc

24 4. LBH and BTZ BH under temperature matching?[not allowed] F ( r ) F ( r ) for r 1.5 and F ( T ) F ( T ) for T T 0.9 BTZ Lif BTZ H Lif H H c F ( r ) F ( r ) for r 1.5 and F ( T ) F ( T ) for T T 0.9 BTZ Lif BTZ H Lif H H c a signal for transition between LBH (T<Tc) and BTZ BH (T>Tc)?

25 Differences between on-shell free energies F on ( r ) 3 Cusp nd order? F on ( r ) 3 4 3r F on ( r ) r 3 6

26 Differences between off-shell free energies[not allowed] 4 off r 6 3 F3 ( r, T) r 4 T( r r ) on F3 ( r ) T<Tc T=Tc T>Tc Minimum on of F3 ( r ) on F3 ( r ) No decaying: BTZLBH No decaying: LBHBTZ

27 Lifshitz brane (LB) and black brane (BB)? [not allowed] Einstein-scalar-Maxwell theory[ ] [ ] d S d x g R e 16 Gd 4 LB solution with z= & d= d z dr dslb L [ r f ( r) dt r dx ], i r f () r i1 r d f r e r r z 1 zd1 ( z d 1)( z d) rt qr 0,, L zd ( ) 1 0 d,, zd, q L ( z 1)( z d) q : Not LB charge 0 0

28 BB solution with d= & z=1 dr ds L [ r f ( r) dt r dx ], BB d i r f () r i1 d 1 r0 f( r) 1, 1 0 const, d r dd ( 1) rt 0, L It is clear that dilaton and Maxwell field play the important roles in order to obtain the LB.

29 Thermodynamics of LB (z=) and BB (z=1) T z H M S C z BH z z z r, 4 dl V r 16 z LV G 4 16G L V r dm LV 4G 4 16G r r V r S z BH z dth zg4 zl 0 z

30 Different definitions of LB free energy d zd on L Vd r z Fz ( r) d r [Euclidean action] LV 16 d Gd 1 16Gd on z z z z z Fz ( r ) M z Q TH SBH d r ( z 1) r ( z ) r r LV d 1 16Gd on z z z z Fz ( r ) M z TH SBH d r ( z ) r zr LV d [ Q is not a charge] 1 16Gd on on st on F ( r ) & F ( r ) not compatibe with the 1 law, but F ( r ) is compatibe with it. z z z

31 On-shell free energy and its off-shell free energies Not compatible with 1 st law off dflb ( r, T) dr 0 T TH Compatible with 1 st law df off BB ( r, T) 0 T T dr H

32 Coordinate and Temperature matching Coordinate Matching : off-shell free energy is not defined Temperature Matching : Consistency but isotropic scaling anisotropic scaling LB BB 4 TH TH r( BB) r 3 4 r TH BB r T LB 3 ( ), H( )

33 On-shell free energies cusp on on F ( T ) F ( T ) for T T BB H LB H H c on on F ( T ) F ( T ) for T T BB H LB H H c 1 FLB ( TC ) FBB ( TC ) at r 0.65& TC on F4 ( r ) r r 3

34 Off-shell free energies off F4 ( r, T) 3r r 4 T r r T<Tc T=Tc T>Tc No decaying: BBLB on F4 ( r ) on F4 ( r ) No decaying: LBBB

35 Any phase transition unlikely occurs between Lifshitz black hole and other configurations. Mainly due to different asymptotes

36 Boundary condition for QNMs (QNFs) r r ( y 1) [asymptotically Lifshitz] r ( y 0)

37 Wave propagations in the black brane spacetimes u=r u Waves falling across the black brane at u=r capture the dissipation of CFT with T 0 at the boundary on u.

38 3. Quasinormal modes and frequencies [ ] m 0 3D ( t, y, ) R( y) e it i, r y r 4 y 3 y m R R R 3 y(1 y ) 1 y M (1 y ) y M 0.

39 Solutionconfluent Huen (HeunC) functions R( y) C1 y (1 y ) HeunC 0,,,, ; y 4 4 4M C y (1 y ) HeunC 0,,,, ; y 4 4 4M m 1, i i 3/ 4 M TH

40 Requiring the Dirichlet condition at infinity (y=0) leads to C_=0. In order to impose the ingoing mode at horizon (y=1), one uses the connection formula ( 1) ( ) HeunC 0,,,, ; z HeunC 0,,,, ;1 z (1 K) ( K) [ ] [ ] ( 1) ( ) (1 z) HeunC 0,,,, ;1 z (1 S) ( S) [ ] where K (1 ) K 0, S (1 ) S 0 with [1 ( 1)( 1)] /

41 y1 1 1(1 ) (1 ) (1 ) ( ) (1 ) ( ), ( K) (1 K) ( S) (1 S) 1 ~ it it 1e (1 y ) e (1 y ) 1 1 i[ t ln(1 y)] i[ t ln(1 y)] TH TH 1e e [ ingoing] [outgoing] y1 [ ] R C y y Near the horizon: ingoing wave y1

42 Purely imaginary QNFs Imposing 0 ( S), (1 S) S n,1 S n, n Z purely imaginary QNFs of Lifshitz black hole 3m i 1 n (4 m ) 7 (3 6 n)(4 m ) 6 n( n 1) 4T H M asymptotic QNFs: n 4T QNFs of BTZ black hole: BTZ 4 H (1 1 ) ( ) 1/ 1/ 1/ i( 6 ) n 1 i T n m H [ ]

43 Thermodynamics and QNFs for Lifshitz black holes 3D Lifshitz BH: C>0, F<0 (Globally stable black hole) 3m i 1 n (4 m ) 7 (3 6 n)(4 m ) 6 n( n 1) 4T M [ ( ) ] 3D 1/ 1/ 1/ H D Lifshitz BH: C>0, F<0 (GSBH) ds 3 r M D 1 dt r 1 r M dr m 9 i 1 n m (3 6 n) m 6 n( n 1) 4T [ ( ) ( ( ) ) ] D 1/ 1/ 1/ H

44 GSBH purely imaginary QNFs z Lifshitz black hole from Einstein-scalar-massive vector theory ( ) dr f () r r z 4D1 () 1 ds r f r dt r dx dx e r f () r r dt A r f r QNFs: 1,, ( ) 1 k k (n 1) ((4n 4n 4 m ) ) ((4n 4n m ) ) 4 m 4D1 i TH TH + TH (4n 4n 3 m ) (4n 4n 3 m ) aysmptotic QNFs: T n 4D1 H in, r r

45 GSBH purely imaginary QNFs z Lifshitz brane (hole) from Einstein-scalar-Maxwell theory C 0, F 0(GSBH) s-mode QNFs: ml 1 1 k 0 i n 4 4TH k 0 QNFs are not derived.

46 Under-damping versus over-damping cases Scalar perturbation AdS BH with oscillation stage Scalar perturbation Lifshitz BH without oscillation stage

47 4. Absorption cross section: Review Classical approach to capture cross-section for light by using null geodesic around the Schwarzschild black hole d gs pp 0 0 dr L ph m Eph Vph( r), Vph( r) 1 d r r

48 Relativistic effective potential for light rays Captured scattered Maximum Vmax Lph / 7m at r=3m (photon sphere)

49 Classical capture (absorption) cross-section for photon For E V, an incoming photon enters r m ph max max so that it is captured by the black hole. For E V, an incoming photon is scattered by the potential E V ph back to infinity so that it is not captured by the blac hole. The condition for capture is L / E 7m ph ph If the impact parameter of the photon that is just captured by the hole is b, the capture cross-section of the area is = ph / b b L E geo = b 7 m : high-energy limit of wave equation approach ph ph max

50 s * Wave equation approach for EM waves Wave equation Schrodinger equation d dr s [ V ] s 0, s m ( 1/ ) 1/ 4 (1 s ) m V ( r) 1, s 3 r r r For s 1(EM wave), its potential is given by V EM ( r) 1 m ( 1/ ) 1/ 4 r r

51 Potential for EM waves around the black hole Transmitted waves Reflected wave Incident wave EM ( 1/ ) Maximum V at r 3 m(photon sphere) 7m

52 Absorption cross-section for EM waves ir* T ( ) e, r s1 * ( r* ) ir* ir* e R ( ) e, r* Absorption probability for EM waves with energy and angular momentum ( ) T ( ) Absorption cross section abs ( ) abs 1 Two limits of ( ) ( 1) ( ) geo A r m 7 m : classical picture, (high-energy limit) 4 16 : universal picture, 0 (low-energy limit)

53 Absorption cross section in Lifshitz black hole it ik [ Lif m ] 0 for ( r) e * 8 5r 3 r r ( k m r )( r r ) ( r r ) ( r) ( r) ( r) 0 4 r r ( r r ) tortoise coordinate r r 4 arccoth 3 r r r r r r * [, ] [,0] r Schrodinger equation with r( r) d * [ V( r )] 0 * dr with 7 4 m 4k 10 M 4 m M 4k M 3 M V () r r r r

54 Potentials for a scalar field with k 0 Ingoing mode Outgoing mode Ingoing mode V1~7 m 9, 4[Lif ],, 1[BF], 1/, 0, Stability of Lifshitz black hole is guaranteed for m 4 Regularity condition at r 4 m 0

55 Ingoing modes near the horizon Using x ( r r ) / r ( (1 x) m M x) k M (1 x) ( x) ( x) ( x) 0 3 x(1 x) 4 M (1 x) x r[ r, ) x [0,1) solution HeunC functions Near the horizon (Ingoing mode), 0 C 1 i [ t ln ] 4 T x H 1e 1 i [ t ln x] 4 TH Ce ( ) ( )

56 Ingoing and outgoing modes at infinity Ingoing and outgoing modes at r 5 ( r) ( r) r r r solution Bessel functions k r m s 1 1 () r C3 C4 r r Ingoing and outgoing modes at infinity by choosing ( r) 0 s ( r) C 3 3J (1 ) C 3 4J (1 ) 3 r 3r 3r [ ] ( ), ( ) C C C C C icc C C 3 in out 4 out in

57 conserved flux rr gg * * Fr ( ) ( r r ) i Absorption coefficient A F in 4 r r in 4 ( 1) in F c C Absorption cross section A 4 abs 4 c( 1) Cin In the low-energy and massless limis t Low-energy and massless limits area law 0, m0 abs r r Area of horizon universality of black holes

58 5. Discussions 1) Lifshitz black holes are similar to AdS black holes ) All Lifshitz black holes GSBH with C>0 and F<0 3) Purely imaginary QNFs for Lifshitz black holes more stable than AdS black hole 4) Asymptotic Lifshitz is slightly different from asymptotic AdS 5) What does the Lif/CFT correspondence predict?