How Fuzzballs solve the black hole information paradox Work done with B. Chowdhury, S. Giusto, O. Lunin, A. Saxena, Y. Srivastava
The entropy problem Black holes behave as if they have an entropy given by their surface area (Bekenstein, 72) S bek = A 4 G But statistical mechanics then says that there should be 4 e S bek states of the hole for the same mass and charge (Classical relativity finds 4 Can we show that there are that black holes have no hair, e S bek states of the hole ? so there is only one state)
The information problem Hawking radiation
Large distance (much bigger than planck length) How can the Hawking radiation carry the information of the initial matter ? If the radiation does not carry the information, then the final state cannot be determined from the initial state, and there is no Schrodinger type evolution equation for the whole system. So we lose quantum theory ...
A simple example: 2-charge holes (Susskind, Sen, Vafa ’94-’95) In string theory, we must make black holes from the objects present in string theory. ∼ Let us compactify spacetime as M 9 , 1 → M 4 , 1 × T 4 × S 1 momentum winding mode mode P NS1
Winding charge n 1 Mass = Charge Momentum charge n p Mass = Charge �
A black hole with winding charge only S micro = ln[256] ∼ 0 (Does not grow with ) n 1 n 1 √ √ Horizon is singular A = 0 Bekenstein entropy vanishes S micro = S bek = 0
A black hole with momentum charge only S micro = ln[256] ∼ 0 ∼ ∼ (Does not grow with ) n p n p � � Horizon is singular A = 0 Bekenstein entropy vanishes S micro = S bek = 0
Computing the entropy × Each quantum of harmonic k carries momentum 2 π k L T Total momentum L T = n 1 L P = 2 π n p = 2 π ( n 1 n p ) L L T So we have to count ‘partitions’ of = n 1 n p − ∼ � � k n k = n 1 n p � √ 2 √ n 1 n p = S micro = 2 π T 4 × S 1 8 bosonic + 8 fermionic degrees of freedom √ S micro = 4 π √ n 1 n p √ 2 √ n 1 n p K 3 × S 1 e 2 π states (Susskind ’93, Sen ’94)
Now let us make a black hole with these charges ... n 1 n p For compactification, K 3 × S 1 √ � geometry gives a Bekenstein - Wald entropy × × S bek = A 2 G = 4 π √ n 1 n p = S micro (Dabholkar ’04) So the 2-charge hole (‘Sen-Vafa hole’) gives a complete story for black hole entropy in string theory ......
The 2-charge black hole is called the ‘small black hole’ since R corrections to the action affect its horizon area To get a black hole whose area is given by just the usual Einstein action R, we need 3 charges ... Recall that we had compactified spacetime as ∼ M 9 , 1 → M 4 , 1 × T 4 × S 1 T 4 Make a bound state + + of 3 kinds of charges S 1 5-branes
T 4 S 1 + n 5 ‘Effective string’ with total winding n 1 n 1 n 5 number + Vibrations of effective string are partitioned into harmonics in S micro = 2 π √ n 1 n 5 n p usual way
Make a black hole with momentum, winding, 5-brane charges .... S bek = A 4 G = 2 π √ n 1 n 5 n p = S micro (Strominger and Vafa, 1996) Thus at least for extremal black holes (mass=charge) we understand something about the entropy from a microscopic viewpoint ... Near-extremal holes have also been understood ... (Callan and Maldacena 1996)
Recall the way we made the 2-charge black hole ... + L n 1 n p √ � This allowed us to count the L T = n 1 L states of the black hole, so we solve the entropy problem, but what about the information puzzle?
A key point The elementary string (NS1) does not have any LONGITUDINAL vibration modes This is because it is not made up of Not a mode for the ‘more elementary particles’ elementary string Thus only transverse oscillations are Momentum is carried permitted by transverse oscillations This causes the string to spread over a nonzero transverse area
L T = n 1 L L L ‘Naive An ‘actual geometry’ geometry’
Making the geometry We know the metric of one straight strand of string We know the metric of a string carrying a wave -- ‘Vachaspati transform’ We get the metric for many strands by superposing harmonic functions from each strand (Dabholkar, Gauntlett,Harvey, Waldram ’95, Callan,Maldacena,Peet ’95) In our present case, we have a large number of strands, so we ‘smear over them to make a continuous ‘strip’ (Lunin+SDM ’01)
4 4 H [ − dudv + Kdv 2 + 2 A i dx i dv ] + ds 2 � � = dx i dx i + dz a dz a string i =1 a =1 1 = 2[ H − 1] , B vi = HA i B uv e 2 φ = H � L T � 1 + Q 1 dv H − 1 = x − � L T F ( v ) | 2 | � 0 � F ( x − t ) � L T dv ( ˙ F ( v )) 2 Q 1 = K x − � L T F ( v ) | 2 L T = n 1 L | � 0 � L T dv ˙ F i ( v ) − Q 1 = A i x − � L T F ( v ) | 2 | � 0
A final step: Use dualities to map to D1 -D5 − ∼ � k n k = n 1 n p � T 4 � � k n k = n ′ 1 n ′ + S 1 5 ‘Effective string’ with n ′ 1 = n p n ′ 5 = n 1 total winding number D1 branes D5 branes n ′ 1 n ′ 5 = n 1 n p ....
Geometry for D1 -D5 � H 1 + K [ − ( dt − A i dx i ) 2 + ( dy + B i dx i ) 2 ] ds 2 = � 1 + K � + dx i dx i + H (1 + K ) dz a dz a H � | − � L T H − 1 = 1 + Q dv dB = − ∗ 4 dA x − � L T F ( v ) | 2 | � 0 � L T dv ( ˙ F ( v )) 2 K = Q x − � L T | � F ( v ) | 2 0 � L T (Lunin+SDM ’01, dv ˙ F i ( v ) A i = − Q also x − � L T F ( v ) | 2 | � ‘Supergravity supertubes’ 0 (Emparan+Mateos+T ownsend ’01) dB = − ∗ dA
Scale of the ‘fuzzball’ 4 G A √ 4 G ∼ √ n 1 n 5 ∼ S micro = 2 π 2 √ n 1 n 5 √ � � (Lunin+SDM ’02) 4 G ∼ ∼ A √ � � n 1 n 5 − J ∼ S micro = 2 π 2 n 1 n 5 − J 4 G ∼
� F ( y − ct ) = NS1-P D1-D5 − state CFT state − 2 ) n 2 . . . | 0 � ( α i 1 − 1 ) n 1 ( α i 2 S,T D1-D5 dualities gravity dual NS1-P geometry
A simple example D1-D5: CFT state has all loops NS1- P : one turn ‘singly wound’, of a uniform helix and all spins aligned − AdS 3 × S 3 Make metric from going over to profile function F flat space at infinity
The horizon of 2-charge holes needed higher derivative corrections Consider D1-D5-P , which does not need such corrections at leading order | k � total = ( J − ,total − (2 k − 4) ) n 1 n 5 . . . ( J − ,total ) n 1 n 5 | 1 � total − (2 k − 2) ) n 1 n 5 ( J − ,total − 2 Piece of many ‘bits’ global Field theory representation AdS 3 × S 3 × T 4 of brane state Geometry created by this state
dr 2 − 1 � � h ( dt 2 − dy 2 ) + Q p hf ( dt − dy ) 2 + hf ds 2 N + a 2 η + d θ 2 N = r 2 N − na 2 η + (2 n + 1) a 2 η Q 1 Q 5 cos 2 θ � � cos 2 θ d ψ 2 r 2 + h h 2 f 2 N + ( n + 1) a 2 η − (2 n + 1) a 2 η Q 1 Q 5 sin 2 θ � � sin 2 θ d φ 2 r 2 + h h 2 f 2 a 2 η 2 Q p cos 2 θ d ψ + sin 2 θ d φ � 2 � + hf 2 a √ Q 1 Q 5 n cos 2 θ d ψ − ( n + 1) sin 2 θ d φ � � + ( dt − dy ) hf 2 a η √ Q 1 Q 5 4 � H 1 cos 2 θ d ψ + sin 2 θ d φ � � � dz 2 dy + − i hf H 5 i =1 √ Q 1 Q 5 N − a 2 η n sin 2 θ + a 2 η ( n + 1) cos 2 θ r 2 = f η ≡ Q 1 Q 5 + Q 1 Q p + Q 5 Q p H 1 H 5 , H 1 = 1 + Q 1 f , H 5 = 1 + Q 5 � = h f
Non-extremal hole (Jejalla, Madden, Ross Titchener ’05) Geometry has a classical ‘ergoregion instability’ (Cardoso, Dias, Jordan, Hovdebo, Myers, ’06) One (non-typical) microstate of a non-extremal hole The same process that gives Hawking radiation from the black hole now gives us the exact frequency of the instability (Chowdhury + SDM (2007)
Summary of the fuzzball picture small g ∼ ∼ 1 1 4 l p 4 l p large g N α l p Conventional Fractionation picture ‘Fuzzball’
The structure of extremal holes Infinite throat horizon ‘ fuzzball cap’ singularity
Classical states and quantum fuzzballs Fix Total Energy F ( y − ct ) Put energy in a few harmonics, large Coherent occupation number state for each harmonic Generic Energy in quantum many harmonics, state occupation number order unity in each ‘Fuzzball’ Size depends on mean harmonic order, fluctuations depend on occupation number
Summary (A) Fuzzball picture offers a resolution of the information paradox ... N α l p l p (B) Lessons must be very general, with applications to other places ...e.g. Cosmology Can we (should we) ask the question: What is the most entropic state in string theory when the energy density is taken to infinity?
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