Therm odynam ics Therm odynam ics and and Fabric of Spacetim e Fabric of Spacetim e Dm itri V. Fursaev Dm itri V. Fursaev Dubna U & JI NR Dubna U & JI NR “Bogoliubov Readings “ Bogoliubov Readings” ” Dubna, Septem ber 2 2 , 2 0 1 0 Dubna, Septem ber 2 2 , 2 0 1 0
Gravity as an emergent phenomenon
A. Sakharov’s suggestion (1968): the Einstein theory can be induced at one-loop 1 ∫ µ 2 4 2 ∇ ∇ + − Λ + + + ln det ( m ) d x g ( R a " R " ...) µ eff eff π 16 G eff Λ eff 4 − M , M UV cutoff G eff 1 2 M G eff Gravitons = collective excitations of underlying degrees of freedom analogy: phonons in solid state physics - Young’s modulus Λ , G eff eff
String theory: “Tree-level” diagram “one-loop” diagram (closed strings) (open strings) “Sakharov’s picture” low-energy limit (10D (super)gravity, …)
Consequences of an emergent nature of gravity? - entropy of a black hole (since 1970’s) ? - T.Jacobson: laws of gravity can be inferred from ‘thermodynamical’ properties of event horizons - E.Verlinde: laws of gravity have a thermodynamical form (horizons are replaced with a more general concept of ‘holographic’ screens)
Entropic origin of gravity (E. Verlinde) Consider a massive source and a holographic screen around it; 1 st postulate the screen is equipotential surface which carries certain entropy: σ d = − σ dN d number of degrees of freedom on the screen on the area G δ = π S 2 ml 2d postulate: - change of the entropy under the movement of a test particle toward the screen; 3d postulate: the energy takes an 'equipartition' form on the screen 1 1 ∫ ∫ σ ∂ φ = d TdN = M (T. Padmanabhan) n π 4 2 G ∂ φ = n − T E.Verlinde a local temperature on the screen π arXiv:1001.0785 [hep-th] 2
Consequences δ = T S W use an analog of the 1st law = − W Fl a work done by the system, = F mw - force acting on the test particle; w - acceleration of the particle ∂ mMG S = = ∂ φ = F mw m = - the Newton law, F T 2 ∂ r l gravity is an emergent phenomenon; the force of gravity has an entropic origin direction of the force – gradients of the entropy
Main problem: mechanism of generation of the entropy?
quantum entanglement: states of subsystems cannot be described independently 1 2 entanglement has to do with quantum gravity: ● possible source of the entropy of a black hole (states inside and outside the horizon); ● d=4 supersymmetric BH’s are equivalent to 2, 3,… qubit systems ● entanglement entropy allows a holographic interpretation for CFT’s with AdS duals
Holographic Formula for the Entropy Ryu and Takayanagi, hep-th/0603001, 0605073 AdS (bulk space) 5 minimal (least area) % B surface in the bulk 1 2 4d space-time manifold (asymptotic boundary of AdS) B separating surface % A entropy of entanglement is measured in terms of the = S + area of ( d 1) 4 G � + ( d 1) is the gravity coupling in AdS G
Holographic formula enables one to compute entanglement entropy in strongly correlated systems with the help of classical methods (the Plateau problem) What about entanglement in quantum gravity?
Can one define an entanglement entropy, S(B), of fundamental degrees of freedom spatially separated by a surface B? How can the fluctuations of the geometry be taken into account? the hypothesis ● S(B) is a macroscopical quantity (like thermodynamical entropy); ● S(B) can be computed without knowledge of a microscopical content of the theory (for an ordinary quantum system it can’t) ● the definition of the entropy is possible at least for a certain type of boundary conditions
Suggestion (DF, 06,07): EE in quantum gravity � between degrees of freedom separated by ∂ � a surface B is 1 A B ( ) = ( ) S B 4 G 2 D 1 D 2 B is a least area minimal hypersurface � in a constant-time slice the system is determined by conditions: a set of boundary conditions; ● static space-times subsets, “1” and “2” , in the bulk are specified by the division of the boundary
The shape of the separating surface is formed under fluctuations of the geometry; As a result the surface is minimal, i.e. has a least area Details: D.V. Fursaev, Phys. Rev. D77 (2008) 124002, e-Print: arXiv:0711.1221 [hep-th]
If the entanglement entropy in QG is a macroscopic quantity, does it allows a thermodynamic interpretation
simple variational formulae (weak field approximation) δ = π S ml − m mass of a particle − l shift (toward of the surface) 37 δ = = S 10 if m 1 , g l 1 cm δ = S O (1) if is a Compton wavelength l δ πµ δ S l z µ − string tension δ − z lenght of the segment
aim of the talk to study simplest dynamics of a minimal surface; to look for its thermodynamic analogy; to relate this analysis to a hypothesis about an entropic origin of gravity (as suggested by E.Verlinde) see D.V. Fursaev, arXiv:1006.2623 [hep-th]
Minimal surfaces may play a role of holographic screens 2-component ‘screen’ around a massive source in weak field approximation screen = 2 parallel planes 2 MG φ = 2 2 2 2 2 - = + φ + φ + + ds - (1 2 ) dt (1 - 2 ) ( dx dy dz ), r 1 ∫ = σ = ∂ φ − E B ( ) d w , w acceleration on the screen k n n n π 4 MG = + = − E ( E B ) ( E B ) M the Komar energy of the source 1 2
Dynamics in the weak field approximation A B ( ) 1 ∫ = − φ − k S = S ( ) r dN ; S entropy for a plane k k k k 4 G 2 MG φ = − ( ) r - potential of the massive source r ∫ = − φ σ A B '( ) 2 '( ' ) r d - modification of the area by a test particle k k mG φ = − '( ' ) r - potential of the test parti cle on the screen k r ' k r ' - distance from a point on the screen to the particle k ∫ δ = − δφ σ = − π A B ( ) 2 '( ' ) r d 4 mGl k k
Notes on the computation φ = π − '( ) r 4 mG D x ( x ) r 0 x - , x - position of the particle position of a point on a screen 0 r = δ D x ( ) ( x ) Δ 0 k k = x ) l δ shift of the particle ( results in the variation 0 k δ − = − ∂ − D x ( x ) l D x ( x ) 0 r 0 k r 0 ∫ δφ σ → '( ') r d l ∫ ∫ k 3 ∂ − σ = = l D x ( x ) d l D x d x ( ) Δ ⊥ k r 0 2 ≡ l l shift in the direction orthogonal to the screen ⊥ shifts along the screen (plane) do not change the area
`Thermodynamics’ δ = δ = − π S S B ( ) ml - for a particle moving out of the surface k k δ = S 0 - for a particle moving inside the screen δ = δ + δ = − π S S S 2 ml - can be derived, is not a postulate! 1 2 single surface = half of the screen: 1 M ∫ = σ = E B ( ) d w n π 4 MG 2 B energy balance: w 1 1 ∫ δ = − δ → = n → = T x ( ) S B ( ) W x ( ) T x ( ) E B ( ) TdN π 2 2 2 B δ − W x ( ) work done by an external force to drag the test particle with coordinates out of the surface x
Static space-time backgrounds (which are solutions to the Einstein equations) 2 2 a b = + ds g ( ) x dt g ( ) x dx dx 00 ab ` holographic screen` is a minimal surface (with a topology of a hyperplane) in a constant-time slice → + g g h - perturbation caused by a test particle µν µν µν → + A B ( ) A B ( ) A B '( ) - perturbation of the area of a minimal surface 1 ∫ ij a b = σ γ A B '( ) d X X h , i , j ab 2 B a a = X X ( )- y change in position of t he surface does not count in the linear approximation ij a b γ X X h - variation of the metric induced on the surface , i , j ab
perturbations: 1 µ µ µ µ µ = π = − δ Lh 16 Gt , h h h ν ν ν ν ν 2 µ ∇ = h 0 µ ν Λ 8 µ µ µλ ρ µ 2 = −∇ − − Lh h 2 R h h ν ν νρ λ ν − n 2 Λ − n - cosmological constant, number of dimensions 2 0 0 0 2 0 a a b ∇ = Δ − ∂ + − h h w h 2 w h 2 w w h 0 0 a 0 0 b a Δ - Laplacian on constant-time slice µ µ − ( n 1) = δ ( ) ( , ) , t x mu u x x m - mass of the particle ν ν 0 µ µ µ = ∇ u , w u - velocity of the particle - acceleration
approximation: curvature terms, lambda term, and acceleration terms are “slowly” changing, perturbations caused by the particle are rapidly changing ; curvature-, lambda-, and acceleration terms can be neglected 1 ∫ 0 = σ − A B '( ) d h area perturbation 0 2 B 0 = π h 16 Gm D x x ( , ) 0 0 − ( n 1) Δ = δ D x x ( , ) ( , x x ) x 0 0
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