Therm odynam ics Therm odynam ics and and Fabric of Spacetim e - - PowerPoint PPT Presentation

Therm odynam ics Therm odynam ics and and Fabric of Spacetim e Fabric of Spacetim e

“ “Bogoliubov Readings Bogoliubov Readings” ” Dubna, Septem ber 2 2 , 2 0 1 0 Dubna, Septem ber 2 2 , 2 0 1 0

Dm itri V. Fursaev Dm itri V. Fursaev

Dubna U & JI NR Dubna U & JI NR

Gravity as an emergent phenomenon

2 4 2 4 2

1 ln det ( ) ( " " ...) 16 , UV cutoff 1

eff eff eff eff eff eff

m d x g R a R G M M G M G

µ µ

π ∇ ∇ + − Λ + + + Λ −

∫

฀ ฀ ฀

,

eff eff

G Λ

• A. Sakharov’s suggestion (1968):

the Einstein theory can be induced at one-loop Gravitons = collective excitations of underlying degrees of freedom analogy: phonons in solid state physics

• Young’s modulus

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

• f 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)

number of degrees of freedom on the screen on the area 2d postulate:

• change of the entropy under the movement
• f a test particle toward the screen;

3d postulate: the energy

2 d dN d G S ml σ σ δ π = − =

takes an 'equipartition' form on the screen (T. Padmanabhan) a local temperature on the screen

1 1 4 2 2

n n

M

d TdN G T σ φ π φ π

=

∂ = ∂ = −

∫ ∫

Consider a massive source and a holographic screen around it; 1st postulate the screen is equipotential surface which carries certain entropy:

E.Verlinde arXiv:1001.0785 [hep-th]

Consequences

gravity is an emergent phenomenon; the force of gravity has an entropic origin direction of the force – gradients of the entropy

2

use an analog of the 1st law a work done by the system, force acting on the test particle; acceleration of the particle

• the Newton law,
• F

T

T S W W Fl F mw w mMG S F mw m r l δ φ

=

= = − = ∂ = = ∂ = ∂

Main problem: mechanism of generation of the entropy?

quantum entanglement: states

• f 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

1

2

B

5

AdS

B %

4d space-time manifold (asymptotic boundary of AdS) (bulk space) separating surface minimal (least area) surface in the bulk Ryu and Takayanagi, hep-th/0603001, 0605073

• entropy of entanglement

is measured in terms of the area of

( 1) d

G

+

is the gravity coupling in AdS

( 1)

4

d

A S 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?

• 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

the hypothesis

Can one define an entanglement entropy, S(B),

• f fundamental degrees of freedom spatially separated by a surface B?

How can the fluctuations of the geometry be taken into account?

Suggestion (DF, 06,07): EE in quantum gravity between degrees of freedom separated by a surface B is ∂

• D

D

1 2

conditions:

• static space-times

B is a least area minimal hypersurface in a constant-time slice

1 2 ( ) ( ) 4 A B S B G =

the system is determined by a set of boundary conditions; 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)

37

mass of a particle shift (toward of the surface) 10 if 1 , 1 (1) if is a Compton wavelength string tension lenght of the segment S ml m l S m g l cm S O l S l z z δ π δ δ δ πµ δ µ δ = − − = = = − − ฀ ฀

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

• rigin of gravity (as suggested by E.Verlinde)

see D.V. Fursaev, arXiv:1006.2623 [hep-th]

2 2 2 2 2

1 2

• (1

2 ) (1 - 2 ) ( ), the Komar energy of the source

acceleration on the screen

2

• 1

( ) , 4 ( ) ( )

k n n n

ds dt dx dy dz

MG r E B d w w MG E E B E B M

φ φ

φ σ φ π

= + + + +

= = = ∂ − = + = −

∫

Minimal surfaces may play a role of holographic screens 2-component ‘screen’ around a massive source

in weak field approximation screen = 2 parallel planes

Dynamics in the weak field approximation

entropy for a plane potential of the massive source modification of the area by a test particle potential of the test parti

( ) 1 = ( ) ; 4 2 ( )

• '(

) 2 '( ' )

• '( ' )
• '

k k k k k k k k k

A B S S r dN S G MG r r A B r d mG r r φ φ φ σ φ = − − = − = − = −

∫ ∫

cle on the screen distance from a point on the screen to the particle

' - ( ) 2 '( ' ) 4

k k k

r A B r d mGl δ δφ σ π = − = −

∫

position of the particle position of a point on a screen shift of the particle ( results in the variation

'( ) 4 ( )

• ,
• ( )

( ) ) ( ) ( ) '( ') ( )

r r k k k r k r k k r

r mG D x x x x D x x x l D x x l D x x r d l D x x

δ

φ π δ δ δφ σ

Δ

= − = = − = − ∂ − → ∂ −

∫

3

shift in the direction orthogonal to the screen shifts along the screen (plane) do not change the area

( ) 2 l d l D x d x l l σ

⊥ ⊥

Δ

= = ≡

∫ ∫

Notes on the computation

`Thermodynamics’

1 2

for a particle moving out of the surface for a particle moving inside the screen can be derived, is not a postulate! single surface = half of the screen:

( )

• 0 -

2

• 1

( )

k k

S S B ml S S S S ml E B δ δ π δ δ δ δ π = = − = = + = − =

energy balance: work done by an external force to drag the test particle with coordinates out of the surface

4 2 1 1 ( ) ( ) ( ) ( ) ( ) 2 2 2 ( )

n B n B

x

M d w MG w T x S B W x T x E B TdN W x σ π δ δ π δ = = − → = → = −

∫ ∫

Static space-time backgrounds

(which are solutions to the Einstein equations)

`holographic screen` is a minimal surface (with a topology of a

hyperplane) in a constant-time slice

2 2 00 , ,

perturbation caused by a test particle perturbation of the area of a minimal surface change in position of t

( ) ( )

• ( )

( ) '( ) - 1 '( ) 2 ( )-

a b ab ij a b i j ab B a a

ds g x dt g x dx dx g g h A B A B A B A B d X X h X X y

µν µν µν

σ γ = + → + → + = =

∫

, ,

he surface does not count in the linear approximation variation of the metric induced on the surface

• ij

a b i j ab

X X h γ

perturbations:

2 2 2 ( 1)

• cosmological constant,

number of dimensions

• Laplacian on constant-time slice

1 16 , 2 8 2 2 2 2 ( ) (

a a b a b a n

Lh Gt h h h h Lh h R h h n n h h w h w h w w h t x mu u

µ µ µ µ µ ν ν ν ν ν µ µ ν µ µ µλ ρ µ ν ν νρ λ ν µ µ ν ν

π δ δ

−

= = − ∇ = Λ = −∇ − − − Λ − ∇ = Δ − ∂ + − Δ =

• mass of the particle
• velocity of the particle
• acceleration

, ) , , x x m u w u

µ µ µ

= ∇

approximation:

( 1)

area perturbation

1 '( ) 2 16 ( , ) ( , ) ( , )

B n x

A B d h h Gm D x x D x x x x σ π δ

−

= − = Δ =

∫

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

in a thin layer near a minimal surface the space is “flat” in the direction orthogonal to the surface (z-direction)

2 2

position of the surface extrinsic curvature of the minimal surface for the shift of the particle out

• f the surface by distance

( , ) (0, ) ( ) '( ) 4

i j ij ij z ij

l

dl dz z y dy dy z k y A B A B mGl γ γ δ δ π = + = − = − ∂ = = = − −

a universal formula, does not depend on the background and its dimensionality

`Thermodynamical’ parameters of a minimal surfaces

1/ 2 00 1/2 00

entropy energy (Komar mass) number of states on the area acceleration at the surface (part normal to the surface) local temperature on the sur

( ) 4 1 1 4 2 2

n B B n n

A B S G E g w d TdN G d dN G w g w T σ π σ π = − = = − = − − = − −

∫ ∫

face

temperature coincides with the Hawking temperature for the surface located near a back hole horizon

`Thermodynamics’

the surface is located between a gravitating body and a test particle

1/2 00

• ne obtains "1st law"

entropy change when dragging particle out of the surface force applied by an observer at infinity (for asymptotically flat spacetimes)

1 ( ) ( ) 2 ( )

n

T x S W x S ml W x Fl m g w F δ δ δ π δ = − = − − = = −

Summary

• support to and developing hypothesis about entropic origin of

gravity (E.Verlinde):

• minimal surfaces as holographic screens
• reducing the number of postulates
• interpretation of the entropy: entanglement of fundamental

degrees of freedom = fabric of spacetime

• a universal variational formula for minimal surfaces
• a simple version of `thermodynamics’ of minimal surfaces

(entropy, temperature, energy)

thank you for attention