GRAVITY: THE INSIDE STORY

Transcription

1 GRAVITY: THE INSIDE STORY T. Padmanabhan (IUCAA, Pune, India) VR Lecture, IAGRG Meeting Kolkatta, 28 Jan 09

2 CONVENTIONAL VIEW GRAVITY AS A FUNDAMENTAL INTERACTION

3 CONVENTIONAL VIEW GRAVITY AS A FUNDAMENTAL INTERACTION SPACETIME THERMODYNAMICS

4 NEW PERSPECTIVE SPACETIME THERMODYNAMICS GRAVITY IS AN EMERGENT PHENOMENON

5 NEW PERSPECTIVE SPACETIME THERMODYNAMICS GRAVITY IS AN EMERGENT PHENOMENON GRAVITY IS THE THERMODYNAMIC LIMIT OF THE STATISTICAL MECHANICS OF ATOMS OF SPACETIME

6 GRAVITY AS AN EMERGENT PHENOMENON SAKHAROV PARADIGM

7 GRAVITY AS AN EMERGENT PHENOMENON SAKHAROV PARADIGM SOLIDS SPACETIME Mechanics; Elasticity (ρ, v...) Einstein s Theory (g ab...) Statistical Mechanics of atoms/molecules Statistical mechanics of atoms of spacetime

8 GRAVITY AS AN EMERGENT PHENOMENON SAKHAROV PARADIGM SOLIDS SPACETIME Mechanics; Elasticity (ρ, v...) Einstein s Theory (g ab...) Thermodynamics of solids Statistical Mechanics of atoms/molecules Statistical mechanics of atoms of spacetime

9 GRAVITY AS AN EMERGENT PHENOMENON SAKHAROV PARADIGM SOLIDS SPACETIME Mechanics; Elasticity (ρ, v...) Einstein s Theory (g ab...) Thermodynamics of solids Statistical Mechanics of atoms/molecules Thermodynamics of spacetime Statistical mechanics of atoms of spacetime

10 BOLTZMANN AND THE POSTULATE OF ATOMS

11 BOLTZMANN AND THE POSTULATE OF ATOMS Temperature, Heat etc. demands microscopic degrees for freedom for their proper description.

12 BOLTZMANN AND THE POSTULATE OF ATOMS Temperature, Heat etc. demands microscopic degrees for freedom for their proper description. Exact nature of these degrees of freedom is irrelevant; their existence is vital. Entropy arises from the ignored degrees of freedom.

13 BOLTZMANN AND THE POSTULATE OF ATOMS Temperature, Heat etc. demands microscopic degrees for freedom for their proper description. Exact nature of these degrees of freedom is irrelevant; their existence is vital. Entropy arises from the ignored degrees of freedom. EXAMPLE: Most of fluid/gas dynamics can be understood phenomenologically. But fundamental explanation for temperature comes from the molecular/atomic structure of matter.

14 BOLTZMANN AND THE POSTULATE OF ATOMS Temperature, Heat etc. demands microscopic degrees for freedom for their proper description. Exact nature of these degrees of freedom is irrelevant; their existence is vital. Entropy arises from the ignored degrees of freedom. EXAMPLE: Most of fluid/gas dynamics can be understood phenomenologically. But fundamental explanation for temperature comes from the molecular/atomic structure of matter. Thermodynamics offers a connection between the two though the form of entropy functional, S[ξ]. No microstructure, no thermodynamics!

15 BOLTZMANN AND THE POSTULATE OF ATOMS Temperature, Heat etc. demands microscopic degrees for freedom for their proper description. Exact nature of these degrees of freedom is irrelevant; their existence is vital. Entropy arises from the ignored degrees of freedom. EXAMPLE: Most of fluid/gas dynamics can be understood phenomenologically. But fundamental explanation for temperature comes from the molecular/atomic structure of matter. Thermodynamics offers a connection between the two though the form of entropy functional, S[ξ]. No microstructure, no thermodynamics! You never took a course in quantum thermodynamics.

16 THE HISTORICAL SEQUENCE

17 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908).

18 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908). There is no guiding principle to obtain the field equations G ab = 8πT ab! We accepted it anyway (1915).

19 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908). There is no guiding principle to obtain the field equations G ab = 8πT ab! We accepted it anyway (1915). The solutions had horizons (like e.g., Schwarschild black hole, 1916). Inevitable and observer dependent.

20 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908). There is no guiding principle to obtain the field equations G ab = 8πT ab! We accepted it anyway (1915). The solutions had horizons (like e.g., Schwarschild black hole, 1916). Inevitable and observer dependent. Wheeler ( 1971): Can one violate second law of thermodynamics by hiding entropy behind a horizon?

21 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908). There is no guiding principle to obtain the field equations G ab = 8πT ab! We accepted it anyway (1915). The solutions had horizons (like e.g., Schwarschild black hole, 1916). Inevitable and observer dependent. Wheeler ( 1971): Can one violate second law of thermodynamics by hiding entropy behind a horizon? Bekenstein (1972): No! Horizons have entropy which goes up when you try this.

22 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908). There is no guiding principle to obtain the field equations G ab = 8πT ab! We accepted it anyway (1915). The solutions had horizons (like e.g., Schwarschild black hole, 1916). Inevitable and observer dependent. Wheeler ( 1971): Can one violate second law of thermodynamics by hiding entropy behind a horizon? Bekenstein (1972): No! Horizons have entropy which goes up when you try this. Black hole horizons have a temperature (1975)

23 THE HISTORICAL SEQUENCE Principle of Equivalence Gravity can be described by g ab ( 1908). There is no guiding principle to obtain the field equations G ab = 8πT ab! We accepted it anyway (1915). The solutions had horizons (like e.g., Schwarschild black hole, 1916). Inevitable and observer dependent. Wheeler ( 1971): Can one violate second law of thermodynamics by hiding entropy behind a horizon? Bekenstein (1972): No! Horizons have entropy which goes up when you try this. Black hole horizons have a temperature (1975) Rindler horizons have a temperature ( )

24 TEMPERATURE = T g 2π ( ck B ) HORIZON OBSERVER X

25 TEMPERATURE = T g 2π ( ck B ) HORIZON OBSERVER X Works for Blackholes, desitter, Rindler

26 WHY ARE HORIZONS HOT? (Where does Temperature spring from?!)

27 WHY ARE HORIZONS HOT? (Where does Temperature spring from?!) PERIODICITY IN IMAGINARY TIME } { FINITE TEMPERATURE exp( i t H) exp( β H)

28 WHY ARE HORIZONS HOT? (Where does Temperature spring from?!) PERIODICITY IN IMAGINARY TIME } { FINITE TEMPERATURE exp( i t H) exp( β H) SPACETIMES WITH HORIZONS EXHIBIT PERIODICITY IN IMAGINARY TIME = TEMPERATURE

29 ds 2 = dy 2 + dx 2 = r 2 dθ 2 + dr 2 Y θ = const θ X 0 θ < 2π X = r cos θ Y = r sin θ r = const

30 ds 2 = dt E 2 + dx 2 = g 2 r 2 dτ E 2 + dr 2 T E τ E = const gτ E X 0 τ E < 2π g X = r cos gτ E T E = r sin gτ E r = const

31 T E = it ds 2 = dt 2 + dx 2 = g 2 r 2 dτ 2 + dr 2 T τ E = iτ τ = const X X = r cosh gτ T = r sinh gτ r = const

32 SO, WHY FIX IT WHEN IT WORKS?

33 SO, WHY FIX IT WHEN IT WORKS? THIS CONVENTIONAL APPROACH HAS NO EXPLANATION FOR SEVERAL PECULIAR FEATURES WHICH NEED TO THE THOUGHT OF AS JUST ALGEBRAIC ACCIDENTS

34 SO, WHY FIX IT WHEN IT WORKS? THIS CONVENTIONAL APPROACH HAS NO EXPLANATION FOR SEVERAL PECULIAR FEATURES WHICH NEED TO THE THOUGHT OF AS JUST ALGEBRAIC ACCIDENTS PHYSICS PROGRESSES BY EXPLAINING FEATURES WHICH WE NEVER THOUGHT NEEDED ANY EXPLANATION!! EXAMPLE: m inertial = m grav

35 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons?

36 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π

37 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a 2 2 2

38 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a Multiply da to write: c ( g ) c 3 2π G d ( ) 1 4 4πa2 1 2 c 4 da G = P d ( ) 4π 3 a3

39 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a Multiply da to write: c ( g ) c 3 2π G d ( ) 1 4 4πa2 1 2 c 4 da G ( ) 4π = P d 3 a3 } {{ } P dv

40 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a Multiply da to write: ( g ) } c {{ 2π } k B T c 3 ( ) 1 G d 4 4πa2 1 2 c 4 da G ( ) 4π = P d 3 a3 } {{ } P dv

41 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a Multiply da to write: ( g ) } c {{ 2π } k B T c 3 ( ) 1 G d 4 4πa2 } {{ } k 1 B ds 1 c 4 da } 2{{ G } de ( ) 4π = P d 3 a3 } {{ } P dv

42 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a Multiply da to write: ( g ) } c {{ 2π } k B T c 3 ( ) 1 G d 4 4πa2 } {{ } k 1 B ds 1 c 4 da } 2{{ G } de ( ) 4π = P d 3 a3 } {{ } P dv Read off (with L 2 P G /c3 ): [TP, 2002] S = 1 (4πa 2 ) = 1 A H ; E = c4 4L 2 P 4 L 2 P 2G a = c4 G ( ) 1/2 AH 16π

43 1. Why do Einstein s equations reduce to a thermodynamic identity for virtual displacements of horizons? Spherically symmetric spacetime with horizon at r = a; surface gravity g: Temperature: k B T = ( ) g c 2π Einstein s equation at r = a is (textbook result!) [ c 4 ga G c 1 ] = 4πP a Multiply da to write: ( g ) } c {{ 2π } k B T c 3 ( ) 1 G d 4 4πa2 } {{ } k 1 B ds 1 c 4 da } 2{{ G } de ( ) 4π = P d 3 a3 } {{ } P dv Read off (with L 2 P G /c3 ): [TP, 2002] S = 1 (4πa 2 ) = 1 A H ; E = c4 4L 2 P 4 L 2 P 2G a = c4 G ( AH Works for Kerr, FRW,... [D. Kothawala et al., 06; Rong-Gen Cai, 06, 07] 16π ) 1/2

44 2. Why is Einstein-Hilbert action holographic?

45 2. Why is Einstein-Hilbert action holographic? Example: The standard action in from classical mechanics is: A q = dt L q (q, q); δq = 0 at t = (t 1, t 2 )

46 2. Why is Einstein-Hilbert action holographic? Example: The standard action in from classical mechanics is: A q = dt L q (q, q); δq = 0 at t = (t 1, t 2 ) But you can get the same equations from an action with second derivatives: A p = dt L p (q, q, q); δp = 0 at t = (t 1, t 2 ) L p = L q d dt ( q L ) q q

47 2. Why is Einstein-Hilbert action holographic? Example: The standard action in from classical mechanics is: A q = dt L q (q, q); δq = 0 at t = (t 1, t 2 ) But you can get the same equations from an action with second derivatives: A p = dt L p (q, q, q); δp = 0 at t = (t 1, t 2 ) L p = L q d dt ( q L ) q q Action for gravity has exactly this structure! [TP, 02, 05] A grav = d 4 x g R = d 4 x g [L bulk + L sur ] glsur = a (g ij gl bulk ( a g ij ) )

48 3. Why does the surface term give the horizon entropy?

49 3. Why does the surface term give the horizon entropy? In the gravitational action [TP, 02, 05] A grav = d 4 x g R = d 4 x g [L bulk + L sur ] glsur = a (g ij gl bulk ( a g ij ) throw away the A sur, vary the rest of the action, solve the field equation for a solution with the horizon. Then... )

50 3. Why does the surface term give the horizon entropy? In the gravitational action [TP, 02, 05] A grav = d 4 x g R = d 4 x g [L bulk + L sur ] glsur = a (g ij gl bulk ( a g ij ) throw away the A sur, vary the rest of the action, solve the field equation for a solution with the horizon. Then... You find that the part you threw away, the A sur, evaluated on any horizon gives its entropy! )

51 3. Why does the surface term give the horizon entropy? In the gravitational action [TP, 02, 05] A grav = d 4 x g R = d 4 x g [L bulk + L sur ] glsur = a (g ij gl bulk ( a g ij ) throw away the A sur, vary the rest of the action, solve the field equation for a solution with the horizon. Then... You find that the part you threw away, the A sur, evaluated on any horizon gives its entropy! In fact, one can develop a theory with A total = A sur + A matter using the virtual displacements of the horizon as key. [TP, 2005] )

52 ASIDE: SURFACE TERM AND ENTROPY

53 ASIDE: SURFACE TERM AND ENTROPY In a Riemann normal coordinates around any event P, we have g η + R x x and L Γ 2 + Γ Γ - Action is pure surface term! And a P b R ab.

54 ASIDE: SURFACE TERM AND ENTROPY In a Riemann normal coordinates around any event P, we have g η + R x x and L Γ 2 + Γ Γ - Action is pure surface term! And a P b R ab. Matter moving across local horizon = Horizon being shifted virtually to engulf matter.

55 ASIDE: SURFACE TERM AND ENTROPY In a Riemann normal coordinates around any event P, we have g η + R x x and L Γ 2 + Γ Γ - Action is pure surface term! And a P b R ab. Matter moving across local horizon = Horizon being shifted virtually to engulf matter. Insist that: Change in the gravitational entropy due to surface term = Change in the matter entropy outside for all local horizons.

56 ASIDE: SURFACE TERM AND ENTROPY In a Riemann normal coordinates around any event P, we have g η + R x x and L Γ 2 + Γ Γ - Action is pure surface term! And a P b R ab. Matter moving across local horizon = Horizon being shifted virtually to engulf matter. Insist that: Change in the gravitational entropy due to surface term = Change in the matter entropy outside for all local horizons. This leads to n a a (n b P b ) = n a n b T ab which leads to Einstein s equations! [TP, 2005,08]

57 4. Why do all these work for a much wider class of theories than Einstein gravity?

58 4. Why do all these work for a much wider class of theories than Einstein gravity? The connection between T ds = de + P dv and field equations works for Lanczos-Lovelock gravity in D dimensions. [A. Paranjape, S. Sarkar, TP, 06; Kothawala, TP, 08]

59 4. Why do all these work for a much wider class of theories than Einstein gravity? The connection between T ds = de + P dv and field equations works for Lanczos-Lovelock gravity in D dimensions. [A. Paranjape, S. Sarkar, TP, 06; Kothawala, TP, 08] The Lanczos-Lovelock Lagrangian has the same holographic redundancy.

60 4. Why do all these work for a much wider class of theories than Einstein gravity? The connection between T ds = de + P dv and field equations works for Lanczos-Lovelock gravity in D dimensions. [A. Paranjape, S. Sarkar, TP, 06; Kothawala, TP, 08] The Lanczos-Lovelock Lagrangian has the same holographic redundancy. In all these cases, the surface term gives rise to entropy of horizons. [A. Mukhopadhyay, TP, 06]

61 4. Why do all these work for a much wider class of theories than Einstein gravity? The connection between T ds = de + P dv and field equations works for Lanczos-Lovelock gravity in D dimensions. [A. Paranjape, S. Sarkar, TP, 06; Kothawala, TP, 08] The Lanczos-Lovelock Lagrangian has the same holographic redundancy. In all these cases, the surface term gives rise to entropy of horizons. [A. Mukhopadhyay, TP, 06] One can again develop a theory with A total = A sur + A matter using the virtual displacements of the horizon as key. [TP, 2005]

62 PRIMER ON LANCZOS-LOVELOCK GRAVITY T.P (2006); A.Mukhopadhyay and T.P (2006)

63 PRIMER ON LANCZOS-LOVELOCK GRAVITY T.P (2006); A.Mukhopadhyay and T.P (2006) A very natural, geometrical generalization of Einstein s theory in D-dimensions.

64 PRIMER ON LANCZOS-LOVELOCK GRAVITY T.P (2006); A.Mukhopadhyay and T.P (2006) A very natural, geometrical generalization of Einstein s theory in D-dimensions. The D-dimensional Lanczos-Lovelock Lagrangian is a polynomial in the curvature tensor: K L (D) = Q bcd a R a bcd = c m L (D) m m=1 ; L(D) m = 1 16π 2 m δ a 1a 2...a 2m b 1 b 2...b 2m R b 1b 2 a 1 a 2...R b 2m 1b 2m a 2m 1 a 2m,

65 PRIMER ON LANCZOS-LOVELOCK GRAVITY T.P (2006); A.Mukhopadhyay and T.P (2006) A very natural, geometrical generalization of Einstein s theory in D-dimensions. The D-dimensional Lanczos-Lovelock Lagrangian is a polynomial in the curvature tensor: K L (D) = Q bcd a R a bcd = c m L (D) m m=1 ; L(D) m = 1 16π 2 m δ a 1a 2...a 2m b 1 b 2...b 2m R b 1b 2 a 1 a 2...R b 2m 1b 2m a 2m 1 a 2m, The Lanczos-Lovelock Lagrangian separates to a bulk and surface terms [ ] gl = 2 c gq bcd a Γ a bd + 2 gq bcd a Γ a dkγ k bc L sur + L bulk and is holographic : [(D/2) m]l sur = i [ g ab δl bulk δ( i g ab ) + jg ab ] L bulk ( i j g ab )

66 PRIMER ON LANCZOS-LOVELOCK GRAVITY T.P (2006); A.Mukhopadhyay and T.P (2006) A very natural, geometrical generalization of Einstein s theory in D-dimensions. The D-dimensional Lanczos-Lovelock Lagrangian is a polynomial in the curvature tensor: K L (D) = Q bcd a R a bcd = c m L (D) m m=1 ; L(D) m = 1 16π 2 m δ a 1a 2...a 2m b 1 b 2...b 2m R b 1b 2 a 1 a 2...R b 2m 1b 2m a 2m 1 a 2m, The Lanczos-Lovelock Lagrangian separates to a bulk and surface terms [ ] gl = 2 c gq bcd a Γ a bd + 2 gq bcd a Γ a dkγ k bc L sur + L bulk and is holographic : [(D/2) m]l sur = i [ g ab δl bulk δ( i g ab ) + jg ab ] L bulk ( i j g ab ) The surface term is closely related to horizon entropy in Lanczos-Lovelock theory.

67 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP

68 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab.

69 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab. Around any event there exists local inertial frames AND local Rindler frames with a local horizon and temperature.

70 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab. Around any event there exists local inertial frames AND local Rindler frames with a local horizon and temperature. Can flow of matter across the local, hot, horizon hide entropy?

71 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab. Around any event there exists local inertial frames AND local Rindler frames with a local horizon and temperature. Can flow of matter across the local, hot, horizon hide entropy? Equivalently, can virtual displacements of a local patch of null surface, leading to flow of energy across a hot horizon allow you to hide entropy?

72 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab. Around any event there exists local inertial frames AND local Rindler frames with a local horizon and temperature. Can flow of matter across the local, hot, horizon hide entropy? Equivalently, can virtual displacements of a local patch of null surface, leading to flow of energy across a hot horizon allow you to hide entropy? No. The virtual displacement of a null surface should cost entropy, S grav.

73 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab. Around any event there exists local inertial frames AND local Rindler frames with a local horizon and temperature. Can flow of matter across the local, hot, horizon hide entropy? Equivalently, can virtual displacements of a local patch of null surface, leading to flow of energy across a hot horizon allow you to hide entropy? No. The virtual displacement of a null surface should cost entropy, S grav. Dynamics should now emerge from maximising S matter + S grav for all Rindler observers!.

74 REWRITING HISTORY: GRAVITY THE RIGHT WAY UP Principle of Equivalence Gravity can be described by g ab. Around any event there exists local inertial frames AND local Rindler frames with a local horizon and temperature. Can flow of matter across the local, hot, horizon hide entropy? Equivalently, can virtual displacements of a local patch of null surface, leading to flow of energy across a hot horizon allow you to hide entropy? No. The virtual displacement of a null surface should cost entropy, S grav. Dynamics should now emerge from maximising S matter + S grav for all Rindler observers!. Leads to gravity being an emergent phenomenon described by Einstein s equations at lowest order with calculable corrections.

75 A NEW VARIATIONAL PRINCIPLE

76 A NEW VARIATIONAL PRINCIPLE Associate with virtual displacements of null surfaces an entropy/ action which is quadratic in deformation field: [T.P, 08; T.P., A.Paranjape, 07] S[ξ] = S[ξ] grav + S matt [ξ] with S grav [ξ] = V d D x g4p abcd c ξ a d ξ b ; S matt = V d D x gt ab ξ a ξ b

77 A NEW VARIATIONAL PRINCIPLE Associate with virtual displacements of null surfaces an entropy/ action which is quadratic in deformation field: [T.P, 08; T.P., A.Paranjape, 07] with S grav [ξ] = V S[ξ] = S[ξ] grav + S matt [ξ] d D x g4p abcd c ξ a d ξ b ; S matt = Demand that the variation should constrain the background. V d D x gt ab ξ a ξ b

78 A NEW VARIATIONAL PRINCIPLE Associate with virtual displacements of null surfaces an entropy/ action which is quadratic in deformation field: [T.P, 08; T.P., A.Paranjape, 07] with S grav [ξ] = V S[ξ] = S[ξ] grav + S matt [ξ] d D x g4p abcd c ξ a d ξ b ; S matt = Demand that the variation should constrain the background. V d D x gt ab ξ a ξ b This leads to P abcd having a (RG-like) derivative expansion in powers of number of derivatives of the metric: P abcd (g ij, R ijkl ) = c 1 (1) P abcd (g ij ) + c 2 (2) P abcd (g ij, R ijkl ) +, The m-th order term is unique: (m) P abcd = ( L (m) / R abcd );

79 A NEW VARIATIONAL PRINCIPLE Associate with virtual displacements of null surfaces an entropy/ action which is quadratic in deformation field: [T.P, 08; T.P., A.Paranjape, 07] with S grav [ξ] = V S[ξ] = S[ξ] grav + S matt [ξ] d D x g4p abcd c ξ a d ξ b ; S matt = Demand that the variation should constrain the background. V d D x gt ab ξ a ξ b This leads to P abcd having a (RG-like) derivative expansion in powers of number of derivatives of the metric: P abcd (g ij, R ijkl ) = c 1 (1) P abcd (g ij ) + c 2 (2) P abcd (g ij, R ijkl ) +, The m-th order term is unique: (m) P abcd = ( L (m) / R abcd ); Example: The lowest order term is: d D x ( S 1 [ξ] = a ξ b b ξ a ( c ξ c ) 2) 8π V

80 ENTROPY MAXIMIZATION LEADS TO DYNAMICS

81 ENTROPY MAXIMIZATION LEADS TO DYNAMICS Demand that δs = 0 for variations of all null vectors: This leads to Lanczos-Lovelock theory with an arbitrary cosmological constant: [ 16π P ijk b R a ijk 1 ] 2 δa b L m (D) = 8πT a b + Λδb a,

82 ENTROPY MAXIMIZATION LEADS TO DYNAMICS Demand that δs = 0 for variations of all null vectors: This leads to Lanczos-Lovelock theory with an arbitrary cosmological constant: [ 16π P ijk b R a ijk 1 ] 2 δa b L m (D) = 8πT a b + Λδb a, To the lowest order we get Einstein s theory with cosmological constant as integration constant. Equivalent to (G ab 8πT ab )ξ a ξ b = 0 ; (for all null ξ a )

83 ENTROPY MAXIMIZATION LEADS TO DYNAMICS Demand that δs = 0 for variations of all null vectors: This leads to Lanczos-Lovelock theory with an arbitrary cosmological constant: [ 16π P ijk b R a ijk 1 ] 2 δa b L m (D) = 8πT a b + Λδb a, To the lowest order we get Einstein s theory with cosmological constant as integration constant. Equivalent to (G ab 8πT ab )ξ a ξ b = 0 ; (for all null ξ a ) In a derivative coupling expansion, Lanczos-Lovelock terms are calculable corrections.

84 EXTREMUM VALUE OF ENTROPY

85 EXTREMUM VALUE OF ENTROPY The extremum value can be computed on-shell on a solution.

86 EXTREMUM VALUE OF ENTROPY The extremum value can be computed on-shell on a solution. On any solution with horizon, it gives the correct Wald entropy: S H [on shell] = 2π H P abcd n ab n cd ɛ = K m=1 = 1 4 A + (Corrections) 4πmc m H d D 2 x σl (D 2) (m 1),

87 EXTREMUM VALUE OF ENTROPY The extremum value can be computed on-shell on a solution. On any solution with horizon, it gives the correct Wald entropy: S H [on shell] = 2π H P abcd n ab n cd ɛ = K m=1 = 1 4 A + (Corrections) 4πmc m H d D 2 x σl (D 2) (m 1),

88 EXTREMUM VALUE OF ENTROPY The extremum value can be computed on-shell on a solution. On any solution with horizon, it gives the correct Wald entropy: S H [on shell] = 2π H P abcd n ab n cd ɛ = K m=1 = 1 4 A + (Corrections) 4πmc m H d D 2 x σl (D 2) (m 1), In the semiclassical limit, we are led to the result that gravitational (Wald) entropy is quantised S H [on shell] = 2πn. To the lowest order this leads to area quantisation.

89 EXTREMUM VALUE OF ENTROPY The extremum value can be computed on-shell on a solution. On any solution with horizon, it gives the correct Wald entropy: S K H [on shell] = 2π P abcd n ab n cd ɛ = 4πmc m d D 2 (D 2) x σl (m 1), H m=1 = 1 4 A + (Corrections) H In the semiclassical limit, we are led to the result that gravitational (Wald) entropy is quantised S H [on shell] = 2πn. To the lowest order this leads to area quantisation. Comparison with quasi-normal modes approach shows that it is the gravitational entropy which is quantised in Lanczos-Lovelock theories. [Kothawala, Sarkar,TP, 2008]

90 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations.

91 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations. The real trouble with cosmological constant is that gravity seems to be immune to bulk vacuum energy.

92 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations. The real trouble with cosmological constant is that gravity seems to be immune to bulk vacuum energy. The matter sector and its equations are invariant under the shift of the Lagrangian by a constant: L matter L matter ρ.

93 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations. The real trouble with cosmological constant is that gravity seems to be immune to bulk vacuum energy. The matter sector and its equations are invariant under the shift of the Lagrangian by a constant: L matter L matter ρ. But this changes energy momentum tensor by T ab T ab + ρg ab and gravity sector is not invariant under this transformation.

94 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations. The real trouble with cosmological constant is that gravity seems to be immune to bulk vacuum energy. The matter sector and its equations are invariant under the shift of the Lagrangian by a constant: L matter L matter ρ. But this changes energy momentum tensor by T ab T ab + ρg ab and gravity sector is not invariant under this transformation. So after you have solved the cosmological constant problem, if someone introduces L matter L matter ρ, you are in trouble again!

95 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations. The real trouble with cosmological constant is that gravity seems to be immune to bulk vacuum energy. The matter sector and its equations are invariant under the shift of the Lagrangian by a constant: L matter L matter ρ. But this changes energy momentum tensor by T ab T ab + ρg ab and gravity sector is not invariant under this transformation. So after you have solved the cosmological constant problem, if someone introduces L matter L matter ρ, you are in trouble again! The only way out is to have a formalism for gravity which is invariant under T ab T ab + ρg ab.

96 5. How come gravity is immune to ground state energy? A cosmological constant term ρ 0 δj i in Einsteins equation acts like matter with negative pressure; consistent with all dark energy observations. The real trouble with cosmological constant is that gravity seems to be immune to bulk vacuum energy. The matter sector and its equations are invariant under the shift of the Lagrangian by a constant: L matter L matter ρ. But this changes energy momentum tensor by T ab T ab + ρg ab and gravity sector is not invariant under this transformation. So after you have solved the cosmological constant problem, if someone introduces L matter L matter ρ, you are in trouble again! The only way out is to have a formalism for gravity which is invariant under T ab T ab + ρg ab. All these have nothing to do with observations of accelerated universe! Cosmological constant problem existed earlier and will continue to exist even if all these observations go away!

97 A simple theorem

98 Assume: A simple theorem (a) Metric g ab is a dynamical variable that is varied in the action.

99 A simple theorem Assume: (a) Metric g ab is a dynamical variable that is varied in the action. (b) Action is generally covariant.

100 A simple theorem Assume: (a) Metric g ab is a dynamical variable that is varied in the action. (b) Action is generally covariant. (c) Equations of motion for matter sector (at low energy) is invariant under L matter L matter ρ 0.

101 A simple theorem Assume: (a) Metric g ab is a dynamical variable that is varied in the action. (b) Action is generally covariant. (c) Equations of motion for matter sector (at low energy) is invariant under L matter L matter ρ 0. Then cosmological constant problem cannot be solved; that is, gravitational equations cannot be invariant under T ab T ab ρ 0 g ab.

102 A simple theorem Assume: (a) Metric g ab is a dynamical variable that is varied in the action. (b) Action is generally covariant. (c) Equations of motion for matter sector (at low energy) is invariant under L matter L matter ρ 0. Then cosmological constant problem cannot be solved; that is, gravitational equations cannot be invariant under T ab T ab ρ 0 g ab. We have dropped the assumption that g ab is the dynamical variable.

103 A simple theorem Assume: (a) Metric g ab is a dynamical variable that is varied in the action. (b) Action is generally covariant. (c) Equations of motion for matter sector (at low energy) is invariant under L matter L matter ρ 0. Then cosmological constant problem cannot be solved; that is, gravitational equations cannot be invariant under T ab T ab ρ 0 g ab. We have dropped the assumption that g ab is the dynamical variable. It also makes the variational principle for Lanczos-Lovelock theories well-defined.

104 GRAVITY IS IMMUNE TO BULK ENERGY

105 GRAVITY IS IMMUNE TO BULK ENERGY The action/entropy functional is invariant under the shift T ab T ab + ρg ab!

106 GRAVITY IS IMMUNE TO BULK ENERGY The action/entropy functional is invariant under the shift T ab T ab + ρg ab! The field equations have a new gauge freedom and has the form: P ijk b R a ijk 1 2 Lδa b κt a b = (constant)δa b

107 GRAVITY IS IMMUNE TO BULK ENERGY The action/entropy functional is invariant under the shift T ab T ab + ρg ab! The field equations have a new gauge freedom and has the form: P ijk b R a ijk 1 2 Lδa b κt a b = (constant)δa b Introduces a new length scale L H. (Observationally, L P /L H exp( 2π 4 ).) Analogy: Solve G ab = 0 to get Schwarzchild metric with a parameter M.

108 GRAVITY IS IMMUNE TO BULK ENERGY The action/entropy functional is invariant under the shift T ab T ab + ρg ab! The field equations have a new gauge freedom and has the form: P ijk b R a ijk 1 2 Lδa b κt a b = (constant)δa b Introduces a new length scale L H. (Observationally, L P /L H exp( 2π 4 ).) Analogy: Solve G ab = 0 to get Schwarzchild metric with a parameter M. We don t worry about the value of (M/M planck ) because M is an integration constant; not a parameter in the equations.

109 GRAVITY IS IMMUNE TO BULK ENERGY The action/entropy functional is invariant under the shift T ab T ab + ρg ab! The field equations have a new gauge freedom and has the form: P ijk b R a ijk 1 2 Lδa b κt a b = (constant)δa b Introduces a new length scale L H. (Observationally, L P /L H exp( 2π 4 ).) Analogy: Solve G ab = 0 to get Schwarzchild metric with a parameter M. We don t worry about the value of (M/M planck ) because M is an integration constant; not a parameter in the equations. Given L P and L H we have ρ UV = 1/L 4 P and ρ IR = 1/L 4 H. The observed values is: ρ DE ρ UV ρ IR 1 L 2 P L2 H H2 G

110 GRAVITY IS IMMUNE TO BULK ENERGY The action/entropy functional is invariant under the shift T ab T ab + ρg ab! The field equations have a new gauge freedom and has the form: P ijk b R a ijk 1 2 Lδa b κt a b = (constant)δa b Introduces a new length scale L H. (Observationally, L P /L H exp( 2π 4 ).) Analogy: Solve G ab = 0 to get Schwarzchild metric with a parameter M. We don t worry about the value of (M/M planck ) because M is an integration constant; not a parameter in the equations. Given L P and L H we have ρ UV = 1/L 4 P and ρ IR = 1/L 4 H. The observed values is: ρ DE ρ UV ρ IR 1 L 2 P L2 H H2 G The hierarchy: ρ vac = [ 1 L 4 P } {{ } x, 1 L 4 P ( LP L H ) 2 } {{ } x 2 1/2, 1 L 4 P ( LP L H ) 4, ]

111 System Material body Spacetime

112 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature

113 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature Evidence for Existence of Existence of Microstructure Temperature Temperature.

114 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature Evidence for Existence of Existence of Microstructure Temperature Temperature. Why does Due to ignoring Existence of null Entropy arise? microscopic d.o.f surfaces in LRF.

115 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature Evidence for Existence of Existence of Microstructure Temperature Temperature. Why does Due to ignoring Existence of null Entropy arise? microscopic d.o.f surfaces in LRF. Thermodynamic T ds = de + P dv T ds = de + P dv description (aka Einsteins equations!)

116 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature Evidence for Existence of Existence of Microstructure Temperature Temperature. Why does Due to ignoring Existence of null Entropy arise? microscopic d.o.f surfaces in LRF. Thermodynamic T ds = de + P dv T ds = de + P dv description (aka Einsteins equations!) Microscopic description Randomly moving atoms Fluctuations of null surfaces

117 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature Evidence for Existence of Existence of Microstructure Temperature Temperature. Why does Due to ignoring Existence of null Entropy arise? microscopic d.o.f surfaces in LRF. Thermodynamic T ds = de + P dv T ds = de + P dv description (aka Einsteins equations!) Microscopic description Randomly moving atoms Fluctuations of null surfaces Connection with Specify the entropy Specify the entropy thermodynamics

118 System Material body Spacetime Macroscopic description Density, Pressure etc. Metric, Curvature Evidence for Existence of Existence of Microstructure Temperature Temperature. Why does Due to ignoring Existence of null Entropy arise? microscopic d.o.f surfaces in LRF. Thermodynamic T ds = de + P dv T ds = de + P dv description (aka Einsteins equations!) Microscopic description Randomly moving atoms Fluctuations of null surfaces Connection with Specify the entropy Specify the entropy thermodynamics Resulting equation Classical / Quantum Einsteins theory with calculable corrections

119 SUMMARY

120 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x).

121 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x). To go from thermodynamics to statistical mechanics, we have to postulate new degrees of freedom and an entropy functional.

122 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x). To go from thermodynamics to statistical mechanics, we have to postulate new degrees of freedom and an entropy functional. Local Rindler observers around any event attribute entropy to local patch of null surface. This is needed for consistency.

123 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x). To go from thermodynamics to statistical mechanics, we have to postulate new degrees of freedom and an entropy functional. Local Rindler observers around any event attribute entropy to local patch of null surface. This is needed for consistency. Maximizing the entropy associated with all null surfaces gives Einstein s theory with Lanczos-Lovelock corrections [but not, e.g., f(r) gravity].

124 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x). To go from thermodynamics to statistical mechanics, we have to postulate new degrees of freedom and an entropy functional. Local Rindler observers around any event attribute entropy to local patch of null surface. This is needed for consistency. Maximizing the entropy associated with all null surfaces gives Einstein s theory with Lanczos-Lovelock corrections [but not, e.g., f(r) gravity]. The deep connection between gravity and thermodynamics goes well beyond Einstein s theory. Closely related to the holographic structure of Lanczos-Lovelock theories.

125 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x). To go from thermodynamics to statistical mechanics, we have to postulate new degrees of freedom and an entropy functional. Local Rindler observers around any event attribute entropy to local patch of null surface. This is needed for consistency. Maximizing the entropy associated with all null surfaces gives Einstein s theory with Lanczos-Lovelock corrections [but not, e.g., f(r) gravity]. The deep connection between gravity and thermodynamics goes well beyond Einstein s theory. Closely related to the holographic structure of Lanczos-Lovelock theories. Connects with the radial displacements of horizons and T ds = de + P dv as the key to obtaining a thermodynamic interpretation of gravitational theories.

126 SUMMARY Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The g ab (t, x) etc. are like ρ(t, x), v(t, x). To go from thermodynamics to statistical mechanics, we have to postulate new degrees of freedom and an entropy functional. Local Rindler observers around any event attribute entropy to local patch of null surface. This is needed for consistency. Maximizing the entropy associated with all null surfaces gives Einstein s theory with Lanczos-Lovelock corrections [but not, e.g., f(r) gravity]. The deep connection between gravity and thermodynamics goes well beyond Einstein s theory. Closely related to the holographic structure of Lanczos-Lovelock theories. Connects with the radial displacements of horizons and T ds = de + P dv as the key to obtaining a thermodynamic interpretation of gravitational theories. Connects with A sur giving the horizon entropy; leads to quantisation of Wald entropy.

127 REFERENCES 1. Original ideas were developed in: T. Padmanabhan, Class.Quan.Grav. 19, 5387 (2002). [gr-qc/ ] T. Padmanabhan, Gen.Rel.Grav., (2002) [gr-qc/ ] [Second Prize essay; Gravity Research Foundation Essay Contest, 2002] T. Padmanabhan, Gen.Rel.Grav., 35, (2003) [Fifth Prize essay; Gravity Research Foundation Essay Contest, 2003] T. Padmanabhan, Gen.Rel.Grav., Foundation Essay Contest, 2006] 38, (2006) [Third Prize essay; Gravity Research T. Padmanabhan, Gravity: the Inside Story, Gen.Rel.Grav., 40, (2008) [First Prize essay; Gravity Research Foundation Essay Contest, 2008] 2. Summary of the basic approach is in: T. Padmanabhan Phys. Reports, 406, 49 (2005) [gr-qc/ ] T. Padmanabhan Gen.Rel.Grav., 40, (2008) [arxiv: ] T. Padmanabhan Dark Energy and its implications for Gravity (2008) [arxiv: ] 3. Also see: A. Mukhopadhyay, T. Padmanabhan, Phys.Rev., D 74, (2006) [hep-th/ ] T. Padmanabhan, Aseem Paranjape, Phys.Rev.D, 75, (2007). [gr-qc/ ]