gravity: the inside story
TRANSCRIPT
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GRAVITY: THE INSIDE STORY
T. Padmanabhan
(IUCAA, Pune, India)
VR Lecture, IAGRG Meeting
Kolkatta, 28 Jan 09
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CONVENTIONAL VIEW
GRAVITY AS A FUNDAMENTAL
INTERACTION
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CONVENTIONAL VIEW
GRAVITY AS A FUNDAMENTAL
THERMODYNAMICS
INTERACTION
SPACETIME
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NEW PERSPECTIVE
THERMODYNAMICS
SPACETIME
GRAVITY IS AN EMERGENT
PHENOMENON
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NEW PERSPECTIVE
THERMODYNAMICS
SPACETIME
GRAVITY IS AN EMERGENT
PHENOMENON
GRAVITY IS THE THERMODYNAMIC LIMIT OF THE
STATISTICAL MECHANICS OF ‘ATOMS OF SPACETIME’
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GRAVITY AS AN EMERGENT PHENOMENON
SAKHAROV PARADIGM
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GRAVITY AS AN EMERGENT PHENOMENON
SAKHAROV PARADIGM
SOLIDS SPACETIME
Mechanics; Elasticity (ρ, v ...) Einstein’s Theory (gab ...)
Statistical Mechanics Statistical mechanics
of atoms/molecules of “atoms of spacetime”
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GRAVITY AS AN EMERGENT PHENOMENON
SAKHAROV PARADIGM
SOLIDS SPACETIME
Mechanics; Elasticity (ρ, v ...) Einstein’s Theory (gab ...)
Thermodynamics of solids
Statistical Mechanics Statistical mechanics
of atoms/molecules of “atoms of spacetime”
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GRAVITY AS AN EMERGENT PHENOMENON
SAKHAROV PARADIGM
SOLIDS SPACETIME
Mechanics; Elasticity (ρ, v ...) Einstein’s Theory (gab ...)
Thermodynamics of solids Thermodynamics of spacetime
Statistical Mechanics Statistical mechanics
of atoms/molecules of “atoms of spacetime”
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BOLTZMANN AND THE POSTULATE OF ‘ATOMS’
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BOLTZMANN AND THE POSTULATE OF ‘ATOMS’
• Temperature, Heat etc. demands microscopic degrees for freedom
for their proper description.
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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.
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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.
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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!
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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’.
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THE HISTORICAL SEQUENCE
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
• There is no guiding principle to obtain the field equations
Gab = 8πTab! We accepted it anyway (1915).
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
• There is no guiding principle to obtain the field equations
Gab = 8πTab! We accepted it anyway (1915).
• The solutions had horizons (like e.g., Schwarschild black hole,
1916). Inevitable and observer dependent.
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
• There is no guiding principle to obtain the field equations
Gab = 8πTab! 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 ?
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
• There is no guiding principle to obtain the field equations
Gab = 8πTab! 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.
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
• There is no guiding principle to obtain the field equations
Gab = 8πTab! 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)
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THE HISTORICAL SEQUENCE
• Principle of Equivalence ⇒ Gravity can be described by gab (∼1908).
• There is no guiding principle to obtain the field equations
Gab = 8πTab! 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 (1975-76)
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TEMPERATURE =g
2π
(~
ckB
)
X
T
OBSERVER
HORIZON
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TEMPERATURE =g
2π
(~
ckB
)
X
T
OBSERVER
HORIZON
Works for Blackholes, deSitter, Rindler
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WHY ARE HORIZONS HOT ?
(Where does Temperature spring from ?!)
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WHY ARE HORIZONS HOT ?
(Where does Temperature spring from ?!)
PERIODICITY IN
IMAGINARY TIME
}
⇐⇒{
FINITE TEMPERATURE
exp(−i t H) ⇐⇒ exp(− β H)
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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
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X = r cos θ
Y = r sin θ
θ = const
X
Y
θ
r = const
0 ≤ θ < 2π
ds2 = dY 2 + dX2 = r2dθ2 + dr2
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τE= con
st
X
TE
r = const
0 ≤ τE <2π
g
ds2 = dTE2 + dX2 = g2r2dτE
2 + dr2
gτE
X = r cos gτE
TE = r sin gτE
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ds2 = −dT 2 + dX2 = −g2r2dτ 2 + dr2
τ = const
X
T
r = constX = r cosh gτ
T = r sinh gτ
TE = iT
τE = iτ
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SO, WHY FIX IT WHEN IT WORKS ?
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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’
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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: minertial = mgrav
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1. Why do Einstein’s equations reduce to a thermodynamic identity
for virtual displacements of horizons ?
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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: kBT =
(~
c
)g
2π
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
• Multiply da to write:
~
c
( g
2π
) c3
G~d
(1
44πa2
)
− 1
2
c4da
G= Pd
(4π
3a3
)
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
• Multiply da to write:
~
c
( g
2π
) c3
G~d
(1
44πa2
)
− 1
2
c4da
G= Pd
(4π
3a3
)
︸ ︷︷ ︸
P dV
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
• Multiply da to write:
~
c
( g
2π
)
︸ ︷︷ ︸
kBT
c3
G~d
(1
44πa2
)
− 1
2
c4da
G= Pd
(4π
3a3
)
︸ ︷︷ ︸
P dV
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
• Multiply da to write:
~
c
( g
2π
)
︸ ︷︷ ︸
kBT
c3
G~d
(1
44πa2
)
︸ ︷︷ ︸
k−1B dS
− 1
2
c4da
G︸ ︷︷ ︸
−dE
= Pd
(4π
3a3
)
︸ ︷︷ ︸
P dV
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
• Multiply da to write:
~
c
( g
2π
)
︸ ︷︷ ︸
kBT
c3
G~d
(1
44πa2
)
︸ ︷︷ ︸
k−1B dS
− 1
2
c4da
G︸ ︷︷ ︸
−dE
= Pd
(4π
3a3
)
︸ ︷︷ ︸
P dV
• Read off (with L2P ≡ G~/c3): [TP, 2002]
S =1
4L2P
(4πa2) =1
4
AH
L2P
; E =c4
2Ga =
c4
G
(AH
16π
)1/2
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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: kBT =
(~
c
)g
2π
• Einstein’s equation at r = a is (textbook result!)
c4
G
[ga
c2− 1
2
]
= 4πPa2
• Multiply da to write:
~
c
( g
2π
)
︸ ︷︷ ︸
kBT
c3
G~d
(1
44πa2
)
︸ ︷︷ ︸
k−1B dS
− 1
2
c4da
G︸ ︷︷ ︸
−dE
= Pd
(4π
3a3
)
︸ ︷︷ ︸
P dV
• Read off (with L2P ≡ G~/c3): [TP, 2002]
S =1
4L2P
(4πa2) =1
4
AH
L2P
; E =c4
2Ga =
c4
G
(AH
16π
)1/2
• Works for Kerr, FRW, .... [D. Kothawala et al., 06; Rong-Gen Cai, 06, 07]
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2. Why is Einstein-Hilbert action holographic ?
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2. Why is Einstein-Hilbert action holographic ?
• Example: The standard action in from classical mechanics is:
Aq =
∫
dt Lq(q, q̇); δq = 0 at t = (t1, t2)
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2. Why is Einstein-Hilbert action holographic ?
• Example: The standard action in from classical mechanics is:
Aq =
∫
dt Lq(q, q̇); δq = 0 at t = (t1, t2)
• But you can get the same equations from an action with second derivatives:
Ap =
∫
dt Lp(q, q̇, q̈); δp = 0 at t = (t1, t2)
Lp = Lq −d
dt
(
q∂Lq
∂q̇
)
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2. Why is Einstein-Hilbert action holographic ?
• Example: The standard action in from classical mechanics is:
Aq =
∫
dt Lq(q, q̇); δq = 0 at t = (t1, t2)
• But you can get the same equations from an action with second derivatives:
Ap =
∫
dt Lp(q, q̇, q̈); δp = 0 at t = (t1, t2)
Lp = Lq −d
dt
(
q∂Lq
∂q̇
)
• Action for gravity has exactly this structure! [TP, 02, 05]
Agrav =
∫
d4x√−g R =
∫
d4x√−g [Lbulk + Lsur]
√−gLsur = −∂a
(
gij∂√−gLbulk
∂(∂agij)
)
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3. Why does the surface term give the horizon entropy ?
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3. Why does the surface term give the horizon entropy ?
• In the gravitational action [TP, 02, 05]
Agrav =
∫
d4x√−g R =
∫
d4x√−g [Lbulk + Lsur]
√−gLsur = −∂a
(
gij∂√−gLbulk
∂(∂agij)
)
throw away the Asur, vary the rest of the action, solve the field
equation for a solution with the horizon. Then .....
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3. Why does the surface term give the horizon entropy ?
• In the gravitational action [TP, 02, 05]
Agrav =
∫
d4x√−g R =
∫
d4x√−g [Lbulk + Lsur]
√−gLsur = −∂a
(
gij∂√−gLbulk
∂(∂agij)
)
throw away the Asur, 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 Asur, evaluated on any
horizon gives its entropy !
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3. Why does the surface term give the horizon entropy ?
• In the gravitational action [TP, 02, 05]
Agrav =
∫
d4x√−g R =
∫
d4x√−g [Lbulk + Lsur]
√−gLsur = −∂a
(
gij∂√−gLbulk
∂(∂agij)
)
throw away the Asur, 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 Asur, evaluated on any
horizon gives its entropy !
• In fact, one can develop a theory with Atotal = Asur + Amatter using
the virtual displacements of the horizon as key. [TP, 2005]
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ASIDE: SURFACE TERM AND ENTROPY
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ASIDE: SURFACE TERM AND ENTROPY
• In a Riemann normal coordinates around any event P,
we have g → η + R x x and L ∼ Γ2 + ∂Γ → ∂Γ - Action is
pure surface term! And ∂aPb ∼ Rab.
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ASIDE: SURFACE TERM AND ENTROPY
• In a Riemann normal coordinates around any event P,
we have g → η + R x x and L ∼ Γ2 + ∂Γ → ∂Γ - Action is
pure surface term! And ∂aPb ∼ Rab.
• Matter moving across local horizon = Horizon being
shifted virtually to engulf matter.
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ASIDE: SURFACE TERM AND ENTROPY
• In a Riemann normal coordinates around any event P,
we have g → η + R x x and L ∼ Γ2 + ∂Γ → ∂Γ - Action is
pure surface term! And ∂aPb ∼ Rab.
• 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.
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ASIDE: SURFACE TERM AND ENTROPY
• In a Riemann normal coordinates around any event P,
we have g → η + R x x and L ∼ Γ2 + ∂Γ → ∂Γ - Action is
pure surface term! And ∂aPb ∼ Rab.
• 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 na∂a(nbPb) = nanbTab which leads to
Einstein’s equations! [TP, 2005,08]
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4. Why do all these work for a much wider class
of theories than Einstein gravity ?
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4. Why do all these work for a much wider class
of theories than Einstein gravity ?
• The connection between TdS = dE + PdV and field
equations works for Lanczos-Lovelock gravity in D
dimensions. [A. Paranjape, S. Sarkar, TP, 06; Kothawala, TP, 08]
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4. Why do all these work for a much wider class
of theories than Einstein gravity ?
• The connection between TdS = dE + PdV 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’.
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4. Why do all these work for a much wider class
of theories than Einstein gravity ?
• The connection between TdS = dE + PdV 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]
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4. Why do all these work for a much wider class
of theories than Einstein gravity ?
• The connection between TdS = dE + PdV 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
Atotal = Asur + Amatter using the virtual displacements of
the horizon as key. [TP, 2005]
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PRIMER ON LANCZOS-LOVELOCK GRAVITY
T.P (2006); A.Mukhopadhyay and T.P (2006)
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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.
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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:
L(D) = Q bcda Ra
bcd =
K∑
m=1
cmL(D)m ; L(D)
m =1
16π2−mδa1a2...a2m
b1b2...b2mRb1b2
a1a2....Rb2m−1b2m
a2m−1a2m,
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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:
L(D) = Q bcda Ra
bcd =
K∑
m=1
cmL(D)m ; L(D)
m =1
16π2−mδa1a2...a2m
b1b2...b2mRb1b2
a1a2....Rb2m−1b2m
a2m−1a2m,
• The Lanczos-Lovelock Lagrangian separates to a bulk and surface terms
√−gL = 2∂c
[√−gQ bcd
a Γabd
]+ 2
√−gQ bcd
a ΓadkΓ
kbc ≡ Lsur + Lbulk
and is ‘holographic’:
[(D/2) − m]Lsur = −∂i
[
gabδLbulk
δ(∂igab)+ ∂jgab
∂Lbulk
∂(∂i∂jgab)
]
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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:
L(D) = Q bcda Ra
bcd =
K∑
m=1
cmL(D)m ; L(D)
m =1
16π2−mδa1a2...a2m
b1b2...b2mRb1b2
a1a2....Rb2m−1b2m
a2m−1a2m,
• The Lanczos-Lovelock Lagrangian separates to a bulk and surface terms
√−gL = 2∂c
[√−gQ bcd
a Γabd
]+ 2
√−gQ bcd
a ΓadkΓ
kbc ≡ Lsur + Lbulk
and is ‘holographic’:
[(D/2) − m]Lsur = −∂i
[
gabδLbulk
δ(∂igab)+ ∂jgab
∂Lbulk
∂(∂i∂jgab)
]
• The surface term is closely related to horizon entropy in Lanczos-Lovelock theory.
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
• Around any event there exists local inertial frames AND local
Rindler frames with a local horizon and temperature.
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
• 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 ?
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
• 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 ?
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
• 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,
Sgrav.
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
• 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,
Sgrav.
• Dynamics should now emerge from maximising Smatter + Sgrav for all
Rindler observers!.
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REWRITING HISTORY: GRAVITY – THE ‘RIGHT WAY UP’
• Principle of Equivalence ⇒ Gravity can be described by gab.
• 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,
Sgrav.
• Dynamics should now emerge from maximising Smatter + Sgrav for all
Rindler observers!.
• Leads to gravity being an emergent phenomenon described by
Einstein’s equations at lowest order with calculable corrections.
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A NEW VARIATIONAL PRINCIPLE
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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 + Smatt[ξ]
with
Sgrav[ξ] =
∫
VdDx
√−g4P abcd∇cξa∇dξb; Smatt =
∫
VdDx
√−gT abξaξb
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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 + Smatt[ξ]
with
Sgrav[ξ] =
∫
VdDx
√−g4P abcd∇cξa∇dξb; Smatt =
∫
VdDx
√−gT abξaξb
• Demand that the variation should constrain the background.
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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 + Smatt[ξ]
with
Sgrav[ξ] =
∫
VdDx
√−g4P abcd∇cξa∇dξb; Smatt =
∫
VdDx
√−gT abξaξb
• Demand that the variation should constrain the background.
• This leads to P abcd having a (RG-like) derivative expansion in powers of number
of derivatives of the metric:
P abcd(gij, Rijkl) = c1
(1)
P abcd(gij) + c2
(2)
P abcd(gij, Rijkl) + · · · ,
• The m-th order term is unique:(m)
P abcd = (∂L(m)/∂Rabcd);
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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 + Smatt[ξ]
with
Sgrav[ξ] =
∫
VdDx
√−g4P abcd∇cξa∇dξb; Smatt =
∫
VdDx
√−gT abξaξb
• Demand that the variation should constrain the background.
• This leads to P abcd having a (RG-like) derivative expansion in powers of number
of derivatives of the metric:
P abcd(gij, Rijkl) = c1
(1)
P abcd(gij) + c2
(2)
P abcd(gij, Rijkl) + · · · ,
• The m-th order term is unique:(m)
P abcd = (∂L(m)/∂Rabcd);
• Example: The lowest order term is:
S1[ξ] =
∫
V
dDx
8π
(∇aξ
b∇bξa − (∇cξ
c)2)
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ENTROPY MAXIMIZATION LEADS TO DYNAMICS
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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 ijkb Ra
ijk −1
2δa
bL(D)m
]
= 8πT ab + Λδa
b ,
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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 ijkb Ra
ijk −1
2δa
bL(D)m
]
= 8πT ab + Λδa
b ,
• To the lowest order we get Einstein’s theory with cosmological
constant as integration constant. Equivalent to
(Gab − 8πTab)ξaξb = 0 ; (for all null ξa)
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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 ijkb Ra
ijk −1
2δa
bL(D)m
]
= 8πT ab + Λδa
b ,
• To the lowest order we get Einstein’s theory with cosmological
constant as integration constant. Equivalent to
(Gab − 8πTab)ξaξb = 0 ; (for all null ξa)
• In a derivative coupling expansion, Lanczos-Lovelock terms are
calculable corrections.
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EXTREMUM VALUE OF ENTROPY
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EXTREMUM VALUE OF ENTROPY
• The extremum value can be computed on-shell on a solution.
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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
∮
HP abcdnabncdε̃ =
K∑
m=1
4πmcm
∫
HdD−2x⊥
√σL(D−2)
(m−1) ,
=1
4A⊥ + (Corrections)
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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
∮
HP abcdnabncdε̃ =
K∑
m=1
4πmcm
∫
HdD−2x⊥
√σL(D−2)
(m−1) ,
=1
4A⊥ + (Corrections)
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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
∮
HP abcdnabncdε̃ =
K∑
m=1
4πmcm
∫
HdD−2x⊥
√σL(D−2)
(m−1) ,
=1
4A⊥ + (Corrections)
• 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.
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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
∮
HP abcdnabncdε̃ =
K∑
m=1
4πmcm
∫
HdD−2x⊥
√σL(D−2)
(m−1) ,
=1
4A⊥ + (Corrections)
• 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]
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5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij in Einsteins equation acts like matter with
negative pressure; consistent with all dark energy observations.
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5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij 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.
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5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij 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: Lmatter → Lmatter − ρ.
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5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij 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: Lmatter → Lmatter − ρ.
• But this changes energy momentum tensor by Tab → Tab + ρgab and gravity sector
is not invariant under this transformation.
![Page 94: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/94.jpg)
5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij 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: Lmatter → Lmatter − ρ.
• But this changes energy momentum tensor by Tab → Tab + ρgab and gravity sector
is not invariant under this transformation.
• So after you have “solved” the cosmological constant problem, if someone
introduces Lmatter → Lmatter − ρ, you are in trouble again!
![Page 95: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/95.jpg)
5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij 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: Lmatter → Lmatter − ρ.
• But this changes energy momentum tensor by Tab → Tab + ρgab and gravity sector
is not invariant under this transformation.
• So after you have “solved” the cosmological constant problem, if someone
introduces Lmatter → Lmatter − ρ, you are in trouble again!
• The only way out is to have a formalism for gravity which is invariant under
Tab → Tab + ρgab.
![Page 96: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/96.jpg)
5. How come gravity is immune to ground state energy?
• A cosmological constant term ρ0δij 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: Lmatter → Lmatter − ρ.
• But this changes energy momentum tensor by Tab → Tab + ρgab and gravity sector
is not invariant under this transformation.
• So after you have “solved” the cosmological constant problem, if someone
introduces Lmatter → Lmatter − ρ, you are in trouble again!
• The only way out is to have a formalism for gravity which is invariant under
Tab → Tab + ρgab.
• 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!
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A simple ‘theorem’
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A simple ‘theorem’
• Assume:
(a) Metric gab is a dynamical variable that is varied in the action.
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A simple ‘theorem’
• Assume:
(a) Metric gab is a dynamical variable that is varied in the action.
(b) Action is generally covariant.
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A simple ‘theorem’
• Assume:
(a) Metric gab 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 Lmatter → Lmatter − ρ0.
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A simple ‘theorem’
• Assume:
(a) Metric gab 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 Lmatter → Lmatter − ρ0.
• Then cosmological constant problem cannot be solved; that is,
gravitational equations cannot be invariant under Tab → Tab − ρ0gab.
![Page 102: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/102.jpg)
A simple ‘theorem’
• Assume:
(a) Metric gab 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 Lmatter → Lmatter − ρ0.
• Then cosmological constant problem cannot be solved; that is,
gravitational equations cannot be invariant under Tab → Tab − ρ0gab.
• We have dropped the assumption that gab is the dynamical variable.
![Page 103: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/103.jpg)
A simple ‘theorem’
• Assume:
(a) Metric gab 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 Lmatter → Lmatter − ρ0.
• Then cosmological constant problem cannot be solved; that is,
gravitational equations cannot be invariant under Tab → Tab − ρ0gab.
• We have dropped the assumption that gab is the dynamical variable.
• It also makes the variational principle for Lanczos-Lovelock theories
well-defined.
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GRAVITY IS IMMUNE TO BULK ENERGY
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GRAVITY IS IMMUNE TO BULK ENERGY
• The action/entropy functional is invariant under the shift Tab → Tab + ρgab !
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GRAVITY IS IMMUNE TO BULK ENERGY
• The action/entropy functional is invariant under the shift Tab → Tab + ρgab !
• The field equations have a new ‘gauge freedom’ and has the form:
P ijkb Ra
ijk −1
2Lδa
b − κT ab = (constant)δa
b
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GRAVITY IS IMMUNE TO BULK ENERGY
• The action/entropy functional is invariant under the shift Tab → Tab + ρgab !
• The field equations have a new ‘gauge freedom’ and has the form:
P ijkb Ra
ijk −1
2Lδa
b − κT ab = (constant)δa
b
• Introduces a new length scale LH. (Observationally, LP /LH ≈ 10−60 ≈ exp(−√
2π4).)
• Analogy: Solve Gab = 0 to get Schwarzchild metric with a parameter M .
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GRAVITY IS IMMUNE TO BULK ENERGY
• The action/entropy functional is invariant under the shift Tab → Tab + ρgab !
• The field equations have a new ‘gauge freedom’ and has the form:
P ijkb Ra
ijk −1
2Lδa
b − κT ab = (constant)δa
b
• Introduces a new length scale LH. (Observationally, LP /LH ≈ 10−60 ≈ exp(−√
2π4).)
• Analogy: Solve Gab = 0 to get Schwarzchild metric with a parameter M .
• We don’t worry about the value of (M/Mplanck) because M is an integration
constant; not a parameter in the equations.
![Page 109: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/109.jpg)
GRAVITY IS IMMUNE TO BULK ENERGY
• The action/entropy functional is invariant under the shift Tab → Tab + ρgab !
• The field equations have a new ‘gauge freedom’ and has the form:
P ijkb Ra
ijk −1
2Lδa
b − κT ab = (constant)δa
b
• Introduces a new length scale LH. (Observationally, LP /LH ≈ 10−60 ≈ exp(−√
2π4).)
• Analogy: Solve Gab = 0 to get Schwarzchild metric with a parameter M .
• We don’t worry about the value of (M/Mplanck) because M is an integration
constant; not a parameter in the equations.
• Given LP and LH we have ρUV
= 1/L4P and ρ
IR= 1/L4
H . The observed values is:
ρDE
≈ √ρ
UVρ
IR≈ 1
L2P L2
H
≈ H2
G
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GRAVITY IS IMMUNE TO BULK ENERGY
• The action/entropy functional is invariant under the shift Tab → Tab + ρgab !
• The field equations have a new ‘gauge freedom’ and has the form:
P ijkb Ra
ijk −1
2Lδa
b − κT ab = (constant)δa
b
• Introduces a new length scale LH. (Observationally, LP /LH ≈ 10−60 ≈ exp(−√
2π4).)
• Analogy: Solve Gab = 0 to get Schwarzchild metric with a parameter M .
• We don’t worry about the value of (M/Mplanck) because M is an integration
constant; not a parameter in the equations.
• Given LP and LH we have ρUV
= 1/L4P and ρ
IR= 1/L4
H . The observed values is:
ρDE
≈ √ρ
UVρ
IR≈ 1
L2P L2
H
≈ H2
G
• The hierarchy:
ρvac =
[1
L4P
︸ ︷︷ ︸
〈x〉
,1
L4P
(LP
LH
)2
︸ ︷︷ ︸
〈x2〉1/2
,1
L4P
(LP
LH
)4
, · · ·]
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System Material body Spacetime
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System Material body Spacetime
Macroscopic description Density, Pressure etc. Metric, Curvature
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System Material body Spacetime
Macroscopic description Density, Pressure etc. Metric, Curvature
Evidence for Existence of Existence of
Microstructure Temperature Temperature.
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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.
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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!)
![Page 116: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/116.jpg)
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
![Page 117: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/117.jpg)
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
![Page 118: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/118.jpg)
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
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SUMMARY
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SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(t,x) etc. are like ρ(t,x),v(t,x).
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SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(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.
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SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(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.
![Page 123: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/123.jpg)
SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(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].
![Page 124: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/124.jpg)
SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(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.
![Page 125: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/125.jpg)
SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(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 TdS = dE + PdV as the
key to obtaining a thermodynamic interpretation of gravitational theories.
![Page 126: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/126.jpg)
SUMMARY
• Gravity is an emergent, long-wavelength phenomenon like fluid mechanics. The
gab(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 TdS = dE + PdV as the
key to obtaining a thermodynamic interpretation of gravitational theories.
• Connects with Asur giving the horizon entropy; leads to quantisation of Wald
entropy.
![Page 127: GRAVITY: THE INSIDE STORY](https://reader034.vdocuments.net/reader034/viewer/2022052001/62859c723b5b2b095e165202/html5/thumbnails/127.jpg)
REFERENCES
1. Original ideas were developed in:
• T. Padmanabhan, Class.Quan.Grav. 19, 5387 (2002). [gr-qc/0204019]
• T. Padmanabhan, Gen.Rel.Grav., 34 2029-2035 (2002) [gr-qc/0205090] [Second Prize
essay; Gravity Research Foundation Essay Contest, 2002]
• T. Padmanabhan, Gen.Rel.Grav., 35, 2097-2103 (2003) [Fifth Prize essay; Gravity Research
Foundation Essay Contest, 2003]
• T. Padmanabhan, Gen.Rel.Grav., 38, 1547-1552 (2006) [Third Prize essay; Gravity Research
Foundation Essay Contest, 2006]
• T. Padmanabhan, Gravity: the Inside Story, Gen.Rel.Grav., 40, 2031-2036 (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/0311036]
• T. Padmanabhan Gen.Rel.Grav., 40, 529-564 (2008) [arXiv:0705.2533]
• T. Padmanabhan Dark Energy and its implications for Gravity (2008) [arXiv:0807.2356]
3. Also see:
• A. Mukhopadhyay, T. Padmanabhan, Phys.Rev., D 74, 124023 (2006) [hep-th/0608120]
• T. Padmanabhan, Aseem Paranjape, Phys.Rev.D, 75, 064004 (2007). [gr-qc/0701003]