Transcript Jacobson's: Einstein equations can be derived from

Gravity field equations
can be derived from
the thermodynamic relation
 Q  T S
for
any metric theory of gravity
Merav Hadad
Ram Brustein, M.H. 0903.0823
Einstein equations can be derived from the
thermodynamic relation (Jacobson,’95)
T - Unruh temperature
bb a  a
Observer acceleration:
  2 T
 E   Tab  a b    Tab  a b
H
S 
A
4G
 Q  T S
H
 A  Rab  a  b 

(Raychauduri equation)
8 GT  R  g R
Ricci scalar theories
D
d
 x  g  R  f ( R) 
the equation of motion can be derived from the
thermodynamic relation if the entropy is
a Noether charge entropy.
(E. Elizalde & P. Silva, 2008)
Any effective action
D
d
 x  g  R  f ( Rabcd , f Rabcd , ....,  ,  f  ,...)  ,
We have generalized Jacobson's
derivation to any theory of gravity
keeping the volume fixed
during variation
 E  T  S  PPVV
The energy measured by the observer
E   Tab  
a b
H
  
 b a (D−1) volume form
a
a
Integration over a short segment of a thin pencil of
horizon generators
cc
c
a b
a b aa b b


T


 E   TabccTabab     Tab  c  
HH
H
In agreement with Jacobson
The entropy in generalized theories of
gravity with bifurcate Killing horizon


2 1 c
ab
ab cd d
S S   AAabcd
ˆˆ  
abcd
T HT H
LWcd  e  2L Wcd 
 Vabcd  
 ....
eV abcd
R
 R
cd
Aabcd
c
d
The entropy variation
ˆ    bi-normal vector to the area element
2
cd
m
c
ab
d
form
ˆ
  S (D-2)
  volume



A


 m  abcd 
ab
a
b
T
H
 Q  T S



H
a
b


L

cL
m
b ab
ab d
2 T2ab pq2   m  ARabcd
ˆ
f
pqr   g  LG
HR pabq R pqra



Gravity field equations for any
theory
of
gravity.
Conservation
of
mab n
a
bn ˆ i b
m nm  n a m 
ˆnmˆcˆab
nmnR
abci
Tab

We have generalized Jacobson's
derivation to any general theory
of gravity.
Are all the gravitation equations
actually a macroscopic description
of an underlying microscopic
dynamical system?