Entropic Gravity
Miao Li
中国科学院理论物理研究所
Institute of Theoretical Physics CAS
兩岸粒子物理與宇宙學研討會
2011.04.02
2016年5月28日星期六
Based on work done with Rong-Xin Miao and Wei Gu
And work done with Rong-Xin Miao and Jun Meng
1. A New Entropic Force Scenario and Holographic
Thermodynamics
arXiv:1011.3419
2. f(R) Gravity and Maxwell Equations from the
Holographic Principle
arXiv:1102.1166
2016年5月28日星期六
1. Verlinde’s entropic force scenario
Fdx=TdS
Newton’s second law
2016年5月28日星期六
Newton’s law of gravitation
2016年5月28日星期六
Verlinde’s derivation of Einstein Equations
Temperature
Holography, namely the bit number
2016年5月28日星期六
Equipartition
Thus
2016年5月28日星期六
And
From Tolman-Komar
mass
2016年5月28日星期六
From the equipartition
theorem
2. Our derivation of Einstein Equations
Verlinde uses a closed holographic screen
We use an open screen
2016年5月28日星期六
Through the screen, there is an energy flow
This is a bulk flow.
2016年5月28日星期六
According to holography, this flow can be written
using only the physical quantities on the screen
2016年5月28日星期六
Naturally, we assume the surface stress tensor
be given by local geometry
Using the Gauss-Codazzi equation
2016年5月28日星期六
We have
Compare to the bulk flow, we find
2016年5月28日星期六
We almost obtain the Einstein equations.
Note that
We deduce
2016年5月28日星期六
3. Comparison with Verlinde and Jacobson
Verlinde
Closed holographic screen
Temperature T
Our proposal
Open or closed screen
Without or with T
Tolman-Komar mass
Brown-York Energy
Equipartition
Surface stress tensor
2016年5月28日星期六
The Brown-York semi-local energy has a form
or
2016年5月28日星期六
We see that the second term is an extra compared
with Verlinde.
The equipartition theorem does not have to be true
since it is very peculiar.
We have extra datum p, which is important in
studying thermodynamics.
2016年5月28日星期六
Jacobson
Open null screen
Our proposal
Open or closed time-like
T only
T, p chemical potential
First law
First law
We have more information.
2016年5月28日星期六
4. Holographic thermodynamics
Consider a screen adiatically moves in
space-time
r
2016年5月28日星期六
r+dr
The first law
E and p are defined (to be substracted), we need
To know
2016年5月28日星期六
For a static and spherically symmetric metric
we have
2016年5月28日星期六
and
We deduce
2016年5月28日星期六
To derive the chemical potential, we notice that
for a black hole (or a region of vacuum) Nh=1 and
dS=0, so
2016年5月28日星期六
Assume the above formula be generally true for
other N and h, we can compute the holographic
entropy for a gas with weak gravity.
where for example
2016年5月28日星期六
We find in general
and for the gas in particular
2016年5月28日星期六
To make the area term absent, x=0 thus
This is the same form of the Bekenstein bound
2016年5月28日星期六
Indeed we also have a bound, when
S reaches its maximum, and agrees with the
Bekenstein bound if
2016年5月28日星期六
5. Derivation of f(R) gravity
I and Pang Yi showed that it is impossible to accomodate f(R) gravity in the Verlinde proposal.
We show that it is rather straightforward to include
it in our program.
We need to simply use a different surface stress
tensor.
2016年5月28日星期六
The new surface stress is postulated to be
The first term is similar to the Einstein gravity,
proportional to the extrinsic curvature. The scond
term is to be determined by consistency.
2016年5月28日星期六
Thus, the screen energy change is
2016年5月28日星期六
We deduce
So q can be determined. To determine F, we use
The Bianchi identity and find
2016年5月28日星期六
Thus, the f(R) gravity equation of motion:
and the surface stress tensor
2016年5月28日星期六
6. The Maxwell equations from holography
Charge flow replaces energy flow in this case.
The bulk charge flow:
2016年5月28日星期六
The charge change on the open screen:
Equating these two we have
2016年5月28日星期六
We postulate
and
2016年5月28日星期六
Solving these conditions, we find A be asymmetric
and
These are Maxwell equation. To show that A is F
given in terms of the gauge potential, we consider
the magnetic charge flow which is actually zero. So
2016年5月28日星期六
To conclude:
1. We make a different proposal from Verlinde
2. Our proposal makes derivation of the Einstein
equations more complete.
3. Our proposal has a reasonable thermodynamics
while Verlinde’s doen’t.
4. We predict a holographic entropy for a gas.
5. More flexible, F(R) and Maxwell theory are derived
2016年5月28日星期六
Future work:
1. Derive a general formula for the chemical
potential.
2. Discuss various situations such as anti-de
Sitter and cosmology (about holographic
entropy).
3. Apply it to study dark energy.
We are already working in these directions.
2016年5月28日星期六
Thank You !
2016年5月28日星期六