Entropic Gravity Miao Li 中国科学院理论物理研究所 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 2019年5月2日星期四

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 2019年5月2日星期四

1. Verlinde’s entropic force scenario Fdx=TdS Newton’s second law 2019年5月2日星期四

Newton’s law of gravitation 2019年5月2日星期四

Verlinde’s derivation of Einstein Equations Temperature Holography, namely the bit number 2019年5月2日星期四

Equipartition Thus 2019年5月2日星期四

From the equipartition And From Tolman-Komar mass From the equipartition theorem 2019年5月2日星期四

2. Our derivation of Einstein Equations Verlinde uses a closed holographic screen We use an open screen 2019年5月2日星期四

Through the screen, there is an energy flow This is a bulk flow. 2019年5月2日星期四

According to holography, this flow can be written using only the physical quantities on the screen 2019年5月2日星期四

Naturally, we assume the surface stress tensor be given by local geometry Using the Gauss-Codazzi equation 2019年5月2日星期四

Compare to the bulk flow, we find We have Compare to the bulk flow, we find 2019年5月2日星期四

We almost obtain the Einstein equations. Note that We deduce 2019年5月2日星期四

3. Comparison with Verlinde and Jacobson Verlinde Our proposal Closed holographic screen Open or closed screen Temperature T Without or with T Tolman-Komar mass Brown-York Energy Equipartition Surface stress tensor 2019年5月2日星期四

The Brown-York semi-local energy has a form 2019年5月2日星期四

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. 2019年5月2日星期四

Open null screen Open or closed time-like Jacobson Our proposal Open null screen Open or closed time-like T only T, p chemical potential First law First law We have more information. 2019年5月2日星期四

4. Holographic thermodynamics Consider a screen adiatically moves in space-time r r+dr 2019年5月2日星期四

E and p are defined (to be substracted), we need To know The first law E and p are defined (to be substracted), we need To know 2019年5月2日星期四

For a static and spherically symmetric metric we have 2019年5月2日星期四

and We deduce 2019年5月2日星期四

To derive the chemical potential, we notice that for a black hole (or a region of vacuum) Nh=1 and dS=0, so 2019年5月2日星期四

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 2019年5月2日星期四

and for the gas in particular We find in general and for the gas in particular 2019年5月2日星期四

To make the area term absent, x=0 thus This is the same form of the Bekenstein bound 2019年5月2日星期四

Indeed we also have a bound, when S reaches its maximum, and agrees with the Bekenstein bound if 2019年5月2日星期四

5. Derivation of f(R) gravity I and Pang Yi showed that it is impossible to acco- modate 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. 2019年5月2日星期四

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. 2019年5月2日星期四

Thus, the screen energy change is 2019年5月2日星期四

So q can be determined. To determine F, we use We deduce So q can be determined. To determine F, we use The Bianchi identity and find 2019年5月2日星期四

Thus, the f(R) gravity equation of motion: and the surface stress tensor 2019年5月2日星期四

6. The Maxwell equations from holography Charge flow replaces energy flow in this case. The bulk charge flow: 2019年5月2日星期四

The charge change on the open screen: Equating these two we have 2019年5月2日星期四

We postulate and 2019年5月2日星期四

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 2019年5月2日星期四

We make a different proposal from Verlinde To conclude: We make a different proposal from Verlinde 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 2019年5月2日星期四

Derive a general formula for the chemical potential. Future work: 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. 2019年5月2日星期四

Thank You ! 2019年5月2日星期四