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Entropic Gravity

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. 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

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Entropic Gravity

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  1. 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

  2. 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

  3. 1. Verlinde’s entropic force scenario Fdx=TdS Newton’s second law

  4. Newton’s law of gravitation

  5. Verlinde’s derivation of Einstein Equations Temperature Holography, namely the bit number

  6. Equipartition Thus

  7. And From Tolman-Komar mass From the equipartition theorem

  8. 2. Our derivation of Einstein Equations Verlinde uses a closed holographic screen We use an open screen

  9. Through the screen, there is an energy flow This is a bulk flow.

  10. According to holography, this flow can be written using only the physical quantities on the screen

  11. Naturally, we assume the surface stress tensor be given by local geometry Using the Gauss-Codazzi equation

  12. We have Compare to the bulk flow, we find

  13. We almost obtain the Einstein equations. Note that We deduce

  14. 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

  15. The Brown-York semi-local energy has a form or

  16. 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.

  17. 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.

  18. 4. Holographic thermodynamics Consider a screen adiatically moves in space-time r r+dr

  19. The first law E and p are defined (to be substracted), we need To know

  20. For a static and spherically symmetric metric we have

  21. and We deduce

  22. To derive the chemical potential, we notice that for a black hole (or a region of vacuum) Nh=1 and dS=0, so

  23. 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

  24. We find in general and for the gas in particular

  25. To make the area term absent, x=0 thus This is the same form of the Bekenstein bound

  26. Indeed we also have a bound, when S reaches its maximum, and agrees with the Bekenstein bound if

  27. 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.

  28. 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.

  29. Thus, the screen energy change is

  30. We deduce So q can be determined. To determine F, we use The Bianchi identity and find

  31. Thus, the f(R) gravity equation of motion: and the surface stress tensor

  32. 6. The Maxwell equations from holography Charge flow replaces energy flow in this case. The bulk charge flow:

  33. The charge change on the open screen: Equating these two we have

  34. We postulate and

  35. 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

  36. 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

  37. 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.

  38. Thank You !

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