Correlations of the stress energy tensor in AdS spaces via the zeta-function method

1. Correlations of the stress energy tensor in AdS spaces via the zeta-function method Hing Tong Cho Tamkang University (Work done in collaboration with Bei-Lok Hu) NTNU June 2010

2. Outline • Introduction • Zeta-function method for correlations • Mode functions and the renormalized stress tensor • Bitensors in AdS spaces and the stress tensor correlations • Discussions

3. Introduction • Fluctuations of the vacuum energy like the Casimir energy. • Fluctuations of the stress energy tensor can back-react onto the background spacetime. In the theory of stochastic gravity, this is in the form of a stochastic force on the right hand side of the Einstein equation. • Anti-de Sitter spacetime has attracted a lot of attention as vacua of supergravity theories and also due to the AdS/CFT correspondence.

4. (A)dS spacetimes are maximally symmetric and are natural generalizations of the Minkowski spacetime. • Due to Phillips and Hu, the zeta-function method has been extended to the consideration of fluctuations of physical quantities like the stress energy tensor.

5. Zeta-function method for correlations Suppose the action of a scalar field in Euclidean signature is given by Formally, the effective action is where is a dimensional renormalization parameter.

6. The zeta-function is defined by where is the heat kernel.

7. Under the zeta-function regularization the renormalized effective action becomes

8. The renormalized stress energy tensor where with is an eigenfunction of

9. For example, for a scalar field we have

10. The stress energy correlation which involves the second functional derivative.

11. Using the Schwinger method (Phillips and Hu 1997) where

12. Hence, the second variation of the zeta-function can be written as, Consider a change of variables,

13. Note that a further regularization has been introduced:

14. Mode functions and the renormalized stress tensor Euclidean section of the AdS space is a N-dimensional hyperbolic space with the metric where measures the geodesic distance from the origin and is a dimensional parameter of the AdS space.

15. The mode function for the Laplacian operator with

16. Since is maximally symmetric, where

17. Considering the asymptotic behavior of the mode function as Taking the limit , only the terms with will survive.

18. The sum over is facilitated by the addition theorem of the hyperspherical harmonics, where is the Gegenbauer polynomial.

19. Hence,

20. The normalization factor where the sum is finite for any particular dimension.

21. In the massless minimally coupled cases, which is the same as the result by Caldarelli (1999).

22. For example,

23. Bitensors in AdS spaces and the stress tensor correlations A set of bitensors (Allen and Jacobson 1986) geodesic distance between and parallel propagator

24. For the hyperbolic space the bitensors are shown along a geodesic.

25. The stress energy tensor correlation can be written as since is maximally symmetric.

26. Under the limit

27. In the same limit,

28. The C’s could be obtained by evaluating various components of the correlations The calculation is simplified again by taking the limits and For simplicity, we shall work on the massless, minimally coupled case.

29. For example, take As only the term with survives.

30. For the angular part,

31. Hence, as and

32. We obtain where

33. For the massless minimally coupled scalar in even dimensions,

35. The integrals are defined as

36. Small geodesic distance limit,

37. For example, for

38. Taking For

39. In general, This is consistent with the results of Phillips and Hu for

40. Large geodesic distance limit,

41. Therefore, in the this limit There is a natural infrared cutoff of the value related to the size of the hyperbolic space.

42. Discussions • Because the hyperbolic spaces are homo-geneous, one can express the stress energy correlators by 5 functions of the geodesic distance. • Via the zeta-function method these 5 functions can be expressed in closed form using a integral of the associated Legendre function. • In the coincident limit, only two functions survive, while the 5 functions have different large distance limits.

43. There are sizable fluctuations of the stress energy tensor and they increase with dimensions • It would be interesting to see how the fluctuations are affected by temperature by considering thermal AdS cases.