Holographic (perfect fluid) stars

1. Holographic (perfect fluid) stars June 11 (Mon) - June 22 (Fri), 2012, APCTP-IEU Focus Program “Cosmology and Fundamental Physics III”, APCTP, Pohang, Korea Kyung Kiu Kim(Kyung-Hee University, CQUeST) With Youngman Kim(APCTP) and Ik-jae Sin(APCTP)

2. Outline • Motivation and introduction • Short note for AdS/CFT • AdS/CFT and Gravity( RS-II model) • Simple compact stars in general relativity • Compact stars from holography • Summary and discussion

3. Motivation and introduction • The perturbative calculation of quantum field theory has been so successful and it made many important progresses in physics. • However, if we study a field theory whose couplings are in the non-perturbative region, then we cannot do anything except for some cases( ex, integrable system(Changrim’s main research subject),… ) • Strong coupling behavior, density and temperature make field theory calculation very difficult. We haven’t found complete method to control these situations yet in field theory framework. • AdS/CFT gave us possibility for dealing with the situations. • At least, for some super conformal field theories, we can overcome these difficulties. • Because of these successful examples, people have tried a lot of attempts to extend AdS/CFT for application.

4. Motivation and introduction • However most of the works are devoted to homogeneous structures of system(uniform density, uniform chemical potential, flat space time, uniform condensation, uniform energy and so on). • In our works, we try to realize inhomogeneous structure in models of holographic approach. • One interesting topic is compact star. • The compact star(neutron star, white dwarfs, quark stars, strange stars,…) is important research subject, because they can tell us the equation of state for some region in the phase space of QCD or other theory. • There have been many observations from compact stars,sopeople are interested in whether such observations can be a guide line or not. • Since these objects are self-bound objects with nuclear force and gravity force, we have to consider a field theory system coupled to gravity.

5. Motivation and introduction • So the construction of gravity degrees of freedom at the boundary of AdS space is very important to realize compact star configuration . • There are many ways to introduce gravity in AdS/CFT -UV cutoff-IR cutoff • One example of UV cut-off model is the Randal-Sundrum II model from AdS/CFT point of view. • In order to explain our work, I will start with basic knowledge of AdS/CFT.

6. Short note for AdS/CFT • Type IIB string theory

7. Short note for AdS/CFT • Low energy effective theory.-- Type IIB super gravity theory

8. Short note for AdS/CFT • black 3-brane solution • Actually this black 3 brane solution is same physical object with D3 brane on end point of open strings. (Polchinski)

9. Short note for AdS/CFT • Let’s consider N D3-brane. The D-brane has two description, D-brane in string theory and Black brane in supergravity. • D-brane in stringy picture

10. Short note for AdS/CFT • D3 brane action becomes Non-abelian DBI action. • The perturbation theory is well defined, when

11. Short note for AdS/CFT • Let us consider low energy limit. • The higher derivative terms can be ignored. • The SUGRA fields are decoupled from D-brane. • Then the action governing D3 brane turns out to be U(N) Super Yang-Mill theory with 4 super-charges.

12. Short note for AdS/CFT • Super gravity point of view • Near horizon limit(r -> 0)-Low energy string limit by Red shift factor • The SUGRA is valid when two following quantities are large.

13. Short note for AdS/CFT There are two limit for same object N D3 brane This is the AdS/CFT correspondence. The strong coupling limit of SYM could be defined by SUGRA or string theory in AdS5. 5d classical gravity ~ 4d boundary QFT => Holography

14. Short note for AdS/CFT The AdS/CFT correspondence. Quantum field theory of with U(N) gauge group , where N = ∞. It’s strong coupling limit is described by type IIB string theory on AdS5 X S^5 Or type IIB supergravity on AdS5 X S^5(N = ∞, (g_YM)^2 N is large )

15. AdS/CFT and Gravity • For compact stars, AdS/CFT or gauge/gravity correspondence is not enough duality. • We need (gauge+ gravity)/gravity correspondence to consider an self-bound object with field theory interaction and gravity interaction. • So we require a model to have an effective action as follows. • Where first term is a generating functional for connected green function for field theory. • We don’t give any restriction for dots. • In fact, there is an example which is same with this model .

16. One example: RS-II model • RS-II model as a CFT system coupled to gravity has been studied by following persons. • For summary of the works, we define AdS/CFT in field theory language. Usually, AdS/CFT is defined by the equivalence between partition functions for 4d theory and 5d gravity theory as follows. • Where \gamma is induced metric at the boundary.

17. One example: RS-II model • There are counter terms to cancel the divergence of the action. • For well defined result, first we has to introduce UV cut-off z_0 To obtain finite value, we have to take z_0 -> 0 limit.

18. One example: RS-II model • The boundary terms can go out from the integration.

19. One example: RS-II model • Now let us consider RS-II model defined by following action. • The brane tension is given by . • And the brain position is z_0. • Now we may consider the path-integral of RS-II model.

20. One example: RS-II model • Then the result is • If we ignore matter degrees of freedom living on the boundary, then the effective action is given by • So this is a CFT system coupled to gravity.

21. One example: RS-II model • The effective Einstein equation is • The effective energy momentum tensor is

22. Simple compact stars in general relativity • Now I review how to study compact star in elementary level to compare our approach. • In this usual way, a hydrostatic equation, Tolman –Oppenheimer-Volkoff (TOV) equation, plays a very important role. • To get intuition for this equation, let us consider following situation in the Newtonian limit. • Newtonian TOV equation

23. Simple compact stars in general relativity • To derive similar quantity in general relativity, one has to consider metric for static stellar object first . • 4d Einstein equation • where the energy momentum tensor is perfect fluid. • From (t,t) component of the Einstein equation , the metric becomes

24. Simple compact stars in general relativity • From (r,r) component . • And if we consider hydrostatic equilibrium, • Finally we can get TOV equation

25. Simple compact stars in general relativity • TOV equation. • And mass and density relation • If we want to solve stellar structure, we need one more equation. The equation of state can play such a role. • Except for EOS, the only parameter is

26. Simple compact stars in general relativity • So the assumption for EOS is very important. • If we know the equation of state, one can solve the two equations. • Solving them all, we can see full functions of mass, pressure and energy density. • From these, we can read the radius and the mass of a compact star. • This computation can be related to observation.

27. Simple compact stars in general relativity • Usually the EOS is assumed by the microscopic nature of the nuclear matter, the radius and the mass are from the macroscopic observation. • Thus the compact stars are good object to understand microscopic physics from observation.

28. The simplest holographic model without matter • Come back to holographic star. • Let us consider holographic model in a curved spacetime. • In order to obtain first intuition, we had better take a simplest toy model. So we are going to consider the Einstein-Hilbert action with negative cosmological constant in 5-dimension. • The general behavior in terms of haolographic renormalization was studied by Skenderis et al(2000).

29. The simplest holographic model without matter • The summary of their work is as follows. • If we take the Fefferman-Graham coordinates,The general behavior of metric is • The expectation value of energy-momentum tensor is

30. The simplest holographic model without matter • Again, the bulk metric is • The first term is a scheme dependent term, when we consider holographic renormalization. For simplicity, we assume that this term vanishes(Polynomial Ansatz). • The second term is interpreted as the metric of boundary system. Usually, this is taken as flat metric. In our case we will put nontrivial non-flat metric into this boundary metric.

31. The simplest holographic model without matter • Since we will take star object into account, we assume that our metric has spherical symmetry. Then our ansatz for bulk metric is • From this ansatz, we can solve the bulk equation.

32. The simplest holographic model without matter • Now we take a star metric as the boundary metric. • Then we consider the bulk Einstein equation order by order in z. • Up to zeroth order equation, g^(2) is given by g^(0).

33. The simplest holographic model without matter • Up to second order, the metric must satisfy following constraint equation. • In other words, If the boundary metric satisfies this equation, then the full bulk metric could be a solution of the bulk Einstein equation up to second order.

34. The simplest holographic model without matter • We take one more assumption. If the boundary metric is a perfect fluid star, we have to consider following constraints. • These come from the boundary Einstein equation and TOV equation. • Then previous complicate constraint becomes simple.

35. The simplest holographic model without matter • If we solve the following equationswith initial condition P(0),rho(0) and m(0)=0, then we can obtain pressure and energy density as functions of the radial coordinate.

36. The simplest holographic model without matter • Indeed, these equations can be solved, the solution is well known uniform density solution. • our model was a simplest toy model, so we expect that a model with matter can give more realistic equation of state.

37. Adding neutral scalar to the model • Now we move to more realistic configuration. • First one we can think is giving a simplest matter to the system. • So we may add a neutral scalar bulk field as follows. • Eom for matter fields

38. Adding neutral scalar to the model • The ansatz(Polynomial type) for the matter field is as follows. • The metric ansatz is same with the previous case. • We can solve the Einstein equation and the equations of motion for matter field near boundary(z=small value) of the AdS5 order by order in z.

39. Adding neutral scalar to the model • For the matter fields, we can obtain the result • Again a complicate equation for metric

40. Adding neutral scalar to the model • And for the scalar field • If we give perfect fluid metric condition and TOV equation to these solution

41. Adding neutral scalar to the model • The two equations become much simpler

42. Adding neutral scalar to the model • Thus the equations we have to solve are

43. Adding neutral scalar to the model • By regularity condition of the star metric, the relevant parameter for this solution are P(0), rho(0), phi_1(0) • If we give three parameters, we can obtain holographic star solutions.

44. Adding neutral scalar to the model

45. Adding neutral scalar to the model

46. Adding neutral scalar to the model

47. Adding neutral scalar to the model • By holography, we have obtained equation of state depending on some parameters. • The back reaction was included in our construction. • And we can make a situation whose surface energy density does not vanish. • In this case, we can interpret the scalar field parameter as a parameter related to surface energy density. So the scalar field has clear physical meaning.

48. Summary and discussion • Inhomogeneous structure in AdS/CFT and extended gauge/gravity correspondence are very interesting topics. In this work, we considered compact stars in holographic setup. • In order to consider compact stars, we have deformed boundary metric which is not flat. • To embed the compact star geometry, we have taken spherical symmetry on the boundary metric and polynomial ansatz. • Since our ansatz and bulk Einstein equation gives one constraint than the boundary Einstein equation, our system gives one more differential equation. • By taking perfect fluid star as the boundary metric, the constraint became very simple equation. • The simplest solution is well-known uniform density star.

49. Summary and discussion • We introduced a scalar field as a matter field. • It turns out that this scalar field has a meaning related to surface energy density. • We can obtain EOS for each parameter point(center values of Pressure and density) • We can consider more complex structure by more matter fields. • As a future direction, we may study full bulk geometry by solving partial differential equation with boundary metric, then we can understand more on this holographic stars. • Formation of BH in holographic approach is another interesting direction.

50. Thank you!