Holography for Rotating Black Holes

In 1993, 't Hooft (1999 Nobel Prize winner in physics) proposed that quantum gravity requires that the information in a three dimensional world can be stored on a two dimensional manifold much like a hologram. This is known as the holographic principle, and since then this idea has changed the direction of researches in quantum gravity. A concrete realization of this idea in string theory was first discovered in 1997 by Maldacena in his famous anti de-Sitter/Conformal Field Theory\footnote{AdS/CFT for short. AdS stands for anti de-Sitter, and CFT is the acronym for Conformal Field Theory.} correspondence conjecture. The AdS/CFT correspondence states that some string theories on a certain manifold that contains AdS space, in some limits, are dual to a CFT living on the boundary of this manifold. Despite the rapid progress in studying the AdS/CFT, this proposal is still away from practical applications. Some of the reasons are the fact that the AdS (anti-de Sitter) spacetime is not likely the spacetime where we are living nowadays and the existence of extra dimensions (as one of the ingredients in string theory) is still under question. The Kerr/CFT correspondence which was proposed in 2008 by Strominger et al appears to be a more ``down to earth'' duality, compared to the AdS/CFT correspondence. Originally, this new correspondence states that the physics of extremal Kerr black holes which are rotating by the maximal angular velocity can be described by a two dimensional CFT living on the near horizon. In this thesis, after reviewing some concepts in Kerr/CFT correspondence, I present some of my research results which extend and support the correspondence for non-extremal rotating black holes. I discuss the extension of the Kerr/CFT correspondence for the rotating black holes in string theory, namely Kerr-Sen black holes, and the Kerr/CFT analysis for vector field perturbations near the horizon of Kerr black holes. It is recently conjectured that a generic non-extremal Kerr black hole could be holographically dual to a hidden conformal field theory in two dimensions. Furthermore, it is known that there are two CFT duals (pictures) to describe the charged rotating black holes which correspond to angular momentum $J$ and electric charge $Q$ of the black hole. Furthermore these two pictures can be incorporated by the CFT duals (general picture) that are generated by $SL(2,\mathbb{Z})$ modular group. The general conformal structure can be revealed by looking at a charged scalar wave equation with some appropriate values of frequency and charge. In this regard, we consider the wave equation of a charged massless scalar field in the background of Kerr-Sen black hole and show in the ``near region", the wave equation can be reproduced by the squared Casimir operator of a local $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R$ hidden conformal symmetry. We can find the exact agreement between macroscopic and microscopic physical quantities like entropy and absorption cross section of scalars for Kerr-Sen black hole. We then find an extension of the vector fields that in turn yields an extended local family of $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R$ hidden conformal symmetries, parameterized by one parameter. For some special values of the parameter, we find a copy of $SL(2,\mathbb{R})$ hidden conformal algebra for the charged Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole in the strong deflection limit. The generic non-extremal Kerr-Newman black holes are holographically dual to hidden conformal field theories in two different pictures. The two pictures can be merged together to the CFT duals in the general picture that are generated by $SL(2,\mathbb{Z})$ modular group. We find some extensions of the conformal symmetry generators that yield an extended local family of $SL(2,\mathbb{R})_L \times SL(2,\mathbb{R})_R$ hidden conformal symmetries for the Kerr-Newman black holes, parameterized by one deformation parameter. The family of deformed hidden conformal symmetry for Kerr-Newman black holes also provides a set of deformed hidden conformal generators for the charged Reissner-Nordstrom black holes. The set of deformed hidden conformal generators reduce to the hidden $SL(2,\mathbb{R})$ conformal generators for the Reissner-Nordstrom black hole for specific value of deformation parameter. We also find agreement between the macroscopic and microscopic entropy and absorption cross section of scalars for the Kerr-Newman black hole by considering the appropriate temperatures and central charges for the deformed CFTs. Also in this thesis, we derive an appropriate boundary action for the vector fields near the horizon of near extremal Kerr black hole. We then use the obtained boundary action to calculate the two-point function for the vector fields in Kerr/CFT correspondence. In performing this analysis we borrow a formula proposed in AdS/CFT, namely the equality between the bulk and boundary theories partition functions. We show the gauge-independent part of the two-point function is in agreement with what is expected from CFT.

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