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Research Activities: Introduction
Black Hole Thermodynamics:
Typically black holes are gravitational configurations
which are physically realized by end points of the gravitational collapse of supermassive
stars. They are characterized by an event horzion which is a trapped lightlike surface shielding
an inner curvature singularity. In classical general relativity black holes are cold and dark.
However it was shown almost two decades back by Yacov Bekenstein and Stephen Hawking
that Black Holes must have a thermodynamic entropy that is proportional to the area of their event horizon in Planck units. In particular the laws of Black Hole Mechanics closely followed the laws of thermodynamics. Although the event horizon or the surface traced out by the last light ray leaving the black hole classicaly does not allow any radiation to escape from the inside. However if quantum effects are taken into account Hawking showed from a semi classical treatment that black holes radiate by quantum tunneling across the event horizon
and this gives them a certain temperature called the Hawking temperature and the radiation is called Hawking radiation. In fact the Hawking Temperature may be identified with the surface
gravity of the black hole giving it a status of a true thermodynamic temperature. Hence Black Holes posess very typical thermodynamics.
Yet the thermodynamic entropy of black holes did not posess a microscopic statistical interpretation as a black hole is characterized by only a few parameters like Mass,
Charge and Angular Momenta which were unable to accommodate for the rather large values of entropy.
One of the successes of modern string theory has been to describe certain special class of black holes called extremal black holes as bound states of collective string theory excitations called D branes in a certain limit. An exact match to the black hole entropy could be obtained by counting microscopic states in the conformal field theory for these D brane systems or directly from perturbative string states. However the Bekenstein-Hawking area entropy relation was true only for restricted theories of gravity as in general relativity. String Theory being more rich in structure involves supergravity theories as their low energy descriptions. These often include higher derivative corrections to the usual Einstein-Hilbert action. It was hence imperative to understand how such higher derivative contributions modify the usual black hole entropy and also to have a microscopic description of such corrected entropies.
Black Hole Attractors:
Black Hole solutions occur in low energy effective supergravity theories arising from string theory compactifications on Calabi-Yau spaces. Amongst these there are a special class of black holes which have zero Hawking temperature known as extremal black holes. They directly violate the third law of thermodynamics in the Nernst form as despite zero temperature they posess a finite non zero entropy. These supergravity black holes are often characterized by their mass, charge and angular momentum and also by the complex structure moduli of the underlying Calabi-Yau manifolds. It is observed that a radial flow from asymptotic infinity to
the horizon for these black holes continuously changes the moduli parameters. At the horizon
the moduli show a fixed point behaviour and are functions of the black hole charges. This is called attractor mechanism and the fixed behaviour is termed attractor flow. The entropy for these
attractor black holes may be obtained through extremization of an entropy function which determines the attractor fixed point. This has enabled a microscopic statistical interpretation of their entropy and also corrections due to higher derivative terms in the effective action of the
low energy supergravity theory. This is an exciting area of current research in string theory.
Summary of Recent Research Activities:
Black Hole Entropy in String Theory
The subleading corrections to the macroscopic entropy of extremal black holes arise from higher curvature terms in the low energy effective supergravity and also from perturbative string loop corrections as well as non perturbative corrections. The higher derivative corrections to the black hole entropy have been
understood in the recent past in the framework of the Wald formalism for generally covariant theories of gravity. In the supergravity limit of string theories
this involves the black hole attractor mechanism which ensures that the macroscopic entropy is independent of the asymptotic values of the moduli fields. Thus allowing for compatibility with a micrsocopic statistical basis. The entropy function approach to the attractor mecahnism has been a important ingredient to compute such subleading corrections and matching with the microscopic entropy from string theories.
Since the discovery of a Riemannian geometric structure incorporated in the Hessian matrix of the Internal Energy U or the Entropy S with respect to the extensive thermodynamic parameters, by Weinhold and subsequently by Ruppeiner, associated with the equilibrium thermodyamic state space of thermodynamic systems, it has been applied to the study of phase transitions and critical phenomena in condensed matter systems. Later this approach has been used by some investigators to study black holes as thermodynamic systems. In this connection we have earlier studied the geometry of the state space of ( 2+1) dimensional rotating BTZ and BTZ-CS black holes. It was shown that both the state space geometries were flat
and insensitive to the higher derivative corrections ruling out any phase transitions away from extremality. We further showed the universality of corrections due to thermal fluctuations in the canonical ensemble and that it leads to a small but non zero thermodynamic scalar curvature and consequently an interacting microscopic statistical basis.
In continuation of this programme we have applied the formalism of thermodynamic geometry to the entropy of charged extremal black holes
in N=2 supergravity limit of Type II string theories. Although extremal black holes have zero Hawking temperatures at which
conventional thermodynamics is invalid, the non zero entropy of extremal black holes seem to indicate that some limiting characterization of thermodynamics is valid even at T=0.
The possibility of a zero temperature chracterization of conventional thermodynamics in the context of extremal black holes is also borne out from AdS-CFT or the gauge theory-gravity correspondence. In this connection our construction is a geomterical realization of the state space of zero temperature extremal black holes described as degenerate quantum ground states. We have shown that for certain examples of extremal black holes in Type II string theory the thermodynamic geometry is well defined and non degenerate at extremality with scalar curvatures which are regular everywhere.
In this context we have explored the thermodynamic geometries of 2 charged small black holes in D=10 Type II and Heterotic string theory framework. It is seen that in this case the thermodynamic geometry is well defined only with higher derivative and non holomorphic corrections to the macroscopic entropy at the two derivative level. The effect of higher order
and string loop corrections to the thermodynamic geometry has also been studied. Additionaly we have studied 3 charged and 4 charged extremal black hole solutions in ten dimensional Type II supergravity. These are described by
D-brane systems in string theory as D1-D5-P and D2-D6-NS5-P configurations of D-branes carrying 3 and 4 charges respectively. The scalar curvatures for all these examples are non zero but finite. This indicates that they must be described by thermodynamically stable systems with an interacting statistical basis.
Currently we are exploring the attractor mechanism in the framework of thermodynamic geometries. It was shown earlier by Kallosh that the geometry of the reduced thermodynamic state space at the attractor point captures the moduli space geometry of the
complex structure moduli for the underlying compact Calabi-Yau manifold. We are studying this issue through the
entropy function approach in the context of certain examples of charged extremal black holes in string theory. This investigation
would allow the description of the attractor mechanism for extremal black holes in string theory through renormalization group like flows in the space of thermodynamic geometries. Additionaly we are also interested in studying flux compactifications and the string
theory landscape in the frame work of the Entropic Principle (OVV Conjecture) and its relation to thermodynamic geometries.
Brane World Black Holes:
An interesting direction of current research involves the description of our visible four dimensional world as being trapped on the four dimensional hypersurface of a topological defect (3-brane) embedded in a higher dimensional slice of an Anti deSitter space-time.
In this scenario the gauge theory for the Standard Model of Elementary
Particles is restricted on the 3-brane world volume and gravity propagates in the full five dimensional bulk. This brane world scenario leads to the exciting possibility of observable Low Scale Quantum Gravity Effects in future accelarators. Randall-Sundrum provided a
a five dimensional brane world model with a warped geometry. For consistency four dimensional
cosmology and black holes on the 3-brane hypersurface required to arise from certain bulk configurations.
In this context Chamblin, Hawking and Reall described a Schwarzschild black hole to arise from
a five dimensional bulk black string. Following this we showed that a rotating Kerr black hole
may be described in the bulk as a five dimensional rotating black string. We further generalized this construction to the description of a N dimensional rotating Myers-Perry black hole on a
(N-1) brane hypersurface as a ( N+1) dimensional bulk rotating black string in a Randall-Sundrum
brane world. We have in the recent past also completed studying the five dimensional rotating neutral black ring in a six dimensional Randall-Sundrum braneworld. The black ring configuration on the 4 brane in this case arises from a six dimensional bulk rotating black string version of the five dimensional black ring. Following the description of a rotating black ring as a periodically identified boosted black string we have establsihed that in the Randall-Sundrum scenario the rotating black ring arises from a six dimensional boosted bulk black 2-brane with
periodic identifications.
AdS Hydrodynamics and Black Hole Viscosity Bound:
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Research Interests |
Past Research
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