Sumit Dey

Bio: Sumit Dey is an academic researcher from Indian Institute of Technology Guwahati. The author has contributed to research in topics: Null (mathematics) & Covariant transformation. The author has an hindex of 3, co-authored 8 publications receiving 42 citations.

General framework to study the extremal phase transition of black holes

Abstract: We investigate the universality of some features for the extremal phase transition of black holes and unify all the approaches which have been applied in different spacetimes. Unlike the other existing approaches where the information of the spacetime and its dimension is directly used to get various results, we provide a general formulation in which those results are obtained for any arbitrary black hole spacetime having an extremal limit. Calculating the second order moments of fluctuations of some thermodynamic quantities we show that, the phase transition occurs only in the microcanonical ensemble. Without considering any specific black hole we calculate the values of critical exponents for this type of phase transition. These are shown to be in agreement with the values obtained earlier for metric specified cases. Finally we extend our analysis to the geometrothermodynamics (henceforth GTD) formulation. We show that for any black hole, if there is an extremal point, the Ricci scalar for the Ruppeiner metric must diverge at that point.

Covariant approach to the thermodynamic structure of a generic null surface

Abstract: We readdress the thermodynamic structure of geometrical relations on a generic null surface. Among three potential candidates, originated from different components of $R_{ab}$ along the null vectors for the surface (i.e. $R_{ab}q^a_cl^b$, $R_{ab}l^al^b$ and $R_{ab}l^ak^b$ where $q_{ab}$ is the projector on the null surface and $l^a$, $k^a$ are null normal and corresponding auxiliary vector of it, respectively), the first one leads to Navier-Stokes like equation. Here we devote our investigation on the other two members. We find that $R_{ab}l^al^b$, which yields the evolution equation for expansion parameter corresponding to $l^a$ along itself, can be interpreted as a thermodynamic relation when integrated on the two dimensional transverse subspace of the null hypersurface along with a virtual displacement in the direction of $l^a$. Moreover for a stationary background the integrated version of it yields the general form of Smarr formula. Although this is more or less known in literature, but a similar argument for the evolution equation of the expansion parameter corresponding to $k^a$ along $l^a$, provided by $R_{ab}l^ak^b$, leads to a more convenient form of thermodynamic relation. In this analysis, contrary to earlier approaches, the identified thermodynamic entities come out to be in covariant forms and also are foliation independent. Hence these can be applied to any coordinate system adapted to the null hypersurface. Moreover, these results are not restricted to any specific parametrisation of $k^a$ and also $k^a$ need not be hypersurface orthogonal. In addition, here any particular dynamical equation for metric is not being explicitly used and therefore we feel that our results are solely based on the properties of the null surface.

Effective metric in fluid-gravity duality through parallel transport: A proposal

Abstract: The incompressible Navier-Stokes (NS) equation is known to govern the hydrodynamic limit of essentially any fluid, and its rich nonlinear structure has critical implications in both mathematics and physics. The employability of the methods of Riemannian geometry to the study of hydrodynamical flows has been previously explored from a purely mathematical perspective. In this work, we propose a bulk metric in ($p+2$) dimensions with the construction being such that the induced metric is flat on a timelike $r={r}_{c}$ (constant) slice. We then show that the equations of parallel transport for an appropriately defined bulk velocity vector field along its own direction on this manifold when projected onto the flat timelike hypersurface requires the satisfaction of the incompressible NS equation in ($p+1$) dimensions. Additionally, the incompressibility condition of the fluid arises from a vanishing expansion parameter $\ensuremath{\theta}$, which is known to govern the convergence (or divergence) of a congruence of arbitrary timelike curves on a given manifold. In this approach Einstein's equations do not play any role, and this can be regarded as an off-shell description of fluid-gravity correspondence. We argue that our metric effectively encapsulates the information of forcing terms in the governing equations as if a free fluid is parallel transported on this curved background. We finally discuss the implications of this interesting observation and its potentiality in helping us to understand hydrodynamical flows in a probable new setting.

Gravity dual of Navier-Stokes equation in a rotating frame through parallel transport

Abstract: The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics. This striking connection has been explored in several dynamics-based approaches and has surfaced in various forms over the past four decades. In a recent construction, it has been shown that the manifold properties of geometric duals are in fact intimately connected to the dynamics of incompressible fluids, thus bypassing the conventional on-shell standpoints. Following such a prescription, we construct the geometrical description that effectively captures the dynamics of an incompressible NS fluid with respect to a uniformly rotating frame. We propose the gravitational dual(s) described by bulk metric(s) in $(p+2)$-dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in $(p+1)$-dimensions. We argue that free fluid flows on manifold(s) described by the proposed metric(s) can be effectively considered as an equivalent theory of non-relativistic viscous fluid dynamics with respect to (w.r.t) a uniform rotating frame. We also present suggestive insights as to how space-time rotation parameters encode information pertaining to the inertial effects in the corresponding fluid dual.

Thermodynamic structure of a generic null surface and the zeroth law in scalar-tensor theory

Abstract: We show that the equation of motion of scalar-tensor theory acquires thermodynamic identity when projected on a generic null surface. The relevant projection is given by $E_{ab}l^ak^b$, where $E_{ab} =8\pi T_{ab}^{(m)}$ represents the equation motion for gravitational field in presence of external matter, $l^a$ is the generator of the null surface and $k^a$ is the corresponding auxiliary null vector. Our analysis is done completely in a covariant way. Therefore all the thermodynamic quantities are in covariant form and hence can be used for any specific form of metric adapted to a null surface. We show this both in Einstein and Jordan frames and find that these two frames provide equivalent thermodynamic quantities. This is consistent with the previous findings for a Killing horizon. Also, a concrete proof of the zeroth law in scalar-tensor theory is provided when the null surface is defined by a Killing vector.

A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics

Abstract: This new textbook is intended for students familiar with general relativity at the introductory level of Bernard Schutz's book A First Course in General Relativity (1985 Cambridge: Cambridge University Press) and not yet accomplished at the advanced level of Robert Wald's book General Relativity (1984 Chicago, IL: University of Chicago Press), upon which it nevertheless draws rather heavily. What is distinctively new in this book is that it is a real toolkit, and yet it is not short of detailed applications. As such, it is a helpful book to recommend to students making the transition for which it is intended. The idea of a new textbook on general relativity usually delights me, as the field is still changing rapidly. New perspectives find new ways to present old things to new students. They also have totally new things to present to us all, based on the interests of the current research from which they have grown. This new book presents a wealth of useful tools to students in just five, well integrated chapters, starting with a quick review of the fundamentals and ending with an extensive application of general relativity to black hole spacetimes. In his own words, Eric Poisson has striven to present interesting topics and common techniques not adequately covered in readily available existing texts. This has certainly been accomplished, in a synthesis extracted from many sources. Congruences of geodesics, a staple analytical tool, occupy a whole chapter, and in greater depth and clarity than can be found elsewhere. A thorough, and lengthy, presentation on hypersurfaces, including a careful treatment of the null case, carries the author's unique perspective. This treatment of hypersurfaces is put to practical use in the chapter on Lagrangian and Hamiltonian formulations, which also leans on recent quasilocal energy discussions and includes an elegant treatment of the Bondi-Sachs mass in a unified context. Many of us have become familiar with the careful, well thought out, pedagogical style of the author, and this book certainly lives up to that reputation. It has developed from a course originating from Poisson, but now already given a number of times by several different instructors, so it is well battle-tested. Since Poisson has worked extensively in many of the areas he covers, the book also carries a personal touch, with an emphasis on clarity. As intended, the influence of Werner Israel, to whom the book is dedicated, shows through, implicit in many places, and at times explicit as well. Probably my strongest quibble with the content is the absence of a comprehensive discussion of isolated and dynamical horizons, which Ashtekar and coworkers have done so much to develop recently. Students equipped with the skills Poisson intends to impart would do well to be prepared in this one particular complementary area too. The potential reader should also be cautioned that there is no treatment of black hole perturbations. Though their role in gravitational wave discussions is becoming increasingly significant, their absence from this book is justified on the grounds of space and compatibility with the techniques presented. Typographically, the book uses a clear, adequately sized font, an essentially uniform notation, and includes 40 line drawings which helpfully illustrate the text. It is exceptionally well proofread, as one might expect from the publisher concerned. I believe it will give a thorough, advanced preparation to any suitably prepared student using it, and I will definitely recommend it for students matching its intended audience.

Ruppeiner thermodynamic geometry for the Schwarzschild-AdS black hole

Abstract: Due to the nonindependence of entropy and thermodynamic volume for spherically symmetric black holes in the anti--de Sitter (AdS) spacetime, when applying the Ruppeiner thermodynamic geometry theory to these black holes, we often encounter an unavoidable problem of the singularity about the line element of thermodynamic geometry. In this paper, we propose a basic and natural scheme for dealing with the thermodynamic geometry of spherically symmetric AdS black holes. We point out that enthalpy, not internal energy, is the fundamental thermodynamic characteristic function for the Ruppeiner thermodynamic geometry. Based on this fact, we give the specific forms of the line element of thermodynamic geometry for the Schwarzschild AdS (SAdS) black hole in different phase spaces, and the results show that the thermodynamic curvatures obtained in different phase spaces are equivalent. It is shown that the thermodynamic curvature is negative which may be related to the information of attractive interaction between black hole molecules for the SAdS black hole. Meanwhile we also give an approximate expression of the thermodynamic curvature of the Schwarzschild black hole which shows that the black hole may be dominated by repulsion on the low temperature region and by attraction on the high temperature region phenomenologically or qualitatively.

Hydrodynamics of simply spinning black holes & hydrodynamics for spinning quantum fluids

Abstract: We find hydrodynamic behavior in large simply spinning five-dimensional Anti-de Sitter black holes. These are dual to spinning quantum fluids through the AdS/CFT correspondence constructed from string theory. Due to the spatial anisotropy introduced by the angular momentum, hydrodynamic transport coefficients are split into groups longitudinal or transverse to the angular momentum, and aligned or anti-aligned with it. Analytic expressions are provided for the two shear viscosities, the longitudinal momentum diffusion coefficient, two speeds of sound, and two sound attenuation coefficients. Known relations between these coefficients are generalized to include dependence on angular momentum. The shear viscosity to entropy density ratio varies between zero and 1/(4π) depending on the direction of the shear. These results can be applied to heavy ion collisions, in which the most vortical fluid was reported recently. In passing, we show that large simply spinning five-dimensional Myers-Perry black holes are perturbatively stable for all angular momenta below extremality.

From Navier-Stokes to Maxwell via Einstein

Abstract: We revisit the cutoff surface formulation of fluid-gravity duality in the context of the classical double copy. The spacetimes in this fluid-gravity duality are algebraically special, with Petrov type II when the spacetime is four dimensional. We find two special classes of fluids whose dual spacetimes exhibit higher algebraic speciality: constant vorticity flows have type D gravity duals, while potential flows map to type N spacetimes. Using the Weyl version of the classical double copy, we construct associated single-copy gauge fields for both cases, finding that constant vorticity fluids map to a solenoid gauge field. Additionally we find the scalar in a potential flow fluid maps to the zeroth copy scalar.

Thermodynamic geometry of AdS black holes and black holes in a cavity

Abstract: The thermodynamic geometry has been proved to be quite useful in understanding the microscopic structure of black holes. We investigate the phase structure, thermodynamic geometry and critical behavior of a Reissner–Nordstrom-AdS black hole and a Reissner–Nordstrom black hole in a cavity, which can reach equilibrium in a canonical ensemble. Although the phase structure and critical behavior of both cases show striking resemblance, we find that there exist significant differences between the thermodynamic geometry of these two cases. Our results imply that there may be a connection between the black hole microstates and its boundary condition.