arXiv subjects
Publications and source records attributed to Amit Ghosh.
We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems involving sums over classes of such forms weighted by their diophantine invariants.
We investigate the thermodynamics of a charged dilaton black hole arising from Einstein-Maxwell-dilaton theory, where the dilaton couples exponentially to the Maxwell field via a dimensionless parameter a. Treating a as a continuous solution parameter, we extend the black hole first law to include a term Psi^A da, where Psi^A is the thermodynamic potential conjugate to a. We derive Psi^A explicitly through a differential analysis of the mass, charge, and entropy, and confirm its form via an independent Hamiltonian calculation. Additionally, by promoting a to a spacetime-dependent scalar and introducing auxiliary gauge fields, we provide a geometric interpretation of Psi^A as a conserved Noether charge. We further analyze the implications of treating a as a thermodynamic variable within the extended phase space. Despite the modification to the first law, we demonstrate that the Smarr relation remains unaffected due to the dimensionless nature of a, highlighting the distinction between variational and scaling symmetries. Our analysis supports the thermodynamic relevance of coupling constants and enriches the framework of extended black hole thermodynamics.
We investigated the form and implications of the local first law of black hole thermodynamics in relation to an observer located at a finite distance from the black hole horizon. Our study is based on the quasilocal form of the first law for black hole thermodynamics, given by $\delta E=\frac{\bar{\kappa}}{8\pi}\delta A$, where $\delta E$ and $\delta A$ represent the changes in the black hole mass and area, respectively, and $\bar{\kappa}$ denotes the quasilocal surface gravity. We show that even at a finite distance, the quasilocal law still holds. It shows how the first law scales with the observer's location.
In this article, we explore an alternative derivation of Hawking radiation. Instead of the field-theoretic derivation, we have suggested a simpler calculation based on quantum mechanical reflection from a one-dimensional potential. The reflection coefficient shows an exponential fall in energy which, in comparison with the Boltzmann probability distribution, yields a temperature. The temperature is the same as Hawking temperature for spherically symmetric black holes. The derivation gives an exact local calculation of Hawking temperature that involves a region lying entirely outside the horizon. This is a crucial difference from the tunneling calculation, where it is necessary to involve a region inside the horizon.
We show that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $ \mathrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.
Using the quasilocal properties alone we show that the area spectrum of a black hole horizon must be discrete, independent of any specific quantum theory of gravity. The area spectrum is found to be half-integer spaced with values $8\pi \gamma \ell_{p}^{2}j$ where $j\in \mathbb{N}/2$. We argue that if microstate counting is carried out for quantum states residing on the horizon only, correction of $\exp(-\mathcal{A}/4\ell_{p}^{2})$ over the Bekenstein-Hawking area law must arise in black hole entropy.
This paper deals with a detail study of gravitational collapse of dust and viscous fluids under the assumptions of spherical symmetry. Our main goal is to closely analyze the horizons which arise during this gravitational phenomenon. To this end, we examine the formation and evolution of trapped surfaces in these spacetimes, with special attention to trapped regions and cylinders foliated by marginally trapped surfaces. The time evolution of trapped surfaces, collapsing shell as well as the event horizon are identified analytically as well as numerically. Using different density profiles of matter, we analyze, how the nature of the marginally trapped surfaces modify as we change the energy momentum tensor. These studies reveal that depending on the mass function and the mass profile, it is possible to envisage situations where dynamical horizons, timelike tubes or isolated horizons may arise.
We re-explore the symmetries of a weakly isolated horizon (WIH) from the perspective of freedom in the choice of intrinsic data. The supertranslations are realized as additional symmetries. Further, it is shown that all smooth vector fields tangent to the cross-sections are Hamiltonian. We show that joining two WIHs which differ in these Hamiltonians and boundary data, under the action of a supertranslation, necessarily require the inclusion of an intermediate dynamical phase, possibly with the inclusion of a stress energy tensor. This phase of the boundary is non-expanding but not a WIH and invariably leads to a violation of the dominant energy condition. The assumptions made allow us to reconstruct the (classically) pathological stress energy tensor also.
In this paper, we develop an effective quantum theory of black hole horizons using only the local horizon geometry. On the covariant phase space of the Holst action admitting Weak Isolated Horizon as an inner boundary, we construct Hamiltonian charges corresponding to Lorentz symmetries. We show that horizon area is the Hamiltonian charge corresponding to Lorentz boosts as well as that for Lorentz rotation which acts on $2$-sphere cross-sections of the horizon. Using this expression of area as a generator of Lorentz rotation, and the fact that quantum states residing on the horizon cross-sections carry a representation of $ISO(2)$, we derive the spectrum of area operator on the horizon. The eigenstates of this area operator are shown to be labelled by integers or half integers. The entropy is obtained completely in terms of these \emph{area quanta} residing on the horizon, and is shown to have exponentially suppressing corrections to the area law. The formalism is also extended to non-minimally coupled scalar fields, where the area operator gets modified due to the value of the scalar field on the horizon.
In this paper we revisited Hawking radiation in the light of the original calculations of Hawking and Wald and found that some additional insights can be gained. We review a "derivation" of the field theory Hilbert space from the space of solutions, followed by the calculation of Bogoliubov coefficients in a collapsed spacetime from first principles. We show that these calculations can be generalized to the case of local Killing horizons and also to asymptotically non-flat spaces.
For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if $V_{k}(\mathbb{Z}_p)$ is non-empty for all primes $p$, and that there are infinitely many $k$'s for which it fails. The Markoff morphisms act on $V_{k}(\mathbb{Z})$ with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces.
We consider natural polynomial truncations of hypergeometric power series defined over finite fields. For these truncations, we establish asymptotic upper bounds of order $O(p^{11/12})$ on the number of roots in the prime field $\mathbb{F}_p$. We discuss the correspondence to families of elliptic curves and K3 surfaces of certain such hypergeometric polynomials, for which sharp bounds are obtained in some cases. We include some computations to illustrate and supplement our results.
In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindel\"of hypothesis. That was a consequence of a topological argument and known subconvexity estimates, together with new sharp lower-bound restriction theorems for the Maass forms. This paper deals with the same question for general (compact or not) arithmetic surfaces which have a reflective symmetry. The topological argument is extended and representation theoretic methods are needed for the restriction theorems, together with results of Waldspurger. Various explicit examples are given and studied.
Recently, authors have studied inequalities involving expectations of selected functions viz. failure rate, mean residual life, aging intensity function and log-odds rate which are defined for left truncated random variables in reliability theory to characterize some well-known distributions. However, there has been growing interest in the study of these functions in reversed time and their applications. In the present work we consider reversed hazard rate, expected inactivity time and reversed aging intensity function to deal with right truncated random variables and characterize a few statistical distributions.
A quantum isolated horizon can be modeled by an SU(2) Chern-Simons theory on a punctured 2-sphere. We show how a local 2-dimensional conformal symmetry arises at each puncture inducing an infinite set of new observables localized at the horizon which satisfy a Kac-Moody algebra. By means of the isolated horizon boundary conditions, we represent the gravitational fluxes degrees of freedom in terms of the zero modes of the Kac-Moody algebra defined on the boundary of a punctured disk. In this way, our construction encodes a precise notion of CFT/gravity correspondence. The higher modes in the algebra represent new nongeometric charges which can be represented in terms of free matter field degrees of freedom. When computing the CFT partition function of the system, these new states induce an extra degeneracy factor, representing the density of horizon states at a given energy level, which reproduces the Bekenstein's holographic bound for an imaginary Immirzi parameter. This allows us to recover the Bekenstein-Hawking entropy formula without the large quantum gravity corrections associated with the number of punctures.
In loop quantum gravity the quantum states of a black hole horizon are produced by point-like discrete quantum geometry excitations (or {\em punctures}) labelled by spin $j$. The excitations possibly carry other internal degrees of freedom also, and the associated quantum states are eigenstates of the area $A$ operator. On the other hand, the appropriately scaled area operator $A/(8\pi\ell)$ is also the physical Hamiltonian associated with the quasilocal stationary observers located at a small distance $\ell$ from the horizon. Thus, the local energy is entirely accounted for by the geometric operator $A$. We assume that: In a suitable vacuum state with regular energy momentum tensor at and close to the horizon the local temperature measured by stationary observers is the Unruh temperature and the degeneracy of `matter' states is exponential with the area $\exp{(\lambda A/\ell_p^2)}$---this is supported by the well established results of QFT in curved spacetimes, which do not determine $\lambda$ but asserts an exponential behaviour. The geometric excitations of the horizon (punctures) are indistinguishable. In the semiclassical limit the area of the black hole horizon is large in Planck units. It follows that: Up to quantum corrections, matter degrees of freedom saturate the holographic bound, {\em viz.} $\lambda=\frac{1}{4}$. Up to quantum corrections, the statistical black hole entropy coincides with Bekenstein-Hawking entropy $S={A}/({4\ell_p^2})$. The number of horizon punctures goes like $N\propto \sqrt{A/\ell_p^2}$, i.e the number of punctures $N$ remains large in the semiclassical limit. Fluctuations of the horizon area are small while fluctuations of the area of an individual puncture are large. A precise notion of local conformal invariance of the thermal state is recovered in the $A\to\infty$ limit where the near horizon geometry becomes Rindler.
We derive black hole entropy based on the near-horizon symmetries of black hole space-times. To derive these symmetries we make use of an $(R,T)$-plane close to a Killing horizon. We identify a set of vector fields that preserves this plane and forms a Witt algebra. The corresponding algebra of Hamiltonians is shown to have a non-trivial central extension. Using the Cardy formula and the central charge we obtain the Bekenstein-Hawking entropy.
This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.