arXiv subjects

Gregory Eskin

Publications and source records attributed to Gregory Eskin.

At least 19 recordsLinked to original sources

arXiv2025

Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesics

We consider a Lorentzian metric in $\mathbb{R}\times\mathbb{R}^n$. We show that if we know the lengths of the space-time geodesics starting at $(0,y,η)$ when $t=0$, then we can recover the metric at $y$. We prove the rigidity of Lorentzian metrics. We also prove a variant of the rigidity property for the case of null-geodesics: if two metrics are close and if corresponding null-geodesics have equal Euclidian lengths then the metrics are equal.

math.AP↗

arXiv2024

Rigidity for Lorentzian metrics with the same length of null-geodesics

We study the Lorentzian metric independent of the time variable in the cylinder $\mathbb{R}\timesΩ$ where $x_0\in\mathbb{R}$ is the time variable and $Ω$ is a bounded smooth domain in $\mathbb{R}^n$. We consider forward null-geodesics in $\mathbb{R}\times Ω$ starting on $\mathbb{R}\times\partialΩ$ at $t=0$ and leaving $\mathbb{R}\timesΩ$ at some later time. We prove the following rigidity result: If two Lorentzian metrics are close enough in some norm and if corresponding null-geodesics have equal lengths then the metrics are equal.

math.AP↗

arXiv2020

Hawking type radiation from acoustic black holes with time-dependent metric

We consider a time-dependent acoustic metric in 2+1 dimensions having a black hole and we study the Hawking type radiation from such black hole. We construct eikonals of special form with the support close to the black hole. We compute the average number of created particles where the average is taken with respect to the Unruh type vacuum.

math-ph↗

arXiv2020

Determination of black holes by boundary measurements

For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in $\mathbb{R}\times Ω$, where $x_0\in \mathbb{R}$, is the time variable and $Ω$ is a bounded domain in $\mathbb{R}^n$. Let $Γ\subset\partialΩ$ be a subdomain of $\partialΩ$. We say that the boundary measurements are given on $\mathbb{R}\timesΓ$ if we know the Dirichlet and Neumann data on $\mathbb{R}\times Γ$. The inverse boundary value problem consists of recovery of the metric from the boundary data. In author's previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of $Ω$ where the spatial part of the wave operator is elliptic. This allow the recovery of the metric in the exterior of the ergoregion. Our goal is to recover the black hole. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.

math.AP↗

arXiv2020

Hawking radiation from not-extremal and extremal Reissner-Nordstrom black holes

We consider the non-extremal Reissner-Nordstrom black hole and construct a wave packet that exhibits the Hawking radiation. We find the average of the number of the created particles with respect to the $|0\rangle$ vacuum state and with respect to Unruh type vacuum state. The average of the number operator in the $|0\rangle$ vacuum state consists of two terms: one is related to the Hawking radiation and the second is not related. We use the same construction for the extremal RN black hole and get that the average of the number operator with respect to the $|0\rangle$ vacuum state is also a sum of two term, where the one related to the Hawking radiation is equal to zero. This result is consistent with other works on the Hawking radiation for the extremal RN black hole.

gr-qc↗

arXiv2019

New examples of Hawking radiation from acoustic black holes

Rotating acoustic metrics may have black holes inside the ergosphere. Simple cases of acoustic black holes were studied in the references [21], [4]. In the present paper we study the Hawking radiation for more complicated cases of acoustic black holes including black holes that have corner points.

math-ph↗

arXiv2018

Behavior of null-geodesics in the interior of Reissner-Nordstrom black hole

We show that an incoming null-geodesic belonging to a plane passing through the origin and starting outside the outer horizon crosses the outer and the inner horizons. Then it turns at some point inside the inner horizon and approaches the inner horizon when the time tends to the infinity. We also construct a geometric optics solution of the Reissner-Nordstrom equation that has support in a neighborhood of the null-geodesic.

math.AP↗

arXiv2018

Hawking radiation from acoustic black holes in two space dimensions

We study the Hawking radiation for the acoustic black hole. In the beginning we follow the outline of T.Jacobson but then we use a different $2+1$ vacuum state similar to the vacuum state constructed by W.Unruh. We also use a special form of the wave packets. The focus of the paper is to treat the 2 dimensional case, in particular, the case when the radial and angular velocity are variable.

gr-qc↗

arXiv2018

A simple approach to temporal cloaking

In recent years a remarkable progress was made in the construction of spatial cloaks using the methods of transformation optics and metamaterials. The temporal cloaking, i.e. the cloaking of an event in spacetime, was also widely studied by using transformations on spacetime domains. We propose a simple and general method for the construction of temporal cloaking using the change of time variables only.

math-ph↗

arXiv2018

Ergoregions between two ergospheres

For a stationary spacetime metric, black holes are spatial regions which disturbances may not propagate out of. In our previous work an existence and regularity theorem was proven for black holes in two space dimensions, in the case where the boundary of the ergoregion is a simple closed curve surrounding a singularity. In this paper we study the case of an annular ergoregion, whose boundary has two components.

math.AP↗

arXiv2017

Inverse problems for general second order hyperbolic equations with time-dependent coefficients

We study the inverse problems for the second order hyperbolic equations of general form with time-dependent coefficients assuming that the boundary data are given on a part of the boundary. The main result of this paper is the determination of the time-dependent Lorentzian metric by the boundary measurements. This is achieved by the adaptation of a variant of the Boundary Control method developed by the author in [E1], [E2].

math.AP↗

arXiv2016

Superradiance initiated inside the ergoregion

We consider the stationary metrics that have both the black hole and the ergoregion. The class of such metric contains, in particular, the Kerr metric. We study the Cauchy problem with highly oscillatory initial data supported in a neighborhood inside the ergoregion with some initial energy $E_0$. We prove that when the time variable $x_0$ increases this solution splits into two parts: one with the negative energy $-E_1$ ending at the event horizon in a finite time, and the second part, with the energy $E_2=E_0+E_1>E_0$, escaping, under some conditions, to the infinity when $x_0\rightarrow +\infty$. Thus we get the superradiance phenomenon. In the case of the Kerr metric the superradiance phenomenon is "short-lived", since both the solutions with positive and negative energies cross the outer event horizon in a finite time (modulo $O(\frac{1}{k})$) where $k$ is a large parameter. We show that these solutions end on the singularity ring in a finite time. We study also the case of naked singularity.

math-ph↗

arXiv2016

Stationary Black Hole Metrics and Inverse Problems in Two Space Dimensions

We study the wave equation for a stationary Lorentzian metric in the case of two space dimensions. Assuming that the metric has a singularity of the appropriate form, surrounded by an ergosphere which is a smooth Jordan curve, we prove the existence of a black hole with the boundary (called the event horizon) that is piece-wise smooth, generally having corners. We consider a physical model of acoustic black hole whose event horizon has corners. In the end of the paper we consider the determination of a black hole by the boundary measurements.

math-ph↗

arXiv2015

Aharonov-Bohm effect revisited

Aharonov-Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov-Bohm effect such as magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang-Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect.

math-ph↗

arXiv2014

Nonstationary analogue black holes

We study the existence of analogue nonstationary spherically symmetric black holes. The prime example is the acoustic model (cf. [V], [U]). We consider also a more general class of metrics that could be useful in other physical models of analogue black and white holes. We give examples of the appearance of black holes and of disappearance of white holes. We also discuss the relation between the apparent and the event horizons for the case of analogue black holes. In the end we study the inverse problem of determination of black or white holes by boundary measurements for the spherically symmetric nonstationary metrics.

math-ph↗

arXiv2014

Inverse Problems for Hyperbolic Equations

We give a survey of author's results on the inverse hyperbolic problems with time-dependent and time-independent coefficients. We consider the case of hyperbolic equations with Yang-Mills potentials and the case of domains with obstacles. In particular, we gave a stability estimate for the broken X-ray transform in the case of one convex obstacle in $\mathbb{R}^2$.

math.AP↗

arXiv2014

A Simple Proof of Magnetic and Electric Aharonov-Bohm Effects

Magnetic Aharonov-Bohm effect (AB effect) was studied in hundreds of papers starting with the seminal paper of Aharonov and Bohm [AB] published in 1959. We give a new proof of the magnetic Aharonov-Bohm effect without using the scattering theory and the theory of inverse boundary value problems. We consider separately the cases of one and several obstacles. The electric AB effect was studied much less. We give the first proof of the electric AB effect in domains with moving boundaries. When the boundary does not move with the time the electric AB effect is absent.

math-ph↗

arXiv2014

Uniqueness and Nonuniqueness in Inverse Hyperbolic Problems and The Black Hole Phenomenon

This paper consists of two parts. In the first part we describe the recent works on the inverse problems for the wave equation in $(n+1)$-dimensional space equipped with pseudo-Riemannian metric with Lorentz signature. We study the conditions of the existence of black (or white) holes for these wave equations. In the second part we prove energy type estimates on a finite time interval in the presence of black or white holes. We use these estimates to prove the nonuniqueness of the inverse problems.

math.AP↗