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2022 | Buch

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Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity

Domoschool 2019

herausgegeben von: Sergio Luigi Cacciatori, Alexander Kamenshchik

Verlag: Springer International Publishing

Buchreihe : Tutorials, Schools, and Workshops in the Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume guides early-career researchers through recent breakthroughs in mathematics and physics as related to general relativity. Chapters are based on courses and lectures given at the July 2019 Domoschool, International Alpine School in Mathematics and Physics, held in Domodossola, Italy, which was titled “Einstein Equations: Physical and Mathematical Aspects of General Relativity”. Structured in two parts, the first features four courses from prominent experts on topics such as local energy in general relativity, geometry and analysis in black hole spacetimes, and antimatter gravity. The second part features a variety of papers based on talks given at the summer school, including topics like:

Quantum ergosphereGeneral relativistic Poynting-Robertson effect modellingNumerical relativityLength-contraction in curved spacetimeClassicality from an inhomogeneous universe

Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity will be a valuable resource for students and researchers in mathematics and physicists interested in exploring how their disciplines connect to general relativity.

Inhaltsverzeichnis

Frontmatter

Main Lectures

Frontmatter

Introduction to the Wang–Yau Quasi-local Energy

Abstract

This chapter is planned as a short introduction to the Wang–Yau quasi-local energy of closed spacelike surfaces in spacetimes. We start with discussing a geometric problem of isometric embedding a surface into the Minkowski spacetime. Then, we review the formula of surface Hamiltonian and its properties. In the explanation of the definition of the Wang–Yau energy, we focus on its physical and variational feature and relate it to some previously known quasi-local quantities. In the end, we outline the ideas behind the proof of its positivity.

Pengzi Miao

Gravitational Self-force in the Schwarzschild Spacetime

Abstract

Gravitational self-force techniques will be shortly reviewed along the lines of the two lectures presented by D. Bini at the 2019 edition of the “Domoschool.” The most important application of gravitational self-force concerns metric and curvature perturbations in black hole spacetimes due to moving particles or evolving fields. However, from a practical point of view (and for teaching purposes) we have chosen to perform the whole discussion at the level of a (massless) scalar field. In fact, in this simple case one can look at the various steps implicit in any self-force computation without facing with the additional difficulties of implementing them in a more involved tensorial background.

Donato Bini, Andrea Geralico

Geometry and Analysis in Black Hole Spacetimes

Abstract

These notes, based on lectures given at the summer school on Einstein Equations at Domodossola 2019, collect material on the Einstein equations, the geometry of black hole spacetimes, and the analysis of fields on black hole backgrounds. The Kerr model of a rotating black hole in vacuum is expected to be unique and stable. The problem of proving these fundamental facts provides the background for the material presented in these notes.

Lars Andersson

Study of Fundamental Laws with Antimatter

Abstract

I will review the motivation for research on antimatter particles related to fundamental physical symmetries such as the CPT theorem and the Weak Equivalence Principle. The general motivations will be presented, also recalling the role of the violation of such symmetries during the early evolution of the Universe. Antimatter systems in the form of bound neutral states will be considered, because of the possibility of studying gravitational interaction; they include anti-Hydrogen and Positronium as well as the Mu-atom. Anti-Hydrogen studies made at the CERN Antiproton Decelerator will be reviewed, culminating with the recent series of measurements that have opened up the field of Antimatter Spectroscopy. These studies hold the promise of allowing for the first measurement ever of the gravitational behavior of an antiparticle. Positronium studies will also be considered, starting with the interferometric measurement of the positron and proceeding to the preparation of a beam of Positronium with the coherence required to perform interferometry related to gravitational studies.

Marco Giammarchi

Proceedings

Frontmatter

Quantum Ergosphere and Brick Wall Entropy

Abstract

We revisit t’ Hooft’s “brick wall” model for black hole entropy taking into account backreaction effects on the horizon structure. We do so by adopting an evaporating metric in the quasi-static approximation in which departures from the standard Schwarzschild metric are governed by a small luminosity factor. The backreaction leads to an ergosphere-like region which naturally tames the usual divergence in the calculation of the partition function of the field. The black hole luminosity sets the width of such “quantum ergosphere.” We find a finite horizon contribution to the entropy which, for the luminosity associated with the Hawking flux, reproduces remarkably well the Bekenstein–Hawking entropy-area law.

Lennart Brocki, Michele Arzano, Jerzy Kowalski-Glikman, Marco Letizia, Josua Unger

Geodesic Structure and Linear Instability of Some Wormholes

Abstract

We consider a static, spherically symmetric wormhole connecting two asymptotically Anti-de Sitter universes with constant negative curvature (AdS wormhole), firstly introduced by Bronnikov as a solution to Einstein’s equations in presence of a self-interacting phantom scalar field. In this chapter, we construct an embedding diagram for this spacetime and study the structure of its timelike and null geodesics. In the limit case in which the asymptotic constant negative curvature approaches to zero, the AdS wormhole trivially reduces to the wormhole of Ellis, Bronnikov, Morris and Thorne (EBMT); we provide a brief review of the linear stability analysis of the EBMT wormhole as it was performed by Cremona, Pirotta and Pizzocchero, making comparison with some previous works about the same subject.

Francesco Cremona

New Trends in the General Relativistic Poynting–Robertson Effect Modeling

Abstract

The general relativistic Poynting–Robertson (PR) effect is a very important dissipative phenomenon occurring in high-energy astrophysics. Recently, it has been proposed a new model, which upgrades the two-dimensional (2D) description in the three-dimensional (3D) case in Kerr spacetime. The radiation field is considered as constituted by photons emitted from a rigidly rotating spherical source around the compact object. Such dynamical system admits the existence of a critical hypersurface, a region where the gravitational and radiation forces balance and the matter reaches it at the end of its motion. Selected test particle orbits are displayed. We show how to prove the stability of these critical hypersurfaces within the Lyapunov theory. Then, we present how to study such effect under the Lagrangian formalism, explaining how to analytically derive the Rayleigh potential for the radiation force. In conclusion, further developments and future projects are discussed.

Vittorio De Falco

Brief Overview of Numerical Relativity

Abstract

The Einstein Field Equations (EFEs) are nonlinear, coupled, partial differential equations that describe the relation between the geometry of a region of spacetime and its matter/energy content. A severe complication is that, with the exception of a few idealised cases characterised by high degrees of symmetry, the EFEs simply cannot be obtained analytically; we need a computer to get the job done. That being said, computers (for better or worse) lack a sense of humour; if you feed them nonsense, they will calculate nonsense. Therefore, in order to find solutions to realistic (asymmetric) spacetimes, we need to be able to somehow prescribe the right numerical recipe to the machine. This recipe comes in several different flavours (formalisms), of which we shall present two of the most widely known variants: the ADM (a.k.a. ADMY) formalism and the BSSN (a.k.a. BSSNOK) formalism. We then close out this overview by briefly mentioning some further considerations (which are huge topics on their own right) such as the initial data problem, gauge choice, and potential applications to problems in relativistic cosmology.

Mario L. Gutierrez Abed

Length-Contraction in Curved Spacetime

Abstract

We present a 4-vector formalism for length-contraction, which applies locally within curved spacetimes. It converts spatial measurements between different reference frames. This leads to two new volume elements on submanifolds: a length-contracted volume and a “de-contraction” volume, built up using wedge products. This approach conveniently handles the vorticity in the rotating disc scenario. For Schwarzschild spacetime, we derive volumes and radial distance relative to certain families of observers, as functions of their energy and angular momentum.

Colin MacLaurin

Exact Solutions of Einstein–Maxwell(-Dilaton) Equations with Discrete Translational Symmetry

Abstract

The aim of this work is to construct exact solutions of Einstein–Maxwell(-dilaton) equations possessing a discrete translational symmetry. We present two approaches to the problem. The first one is to solve Einstein–Maxwell equations in 4D, and the second one relies on dimensional reduction from 5D. We examine the geometry of the solutions, their horizons and singularities and compare them.

Jiří Ryzner, Martin Žofka

Exact Solutions of the Einstein Equations for an Infinite Slab with Constant Energy Density

Abstract

We find exact static solutions of the Einstein equations in the spacetime with plane symmetry, where an infinite slab with finite thickness and homogeneous energy (mass) density is present. In the first solution, the pressure is isotropic, while in the second solution the tangential components of the pressure are equal to zero. In both cases, the pressure vanishes at the boundaries of the slab. Outside the slab, these solutions are matched with the Rindler spacetime and with the Weyl–Levi-Civita spacetime, which represent special cases of the Kasner solution.

Tereza Vardanyan, Alexander Yu. Kamenshchik

Emergence of Classicality from an Inhomogeneous Universe

Abstract

We give a short account of the quantisation of the Szekeres spacetime by considering the symmetries of a reduced action principle. This is an alternative approach from the one followed in the literature for the study of inhomogeneities, which is usually based on perturbations of an inhomogeneous field on the spacetime background. Here, we examine the emergence of classicality with an exact inhomogeneous solution. We check whether the two criteria for classicality are satisfied, that is, the correlations on the phase space and decoherence. We verify that these two properties indeed hold, and thus the classical behaviour emerges from our considerations. We comment on the connection between the emergence of an inhomogeneous spacetime and the current cosmological observations of a highly homogeneous universe at large scales.

Adamantia Zampeli

Titel: Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity
herausgegeben von: Sergio Luigi Cacciatori
Alexander Kamenshchik
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-21845-3
Print ISBN: 978-3-031-21844-6
DOI: https://doi.org/10.1007/978-3-031-21845-3