We consider the Cauchy problem for the scalar wave equation in the Kerr geometry for smooth initial data supported outside the event horizon. We prove that the solutions decay in time in L^\infty_loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables.Comment: 44 pages, 5 figures, minor correction
We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar's separation of variables. It is proved that for initial data in L ∞ loc near the event horizon with L 2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity. e-print archive:
The international monograph series "Fundamental Theories of Physics" aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields. Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paperThis Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland PrefaceThis book is devoted to explaining how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields. It is the result of an endeavor which I was occupied with for many years. Publishing the methods and results as a book gives me the opportunity to present the material in a coherent and comprehensible way.The four chapters of this book evolved differently. Chapters 1 and 2 are based on the notes of my lecture "The fermionic projector and causal variational principles" given at the University of Regensburg in the summer semester 2014. The intention of this lecture was to introduce the basic concepts. Most of the material in these two chapters has been published previously, as is made clear in the text by references to the corresponding research a...
As was recently pointed out to us by Thierry Daudé, there is an error on the last page of our paper [2]. Namely, the inequality (8.3) cannot be applied to the function (t) because it does not satisfy the correct boundary conditions. This invalidates the last two inequalities of the paper, and thus the proof of decay is incomplete. We here fill the gap using a different method. At the same time, we will clarify in which sense the sum over the angular momentum modes converges in [1, Theorem 1.1] and [2, Theorem 7.1], an issue which in these papers was not treated in sufficient detail. The arguments in these papers certainly yield weak convergence in L 2 loc ; here we will prove strong convergence. Our method here is to split the wave function into the high and low energy components. For the high energy component, we show that the L 2 -norm of the wave function can be bounded by the energy integral, even though the energy density need not be everywhere positive (Sect. 1). For the low energy component we refine our ODE techniques (Sect. 2). Combining these arguments with a Sobolev estimate and the Riemann-Lebesgue lemma completes the proof (Sect. 3).We begin by considering the integral representation of [2, Theorem 7.1], for fixed k and a finite number n 0 of angular momentum modes, The online version of the original article can be found under
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