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NonTuring computations via MalamentHogarth spacetimes
 Int. J. Theoretical Phys
, 2002
"... We investigate the Church–Kalmár–Kreisel–Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church–Turingtype Theses (not only ..."
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Cited by 79 (8 self)
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We investigate the Church–Kalmár–Kreisel–Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church–Turingtype Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church– Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various “obstacles ” to computing a nonrecursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the “design ” of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.
Rodnianski Decay for solutions of the wave equation on Kerr exterior spacetimes III: The case ∣a∣ <M
"... Abstract. This paper contains the first two parts (III) of a threepart series concerning the scalar wave equation 2gψ = 0 on a fixed Kerr background(M, ga,M). We here restrict to two cases: (II1) ∣a ∣ ≪ M, general ψ or (II2)∣a ∣ < M, ψ axisymmetric. In either case, we prove a version of ‘inte ..."
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Cited by 46 (4 self)
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Abstract. This paper contains the first two parts (III) of a threepart series concerning the scalar wave equation 2gψ = 0 on a fixed Kerr background(M, ga,M). We here restrict to two cases: (II1) ∣a ∣ ≪ M, general ψ or (II2)∣a ∣ < M, ψ axisymmetric. In either case, we prove a version of ‘integrated local energy decay’, specifically, that the 4integral of an energytype density (degenerating in a neighborhood of the Schwarzschild photon sphere and at infinity), integrated over the domain of dependence of a spacelike hypersurface Σ connecting the future event horizon with spacelike infinity or a sphere on null infinity, is bounded by a natural (nondegenerate) energy flux of ψ through Σ. (The case (II1) has in fact been treated previously in our Clay Lecture notes: Lectures on black holes and linear waves, arXiv:0811.0354.) In our forthcoming Part III, the restriction to axisymmetry for the general∣a ∣ < M case is removed. The complete proof is surveyed in our companion paper The black hole stability problem for linear scalar perturbations, which includes the essential details of our forthcoming Part III. Together with previous work (see our: A new physicalspace approach to decay for the wave equation with applications to black hole spacetimes, in XVIth International Congress on Mathematical Physics, Pavel Exner ed., Prague 2009 pp. 421– 433, 2009,
Gravitational Lensing from a Spacetime Perspective
, 2004
"... The theory of gravitational lensing is reviewed from a spacetime perspective, without quasiNewtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It i ..."
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Cited by 28 (3 self)
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The theory of gravitational lensing is reviewed from a spacetime perspective, without quasiNewtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary generalrelativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.
Causal properties of AdSisometry groups II: BTZ multiblack holes
"... Abstract. We study the causality relation in the 3dimensional antide Sitter space AdS and its conformal boundary Ein2. To any closed achronal subset Λ in Ein2 we associate the invisible domain E(Λ) from Λ in AdS. We show that if Γ is a torsionfree discrete group of isometries of AdS preserving Λ ..."
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Cited by 14 (9 self)
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Abstract. We study the causality relation in the 3dimensional antide Sitter space AdS and its conformal boundary Ein2. To any closed achronal subset Λ in Ein2 we associate the invisible domain E(Λ) from Λ in AdS. We show that if Γ is a torsionfree discrete group of isometries of AdS preserving Λ and is nonelementary (for example, not abelian) then the action of Γ on E(Λ) is free, properly discontinuous and strongly causal. If Λ is a topological circle then the quotient space MΛ(Γ) = Γ\E(Λ) is a maximal globally hyperbolic AdSspacetime admitting a Cauchy surface S such that the induced metric on S is complete. In a forthcoming paper [8] we study the case where Γ is elementary and use the results of the present paper to define a large family of AdSspacetimes including all the previously known examples of BTZ multiblack holes. 1.
Creation of fermions by rotating charged blackholes
, 2009
"... This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal ..."
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Cited by 11 (2 self)
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This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the spacetime outside the star is given by the KerrNewman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic behavior of certain timelike geodesics in the KerrNewman metric. The Dirac equation is then written using coordinates and a NewmanPenrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields in a globally hyperbolic spacetime is described. We formulate and prove a theorem about the Hawking effect in this setting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result
General relativistic hypercomputing and foundation of mathematics
"... Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, ..."
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Cited by 9 (1 self)
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Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, e.g. which can decide the halting problem of Turing machines or decide whether ZF set theory is consistent (more precisely, can decide the theorems of ZF). Starting from this, we will discuss the impact of recent breakthrough results of relativity theory, black hole physics and cosmology to well established foundational issues of computability theory as well as to logic. We find that the unexpected, revolutionary results in the mentioned branches of science force us to reconsider the status of the physical Church Thesis and to consider it as being seriously challenged. We will outline the consequences of all this for the foundation of mathematics (e.g. to Hilbert’s programme). Observational, empirical evidence will be quoted to show that the statements above do not require any assumption of some physical universe outside of our own one: in our specific physical universe there seem to exist regions of spacetime supporting potential nonTuring computations. Additionally, new “engineering ” ideas will be outlined for solving the socalled blueshift problem of GRcomputing. Connections with related talks at the Physics and Computation meeting, e.g. those of Jerome DurandLose, Mark Hogarth and Martin Ziegler, will be indicated. 1
CARTAN’S STRUCTURAL EQUATIONS FOR DEGENERATE METRIC
, 2011
"... Abstract. Cartan’s structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. On singular semiRiemannian manifolds, because the metric is allowed to be degenerate, there are some obstructions ..."
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Cited by 8 (4 self)
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Abstract. Cartan’s structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. On singular semiRiemannian manifolds, because the metric is allowed to be degenerate, there are some obstructions in constructing the geometric objects normally associated to the metric. We can no longer construct local orthonormal frames and coframes, or define a metric connection and its curvature operator. But we will see that if the metric is radical stationary, we can construct objects similar to the connection and curvature forms of Cartan, to which they reduce if the metric is nondegenerate. We write analogs of Cartan’s first and second structural equations. As a byproduct we will find a compact version of the Koszul formula. Contents
Geodesics on convex regions of stationary spacetimes and Finslerian Randers spaces, arXiv:1112.3892v1 [math.DG
, 2011
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Quantum vacuum effects in gravitational fields: theory and detectability
, 2000
"... This thesis is devoted to the study of quantum vacuum effects in the presence of strong gravitational fields. We shall see how the quantum vacuum interacts with black hole geometries and how it can play an important role in the interpretation of the gravitational entropy. In this respect particular ..."
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Cited by 6 (3 self)
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This thesis is devoted to the study of quantum vacuum effects in the presence of strong gravitational fields. We shall see how the quantum vacuum interacts with black hole geometries and how it can play an important role in the interpretation of the gravitational entropy. In this respect particular attention will be given to the peculiar role of the extremal black hole solutions. From this branch of our research we shall try to collect some important hints about the relation between quantum gravity theories and the semiclassical results. After these investigations we shall move our attention toward possible experimental tests of particle creation from the quantum vacuum which is an indirect confirmation of the Hawking effect. This aim will lead us to study acoustic geometries and their way of “simulating ” General Relativity structures, such as horizons and black holes. We shall study the stability of these structures and the problems related to setting up experimental detection of phonon Hawking flux from acoustic horizons. This research will naturally lead us to propose a new model for explaining the emission of light in the phenomenon of Sonoluminescence, based on the dynamical Casimir effect. Possible experimental tests of this proposal will be discussed. In this way we shall set up one of the few available models of quantum vacuum radiation amenable to observational test within the next few years. After this