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16
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 66 (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.
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 6 (5 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.
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 5 (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
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 4 (0 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 1 (0 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
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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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
THEORETICAL GRAVITATIONAL LENSING – BEYOND THE WEAKFIELD SMALLANGLE APPROXIMATION
, 708
"... An overview is given on those theoretical gravitational lensing results that can be formulated in a spacetime setting, without assuming that the gravitational fields are weak and that the bending angles are small. The first part is devoted to analytical methods considering spacetimes in which the eq ..."
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An overview is given on those theoretical gravitational lensing results that can be formulated in a spacetime setting, without assuming that the gravitational fields are weak and that the bending angles are small. The first part is devoted to analytical methods considering spacetimes in which the equations for light rays (lightlike geodesics) is completely integrable. This includes spherically symmetric static spacetimes, the Kerr spacetime and plane gravitational waves. The second part is devoted to qualitative methods which give some information on lensing properties without actually solving the equation for lightlike geodesics. This includes Morse theory, methods from differential topology and bifurcation theory. 1.
PAINLEVÉGULLSTRAND COORDINATES FOR THE KERR SOLUTION
, 805
"... Abstract. We construct a coordinate system for the Kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the PainlevéGullstrand coordinate system for the Schwarzschild solution. The Kerr metric can then be interpreted as describing space flowing on a ( ..."
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Abstract. We construct a coordinate system for the Kerr solution, based on the zero angular momentum observers dropped from infinity, which generalizes the PainlevéGullstrand coordinate system for the Schwarzschild solution. The Kerr metric can then be interpreted as describing space flowing on a (curved) Riemannian 3manifold. The stationary limit arises as the set of points on this manifold where the speed of the flow equals the speed of light, and the horizons as the set of points where the radial speed equals the speed of light. A deeper analysis of what is meant by the flow of space reveals that the acceleration of freefalling objects is generally not in the direction of this flow. Finally, we compare the new coordinate system with the closely related Doran coordinate system.