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Dimensional Reduction and Spacetime Pathologies

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Dimensional Reduction and Spacetime Pathologies Dimensional Reduction and Spacetime Pathologies by Sergei Slobodov M.Sc., The St. Petersburg Technical University, 1985 M.Sc., The University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2010 c Sergei Slobodov 2010 Abstract Dimensional reduction is a well known technique in general relativity. It has been used to resolve certain singularities, to generate new solutions, and to reduce the computational complexity of numerical evolution. These ad- vantages, however, often prove costly, as the reduced spacetime may have various pathologies, such as singularities, poor asymptotics, negative en- ergy, and even superluminal matter flows. The first two parts of this thesis investigate when and how these pathologies arise. After considering several simple examples, we first prove, using pertur- bative techniques, that under certain reasonable assumptions any asymptot- ically flat reduction of an asymptotically flat spacetime results in negative energy seen by timelike observers. The next part describes the topological rigidity theorem and its consequences for certain reductions to three dimen- sions, confirming and generalizing the results of the perturbative approach. The last part of the thesis is an investigation of the claim that closed timelike curves generically appearing in general relativity are a mathematical arti- fact of periodic coordinate identifications, using, in part, the dimensional reduction techniques. We show that removing these periodic identifications results in naked quasi-regular singularities and is not even guaranteed to get rid of the closed timelike curves. ii Statement of Collaboration The research covered by this thesis is a collection of projects done either alone or in collaboration with or under supervision of Kristin Schleich, Don Witt and Johan Brannlund. Specifically, chapter 3 is largely an unassisted effort, most of the chapter 4 is a presentation of the work done in collab- oration with Kristin Schleich, Don Witt and Johan Brannlund. Chapter 5 describes and extends the results first obtained by Kristin Schleich and Don Witt. Chapter 6 describes my original work, with guidance and feedback provided by Kristin Schleich. iii Table of Contents Abstract ................................. ii Statement of Collaboration ..................... iii Table of Contents ............................ iv List of Figures .............................. vii Acknowledgements ........................... x Dedication ................................ xi 1 Introduction ............................. 1 2 Background ............................. 7 2.1 Causality and Hyperbolicity . 7 2.2 EnergyConditions ........................ 9 2.2.1 Weak Energy Condition . 9 2.2.2 Null Energy Condition . 9 2.2.3 Average Null Energy Condition . 10 2.2.4 Dominant Energy Condition . 11 2.2.5 Timelike Convergence and the Strong Energy Condi- tion ............................ 11 2.3 AsymptoticFlatness . 12 2.4 Total Energy and Angular Momentum . 15 2.5 Axisymmetry ........................... 16 2.6 DimensionalReduction . 17 3 Examples of Dimensional Reduction .............. 22 3.1 Schwarzschild Blackhole . 22 3.2 ChargedBlackhole . .. .. .. .. .. .. 28 3.3 SpinningBlackhole ....................... 29 3.4 Axisymmetric Gravitational Radiation . 31 iv Table of Contents 3.5 Five-dimensional Schwarzschild-like Spacetime . 33 3.6 Higher-dimensional Schwarzschild Spacetime . 33 3.7 Effects of the Cosmological Constant . 34 3.7.1 De Sitter and Anti-de Sitter Spacetimes . 35 3.7.2 Schwarzschild-(Anti-)de Sitter . 35 3.7.3 Four-dimensional AdS Soliton . 36 3.8 Doubling Reduced Spacetime . 38 3.8.1 Doubling Reduced Minkowski Spacetime . 38 3.9 Axisymmetric Conformal Reduction . 40 4 Violations of Energy Conditions in Dimensionally Reduced Spacetimes .............................. 43 4.1 Perturbative Calculation . 43 4.2 Violation of the Dominant Energy Condition in Axisymmetric Reduction ............................ 51 5 Topological Restrictions on the Properties of Dimensionally Reduced Spacetimes ........................ 54 5.1 Introduction ........................... 54 5.2 Topological Censorship . 55 5.3 PTC Corollary for Three-dimensional Spacetimes . 56 5.4 Topological Rigidity of U(1) Reduction to Three Dimensions 58 5.5 No AF U(1) reduction from Four to Three Dimensions . 60 5.6 ExamplesofReduction . 61 5.7 No Blackhole Horizons in Reduction to Three Dimensions . 62 5.8 Considerations for Axisymmetric Numerical Schemes . 63 6 Unwrapping Closed Timelike Curves ............. 66 6.1 Introduction ........................... 66 6.2 SpacetimeswithCTCs . .. .. .. .. .. 68 6.3 Unwrapping ........................... 72 6.3.1 Singular Spacetimes . 73 6.3.2 Quasi-regular Singularities in a Flat Spacetime . 76 6.3.3 Singularities Created by Unwrapping . 79 6.4 Unwrapped Gott Spacetime . 80 6.4.1 Construction ....................... 81 6.4.2 CTCs of the Gott Spacetime . 83 6.4.3 Unwrapping Gott Spacetime . 84 6.5 Unwrapping G¨odel Spacetime . 86 6.5.1 Unwrapping Procedure and Circular CTCs . 87 v Table of Contents 6.5.2 RemainingCircularCTCs . 88 6.6 Multiple Snwrapping of the G¨odel Spacetime . 89 6.6.1 DoubleUnwrapping . 91 6.6.2 Multiple Unwrapping . 92 6.6.3 Sector-like CTCs in the G¨odel Spacetime . 94 6.7 Conclusion ............................ 96 7 Conclusion .............................. 97 Bibliography ............................... 99 Index ...................................108 vi List of Figures 2.1 Conformal diagram of an infinite Minkowski universe. Cour- tesyofWikimediaCommons. 8 3.1 A schematic representation of axisymmetric dimensional re- duction and doubling of the Schwarzschild blackhole. One spatial slice is shown. Arrows show the orbits of the Killing vector φ. Spherical horizon r = 2M in three dimensions be- comes a half-circle after reduction, connected to the boundary at sin θ = 0. After doubling, the spacetime has no boundary and there are non-contractible causal curves starting and end- ing on I in this spacetime. One such curve is shown. 25 6.1 Unwrapping Minkowski space: a) pick a cylindrical chart, b) remove the fixed points of φ at r = 0, c) go to the universal cover by extending the range of φ to all reals. A t = z = 0 slice is shown as a collection of Cartesian charts with y = 0,x 0 ≥ removed and the surfaces φ = 2π of one chart identified with the surface φ = 0 of the next chart. φ¯ is the global angular coordinate, φ¯ R. ........................ 77 ∈ 6.2 A visualization of a two-dimensional spatial slice of the un- wrappedMinkowskispace.. 77 6.3 Conical wedge identification choices in the Gott spacetime. Wedge fill lines indicate the identified points. The strings are shown at the moment of the closest approach. Left: Gott identification, where CTCs are manifest (one CTC is shown). Center: Cutler identification, used to prove the existence of points not lying on any CTCs. Right: Carroll identification, used to visualize the existence of CTC-free spacelike hyper- surfaces, as the opposite sides of the wedge are identified at equaltimes............................. 82 vii List of Figures 6.4 Unwrapping Gott spacetime. a) The x = y = 0 subspace is removed and the universal cover is constructed by patching multiple Cartesian charts together. The former CTCs (in- dicated by the vertical arrows) are now open curves passing from chart to chart. The identifications between charts are in- dicated by the wavy arrows. b) An alternative way to unwrap: a single Gott chart is matched to a collection of Minkowski charts. The former CTCs wrap around just the two strings inthesingleGottchart. 85 6.5 Normalized concentric CTCs of radius R illustrated in the co- ordinates x , t . The CTCs lie inside a square 2R 2√2 tan−1 sinh R with the side equal to two, but outside of a circle inscribed intoit................................ 90 6.6 Constructing the homotopy equivalence of the G¨odel space after double unwrapping. We start with the (t,x,y = 0) sub- space with two points removed (left), then retract the space onto first two circles connected by a line segment, then the “figure 8”, and finally construct the universal cover of the “fig- ure 8” (only the first four levels of nodes are shown). Each edge corresponds to a circle wrapping around one of the two singularities in the original space, so the unwrapped space contains infinitely many singularities. 92 6.7 A tessellation of the two-dimensional quotient space of the G¨odel spacetime. Each vertex corresponds to a timelike line in the full spacetime and each circle is an image of a closed null curve R = Rmin. The tessellation is dense enough to make any circular CTCs wrap around at least one such line, as shown. Once the lines are removed and the resulting non- simply connected spacetime is lifted into the full spacetime and then into its universal cover, no circular CTCs are present in this “multiply unwrapped” spacetime. 93 6.8 A visualization of the two-dimensional quotient space of the G¨odel spacetime unwrapped at three points at once. Only a small patch of this space is shown, as unwrapping around two or more points results in a countable infinity of singular boundaries. ............................ 93 viii List of Figures 6.9 Images of sector-like closed null curves in the (x,t) coordi- nates for a range of values R. Arrows indicate the future null directions. For large R the curve tends to a triangle with a fixed minimum angle.
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