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Quantum fluctuations and thermodynamic processes in the presence of closed timelike curves by Tsunefumi Tanaka A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Montana State University © Copyright by Tsunefumi Tanaka (1997) Abstract: A closed timelike curve (CTC) is a closed loop in spacetime whose tangent vector is everywhere timelike. A spacetime which contains CTC’s will allow time travel. One of these spacetimes is Grant space. It can be constructed from Minkowski space by imposing periodic boundary conditions in spatial directions and making the boundaries move toward each other. If Hawking’s chronology protection conjecture is correct, there must be a physical mechanism preventing the formation of CTC’s. Currently the most promising candidate for the chronology protection mechanism is the back reaction of the metric to quantum vacuum fluctuations. In this thesis the quantum fluctuations for a massive scalar field, a self-interacting field, and for a field at nonzero temperature are calculated in Grant space. The stress-energy tensor is found to remain finite everywhere in Grant space for the massive scalar field with sufficiently large field mass. Otherwise it diverges on chronology horizons like the stress-energy tensor for a massless scalar field. If CTC’s exist they will have profound effects on physical processes. Causality can be protected even in the presence of CTC’s if the self-consistency condition is imposed on all processes. Simple classical thermodynamic processes of a box filled with ideal gas in the presence of CTC’s are studied. If a system of boxes is closed, its state does not change as it travels through a region of spacetime with CTC’s. But if the system is open, the final state will depend on the interaction with the environment. The second law of thermodynamics is shown to hold for both closed and open systems. A similar problem is investigated at a statistical level for a gas consisting of multiple selves of a single particle in a spacetime with CTC’s. QUANTUM FLUCTUATIONS AND THERMODYNAMIC PROCESSES IN THE PRESENCE OF CLOSED TIMELIKE CURVES by Tsunefumi Tanaka A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics MONTANA STATE UNIVERSITY - BOZEMAN Bozeman, Montana April 1997 J)3l% ii APPROVAL of a thesis submitted by Tsunefumi Tanaka This thesis has been read by each member of the thesis committee, and has been found to be satisfactory regarding content, English usage, format, citations, biblio­ graphic style, and consistency, and is ready for submission to the College of Graduate Studies. William A. Hiscock A L zi /n ^ (Signature) Date ^ Approved for the Department of Physics John Hermanson (Signature) Date Approved for the College of Graduate Studies Z? / ? Zv -7 Robert Brown (Signature) Date iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a doctoral degree at Montana State University - Bozeman, I agree that the Library shall make it available to borrowers under rules of the Library. I further agree that copying of this thesis is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U. S. Copyright Law. Requests for extensive copying or reproduction of this thesis should be referred to University Microfilms International, 300 North.Zeeb Road, Ann Arbor, Michigan, 48106, to whom I have granted “the exclusive right to reproduce and distribute my dissertation in and from microform along with the non­ exclusive right to reproduce and distribute my abstract in any format in whole or in part.” Signature _ Date v5~/ QIJ iv ACKNOWLEDGMENTS I wish to express my gratitude and appreciation to my grandparents who had set the goal of my life for which I will be always striving. V TABLE OF CONTENTS 1. Introduction I Closed Timelike Curves ................................................................................... I Chronology Horizons . ....................................................................................... 2 Spacetimes with CTCs ....................................................................................... 3 Van Stockum Space.............................................................................'. 4 The Godel Universe.................................... 5 Gott Space ........................................... 6 Misner S p a c e .................................................... 7 Wormhole Spacetim e................................................................................ 8 Chronology P rotection...................................................................................... 10 Astronomical Observations.............................................. 11 Classical Instability of Fields.................................................................... 12 Weak Energy C o n d itio n ...............................' ......................................... 13 Vacuum Fluctuations............................................................. 13 2. Scalar Fields in Grant Space 17 Grant Space............................................................................................ ... 19 Calculation of (01 ^ | 0) in Grant S pace ................................................. ... 22 vi Stress-Energy Tensor for a Free Scalar F ie ld ............................ 23 Vacuum State of Grant Space . ■........................................... 24 Method of Images . .................................................................... 27 Massive Scalar Field . .................................................................... 29 Self-Interacting Scalar Field (A</>4) . ............................... ..................... 32 Nonzero tem perature....................................., .................................... 37 3. Physics in the Presence of CTC’s 43 Classical Scattering in Wormhole Spacetime .................................. ... 44 Quantum Mechanics in a Nonchronal Region..................................... 46 Path Integral Formulation.......................................................... 47 Density Matrix Formulation . : ........................................ 50 4. Classical Thermodynamic Processes in a Nonchronal Region 55 A pparatus............ ; .........................................................................; . 56 Step I ............................................................................................ 58 Step 2 ...................................................................................... ... 60 Step 3 ............................................................................................ 60 Self-consistency c o n d itio n .................................................................... 60 Closed Systems . ............................................................................. 61 Heat T ransfer................................................................................ 62 Adiabatic Expansion................................................................... 65 Mixing of the Gases 67 ) vii Open Systems .............................................................. 68 Isothermal Expansion . ..................................... 69 Isobaxic E xpansion........................................... 73 5. Statistical Mechanics in a Nonchronal Region 77 Formation of a Multiple-Self G a s ............................ 77 Objectives.................................................................... 80 General Approach....................................................... 81. Self- Consistency Condition........................................ 88 Simplifications .......................................................... 88 Model Spacetim e.............................................. 88 Particle Structure.............................................. 90 Particle Interaction........................................... 91 Next S te p .................................................................... 93 6. Conclusion 94 Quantum Fluctuations . 95 Thermodynamic Processes 97 Multiple-Self Gas ...... 98 BIBLIOGRAPHY 99 viii LIST OF FIGURES 1 Nonchronal region...................................................................................... 3 2 Van Stockum sp a ce ................................................................................... 5 3 Gott space ................................................................................................... 7 4 Misner space...............................' .............................................................. 8 5 Wormhole spacetime .......................................................... 9 6 Wormhole time m ac h in e .......................................................................... 10 7 Defocusing e ff e c t...................................................................................... 12 8 Grant sp a c e .............................................. 21 9 Energy density vs field m a s s ..................... ■............................................. 31 10 Thermal contribution to p ........................................'....................... ... 40 11 Foliation of spacetime................................................................................ 47 12 Quantum computational network . .................................................. 51 13 One time traverse..................................................................................... 56 14 Trajectories for one time tra v e rse ...........................................................
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