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Curvature Invariants for Wormholes and Warped Spacetimes

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Curvature Invariants for Wormholes and Warped Spacetimes ABSTRACT Curvature Invariants for Wormholes and Warped Spacetimes B. Mattingly, Ph.D. Mentor: G. Cleaver, Ph.D. The Carminati and McLenaghan (CM) curvature invariants are powerful tools for probing spacetimes. Henry et al. formulated a method of plotting the CM curva- ture invariants to study black holes. The CM curvature invariants are scalar functions of the underlying spacetime. Consequently, they are independent of the chosen coor- dinates and characterize the spacetime. For Class B1 spacetimes, there are four inde- pendent CM curvature invariants: R, r1, r2, and w2. Lorentzian traversable wormholes and warp drives are two theoretical solutions to Einstein’s field equations, which allow faster-than-light (FTL) transport. The CM curvature invariants are plotted and an- alyzed for these specific FTL spacetimes: (i) the Thin-Shell Flat-Face wormhole, (ii) the Morris-Thorne wormhole, (iii) the Thin-Shell Schwarzschild wormhole, (iv) the arXiv:2103.14725v1 [gr-qc] 26 Mar 2021 exponential metric, (v) the Alcubierre metric at constant velocity, (vi) the Natário metric at constant velocity, and (vii) the Natário metric at an accelerating velocity. Plots of the wormhole CM invariants confirm their traversability and show how to distinguish the wormholes. The warp drive CM invariants reveal key features such as a flat harbor in the center of each warp bubble, a dynamic wake for each warp bubble, and rich internal structure(s) of each warp bubble. Copyright © 2021 by B. Mattingly All rights reserved TABLE OF CONTENTS LIST OF FIGURES ................................................................... vi LIST OF ABBREVIATIONS......................................................... x CHAPTER ONE Introduction ........................................................................ 1 1.1 Historical Foundations................................................... 1 1.2 Wormholes................................................................ 8 1.3 Warp Drives .............................................................. 10 1.4 Purpose of Research ..................................................... 12 CHAPTER TWO Curvature Invariants ............................................................... 15 2.1 Spacetime Manifolds ..................................................... 15 2.2 Curvature Invariants..................................................... 21 CHAPTER THREE Lorentzian Traversable Wormholes ............................................... 33 3.1 Thin-Shell Flat-Face Wormhole ......................................... 34 3.2 Morris-Thorne Wormhole ............................................... 36 3.3 Thin-Shell Schwarzschild Wormhole .................................... 43 3.4 Exponential Metric....................................................... 45 CHAPTER FOUR Warp Drives Moving at a Constant Velocity ..................................... 47 4.1 Warp Drive Spacetimes .................................................. 48 4.2 Alcubierre’s Warp Drive ................................................. 49 4.3 Natário’s Warp Drive.................................................... 66 CHAPTER FIVE Warp Drives Moving at a Constant Acceleration ................................ 78 5.1 The Accelerating Natário Spacetime .................................... 79 5.2 Invariant Plots of Time for Natário .................................... 82 5.3 Invariant Plots of Acceleration for Natário ............................ 88 5.4 Invariant Plots of Skin Depth for Natário ............................. 93 5.5 Invariant Plots of Radius for Natário .................................. 94 CHAPTER SIX Conclusion .......................................................................... 97 6.1 Invariant Research ....................................................... 99 6.2 Wormhole Research ......................................................102 iv 6.3 Warp Drive Research ....................................................104 6.4 Closing Thoughts.........................................................108 APPENDIX A Mathematica Program ............................................................. 111 APPENDIX B Invariants for the Natário Metric at Constant Velocity ......................... 122 APPENDIX C Invariants for the Natário Metric at Constant Acceleration..................... 130 BIBLIOGRAPHY ..................................................................... 133 v LIST OF FIGURES Figure 1.1. Figure 17.2 from [5]. It depicts the 2D analog of the time evolution of Eq. 1.2 rotated about the central vertical axis. ......... 4 Figure 1.2. Figure 17.3 from [5]. It depicts the full time evolution of Eq. 1.2.... 4 Figure 1.3. Figure 8.3 from [5]. It shows the null cone or light cone of events relative to P.............................................................. 6 Figure 1.4. An example of a tipping light cone by a warped underlying spacetime. ............................................................... 7 Figure 1.5. The light cone of an object rotating around the axis of a van Stockum spacetime will tip over........................................ 7 Figure 1.6. Fig. 1 from [11]. (a) is an embedding diagram for a wormhole that connects two different universes. (b) is an embedding diagram for a wormhole that connects two distant regions of our own universe. Each diagram depicts the geometry of an π equatorial (θ = 2 ) slice through space at a specific moment in time (t = constant). .................................................... 10 Figure 2.1. Figure 1.3.1 from [6]. ................................................... 17 Figure 2.2. Figure 1.3.2 from [6]. ................................................... 18 Figure 2.3. Figure 1.3.3 from [6]. ................................................... 19 Figure 3.1. Plots of the non-zero invariants for the MT wormhole. The plots are in radial coordinates with r 2 f0; 4g. Each radial mesh line represents a radial distance of r = 0:26¯. G = M = 1 were normalized for simplicity and r0 = 2 was chosen as the throat. Notice the divergence at the center of each plot is completely inside the r = 2 = r0 radial line. This does not affect the traversability of the wormhole. ........................................ 38 Figure 3.2. Successive plots of the shape function for different powers of r in the Ricci scalar for the MT wormhole. The plots are in radial coordinates with r 2 f0; 4g. Each radial mesh line represents a radial distance of r = 0:26¯. G = M = 1 were normalized for simplicity and r0 = 2 was chosen as the throat. ...................... 41 vi Figure 3.3. Plot of MT w2 for the shape function given in Eq. (3.11). The plots are in radial coordinates with r 2 f0; 4g with G = M = 1 normalized and r0 = 2 chosen. ......................................... 42 Figure 3.4. Plot of Schwarzschild w2. The plot is in radial coordinates with r 2 f0; 4g. Each mesh line represents a radial distance of 0:5. 3 The δ-function can be seen as a thin discontinuity at r = 2 and its value is recorded in Eq. (3.14). ..................................... 45 Figure 3.5. Plots of the non-zero invariants for the exponential metric. The plots are in radial coordinates with r 2 f0; 1:8Mg. Each mesh line represents a radial distance of 0:1M. The throat begins at r = M.................................................................... 46 Figure 4.1. The expansion of the normal volume elements for the Alcubierre warp drive spacetime.................................................... 49 Figure 4.2. Plots of the R invariants for the Alcubierre warp drive while varying a velocity. σ = 8 and ρ = 1 as Alcubierre originally suggested in his paper [32]. ............................................. 55 Figure 4.3. Plots of the r1 invariants for the Alcubierre warp drive while varying velocity. The other variables were chosen as σ = 8 and ρ = 1 to match the variables Alcubierre originally suggested in his paper [32]. ........................................................... 56 Figure 4.4. Plots of the w2 invariants for the Alcubierre warp drive while varying velocity. The radius was chosen as ρ = 1 to match the variables Alcubierre originally suggested in his paper [32]. The skin depth was chosen as σ = 2 to keep the plots as machine size numbers. Both velocity and skin depth have an exponential affect on the magnitude of the invariants.............................. 57 Figure 4.5. Plots of the R invariants for the Alcubierre warp drive while varying skin depth. The variables were chosen as vs = 8 and ρ = 1 to match the variables Alcubierre originally suggested in his paper [32]. ........................................................... 59 Figure 4.6. Plots of the r1 invariants for the Alcubierre warp drive while varying skin depth. The other variables were chosen as vs = 1 and ρ = 1 to match the variables Alcubierre originally suggested in his paper [32]. ........................................................ 60 Figure 4.7. Plots of the w2 invariants for the Alcubierre warp drive while varying skin-depth. The radius and velocity were chosen as ρ = 1 and vs = 1 to match the variables Alcubierre originally suggested in his paper [32]. ............................................. 61 vii Figure 4.8. Plots of the R invariants for the Alcubierre warp drive while varying radius. The other variables were chosen as σ = 8 and vs = 1 to match the variables Alcubierre originally suggested in his paper [32]. ........................................................... 63 Figure 4.9. Plots of the r1 invariants for the Alcubierre warp drive
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