ABSTRACT Curvature Invariants for Wormholes and Warped

ABSTRACT

Curvature Invariants for Wormholes and Warped Spacetimes

B. Mattingly, Ph.D.

Chairperson: 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 coordinates and characterize the spacetime. For Class B1 spacetimes, there are four independent 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 analyzed 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 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 t heir t raversability a nd s how h ow 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. Curvature Invariants for Wormholes and Warped Spacetimes

B. Mattingly, B.A., M.S.

A Dissertation

Approved by the Department of Physics

D. Russell, Ph.D., Chairperson

Submitted to the Graduate Faculty of Baylor University in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

Approved by the Dissertation Committee

G. Cleaver, Ph.D., Chairperson

G. Benesh, Ph.D.

J. Dittmann, Ph.D.

Q. Sheng, Ph.D.

A. Wang, Ph.D.

Accepted by the Graduate School May 2020

J. Larry Lyon, Ph.D., Dean

LIST OF FIGURES ...... vii

LIST OF ABBREVIATIONS...... xii

ACKNOWLEDGMENTS ...... xiii

DEDICATION ...... xiv

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...... 22

CHAPTER THREE Lorentzian Traversable Wormholes ...... 33

3.1 Thin-Shell Flat-Face Wormhole ...... 34

iv 3.2 Morris-Thorne Wormhole ...... 36

3.3 Thin-Shell Schwarzschild Wormhole ...... 42

3.4 Exponential Metric...... 44

CHAPTER FOUR Warp Drives Moving at a Constant Velocity ...... 46

4.1 Warp Drive Spacetimes ...... 47

4.2 Alcubierre’s Warp Drive ...... 48

4.3 Natário’s Warp Drive...... 65

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

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

v APPENDIX C Invariants for the Natário Metric at Constant Acceleration...... 130

BIBLIOGRAPHY ...... 133

vi 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...... 3

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...... 5

Figure 1.4. An example of a tipping light cone by a warped underlying spacetime...... 6

Figure 1.5. The light cone of an object rotating around the axis of a van Stockum spacetime will tip over...... 6

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]...... 16

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 ∈ {0, 4}. 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

vii Figure 3.2. Plot of MT w2 for the shape function given in Eq. (3.11). The plots are in radial coordinates with r ∈ {0, 4} with G = M = 1 normalized and r0 = 2 chosen...... 40

Figure 3.3. 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 ∈ {0, 4}. 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

Figure 3.4. Plot of Schwarzschild w2. The plot is in radial coordinates with r ∈ {0, 4}. 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)...... 44

Figure 3.5. Plots of the non-zero invariants for the exponential metric. The plots are in radial coordinates with r ∈ {0, 1.8M}. Each mesh line represents a radial distance of 0.1 M. The throat begins at r = M...... 45

Figure 4.1. The expansion of the normal volume elements for the Alcubierre warp drive spacetime...... 48

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]...... 54

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]...... 55

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...... 56

viii 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]...... 58

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]...... 59

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]...... 60

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]...... 62

Figure 4.9. Plots of the r1 invariants for the Alcubierre warp drive while varying the 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 4.10. Plots of the w2 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]...... 64

Figure 4.11. The velocity evolution of R, the Ricci scalar for the Natário warp drive at a constant velocity. It is understood that vs is multiplied 1 by c. The other variables are set to σ = 50,000 m and ρ = 100 m. . 70

Figure 4.12. The velocity evolution of the r1 invariant for the Natário warp drive at a constant velocity. It is understood that vs is multiplied 1 by c. The other variables are set to σ = 50,000 m and ρ = 100 m. . 71

ix Figure 4.13. The velocity Evolution of the r2 invariant for the Natário warp drive at a constant velocity. It is understood that vs is multiplied 1 by c. The other variables are set to σ = 50,000 m and ρ = 100 m. . 72

Figure 4.14. The velocity evolution of the w2 invariant for the Natário warp drive at a constant velocity. The other variables are set to σ = 1 50,000 m and ρ = 100 m...... 73

Figure 4.15. The warp bubble skin depth for the Ricci scalar and r1 for the Natário warp drive at a constant velocity. The other variables m were chosen to be v = 1 s , and ρ = 100 m in natural units...... 74

Figure 4.16. The warp bubble skin depth for r2 and w2 for the Natário warp drive at a constant velocity. The other variables were chosen to m be v = 1 s , and ρ = 100 m in natural units...... 75

Figure 4.17. The warp bubble radius for the Ricci scalar and r1 for the Natário warp drive at a constant velocity. The other variables m 1 were chosen to be v = 1 s , and σ = 50, 000 m in natural units...... 76

Figure 4.18. The warp bubble radius for the r2, and w2 for the Natário warp drive at a constant velocity. The other variables were chosen to m 1 be v = 1 s , and σ = 50, 000 m in natural units...... 77

Figure 5.1. Foliation of spacetime into 3D spacelike hypersurfaces at three diﬀerent times from [46]...... 79

Figure 5.2. The time evolution of R, the Ricci scalar. The other variables m 1 were chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 84

Figure 5.3. The time evolution of the invariant r1. The other variables were m 1 chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 85

Figure 5.4. The time evolution of the invariant r2. The other variables were m 1 chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 86

x Figure 5.5. The time evolution for the invariant w2. The other variables m 1 were chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 87

Figure 5.6. Varying the acceleration of R, the Ricci scalar. The other m 1 variables were chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 89

Figure 5.7. Varying acceleration for the invariant r1. The other variables m 1 were chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 90

Figure 5.8. Varying acceleration for the invariant r2. The other variables m 1 were chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 91

Figure 5.9. Varying acceleration for the invariant w2. The other variables m 1 were chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units...... 92

Figure 5.10. The warp bubble skin depth for the Ricci scalar and r1. The m other variables were chosen to be t = 1 s, a = 1 s2 , and ρ = 100 m in natural units...... 93

Figure 5.11. The warp bubble skin depth for r2 and w2. The other variables m were chosen to be t = 1 s, a = 1 s2 , and ρ = 100 m in natural units. 94

Figure 5.12. The warp bubble radius for the Ricci scalar and r1. The other m 1 variables were chosen to be t = 1 s, a = 1 s2 , and σ = 50,000 m in natural units...... 95

Figure 5.13. The warp bubble radius for r2 and w2. The other variables were m 1 chosen to be t = 1 s, a = 1 s2 , and σ = 50,000 m in natural units. . 96

xi LIST OF ABBREVIATIONS

CM Carminati and McLenaghan

CTC Closed-timelike-curves

FTL Faster-than-Light

GR General Relativity

MT Morris-Thorne

NEC Null-energy-condition(s)

PhD Doctor of Philosophy

QM Quantum Mechanics

SP Scalar-Polynomial

SR Special Relativity

TS Thin-Shell

xii ACKNOWLEDGMENTS

Thank you, Dr. Davis for inspiring this research and helping me understand the basics of wormholes and warp drives.

Thank you, Dr. McNutt for helping me to understand SP invariants.

Thank you, Dr. Cleaver for this amazing opportunity to study the boundaries of

Physics.

Thank you, Abinash, Cooper, Mat, Will, and Caleb for helping me at each step to complete this research.

Thank you, A. Alexander Beaujean, Ph.D. (Associate Professor of Psychology &

Neuroscience) for providing the LATEX template for this dissertation.

xiii DEDICATION

To my parents, Ed and Jill. You two supported me every step along this journey.

During the highs, y’all were there to celebrate. During the lows, y’all were there to

encourage. I love both of y’all so much. Godspeed.

xiv CHAPTER ONE

Introduction

Faster-Than-Light (FTL) travel fascinates and delights the imaginations of scientists and science enthusiasts alike. FTL promises a voyage into the final frontier. It tempts us with possibilities and answers beyond what we can either find or conceive here on Earth. Unfortunately, physical reality constrains what is possible. Einstein s and his theories of relativity teach us that the speed of light, c = 299, 792, 458 m , is the speed limit for travel inside of the universe. Matter and energy absolutely cannot move through spacetime faster than c. However, Einstein did leave us a loophole.

Spacetime itself has no such limitation.

1.1 Historical Foundations

General Relativity (GR) successfully describes classical physics on the largest of scales. It precisely lays the theoretical foundation for how the stars dance through the galaxies, conducted by gravity. Its every major prediction has been rigorously tested. Most recently, its prediction of gravitational waves was conﬁrmed by LIGO.

But, there exist solutions in Einstein’s equations

8πG Gµν = T µν (1.1) c4 that allow extraordinary consequences. By considering spacetimes with interesting properties, Eq. (1.1) may be solved in reverse for its possible matter source. In this manner, “exotic matter” sources allow FTL spacetimes in GR without requiring new physics such as the uniﬁcation of GR and Quantum Mechanics (QM).

1 It is emphasized at this point that these FTL solutions are mathematically

reasonable, but physically extreme. The “exotic matter” at the source of each solution

violates every known null-energy-condition (NEC).∗ Moreover, the solutions have

physical implications like closed-timelike-curves (CTC), which may violate causality.

Instead of dismissing FTL solutions as physically unrealistic and impossible, their best

use is in exploring the theoretical limits of GR in the same way that the black-body

radiation experiments and the photoelectric eﬀect probed the limits of classical physics.

In the last one hundred years, physicists identified two main possibilities

for FTL travel. The first is the Einstein-Rosen Bridge, colloquially known as a

worm-hole. The second is the warp drive developed by Alcubierre, Van Den

Broeck, Kras-nikov, Natário and others. To begin this dissertation, two gedanken experiments will be presented for each FTL possibility. Each one will be constructed in GR alone to demonstrate that these possible means of FTL require

no exotic physics.

1.1.1 Einstein-Rosen Bridge

The first hint of FTL travel dates to the early development of GR [2]. It was the

seminal paper by Einstein and Rosen which introduced the first series of mathemat-

ical calculations for a wormhole [3]. In it, they showed a "bridge" which connected distant regions of asymptotically flat spacetime. While traveling through the bridge is fatal, it is a good test example of FTL.

∗ A NEC asserts that for any null vector, kµ, the stress energy tensor, T µν in Eq. (1.1) must µ ν satisfy Tµν k k ≥ 0 [1]. Matter that violates the NEC is called “exotic matter.”

2 The Einstein-Rosen Bridge is a submanifold of the Kruskal black hole in GR

[4, 5]. The Kruskal black hole is a maximal and unique solution to the Schwarzschild

solution. Its line element is

2 2 2 16m − r 02 16m − r 02 2 2 2 2 ds = e 2m dt − e 2m dx − r (dθ + sin θdφ ), (1.2) r r

02 02 − r 0 0 with r being a function determined by t − x = −(r − 2m)e 2m , t and x being

related to the advanced null coordinate, v0, and the retarded null coordinate, w0, by

0 1 0 0 0 1 0 0 0 t = 2 (v + w ) and x = 2 (v − w ). Making the free choice, t = 0, and taking the 1 cross section, θ = 2 π, the line element then becomes

ds2 = −F 2dx02 + r2dφ2. (1.3)

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.

From Fig. 1.1, the chosen cross section of the Kruskal manifold can be de-

scribed as a "bridge" in Einstein-Rosen’s words. In modern terms, it is a wormhole

connecting diﬀerent parts of an asymptotically ﬂat u niverse. I n t he t ime p eriod of

t0 < 1 in Fig. 1.2, two distinct, asymptotically ﬂat Schwarzschild manifolds exist be-

fore connecting the wormhole. At t0 = −1, the previously separate manifolds pinch and connect. During −1 < t0 < 1, the manifolds are connected and particles may pass

3 Figure 1.2: Figure 17.3 from [5]. It depicts the full time evolution of Eq. 1.2. uninhibited from one manifold to the other. At t0 = 1, the manifolds disconnect. For

t0 > 1, the manifolds remain separate, and the particles that transferred during the

pinch will remain in their new manifold. To a distant observer, the particles will ap-

pear to have traveled through vast distances of spacetime, though from the particles’

perspective they traveled the short distance of the pinch.

The EPR bridge naturally leads to many of the terms used to describe a worm-

hole. The narrowest part of the geometry that allows passage is often called the throat

of the wormhole and can be seen in the t0 = 0 slice of Fig. 1.2. The region nearby is

called the bridge or wormhole, where travel from one manifold to the other is possi-

ble. The two asymptotically flat regions of space connected by the bridge are labeled patches.

Due to its construction, the EPR bridge is equivalent to the maximally extended

Schwarzschild solution. It is merely a coordinate artifact due to a special choice of

coordinate patches. An observation of one would appear as a black hole through our

telescopes. While this is a nice toy model of how a wormhole could exist, it is a

highly simplified form of the Schwarzschild equation and requires an extreme amount

of exotic matter to maintain in reality. The discovery and identification of a real

wormhole in our universe remains as an open question in physics.

4 1.1.2 Tipping Light Cones

Figure 1.3: Figure 8.3 from [5]. It shows the null cone or light cone of events relative to P.

In special relativity (SR), the light cone of an event, P, consists of all null

geodesics† passing through the point P [5]. An example of a light cone is presented in Fig. 1.3. The light cone divides spacetime into three distinct regions. The Future can be connected to P by future directed timelike or null geodesics. Points in the future will be mapped to the future possibilities of P by an orthochronous Lorentz transformation, hence the name. The Past can be connected to P by past-directed timelike or null geodesics. Points in the past will be mapped to the past history of P by an orthochronous Lorentz transformation, again hence the name. Geodesics that pass through P from the past into the future, whether null or timelike, are consistent with causality and SR restricts itself to only considering these geodesics physical.

All other geodesics are spacelike and lie in the volume outside the light cone labeled

Elsewhere. They do not adhere with causality. Assume there are two events, P and

Q, connected by a spacelike geodesic in Elsewhere and ordered such that P happens † A geodesic is a curve that is straight and uniformly parameterized as measured by each local Lorentz frame along its length [6]. It is a generalization of Euclid’s statement that “the shortest distance between two points is a straight line” in ﬂat space to curved space. 5 before Q in time. Then, an orthochronous Poincaré transformation is not guaranteed to preserve the ordering of P and Q for all observers. In this sense, spacelike geodesics violate causality. As light travels along null geodesics, the slope of each null cone is c.

For a warp drive to exist, it must travel along a non-physical spacelike geodesic, and

that ends the possibility of warp drives in just SR.

Figure 1.4: An example of a tipping light cone by a warped underlying spacetime.

Figure 1.5: The light cone of an object rotating around the axis of a van Stockum spacetime will tip over.

6 But GR allows another possibility: the light cones themselves may be tipped

over in significantly warped spacetimes as in Fig. 1.4. A sequence of such tipping may

wind a timelike geodesic without even leaving its light cone. An example is the van

Stockum spacetime consisting of a cloud of dust rotating cylindrically about an axis

though it is a general feature of any cylindric spacetime [7, 8]. As an object rotates

about the azimuthal coordinate φ, its light cone will begin to tip as depicted in Fig. 1.5.

For a large enough angular velocity, the light cone will inevitably become a CTC and

contain a causality violating region outside the dust cylinder.

Now, the van Stockum solution is physically unreasonable. Its toy construction of an infinitely long cylinder of dust is present nowhere in our physical universe.

Moreover, it requires non-asymptotically flat spacetime, which matches not a single realistic model of our universe. Instead, it illustrates how light cones may be manipu-

lated by strong gravitational fields. The theoretical limit on the rate at which gravity

may expand and contract spacetime is unknown and potentially greater than c. The possibility of warping spacetime at a rate greater than c sparks the potential for a warp drive.

It is an open question in physics how deeply one can tip a light cone. The

Superluminal-Censorship Theorem predicts that “if an asymptotically flat spacetime, whose domain of outer communication is globally hyperbolic, possesses a FTL null curve γ from past null infinity to future null infinity, then there exists a fastest null geodesic γmax from past null infinity to future null infinity that is in the same homotopy class and does not have any conjugate points” [9]. The Averaged NEC will be violated by γmax. Now, both quantum and classical systems violate the ANEC, so the

Superluminal-Censorship Theorem does not immediately prohibit FTL [10]. It remains to discover whether γmax is above or below c.

7 1.2 Wormholes

A physically more reasonable alternative to the EPR bridge presented in Sec-

tion 1.1.1 is that of the Lorentzian traversable wormholes. They were first described by

Kip Thorne and his collaborators who used Einstein’s general relativistic field equations to explore the possibility of FTL spaceflight without violating SR [4, 11, 12].

Earlier studies demonstrated the possibility of traversable wormholes in GR [13, 14]. A precise definition of a Lorentzian traversable wormhole is a topological opening in spacetime, which manifests traversable intra-universe and/or inter-universe connections, as well as possible diﬀerent chronological connections between distant spacetime points. In simpler terms, it is a shortcut in spacetime created by an extreme warp-ing and/or folding of spacetime due to a powerful gravitational field. The condition for a

Lorentzian wormhole to be traversable is that it is free of both event horizons and singularities [4]. Such a wormhole is fully traversable in both directions, geodesi-cally

complete, and possesses no crushing gravitational tidal forces. Consequently,

Lorentzian traversable wormholes are unlike the non-traversable Schwarzschild worm-

hole, or Einstein-Rosen bridge considered in Section 1.1.1. Exotic matter, which violates the point-wise and averaged energy conditions, is required to open and stabilize a

Lorentzian traversable wormhole. A comprehensive technical overview of this subject is found in [4].

The seminal paper on the EPR Bridge prompted the discovery of the existence of many wormhole solutions in GR. Wheeler introduced the name "wormhole" in his discussion of spacetime foam and the Schwarzschild solution [15, 16]. Morris and Thorne popularized the concept of a wormhole with their exploration of the

Morris-Thorne(MT) wormhole [11, 12]. They looked at Kerr’s solution for a rotating black hole and saw that impassable, intra-universal passageways must exist in it. Continuing, they predicted the basic requirements for a traversable wormhole and

8 then penned their famous metric presented later as Eq. (3.5). Since then, many more

types of wormholes have been presented such as rotating traversable wormholes and

wormholes whose throats change dynamically in time [1, 4, 17]. Furthermore, well

known line elements have been demonstrated to contain wormholes [18]. Wormhole physics is a rich field that probes the limits of GR.

Previous studies of Lorentzian traversable wormholes rely on either calculating

i the elements of the Riemann curvature tensor, R jkl, to “observe” the eﬀects of the wormhole’s spacetime curvature on photons and matter moving through it or by em-

i bedding diagrams. However, the R jkl cannot be calculated in an invariant manner

i because it is a function of the chosen coordinates [19]. Analysis of R jkl can be mis- leading because a different choice of coordinate basis will result in different tensor components. These coordinate mapping distortions arise purely as an artifact of the coordinate choice. Embedding diagrams offer a narrow view of the spacetime manifold. In Fig. 1.6, the embedding diagrams depict the wormhole geometry along an equatorial (θ = π ) slice through space at a specific moment in time [11]. Embedding 2 diagrams offer only a limited view of the physics involved in the wormhole. Curvature invariants allow a manifestly coordinate invariant characterization of certain geometrical properties of spacetime [20, 21]. Using curvature invariants to probe black holes is a fruitful field of interest and research [22, 23, 24, 25, 26, 27, 28, 29, 30]. In a sim-ilar

fashion, the research presented herein seeks the best way to illustrate wormhole spacetimes by plotting their independent curvature invariants [31].

9 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 equatorial (θ = π ) slice through space at a specific moment in time (t = constant). 2

1.3 Warp Drives

A warp drive is a solution to the Einstein field equations that allows a spaceship to make a trip to a distant star in an arbitrarily short proper time [32]. The first propulsion mechanism for a warp drive was a dipole configuration connecting a local contraction of spacetime in front of the spaceship with a local expansion of spacetime behind the ship. While locally the spaceship remains within its own light cone and never exceeds c, globally the relative velocity‡ can be much greater than c. For a distant observer, the eﬀect of the expansion and contraction would cancel each other out and spacetime would be asymptotically flat. While an interesting theoretical concept, the original formulation was shown to violate every NEC, produce tachyonic

‡ deﬁned as proper spatial distance divided by proper time

10 matter, and require an amount of energy greater than the mass of our observable universe. Only naively could such an original idea be considered possible.

Since then, there has been considerable research into realistic FTL warp drives.

In [33], Krasnikov considered a non-tachyonic FTL warp bubble and showed it to be possible mathematically. Van Den Broeck in [34] modified Alcubierre’s warp drive to have a microscopic surface area and a macroscopic volume inside. He demonstrated a modification that allowed a warp bubble to form with energy requirements of only a few solar masses. His geometry also has more lenient violations of the NEC. Later,

Natário improved upon Alcubierre’s work by presenting a warp drive metric such that zero spacetime expansion occurs [35]. Instead of riding a contraction and expansion of spacetime, the warp drive may be observed to be “sliding" through the exterior spacetime at a constant global velocity. Finally, Loup expanded Natário’s work to encompass an accelerating global velocity by modifying the distance between hypersurfaces. All of this research hints at the possibility in GR of a realistic warp drive [36].

FTL travel obeys eight general requirements [37]. First, the rocket equation does not apply. Second, the travel time through the FTL space warp should take less than one year as seen both by the passengers in the warp and by stationary observers outside the warp. Third, the proper time as measured by the passengers will not be dilated by any relativistic eﬀects. Fourth, any tidal-gravity accelerations acting on any passengers should be less than the acceleration of gravity near the Earth’s surface,

m g = 9.81 s2 . Fifth, the local speed of any passengers should be less than c. Sixth, the matter of the passengers must not couple with any material used to generate the

FTL space warp. Seventh, the FTL space warp should not generate an event horizon.

Finally, the passengers should not encounter a singularity inside or outside of the

FTL warp.

11 While the mathematics of a warp drive are well developed, mapping the space-

time around the warp drive remains underdeveloped. Considering that a ship inside of a warp bubble is causally disconnected from the exterior [1, 35], computer simulations of the surrounding spacetime are critical for the ship to map its journey and steer the

warp bubble. In [32], Alcubierre uses the York time§ to map the volume expansion

of a warp drive. He plotted the York time to show how spacetime warped behind and

in front of the spaceship. While the York time is appropriate when the 3-geometry

of the hypersurfaces is flat, it will not contain all information about the curvature of

spacetime in non-flat 3-geometries such as the accelerating Natátio warp drive space-

time. As an alternative to the York time, curvature scalars of the line element may

be considered.

1.4 Purpose of Research In this dissertation, the CM curvature invariants for four diﬀerent wormhole and three warped spacetimes will be derived, plotted and analyzed. The curvature scalars

(or invariants) show the magnitude by which spacetime diﬀers from being flat. They can be used to map the geometric structure of the spacetime independent of coordinate

basis. They have values independent of the choice of coordinates and pro-vide a manifestly coordinate-invariant characterization of spacetime [6, 20, 21]. They are especially helpful in identifying intrinsic curvature singularities of the spacetime, the

Petrov and Segre type of the eigenvalue problem for the Ricci tensor and Weyl tensor, and can answer the equivalence problem between spacetimes. By computing all independent curvature scalars, the geometry of any spacetime may be analyzed and plotted.

§ The York time is deﬁned as Θ = vs x−xs df c rs drs

12 The focus of previous studies of wormholes and warp drives is their stress- energy tensor. The stress-energy tensor answers questions such as how realistic a warped spacetime is and what amount of exotic matter is required to generate and control one. Little attention has been given to what each spacetime looks like, how the spacetime curves, or what singularities are present in them. Computing and plotting the invariants will provide a window to answer these questions. In this dissertation, the independent set of CM curvature invariants for four wormhole spacetimes and four warp drive spacetimes are calculated and plotted. The identity of any intrinsic curvature singularities present will be found. An analysis of each plot will be provided

to help describe the surrounding spacetime.

The order of the research in this dissertation is as follows. Chapter 2 details how to calculate the independent CM curvature invariants. It includes a basic review of

GR, an introduction to the set of CM curvature invariants, and how to calculate the four independent invariants R, r1, r2, and w2 for Class B1 spacetimes. Chap-ter 3 investigates the wormhole spacetimes. The non-zero curvature invariants for four

wormhole spacetimes are presented: the Thin-Shell(TS) Flat-Face wormhole, the MT

wormhole, the TS Schwarzschild wormhole, and the exponential metric. Chap-ter 4

explores warp drive spacetimes at a constant velocity. It starts with a description of a

generic warp drive spacetime in 3 + 1 ADM. Then, the Alcubierre and Natário warp

drives are demonstrated and the spacetimes variables, vs, σ and ρ, are varied to reveal

their eﬀect on the curvature invariants. Chapter 5 expands on Chapter 4 by looking at

the eﬀects of a constant acceleration for the Natário spacetime. The warped drive

spacetimes are expanded from a constant velocity to a changing velocity. Then, it

investigates the Natário warped drive spacetime with a constant accelera-tion and the

eﬀect of the diﬀerent variables t, a, ρ and σ on the curvature invariants. The conclusion

in Chapter 6 looks at the diﬀerent possible directions the research

13 contained within can be expanded and what research is already being undertaken by the EUCOS group.

14 CHAPTER TWO

Curvature Invariants

GR provides the framework to study FTL spacetimes. While it is impossible for an individual particle to exceed a velocity of c, the underlying spacetime has no such restriction. Consequently, the study of FTL line elements examines spacetime manifolds inside the prescription of GR.

To analyze the underlying spacetime manifold, GR uses the language of tensors.

In this chapter, the basic principles of GR such as line elements and the basic tensors used in the dissertation will be presented. Next, the set of CM curvature invariants will be demonstrated. Finally, the minimal set needed for FTL line elements will be shown.

2.1 Spacetime Manifolds

GR is a purely classical theory [4, 5, 6, 19, 20, 38, 39]. It recognizes that the

Newtonian concepts of space and time are a single object called spacetime. Space-

∗ time is a diﬀerentiable manifold endowed with a metric, gµν , which is a symmetric covariant tensor of rank 2. Formally,

Definition 2.1.1. Spacetime is a four-dimensional manifold equipped with a Lorentz- ian (pseudo-Riemannian) metric. The metric should have Lorentzian (pseudo- Riemann-

∗ 3, Indices are purely a choice of convention. In this dissertation, Greek indices range from 0 to including both the single time component and the three space components, of the manifold. Lowercase Latin indices that range from 1 to 3 will denote the space components of the geometry inside the manifold. Uppercase Latin indices denote spinor components.

15 ian) signature (−,+,+,+). The signature is represented as (3 + 1) dimensions with 3 spacelike dimensions and 1 timelike dimension. The manifold is Hausdorﬀ and para- compact.

Locally, a spacetime manifold resembles a small, Lorentzian patch around each point in it. A single point in spacetime represents the physical event that occurs. A basis set of coordinates (eµ = x1, x2, ..., xn) can be labeled at each point in the manifold. The coordinates are not needed in an absolute sense, but they ease the questions of ordering, comparing, and/or relating the events between separate points.

The coordinates take real values. Their possible range is (−∞, ∞) though subsets will be often useful for coordinate patches. In some cases, a unique set of coordinates can be defined at each point in the manifold. In others, the spacetime manifold neces-sitates multiple coordinate patches with each only covering a portion. A coordinate transformation may be defined for these cases in an overlapping region between the coordinate patches.

Figure 2.1: Figure 1.3.1 from [6].

16 The geometry of spacetime is Lorentzian. The intervals between any event and

any other event satisfies the theorems of Lorentz-Minkowski geometry. Consider the worldline of A L and an event, B, not on it as in Fig. 2.1. For B to interact with A L ,

light rays need to be transmitted and/or reflected. Assume one light ray transmitted from A L interacts with B at the event P and a second light ray transmitted from B

interacts with A L at the event L . Then, the proper time, τA B, between A and B is the timelike separation given the diﬀerence between the two light rays. The the proper

2 2 distance sA B is the spacelike separation related to the proper time by sA B ≡ −τA B =

−τA BτA B. Then, the coordinate system is locally Lorentzian, and the geometry of the spacetime is also locally Lorentzian.

As an example, consider the world line A L to be propagating along the x = 0 hypersurface and the event B is separated from the world line by a spatial distance of

2 2 2 x as in Fig. 2.2. Then, the proper time is τA B = t − x = (t + x)(t − x) = τA PτA L . As a second example, choose a coordinate system given by the parametrization xµ = xµ(A ) for any local event A as in Fig. 2.3. The proper time between two locally,

2 2 0 0 2 1 neighboring event pairs A and B is sA B ≡ −τA B = −[x (B)−x (A )] +[x (B)− x1(A )]2 + [x2(B) − x2(A )]2 + [x3(B) − x3(A )]2. The examples above describe in GR

the generalization of Euclid’s famous statement, “The shortest distance between two

points is a straight line.” The gravitational field of an object is the bending of spacetime

near that object. The spacetime metric describes the gravitational potential in any

given region. In components, the spacetime line element is

2 µ ν ds = gµνdx dx . (2.1)

The line element represents the squared length of the inﬁnitesimal displacement be-

µ µ µ tween two neighboring points x and x + dx . The metric, gµν, is a set of ten functions of position that gives the interval between any event and any nearby event as

described above. It deﬁnes the distances and lengths of vectors on the Riemannian 17 Figure 2.2: Figure 1.3.2 from [6].

manifold. For a contravariant vector,† Xµ, the length or norm is deﬁned to be the scalar product

2 µ ν X = gµνX X . (2.2)

The metric components depend on the basis vectors, called a tetrad, of the manifold as well as several conditions. The metric is symmetric gµν = gνµ. In the mixed repre-

ν ν sentation gµ , it is identical to the Kronecker delta, δµ . The inverse metric is deﬁned as

µλ µ µ g gλν = g ν = δ ν, (2.3)

µν −1 ||g || = ||gµν|| . (2.4)

† A contravariant vector is a set of quantities at a point P that transforms under a change of 0µ ∂x0µ ν 0µ 0µ coordinates as X = ∂xν X evaluated at P [5]. It is written as X in the x coordinate system. It is also called a contravariant tensor of rank (order) 1.

18 Figure 2.3: Figure 1.3.3 from [6].

µν... For an arbitrary tensor, T λξ..., the metric and inverse metric can raise and lower the tensor elements. Using the Einstein summation,

ρλ µν... ρµν... g T λξ... = T ξ..., (2.5)

µν... ν... gρµT λξ... = Tρ λξ.... (2.6)

For a Lorentzian spacetime, the line element can be decomposed into a vector ﬁeld that forms its tetrad or basis. An orthonormal tetrad consists of three spacelike vectors

Vi and one timelike vector Vt that satisfy the relationships

Vµ = {Vt,Vi} = {t, x, y, z}, (2.7.a)

gµν = xµxν + yµyν + zµzν − tµtν, (2.7.b)

Vi · vj = δi,j, (2.7.c)

t · t = −1, (2.7.d)

Vi · t = 0. (2.7.e)

In some cases, an orthonormal tetrad may be read directly from the non-zero metric components. Complex null tetrads form a second type of tetrad. They consist of two

19 real null vectors k and l and two complex conjugate null vectors m and m¯ . The scalar

µ µ products of the tetrad vanish apart from k lµ = −1 and m m¯ µ = 1. The easiest way to find a null tetrad for a spacetime is by first finding an orthonormal tetrad and using the relationships

1 l = (V + V ), (2.8.a) µ 2 0 1 1 k = (V − V ), (2.8.b) µ 2 0 1 1 m = (V + iV ), (2.8.c) µ 2 2 3 1 m¯ = (V − iV ). (2.8.d) µ 2 2 3

Computing the null tetrad for a spacetime greatly increases the computational speed for the invariants found in this chapter.

Given a spacetime line element, the covariant derivative may be derived to

i compare the spacetime between any two points in it. The aﬃne connection, Γ jk, parallel transports neighboring events to allow them to be compared. It is deﬁned as

1 Γµ = gµλ (∂ g + ∂ g − ∂ g ) . (2.9) αβ 2 α λβ β λα λ αβ

The connection links the acceleration a free-falling particle experiences with the surrounding gravitational ﬁeld. The particle will follow a geodesic. A geodesic computes the shortest distance between two points in curved space and is an extension of a straight line in Euclidean space. The geodesic equation for an aﬃne parameter is

d2xµ dxα dxβ + Γµ = 0. (2.10) ds2 αβ ds ds

µ dxµ The four-vectors V = ds may be defined to be timelike if gµν < 0, null if gµν = 0, µ and spacelike if gµν > 0. For torsion-free connections, the Riemann tensor R αβγ is related to the commutator of covariant differentiation. It is defined to be

µ µ µ µ ν µ ν R αβγ = ∂βΓ αγ − ∂γΓ αβ + Γ νβΓ αγ − Γ νγΓ αβ. (2.11) 20 The Riemann tensor governs the difference in accelerations that two different, free- falling particles experience. A necessary and sufficient condition for a metric to be flat is for its Riemann tensor to vanish everywhere. Conversely, the curvature tensor is then the Riemann tensor. The Ricci tensor Rαβ, the Ricci scalar R, and the trace-free

Ricci tensor, Sαβ can be obtained from Eq. (2.9) and the contractions in Eqs. (2.5) and (2.6). They are deﬁned as the following

γ γ δ γ δ γ Rαβ = ∂γΓ αβ − ∂βΓ αγ + Γ αβΓ δγ − Γ αγΓ δβ, (2.12)

αβ R = g Rαβ, (2.13) R S = R − g . (2.14) αβ αβ 4 αβ

The Ricci scalar or curvature scalar is the ﬁrst example of an invariant. In dimensions n = 3 + 1, there are 20 independent components of the Riemann tensor: 10 of the independent components of the Ricci tensor and 10 of the independent components of the Weyl tensor. The Weyl tensor or conformal tensor Cαβγδ is

Cαβγδ = 1 1 R + (g R + g R − g R − g R ) + (g g − g g ) R. (2.15) αβγδ 2 αδ βγ βγ αδ αγ βδ βδ αγ 6 αγ βδ αδ βγ

2 ‡ A necessary and suﬃcient condition for a metric to be conformally ﬂat, gµν = Ω ηµν, is that its Weyl tensor vanishes everywhere. There are many additional tensors that may be derived from a given metric such as the Einstein Gµν or the other tensors of the irreducible representation of the full Lorentz group Eαβγδ and Gαβγδ. This includes the set of tensors needed for the following research.

‡ ηµν is the Minkowski metric diag(−1, 1, 1, 1).

21 2.2 Curvature Invariants

The Riemann curvature invariants provide a manifestly coordinate invariant characterization of spacetime [6, 20, 21, 40]. A curvature invariant has a value in-

dependent of the choice of the coordinates. Riemann curvature invariants are scalar

products of the Riemann Eq. (2.12), Ricci Eq. (2.12), the Weyl tensors Eq. (2.15)

and their traces, covariant derivatives, and/or duals. Invariants measure the curva-

ture by which a spacetime geometry diﬀers from being ﬂat. T he p rime e xample of

the invariants are the scalar polynomial (SP) invariants such as the Kretschmann

ijkl invariant, R Rijkl. Other types exist, such as the Cartan invariants which gives a

unique coordinate-independent characterization, but this research focuses on the SP

invariants. The SP preﬁx should be assumed throughout this dissertation.

The complete set of invariants are important in the study of GR. Invariants are

critical for studying curvature singularities, the Petrov type of the Weyl tensor, the

Segre type of the trace-free Ricci tensor, and the equivalence problem. If a singularity

occurs in the curvature invariants, then the curvature singularity must be fundamental to the spacetime, instead of an artifact of the choice of coordinates. Studies of the

Petrov type and the Segre type categorize the solutions of the eigenvalue problem of the Riemann tensor. Investigating the curvature invariants reveals the eigenvalue structure of the spacetime and relates it using the NP components. The equivalence problem asks whether two diﬀerent metrics describe identical spacetimes. Finally, a scalar invariant is the seed of GR. Motivated by Einstein, Hilbert saw that an action principle based on the scalar invariant R gives the geometrodynamic law. Choosing

1 a Lagrangian Lgeom = ( )R gives the law with a very simple correspondence to 16 Newtonian’s Theory of Gravity. Any scalar invariant may be chosen in place of R,

but the simple correspondence to Newton’s theory will be lost as the action will

22 no longer be second order in the derivatives from the metric components. In these manners, the curvature invariants are of fundamental importance in the study of GR.

The number of free parameters in the Riemann tensor determines the number of curvature invariants. The Riemann tensor has 20 independent components after including its symmetries. The first and most famous component is the curvature scalar

Eq. (2.13). Nine independent components appear in the trace-free Ricci ten-sor Eq.

(2.14). Ten independent components appear in the Weyl tensor Eq. (2.15). The

Lorentz transformation represents 6 additional degrees of freedom and entangles these components. Its six parameters further reduce the number of free parameters and curvature invariants to 20 − 6 = 14. In an arbitrary spacetime, the fourteen parameters determine the coordinate independent, local features of the curvature. A specific choice of line element can further reduce the number since the number of independent variables§ might be further reduced.

2.2.1 Scalar Polynomial Invariants In general, there are three classes of Riemann SP invariants [21]. The first is the set of four Weyl invariants. The two complex invariants are

1 1 I = I(Ψ, Ψ) = Ψ ΨABCD, (2.16) 6 6 ABCD 1 J = I(Ψ,Q) = Ψ ΨCD ΨEF AB. (2.17) 6 ABCD EF

where I is the invariant product and Ψ, Q are Ψ-like spinors derived from the Weyl

tensor in Eq. (2.15). The Weyl invariants come from the real and imaginary compo-

§ The independent variables are the four coordinates plus any functions of integration that have physical meaning.

23 nents of these two functions. They can be expressed as the following relationships,

I1 = Re(I), (2.18)

I2 = Im(I), (2.19)

I3 = Re(J), (2.20)

I4 = Im(J). (2.21)

Any other choice of Weyl invariants will be related to this choice. The second set is the set of four Ricci invariants. They are the following

I5 = R, (2.22) 1 I = I(Φ, Φ), (2.23) 6 3 1 I = I(Φ,E), (2.24) 7 6 1 I = I(E,E). (2.25) 8 12 where Φ is a Ψ-like spinor and E is a Φ-like spinor. Any choice of Ricci invariants will

be related to this choice. The ﬁnal type is the Mixed invariants. The Mixed invariants

are the hardest to construct. Most of the diﬀerences between the sets of invariants

occur in the Mixed type. The complete set is

I9 = Re(K), (2.26)

I10 = Im(K), (2.27)

I11 = Re(L), (2.28)

I12 = Im(L), (2.29)

I13 = Re(M), (2.30)

I14 = Im(M), (2.31)

0 I15 = M1 = I(C ,C), (2.32)

I16 = Re(M2), (2.33)

I17 = Im(M2). (2.34)

24 1 ˜ 0 ˜ where K = I(Φ,C), L = I(Q, ξ), M = 4 I(Ψ, ξ), M1, and M2 = I(C , C) are spinors that mix Ψ-like and Φ-like spinors according to the choice of line element. These are

the general sets of CM invariants. Specific choices of line elements and the symmetries inherent to the spacetime will reduce the number of invariants.

The list of invariants presented contains 17 elements, not 14, as certain non- degenerate cases are taken into account. It is stressed that this set of invariants is a

complete set as opposed to an independent set. A complete set of invariants

includes the number of invariants that meet the requirements of the 90 diﬀerent

Petrov and Segre types. In contrast, an independent set of invariants contains only

invariants that independent of each other. The CM invariants in Eq. (2.26) to Eq.

(2.34) will not be linearly independent for a specific choice of a spacetime. Algebraic

and polynomial relationships will reduce the number of invariants in a complete set down to the six required by Lorentz invariance. These relationships are the syzygies of the invariant set.

De inition 2.2.1. Syzygy is a polynomial relationship between functions in a com-

plete set. The independent invariants, I, satisfy

2 3 n c0 + c1I + c2I + c3I + ... + cnI = 0. (2.35)

where the ci’s are polynomials of the other non-independent invariants in the set.

Identifying the non-zero spinor components allows the syzygy relationships to be derived. Solving the syzygies between each spacetime’s invariants will greatly re-

duce the number of independent invariants.

25 2.2.2 Invariants of Sphere

A simple example can help illustrate the use of invariants. Consider a 2-sphere

as given in [28]. The metric of a 2-sphere is   a2 0   gij =   . (A.1) 0 a2 sin2 θ where a is the radius of the sphere. There are two nonzero components of the Riemann

1 2 1 2 tensor, R 221 = sin θ and R 212 = sin θ, computed from Eq. (2.11). The non-zero components fully determine the curvature of the sphere. However, normally we think

of the curvature in terms of the Gaussian curvature computed from Eq. (2.13). For

1 the sphere, the Ricci scalar, R = a2 , is related to equation of the circle bounding the equator. Alternatively, any other invariant for other characteristics of the curvature

ijkl 2 can be computed. For example, the Kretschmann invariant is R Rijkl = a4 and is connected with the surface area of the sphere. Here, the curvature invariants mea-

sure the curvature of the manifold, and not the object’s path through the manifold.

Unfortunately, the invariants in 3, 4, and 5 are not as simple as the invariants of the sphere. To gain a physical insight into the nature of the invariants, they will be plotted.

2.2.3 CM Invariants Carminati and McLenaghan (CM) proposed a set of invariants that is a complete and minimal set for the Einstein-Maxwell and perfect-ﬂuid spacetimes [41]. The CM invariants have highly desirable properties such as linear independence, the lowest possible degree, and containing a minimal set for both any Petrov type and speciﬁc choice of the Ricci tensor. The complete set of invariants is the Ricci scalar from

Eq. (2.13) and

1 r := Φ ΦABA˙B˙ = S βS α, (2.36) 1 ABA˙B˙ 4 α β

26 1 r := Φ ΦB B˙ ΦCAC˙ A˙ = − S βS γS α, (2.37) 2 ABA˙B˙ C C˙ 8 α β γ 1 r := Φ ΦB B˙ ΦC C˙ ΦDAD˙ A˙ = S βS γS δS α, (2.38) 3 ABA˙B˙ C C˙ D D˙ 16 α β γ δ 1 w := Ψ ΨABCD = C¯ C¯αβγδ, (2.39) 1 ABCD 4 αβγδ 1 w := Ψ ΨCD ΨEF AB = − C¯ C¯γδ C¯ζαβ, (2.40) 2 ABCD EF 8 αβγδ ζ 1 m := Ψ ΦCD ΦABC˙ D˙ = C¯ SγδSαβ, (2.41) 1 ABCD C˙ D˙ 4 αγδβ 1 m := Ψ ΦCD ΨAB ΦEF C˙ D˙ = C¯ SγδC¯α βSζ , (2.42) 2 ABCD C˙ D˙ EF 4 αγδβ ζ 1 α m := ΨAB ΦCD Ψ¯ A˙B˙ Φ C˙ D˙ = C¯ SγδC† βSγζ , (2.43) 3 CD A˙B˙ C˙ D˙ AB 4 αγδβ γζ 1 β m := Ψ B ΦDE B˙ Ψ¯ C˙ Φ CD˙ E˙ Φ A A˙ = − C¯ SγδC† Sζ Sαη, (2.44) 4 A DE A˙ B˙ D˙ E˙ B C C˙ 8 αγδβ ζη 1 η m := ΨAB ΨCD ΦEF Ψ¯ E˙ F˙ Φ C˙ D˙ = C¯ C¯α βSγδC† θSζ . (2.45) 5 CD EF E˙ F˙ C˙ D˙ AB 4 αηθβ γδ ζ This set can be related to Eq. (2.26) through Eq. (2.34) by noting that: 1 I = w , (2.46) 6 1 1 J = w , (2.47) 6 2 1 I = r , (2.48) 6 3 1 1 I = r , (2.49) 7 3 2 1 I = Φ ΦB C˙ (ΦAGH˙ D˙ ΦE F˙ + ΦAGH¯ F¯ ΦE D˙ ), (2.50) 8 6 ABC˙ D˙ E F˙ G H˙ G H˙

K = m1, (2.51)

EF AB CDG˙ H˙ L = Ψ(AB ΨCD)EF Φ G˙ H˙ Φ , (2.52)

1 A BEF¯ C G¯ DEF¯H¯ C H¯ DEF¯G¯ M = Ψ Φ ¯ ¯ Φ ¯ (Φ ¯ Φ + Φ ¯ Φ ), (2.53) 2 ABCD EF G H E F E F

M1 = m3, (2.54)

(AB EF )CD ¯ I˙J˙ G˙ H˙ M2 = Ψ CD Ψ ΨG˙ H˙ I˙J˙ ΦAB ΦEF . (2.55)

Several invariants do not have an explicit relationship, but it can be noticed I8 is related to R3, L is the symmetrization of m2, and M2 is the symmetrization of m5. The

27 invariant M is an additional one designed to augment and complete the CM invari-ants for general spacetimes [21]. It is emphasized that the specific set attributed to CM is suﬃcient to compute the invariants for the spacetimes considered in Chapters 3, 4, and 5. The full set of CM invariants have been computed in the NP formalism using the computer program MAPLE. The set is long and complicated, so only the non-zero ones in Class B spacetimes will be presented in the next subsection.

2.2.4 Syzygies of CM Invariants

Each CM of the invariants Eq. (2.36) through Eq. (2.45) may be calculated

for any possible spacetime. But the set will be degenerate due to sets of internal

relationships. To reduce the complete set of invariants to the subset of independent

ones, the syzygies of the CM invariants must be considered. For Class B warped product spacetimes, the syzygies and independent set are known and will be presented in this subsection from [42].

Definition 2 .2.2. Class B Spacetime is the product of two 2-D spaces, one Lorentzian and one Riemannian, subject to a separability condition on the function which couples the spaces. The metric is of the form 2 2 γ 2 ds = dsΣ1 (u, v) + C(x ) dsΣ2 (θ, φ), (2.56)

γ 2 where Σ1 is the Lorentzian manifold and Σ2 is the Riemannian. The restriction C(x ) is of the form C(xγ)2 = r(u, v)2w(θ, φ)2.

Class B spacetimes are a speciﬁc case of the Petrov Class D metric.¶ Class B

spacetimes are further divided into two main categories. Class B1 spacetimes have

¶ Petrov Class D spacetimes have an eigenvalue equation of the Weyl tensor of the form (C + 1 1 2 λI)(C − 2 λI) = 0. The eigenvalues are simple divisors and satisfy λ1 = λ2 6= λ3.

28 sig(Σ1) = 0 and sig(Σ2) = 2( = ±1). A suﬃciently generic metric of Class B1 is

ds2 = −2f(u, v)dudv + r(u, v)2g(θ, φ)2(dθ2 + dφ2). (2.57)

Class B2 spacetimes have sig(Σ1) = 2 and sig(Σ2) = 0. A suﬃciently generic metric of Class B2 is

ds2 = f(u, v)2(du2 + dv2) − 2r(u, v)2g(θ, φ)dθdφ. (2.58)

Class B1 warped product spacetimes include all spherical, planar, hyperbolic, while

Class B2 spacetimes include the non-null EM, Λ-term, or vacuum spacetimes. All spacetimes considered in this dissertation are Class B1.

Next, the number of independent invariants will be considered to restrict the number of needed syzygies. The number of independent degrees of freedom in a spacetime leads to the required number of invariants. General Petrov type D spacetimes allow a choice of our tetrad to be along the principal null directions of the Weyl tensor. With this choice, the remaining degrees of freedom are the Weyl and Ricci freedoms, the Ricci scalar, and the dimension of the invariance group. Consequently, the number of independent invariants is four for Class B1 spacetimes.

There are twelve separate invariants in the CM set between Eqs. (2.13), (2.36) through (2.45), and (2.53). After considering the degrees of freedom, the CM set should have eight syzygies leaving only four real, independent invariants. The eight syzygies are

2 3 0 = 6w2 − w1 , (2.59)

0 = (3m2 − w1r1)w1 − 3m1w2, (2.60)

0 = (3m5 − w1m¯ 1)w1 − 3m3w2, (2.61)

0 = 6m4 + w1r2, (2.62)

0 = m3 − m2, (2.63)

29 2 2 3 0 = (−12r3 + 7r1 )w1m1 − (12r2 − 36r1r3 + 17r1 )w2, (2.64)

2 0 = 2(3m6 − m1r1)w2 + m1 w1, (2.65)

2 3 2 3 2 0 = (−12r3 + 7r1 ) − (12r2 − 36r1r3 + 17r1 ) . (2.66)

The syzygies Eqs. (2.60) through (2.66) are well known for Class D spacetimes. The

remaining syzygy Eq. (2.59) is particular to Class B warped product spacetimes.

The syzygies allow the invariants Eqs. (2.38), (2.39), and (2.41) through (2.45) to be

expressed in terms of the remaining ones. Syzygy Eq. (2.59) allows w2 to be expressed

in terms of w1 or vice versa. Since the sign of w2 changes based on the signature of the metric, it is chosen as part of the independent set. Consequently, the set of

independent invariants for Class B warped product spacetimes is the four

(R, r1, r2, w2). (2.67)

For reference in Chapters 3, 4, and 5, the three invariants r1, r2, and w2 written in terms of the trace-free Ricci tensor, Weyl tensor, and in terms of the NP coordinates

are

1 β α r1 = Sα Sβ 4 (2.68) 2 = 2Φ20Φ02 + 2Φ22Φ00 − 4Φ12Φ10 − 4Φ21Φ01 + 4Φ11 , 1 r = − S βS αS γ 2 8 α γ β

= 6Φ02Φ21Φ10 − 6Φ11Φ02Φ20 + 6Φ01Φ12Φ20 − 6Φ12Φ00Φ21 − 6Φ22Φ01Φ10 + 6Φ22Φ11Φ00,

(2.69) 1 w = − C¯ C¯αβζ C¯γδ = 6Ψ Ψ Ψ − 6Ψ 3 − 6Ψ 2Ψ − 6Ψ 2Ψ + 12Ψ Ψ Ψ , 2 8 αβγδ ζ 4 0 2 2 1 4 3 0 2 1 3 (2.70)

where the Ψ and Φ are the following abbreviations for the tetrad components of the

trace-free Ricci tensor Eq. (2.14) and the Weyl tensor Eq. (2.15):

1 1 Φ ≡ S kαkβ = R = Φ¯ , (2.71.a) 00 2 αβ 2 44 00 30 1 1 Φ ≡ S kαmβ = R = Φ¯ , (2.71.b) 01 2 αβ 2 41 10 1 1 Φ ≡ S mαmβ = R = Φ¯ , (2.71.c) 02 2 αβ 2 11 20 1 1 Φ ≡ S (kαlβ + mαm¯ β) = (R + R ) = Φ¯ , (2.71.d) 11 2 αβ 4 43 12 11 1 1 Φ ≡ S lαmβ = R = Φ¯ , (2.71.e) 12 2 αβ 2 31 21 1 1 Φ ≡ S lαlβ = R = Φ¯ , (2.71.f) 22 2 αβ 2 33 22 α β γ δ Ψ0 ≡ Cαβγδk m k m , (2.71.g)

α β γ δ Ψ1 ≡ Cαβγδk l k m , (2.71.h)

α β γ δ Ψ2 ≡ −Cαβγδk m l m¯ , (2.71.i)

α β γ δ Ψ3 ≡ Cαβγδl k l m¯ , (2.71.j)

α β γ δ Ψ4 ≡ Cαβγδl m¯ l m¯ . (2.71.k)

As some final comments about this set, it is neither complete nor unique. A

complete choice of invariants would cover every possible Petrov and Segre type. Each

CM invariant not contained in Eq. (2.67) may be written as a rational integer multiple

1 2 of the independent invariants. The best example of this statement is how w1 = 63 w2 3 in

Eq. (2.63). A choice was made to include (R, r1, r2) in the set of independent invariants

because of their relative simplicity compared to the other CM invariants. An

alternative set, (R, m2, m4), would have also satisfied the syzygies, but m2 and m4

β ¯ contain more summations of Sα and C αβγδ than r1 and r2. The chosen three are the most direct to calculate from a given metric Eq. (2.1). Next, the remaining CM invariants can be found from the set Eq. (2.67) by solving for them using the syzygies.

Finally, the CM invariants may be found either by use of the standard trace- free Ricci tensor Eq. (2.14) and the Weyl tensor Eq. (2.15) or by computing the NP

indices Eq. (2.71.a) through Eq. (2.71.k). Two formulas for calculating the invariants have been presented in this section because each method has its own advantages.

31 The first set of formulas computes the CM invariants from the Trace-free Ricci tensor and the Weyl tensor. It is the more straightforward procedure as it requires only a single input, the metric, Eq. (2.1), of the spacetime manifold. The wormhole metrics presented in Chapter 3 were computed using the first set. The second set of formulas computes the NP indices from specific functions in that they are individual elements of the Ricci tensor and Weyl tensor as seen in Eq. (2.71.a) through Eq. (2.71.k). The second formulas may be computed more rapidly by a computer as they require fewer summations. The drawback of the NP indices is that they require two inputs, the metric from the spacetime manifold, Eq. (2.1), and a null tetrad, Eq. (2.8.a) through

Eq. (2.8.d). The CM invariants for the warp drive metrics in Chapters 4 and 5 were computed by ﬁnding the NP indices. All CM invariants contained in this dissertation were computed using Wolfram Mathematica 10.4 R . As a test, the output of the program matched exactly the known Riemann, Ricci, Weyl tensors and the Ricci scalar for the Schwarzschild metric, the MT wormhole, and the exponential metric. In addition, derivations by hand of the four CM invariants for each wormhole in Chapter

3 matched the program’s output exactly. The complete program is provided in Ap- pendix A.

32 CHAPTER THREE

Lorentzian Traversable Wormholes

Before 1988, each identified wormhole line element precluded travel through it.

A traveler would encounter many crippling impossibilities such as a requirement of a trip duration lasting an infinite amount of time for an external observer, impassable throats on the order of the Planck length of 10−35m, naked singularities generating permanently destructive tidal forces, or curvature singularities lurking behind the event horizons. Needless to say, scientists dismissed traversable wormholes as more science fiction than science fact.

Morris and Thorne demonstrated traversable wormholes by considering the con- verse situation to previous analysis [11, 12]. Instead of investigating known spacetimes for wormholes, they generated spacetimes with wormholes that obeyed certain traversability requirements. The traversable wormhole should allow a human (or any amount of matter of a similar size) to travel through it without damage and return in a reasonable amount of time. To satisfy this main requirement, the wormhole spacetime should be both free of event horizons and naked singularities. Wormhole spacetimes that contain no curvature singularities satisfy these two requirements.

Deriving the main curvature invariants using the process in the previous chapter and then plotting them will reveal any curvature singularities.

In this chapter, four basic wormholes will be presented: the TS Flat-Face wormhole (3.1), the MT wormhole (Section 3.2), the TS Schwarzschild wormhole (Sec- tion 3.3), and the exponential metric (3.4). The non-trivial CM invariant functions

33 for each wormhole will be presented and certain demonstrative plots for each will

be displayed. The plots will be inspected for places of great curvature and general traversability. Appropriately, the wormholes will be compared and contrasted for similar features.

3.1 Thin-Shell Flat-Face Wormhole

The TS Flat-Face wormhole is one of the simplest wormhole solutions. It consists of two separate regions of Minkowski spacetime. A small portion of each region is sliced out and connected with the other region using the TS Formalism. The formalism to compute the tensors for TS wormholes is outlined in [4]. In brief, two copies of

Minkowski flat space on either side of the wormhole’s throat are assumed, identical regions from each space are removed, and then separate regions along the boundary are connected. This formalism leads to a well-behaved wormhole, with the throat being located at the connecting boundary between the separate regions.

In the TS formalism, the metric is modiﬁed to be:

+ − gµν(x) = Θ (η(x)) g µν(x) + Θ (−η(x)) g µν(x), (3.1)

± where g µν is the metric on the respective sides, Θ (η(x)) is the Heaviside-step function and η(x) is the outward pointing normal from the wormhole’s throat. The radius of the wormhole’s throat is located at the point the regions overlap, x = a (that x ≥ a is important to note in regards to analyzing divergences). This formalism requires the

34 ± second fundamental form Kµν for the analysis at the throat to be:   0 0 0 0      0 1 0 0  ±  R1  Kµν = ±   , (3.2)    0 0 1 0   R2    0 0 0 0 where R1 and R2 are the radii of curvature of the wormhole on either side. The TS formalism modiﬁes the Riemann tensor to become:

+ − Rκλµν = −δ(η)[kκµnλnν + kλνnκnµ − kκνnλnµ − kλµnκnν]+Θ (η) Rκλµν +Θ (−η) Rκλµν . (3.3)

+ − where δ(η) is the delta function, kµν = Kµν −Kµν is the discontinuity in the second fundamental form, and nλ is the unit normal to the boundary of the shell.

For the TS Flat-Face wormhole, the line element on either side of the throat is the Minkowski metric,

ds2 = −dt2 + dx2 + dy2 + dz2. (3.4)

By computing the TS formalism, computing the TS Flat Face wormhole’s CM invariants shows that all invariants vanish. The plot is the same as Minkowski space with zero curvature. The plot forms a single round disk with no divergences, singularities, discontinuities or other artifacts that might prevent travel through a TS wormhole. As the plot is incredibly simple, it was not included. Because the invariants are equivalent to ﬂat space, a TS ﬂat-face wormhole will be traversable.

35 3.2 Morris-Thorne Wormhole

The MT wormhole is a spherically symmetric and Lorentzian spacetime. In the standard Schwarzschild coordinates [11, 12], its line element is:

2 ± dr ds2 = −e2φ (r)dt2 + + r2(dθ2 + sin2 θ dϕ2). (3.5) b±(r) 1 − r

The tetrad for the MT line element uses the spherical coordinates (r: with circumference = 2πr; 0 ≤ θ < π; 0 ≤ ϕ < 2π), and (−∞ < t < ∞) is the proper time of a static observer. φ±(r) is the freely specifiable redshift function that defines the proper time lapse through the wormhole throat. b±(r) is the freely specifiable shape function that defines the wormhole throat’s spatial (hypersurface) geometry. The ± indicates the side of the wormhole. The throat described by Eq. (3.5) is spherical. A

ﬁxed constant, r0, is chosen to deﬁne the radius of the wormhole throat such that

± b (r0) = r0, which is an isolated minimum. Two coordinate patches of the manifold are then joined at r0. Each patch represents either a diﬀerent part of the same universe or another universe, and the patches range from r0 ≤ r < ∞. The condition that

2φ±(r) the wormhole is horizon-free requires that gtt = −e 6= 0. This statement implies that |φ±(r)| must be ﬁnite everywhere [1, 4]. The use of Schwarzschild coordinates in Eq. (3.5) leads to more eﬃcient computations of the Riemann and Ricci curvature tensors, the Ricci scalar, and all four invariants. The four CM invariants for the MT

Wormhole are

1 0 R = (b0(rΦ0 + 2) + 2r(b − r)Φ00 − 2r(r − b)Φ 2 + (3b − 4r)Φ0), (3.6) r2

36 1 0 0 0 r = r2b 2 r2Φ 2 + 2 − 4rb0Φ0 r2Φ00 + r2Φ 2 − 2 1 16r6 + 4r2r2Φ002 + r2Φ04 + 2Φ02 r2Φ00 + 1

0 0 − 2rbb0 −2r3Φ 3 + Φ0 6r − 2r3Φ00 + r2Φ 2 + 2

+ 2r2r3Φ04 + 2Φ02 2r3Φ00 + r

+ 2rΦ00 r2Φ00 − 1 − r2Φ03 + Φ0 2 − r2Φ00

+ b24r4Φ002 + 4r4Φ04 − 4r3Φ03 − 8r2Φ00 − 4rΦ0 r2Φ00 − 3

+ Φ02 8r4Φ00 + r2 + 6, (3.7)

3 r = − b (2rΦ0 + 1) − r (b + 2rΦ0)2 2 64r9 0 0 × r2 b0Φ0 − 2r Φ00 + Φ 2 + b 2r2Φ00 + 2r2Φ 2 − rΦ0 − 2 , (3.8)

1 0 w = r b0 (1 − rΦ0) + 2r rΦ00 + rΦ 2 − Φ0 2 144r9 0 − b 2r2Φ00 + 2r2Φ 2 − 3rΦ0 + 3 3. (3.9)

All the invariants are non-zero and depend only on the radial coordinate, r, implying they are spherically symmetric. The invariants are plotted in Fig. 1 after selecting a redshift function of φ(r) = 0 and the shape function of

r0−r r0−r b(r) = 2GM 1 − e + r0e . (3.10)

These functions satisfy the constraints on the asymptotic behavior and continuity at the wormhole’s throat [4]. At a distant greater than 0.5 r0, all the figures are asymptotically flat. For r → 0, the figures diverge to infinity. The intrinsic singularity at r = 0 is not pathological as the radial coordinate r has a minimum r0 > 0 at the wormhole’s throat. Thus, a traveler passing through the wormhole would not experi-

37 ence the divergence. Any tidal forces on the traveler would be minimal. Consequently, the MT wormhole would be traversable as indicated by the included invariant plots.

(a) Plot of MT R (b) Plot of MT r1

Figure 3.1: Plots of the non-zero invariants for the MT wormhole. The plots are in radial coordinates with r ∈ {0, 4}. 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.

3.2.1 Variation of Shape Function

The MT Wormhole given in Eq. (3.5) has two freely speciﬁable functions, i.e., the redshift function, φ±(r), and the shape function, b±(r). These two functions satisfy the consistency requirements as [4] entails. However, additional constraints can be

38 imposed by choice of the shape function for ease of calculations and temporal and

spatial symmetry. The shape function must have:

(1) existence and ﬁniteness at both limits, limr→±∞ b(r) = b±,

(2) the masses of the wormhole M± on the two sides are given by b± = 2GM±,

0 b(r) (3) ∃r∗k∀r ∈ (r0, r∗), b (r) < r , 0 0 (4) b+(r0) = b−(r0) and b+(r0) = b−(r0) at the throat.

The shape function chosen in Eq. (3.10) obeys the listed conditions, with

limr→±∞ b(r) = 2GM. Thus, the shape function at the throat exists, is ﬁnite, and is continuous, which satisﬁes the second and fourth conditions. At the throat (i.e.

0 r0−r r → r0), b(r) = r0. The derivative, b (r) = (2GM − r0)e = b(r) + 2GM, is in agreement with the third condition mentioned above.∗ The freely speciﬁed shape

function can have a signiﬁcant impact on the form of the invariant functions. By

applying the second derivative test, it is seen that a shape function with a term an

rn term with n ≥ 3 will not have a discontinuity at r = 0. To test this, consider a

series of shape functions rn r0−r b(r) = 3 e , (3.11) r0 for n = (0, 1, 2, 3). Shape functions of this nature satisfy the conditions above.

The successive shape functions for the Ricci scalar are displayed in Fig. 3.3.

While the discontinuity at the center exists for the lower values of n, it disappears

at r = 0 and the shape resembles that of an inverted cone with a ﬁnite depth. While

this is outside the domain of r as discussed above, it demonstrates the eﬀect that the shape function holds over the invariants and how controlling the shape function allows the removal of any singularities as a wormhole is engineered.

∗ 0 b(r) Since G > 0 and M > 0, b (r) < r for all r0 ≤ r ≤ ∞ instead of a speciﬁc range of r∗ as necessitated by the third condition.

39 In fact, the power of the shape function extends further. Fig. 3.2 is very similar to the Ricci scalar from the exponential metric discussed in Section 3.4. This suggests that the exponential metric can be obtained from the MT metric by a suitable choice of the shape function. This is an interesting research topic and is a consequence of the

CM curvature invariant’s ability to distinguish spacetime metrics by solving the equivalence problem. Potentially, a diﬀerent choice of invariants like the Cartan set may be able to answer whether the MT and exponential metrics describe the same spacetime.

Figure 3.2: Plot of MT w2 for the shape function given in Eq. (3.11). The plots are in radial coordinates with r ∈ {0, 4} with G = M = 1 normalized and r0 = 2 chosen.

40 (a) Plot of MT R with n = 0 in Eq. (3.11) (b) Plot of MT R with n = 1 in Eq. (3.11)

(d) Plot of MT R with n = 3 in Eq. (3.11)

Figure 3.3: 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 ∈ {0, 4}. 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 3.3 Thin-Shell Schwarzschild Wormhole

The next wormhole to investigate is the Thin-Shell Schwarzschild wormhole. It is

two Schwarzschild black holes connected at their event horizons to make a

wormhole. The Schwarzschild geometry in natural units is given by the line element:

2M dr2 ds2 = − 1 − dt2 + + r2 dθ2 + sin2 θ dϕ2 , (3.12) r 2M 1 − r where M is the mass of the wormhole. The tetrad of the line element is the set of

spherical coordinates. The TS formalism developed in [4] and used in Section 3.1 is

used to connect the two sides of the wormhole. Each side is described by Eq. (3.12).

q 2M The thin-shell formalism is applied with a unit normal ni = 0, 1 − r , 0, 0 . Re- 3M gions described by Ω1,2 ≡ r1,2 ≤ a | a > 2 are removed from the two spacetimes leaving two separate and incomplete regions with boundaries given by the timelike hy-

3M persurfaces ∂Ω1,2 ≡ r1,2 = a | a > 2 . The boundaries ∂Ω1 = ∂Ω2 at the wormhole 3M throat of r = a are identiﬁed and connected. The boundary at a = 2 is chosen to satisfy the Einstein equations and equation of state in [4]; however, an event horizon is expected. The resulting spacetime manifold is geodesically complete and contains two asymptotically ﬂat regions connected by the wormhole.

Of the four CM invariants computed for the Schwarzschild wormhole, three invariants R, r1, and r2 equal zero. The remaining invariant is r 12M 3 6M 2 2M w = − + 1 − a (r − 2M) + 2M 2M + 2r3 − r δ (r − a) 2 r9 a2r9 a 12M + (a − 2M) 4M 2 (a − 2M)2 + r2 (a − 2M)2 a5r9 − 4Mr (a − 2M)2 − 2M 2r6δ (r − a)2 8 2M 3/2 + 1 − (a − 2M)3 (r − 2M)3 + M 3r9 δ (r − a)3 . (3.13) a6r9 a

12M 3 The w2 invariant is broken into two main portions. The leading term of − r9 is the

Schwarzschild black hole’s w2 invariant. The remaining portions of the function are 42 all proportional to diﬀerent powers of δ (r − a). Consequently, they contribute to the

throat of the wormhole. Evaluating w2 at the throat gives

6 ! 3 2 (a − 2M) 2M 2 w | = −6a5M 3 + 4a8 + M 3 1 − 2 r=a a14 a9 a r 2M + 3a3M 2 4a3M + a2 − 4aM + 4M 2 1 − a − 6M (a − 2M) 2a6M 2 − a4 + 8a3M − 24a2M 2 + 32aM 3 − 16M 4.

(3.14)

1 Since w2 ∝ a14 , the throat will experience greater curvature the smaller the radius is and vice versa. The only nonzero invariant, w2, is plotted in Fig. 3.4. The mass

3 and radius of the throat are normalized to M = 1 and a = 2 in the plot. Its plot has one divergence and one discontinuity. The divergence occurs at r = 0, which is

outside the manifold of Ω1,2. By the same argument for the apparent MT divergence, the ﬁrst Schwarzschild divergence would not impede the traversability of the worm-

3M hole. The discontinuity occurs at r = a = 2 and is located at the throat where the horizons are connected by the Schwarzschild wormholes. In these invariants, it is represented by a discontinuous jump to the value in Eq. (3.14). Since the invariants at the horizon are inversely proportional to a−14, the tidal forces on a traveler is be- nign at the horizon, and the Thin-Shell Schwarzschild wormhole would be traversable.

43 Figure 3.4: Plot of Schwarzschild w2. The plot is in radial coordinates with r ∈ {0, 4}. Each mesh line represents a radial distance of 0.5. The δ-function can be seen as a 3 thin discontinuity at r = 2 and its value is recorded in Eq. (3.14).

3.4 Exponential Metric

The exponential metric was demonstrated recently in [18] to contain a traversable wormhole throat. In natural units, its line element is

2 − 2M 2 + 2M 2 2 2 2 2 ds = −e r dt + e r {dr + r dθ + sin θdϕ }, (3.15)

where M is the mass of the wormhole. The tetrad utilizes the spherical coordinates. It

has a traversable wormhole throat at r = M. The area of the wormhole is a concave

function with a minimum at the throat where it satisﬁes the “ﬂare out” condition. It

does not have a horizon since gtt 6= 0 for all r ≥ 0. The region r < M on the other side of the wormhole is an inﬁnite volume “other universe” that exhibits an “underhill

− 2M eﬀect” where time runs slower since e r > 0 in this region. The four curvature invariants for the exponential metric are

2 2M − 2M R = − e r , (3.16) r4 4 3M − 4M r = e r , (3.17) 1 4r8 6 3M − 6M r = e r , (3.18) 2 8r12 44 3 3 32M (2M − 3r) − 6M w = − e r . (3.19) 2 9r12

Each invariant is nonzero and depends only the radial coordinate r implying spherical

symmetry. In addition, they are ﬁnite at the throat r = M and go to zero as r −→∞

in accordance with [18]. w2 and R have minima near the throat, while r1 and r2 have maxima. The plots are ﬁnite everywhere and completely connected conﬁrming the lack

of a horizon. The encountered tidal forces would be minimal. It can be concluded that

the exponential metric represents a traversable wormhole.

(a) Plot of the exponential metric R (b) Plot of exponential metric r1

Figure 3.5: Plots of the non-zero invariants for the exponential metric. The plots are in radial coordinates with r ∈ {0, 1.8M}. Each mesh line represents a radial distance of 0.1M. The throat begins at r = M.

45 CHAPTER FOUR

Warp Drives Moving at a Constant Velocity

The next set of spacetimes to be inspected are warp drive spacetimes moving at a constant velocity. These examples of FTL spacetime share a recent yet rich history as seen in the following [32, 33, 34, 35, 36, 37, 43, 44]. The key idea governing these theoretical spacetimes is that while motion through spacetime is limited by the speed of light, there is no limit to the rate spacetime itself can expand or contract. The most famous FTL solution to the Einstein field equations demonstrates how a spaceship may make a trip to a distant star in an arbitrarily short proper time [32]. A local contraction of spacetime in front of the spaceship paired with a local expansion of spacetime behind the ship allows the ship to surf along a warp bubble. While locally the spaceship remains within its own light cone as in Section 1.1.2 and never exceeds c, globally the relative velocity∗ may be much greater than c. FTL propulsion mechanisms based on this principle are called “warp drives.”

In the following chapter, a procedure to plot warp drive spacetimes using curvature invariants will be presented. First, the general metric for a warp drive spacetime in the 3 + 1 ADM formalism will be demonstrated with its shift and lapse functions.

Then, the methods to calculate the Alcubierre and Natário warp drive metrics at constant velocity will be presented with their curvature invariants. Each one will have their main variables and functions varied while maintaining all others as constants to see the individual aﬀects on the invariant plots. The Natário warp drive at changing velocities will be presented in the next chapter.

∗ deﬁned as proper spatial distance divided by proper time

46 4.1 Warp Drive Spacetimes

Alcubierre and Natário developed warp drive theory using 3+1 ADM formalism

[4, 43, 45]. In a well deﬁned coordinate patch, a spacetime may be decomposed into spacelike hypersurfaces, denoted as Σ, by use of an arbitrary time coordinate dx0.

0 0 Two nearby hypersurfaces, Σt where x = const and Σt+dt where x + dx = const, are separated by the proper time dτ = N(xα, x0)dx0.

The ADM four-metric is   −N 2 − N N gij N  i j j  gµν =   , (4.1) Ni gij

where N is the lapse function and Nα is the shift vector. These functions describe how to assemble the hypersurfaces to construct the entire spacetime. The lapse function

gives the interval of proper time between nearby hypersurfaces as measured by Eule-

rian† observers. The shift vector relates the geometrical coordinates between diﬀerent

hypersurfaces. It relates the relative velocity between Eulerian observers and the lines

of constant spatial coordinates. The 3-metric, gij, encodes the individual geometry of the 3-space for each hypersurface and measures the proper distance between two

points inside each hypersurface. As long as the 3-metric is positive deﬁnite for all

values of t, the spacetime is guaranteed to be globally hyperbolic. Consequently, the

generic warp drive metric is a class B1 spacetime.

For a warp drive spacetime moving at a constant velocity, the lapse between

inﬁnitesimally displaced hypersurfaces, say Σt and Σt+dt, remains constant. Thus, the lapse function is unitary, N = 1. The shift function forms a time-dependent vector

ﬁeld in Euclidean 3-space given by the equation [1, 32, 35]:

∂ ∂ ∂ ∂ X = Xi = X + Y + Z . (4.2) ∂xi ∂x ∂y ∂z

† These are observers whose four-velocity is normal to the hypersurfaces 47 Eq. (4.2) is important in deﬁning the future pointing normal covector to the Cauchy

α ∂ i ∂ ∂ surface as nα = −dt ⇔ n = ∂t +X ∂xi = ∂t +X. Any observer that travels along this covector is a Eulerian observer and a free-fall observer. A warp drive spacetime is ﬂat wherever X is spatially constant and a Killing vector ﬁeld for the Euclidean metric.

By assuming that the 3-metric is flat with Cartesian (x, y, z) coordinates, Eq. (4.1) simplifies to X ds2 = −dt2 + 3(dxi − Xidt)2. (4.3) i=1 The warp drive spacetimes considered in this chapter correspond to specific choices for the shift function X in Eq. (4.3).

4.2 Alcubierre’s Warp Drive

Figure 4.1: The expansion of the normal volume elements for the Alcubierre warp drive spacetime.

One of the simplest choices for the vector ﬁeld X is for the warp bubble to be moving at a constant velocity in a single cardinal direction. Choosing the cardinal direction to be dx1 = x corresponds mathematically to the vector ﬁeld (X,Y,Z) =

48 dxs(t) (vsf(rs), 0, 0) [32]. The velocity vector is vs(t) = dt = constant represents the

speed the warp bubble travels at to a distant observer. The shape function is f(rs), and Fig. 4.1 is a plot of the volume elements of Alcubierre’s chosen shape function.

It encodes the shape of the warp bubble in a similar manner to the shape functions

for the wormholes discussed in Chapter 3. Then, Eq. (4.3) becomes

2 2 2 2 2 ds = −dt + (dx − vsf(rs)dt) + dy + dz . (4.4)

Eq. (4.4) is in traditional Cartesian coordinates (−∞ < (x, y, z) < ∞) and its

origin is at the beginning of its ﬂight. vs(t) is the arbitrary speed Eulerian observers inside the warp bubble move in relation to Eulerian observers outside the warp bubble.

Thus, it is the speed of travel for the warp bubble itself. The radial distance is given

p 2 2 2 by rs(t) = (x − xs(t)) + y + z . It is the path an Eulerian observer takes starting inside the warp bubble and traveling to the outside of the bubble. Thus, it is the

distance from the center of the warp bubble to any observers outside. The Alcubierre

warp drive makes the choice for its continuous shape function, f(rs), as

tanh σ(r + ρ) − tanh σ(r − ρ) f(r ) = s s . (4.5) s 2 tanh σρ

where σ is the skin depth of the warp bubble and ρ is the radius of the warp bubble.

It is the shape of the warp bubble that can be represented by the expansion of the

volume elements as in Fig. 4.1. The shape function is arbitrary other than f = 1 in the interior of the bubble and f = 0 in the exterior. The chosen shape function is spherically symmetric. The parameters, σ > 0 and ρ > 0, are arbitrary apart from being positive.

49 The orthonormal tetrad may be read from the components in Eq. (4.4). It is     1 f(rs)vs         0  −1      E1 =   ,E2 =   , 0  0          0 0     0 0         0 0     E3 =   ,E4 =   . (4.6) 1 0         0 1

The null tetrad computed from Eq. (4.6) with Eq. (2.8.a) through Eq. (2.8.d) is     1 + f(rs)vs 1 − f(rs)vs         1  −1  1  1      li = √   , ki = √   , 2  0  2  0          0 0     0 0         1  0  1  0      mi = √   , m¯ i = √   . (4.7) 2  1  2  1          i sin θ −i sin θ The comoving null tetrad describes light rays traveling parallel with the warp bubble.

Eqns. (4.4) and (4.7) may be applied to the equations in Chapter 2 to derive the four

CM invariants. After making the speciﬁc choices detailed above, they are 1 R = σ2v 2 coth(ρσ) 2 s

p 2 2 p 2 × (4 tanh(σ(ρ + (x − tvs) ))sech (σ(ρ + (x − tvs) ))

p 2 2 p 2 − 4 tanh(σ( (x − tvs) − ρ))sech (σ( (x − tvs) − ρ)) (4.8)

3 p 2 p 2 − 2 sinh(ρσ) cosh (ρσ)(cosh(2σ( (x − tvs) − ρ)) + cosh(2σ(ρ + (x − tvs) ))

p 2 4 p 2 4 p 2 − 2 cosh(4σ (x − tvs) ) + 4)sech (σ( (x − tvs) − ρ))sech (σ(ρ + (x − tvs) ))) 50 1 r = σ4v 4(cosh4 (ρσ) 1 16 s

p 2 p 2 × (cosh(2σ( (x − tvs) − ρ)) + cosh(2σ(ρ + (x − tvs) ))

p 2 − 2 cosh(4σ (x − tvs) ) + 4) (4.9) 4 p 2 4 p 2 × sech (σ( (x − tvs) − ρ))sech (σ(ρ + (x − tvs) ))

p 2 2 p 2 + 2 coth(ρσ)(tanh(σ( (x − tvs) − ρ))sech (σ( (x − tvs) − ρ))

p 2 2 p 2 2 − tanh(σ(ρ + (x − tvs) ))sech (σ(ρ + (x − tvs) ))))

r2 = 0 (4.10)

1 w = − σ6v 6 2 288 s

p 2 2 p 2 × (2 coth(ρσ)(tanh(σ(ρ + (x − tvs) ))sech (σ(ρ + (x − tvs) ))

p 2 2 p 2 − tanh(σ( (x − tvs) − ρ))sech (σ( (x − tvs) − ρ))) (4.11) 4 p 2 p 2 − cosh (ρσ)(cosh(2σ( (x − tvs) − ρ)) + cosh(2σ(ρ + (x − tvs) ))

p 2 4 p 2 − 2 cosh(4σ (x − tvs) ) + 4)sech (σ( (x − tvs) − ρ))

4 p 2 3 × sech (σ(ρ + (x − tvs) )))

While Eqs. (4.8) through (4.11) are very complicated functions, several features are apparent from inspecting them. First, r2 is zero. It will not be plotted in the following subsections as its plots are ﬂat disks showing no curvature. Next, each non-

zero invariant depends only on the tetrad elements t and x. The axes of the plots are chosen to be the tetrad components to see the eﬀect of the free variables over time. In addition, each of the non-zero invariants is proportional to both the skin depth σn and

n the velocity, vs . It should be expected that the magnitude of the invariants will then increase with the magnitude of both. Finally, each non-zero invariant does not have any recognizable singularities inside the spacetime manifold. In the next subsections,

51 the non-zero CM invariants will be plotted to see any remaining individual aﬀects of vs, ρ and σ.

4.2.1 Invariant Plots of Velocity for Alcubierre

The plots for varying the velocities are included in Figures 4.2, 4.3, and 4.4 at the end of this subsection. The plots are 3D plots between the x coordinate, t coordinate, and the magnitude of the invariants along each axis. As natural units were selected, the plots have been normalized such that c = 1 and a slope of 1 in the x vs. t corresponds to the warp bubble traveling at light speed. The plots show a small range over the possible values of the variables to demonstrate many of the basic features of each invariant. First, the shape of all invariants resembles the “top hat” features of the shape function as discussed in [32]. However, there are some minor variations between each invariant. The Ricci scalar R oscillates from a trough, to a peak, to a flat area, to a peak and back to a trough. The r1 invariant simply has a peak with a flat area followed by another peak. The w2 follows the reverse pattern as the R wavering from a peak, into a trough, into a flat area, into a trough and returning into a peak. In each invariant, a ship could safely surf along in the flat area, which is dubbed the harbor. The central harbor disappears in Figures 4.2d through

4.2f because the plots lack precision. By plotting more points and consequently taking longer computational time, the central features will be recovered. The harbor’s width is much less than the distance covered in these later plots. Inspecting the functions, the harbor remains, and choosing smaller time intervals allows it to reappear in the plots.

Varying the velocity has several eﬀects. First, it increases the amount of distance, the x coordinate, covered per unit time. The warp bubble’s velocity acts exactly like v = d/t and is a good check that the program has been encoded correctly. The

52 warp bubble will cover an increasing amount of distance over time as observed by an Eulerian observer. Second, the velocity causes the magnitude of the invariants to decrease exponentially. This observation is in contrast to what was predicted by inspecting the leading terms of the invariants. It can be concluded that the additional terms overpower the leading term. Next, the shape of the warp bubble remains constant throughout the ﬂight. The only aﬀect of time is the distance covered. Finally, the plots have no holes or discontinuities, agreeing with the inspection of the invariant functions that no intrinsic singularities exist. While not truly practical, the invariants reveal nothing that will prevent a spaceship to surf the center channel.

53 (a) Plot of Alcubierre R with and vs = 0 (b) Plot of Alcubierre R with and vs = 1

(e) Plot of Alcubierre R with vs = 4 (f) Plot of Alcubierre R with vs = 5

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]. 0

54 (a) Plot of Alcubierre r1 with and vs = 0 (b) Plot of Alcubierre r1 with and vs = 1

(e) Plot of Alcubierre r1 with vs = 4 (f) Plot of Alcubierre r1 with vs = 5

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].

55 (a) Plot of Alcubierre w2 with and vs = 0 (b) Plot of Alcubierre w2 with and vs = 1

(e) Plot of Alcubierre w2 with vs = 4 (f) Plot of Alcubierre w2 with vs = 5

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.

56 4.2.2 Invariant Plots of Skin Depth for Alcubierre

The plots for varying the skin depth are included in Figures 4.5, 4.6, and 4.7 at

the end of this subsection. Repeating the procedure of Section 4.2.1, the variable σ has been varied between values of 1 and 8 while maintaining the other variables at the constant values of ρ = 1 and vs = 1. Many of the features in these plots are the same as those discussed at the beginning of Section 4.2.1; thus, the skin depth variation reveals two additional features. First, the plots advance towards the “top hat” function by slowly straightening out any dips. This feature is most notable in the plots of r1 in Figures 4.5a and 4.6a. Multiple ripples occur in these two plots initially, but then gradually smooth out as σ increases. These unforeseen ripples could be the source of a rich internal structure inside of the warp bubbles and potentially aﬀect its flight.

Second, the relative magnitude of the Ricci scalar and r1 is several orders of magnitude

−9 greater than that of w2. This can be seen as σ → 8 the Ricci scalar goes to 10 , r1 goes

−11 −28 to 10 , and w2 goes to the order of 10 . Consequently, the trace terms of the Riemann tensor will have the greatest effect on the curvature as both the Ricci scalar and r1 are members of the Ricci invariants in Eq. (2.22) and Eq. (2.41). The terms of the Weyl tensor will have negligible effects since w2 is a member of the Weyl tensor in Eq. (2.40). This conclusion can help warp drive calculations by focusing on the effects in the easier to calculate Ricci tensor.

The main eﬀects of varying the skin depth is to decrease the magnitude of the warp bubble’s curvature exponentially. This can be seen in each of the invariants as the magnitude decreases from being on the order of 10−3 to 10−28. The exponential decrease implies that thinner values of the warp bubble’s thin depth would propel itself at greater velocities due to the greater amount of curvature. This novel idea needs further exploration.

57 (a) Plot of Alcubierre R with and σ = 1 (b) Plot of Alcubierre R with and σ = 2

(e) Plot of Alcubierre R with σ = 8 (f) Plot of Alcubierre R with σ = 10

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].

58 (b) Plot of Alcubierre r1 with and σ = 2 (a) Plot of Alcubierre r1 with and σ = 1

(e) Plot of Alcubierre r1 with σ = 8 (f) Plot of Alcubierre r1 with σ = 10

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].

59 (a) Plot of Alcubierre w2 with and σ = 1 (b) Plot of Alcubierre w2 with and σ = 2

(e) Plot of Alcubierre w2 with σ = 8 (f) Plot of Alcubierre w2 with σ = 10

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].

60 4.2.3 Invariant Plots of Radius for Alcubierre

The plots for varying the radius ρ of the Alcubierre warp bubble are included in

Figures 4.8, 4.9, and 4.10 at the end of this subsection. Following the procedure of

Section 4.2.1, the variable ρ has been varied between values of 0.1 m and 5 m while maintaining the other variables at the constant values of σ = 8 and vs = 1. Many of the features in these plots are the same as those discussed at the beginning of Section 4.2.1, but the variation of the radius does reveal an additional feature. The spatial size for a spaceship to harbor inside the warp bubble is directly aﬀected by the value of ρ. By inspecting the x-axis of each plot, the size of the harbor is the same value of that of ρ.

While expected of the radius, it is confirmation that the program is encoded correctly and that the invariant functions reveal spacetime’s curvature. Of greater interest, the magnitude of the invariants does not have a clear correlation with ρ. As an example, consider the r1 plots in Fig. 4.9. When ρ = 0 m, the r1 invariants has its lowest magnitude order of 10−13. As the radius increases in the next four plots, the invariant increases to an order of 10−8. At the largest value ρ = 5 m, the invariant decreases to an order of 10−9. Inspecting the invariant function itself in Eq. (4.9), ρ does not have a noticeable relationship that explains this behavior. It can be hypothesized that resonance values of the radius ρ exist that could create massive warp bubbles. In conclusion, the radius ρ defines the size of the harbor and the warp bubble. It must always be chosen large enough for the ship to be unaﬀected by the curvature of the warp bubble itself.

61 (a) Plot of Alcubierre R with and ρ = 0.1 (b) Plot of Alcubierre R with and ρ = 1

(e) Plot of Alcubierre R with ρ = 4 (f) Plot of Alcubierre R with ρ = 5

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].

62 (a) Plot of Alcubierre r1 with and ρ = 0.1 (b) Plot of Alcubierre r1 with and ρ = 1

(e) Plot of Alcubierre r1 with ρ = 4 (f) Plot of Alcubierre r1 with ρ = 5

Figure 4.9: Plots of the r1 invariants for the Alcubierre warp drive while varying the radius. The other variables were chosen as σ = 8 and vs = 1 to match the variables Alcubierre originally suggested in his paper [32].

63 (a) Plot of Alcubierre w2 with and ρ = 0.1 (b) Plot of Alcubierre w2 with and ρ = 1

(e) Plot of Alcubierre w2 with ρ = 4 (f) Plot of Alcubierre w2 with ρ = 5

Figure 4.10: Plots of the w2 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].

64 4.3 Natário’s Warp Drive

The Natário warp drive spacetime describes a spacetime that “slides” the warp

bubble region through space [35]. A region in front of the bubble will be contracting

and balanced by a region behind the bubble, which will be expanding. Their net

eﬀect may propel the warp bubble at arbitrary velocities potentially even greater than lightspeed. Its line element is

2 rs θ 2 2 2 2 2 2 ds = (1 − XrsX − XθX )dt + 2(Xrs drs + Xθrdθ)dt − drs − rs (dθ − sin θ dϕ ). (4.12)

The standard spherical coordinates are (0 ≤ rs < ∞; 0 ≤ θ < π; 0 ≤ ϕ < 2π), and (−∞ < t < ∞) [35, 36]. The vector ﬁeld i n E q. ( 4.3) i s c onverted t o spherical

coordinates and then set to

2 2 0 X ∼ −vs(t)d[n(rs)rs sin θdφ] ∼ −2vsn(rs) cos θers + vs(2n(rs) + rsn (rs)) sin θeθ. (4.13) vs(t) is the constant speed of the warp bubble observed by Eulerian observers and

n(rs) is the shape function of the warp bubble. The shape function, n(rs), is arbitrary

1 other than the conditions n(rs) = for large r and n(rs) = 0 for small r. These 2 conditions on the shape function match the conditions for the “top-hat” function

from Section 4.2. The selected shape function is

1 1 n(r ) = (1 − ( (1 − tanh σ(r − ρ)))). (4.14) s 2 2

where σ is the skin depth of the bubble and ρ is the radius of the bubble [44]. The front

of the warp bubble corresponds to cos θ > 0 and the back vice versa. At the front,

there is a compression in the radial direction and an expansion in the perpendicular

65 direction. The comoving null tetrad is:     1 + Xrs 1 − Xrs         1  −1  1  1      li = √   , ki = √   , 2  0  2  0          0 0     Xθ Xθ         1  0  1  0      mi = √   , m¯ i = √   . (4.15) 2  −r  2  −r          ir sin θ −ir sin θ

Inputting Eqs. (4.12) and (4.15) into the method from Chapter 2, one may derive the four CM invariants. The Ricci scalar is

1 2 2 4 2 2 2 2 R = − σ vs sech (σ(r − ρ))(cos(2θ) + r σ sin (θ) tanh (σ(r − ρ)) 8 , (4.16) − 2rσ sin2 (θ) tanh(σ(r − ρ)) + 2)

The Ricci scalar is included alone in this chapter as a demonstrative example. The

remaining three are included in Appendix B.

Like the Alcubierre invariants, the Natário invariants are exceptionally com-

plicated. By inspecting the functions, similar features may be observed. First, the

Natário invariants do not change with time, but instead with r and θ. The warp bubble

skims along the comoving null tetrad in Eq. (4.15). As a consequence, the coordinates

for the plots are chosen to be r and θ. They will show the shape of the bubble around

n n the ship during flight. Second, each invariant is proportional to both vs and σ like the

Alcubierre invariants. The magnitude of the bubble’s curva-ture then increases

exponentially with both velocity and skin depth. In addition, the Natário invariants are proportional to cosn (θ ) and sechn(σ(r − ρ)). The warp bubble is shaped such that 2 the curvature is at a maximum in front of the ship around θ = 0 and a minimum behind 2 the ship around θ = π . It is at a maximum for r = ρ along 2 2

66 the center of the warp bubble, since there sech(0) = 1. Outside these values, the curvature should then fall oﬀ and go asymptotically to 0. These features match that of the “top-hat” function described in [32]. Finally, there are no intrinsic singulari-ties.

The manifold is asymptotically flat and completely connected. The flight of such a warp bubble should be significantly less aﬀected by any gravitational tidal forces as compared to the Alcubierre metric in Section 4.2. The CM curvature invariants confirm that the Natário warp drive is a more realistic alternative to Alcubierre’s.

Despite the complexity of the invariants, the shape of their plots is very simple.

It forms a very narrow and jagged ring as in Fig. 4.1. Precisely at r = ρ, the CM curvature invariants spike to non-zero magnitudes depending on the invariant. The

Ricci scalar R takes the form of a smooth disc outside the the warp bubble. The shape of the r1 invariant is that of a jagged disc at r = ρ. The disc has jagged edges in the negative direction, with sharp spikes at radial values r = ρ and at polar angle values of

θ = 0 and θ = π. The shape of the r2 invariant is that of a jagged disc at r = ρ. Its edges vary between positive and negative values depending on the polar angle θ. The shape of the w2 invariant is that of a jagged disc at r = ρ. In front of the harbor (θ > 0), the invariant has rapidly changing negative values between 0 and 1. Behind the harbor (θ

< 0), the invariant has positive values rapidly changing. The jagged edges of the plots must mean that the r1, r2 and w2 invariants oscillate rapidly be-tween 0 and 1 along the circumference of the warp bubble. The oscillations are more rapid than the plotted precision. Potentially, the r1 invariant could be replotted at a greater precision but at an extreme loss of computational speed. Outside, the CM in-variants of the Natário warp bubble are asymptotically flat. Inside, the CM invariants of the Natário warp bubble are asymptotically flat, implying there is a safe harbor for a ship to reside. The simplicity of the plots in comparison to the complexity of the invariant functions implies that only a single term (or a very small subset of all the

67 terms) in the invariant functions dominates the size of the warp bubble magnitude. In

the remainder of this section, the eﬀect of the velocity vs, the skin depth, σ, and the

radius ρ is analyzed.

4.3.1 Invariant Plots of Velocity for Natário

Figures 4.11 through 4.14 plot the Natário invariants while changing the veloc-

ity. First, the manifold is completely flat when vs = 0 for each invariant. The warp bubble must be turned oﬀ at this velocity, which confirms that the program encoded

the invariant functions correctly. For the Ricci scalar, a non-zero velocity causes the

invariant’s magnitude to jump to a small negative value at r = ρ. For r1, an increase in

velocity causes the magnitude of the invariant to swap from negative values to positive

values as the velocity increases along the circumferences of the circle with a radius of r

= ρ. For r2, an increase in velocity causes the magnitude of the invariant to swap from

positive values to negative values along the circumferences of the circle with a radius of

r = ρ. For w2, an increase in velocity swap the magnitude of the in-variant between

positive values to negative values as the velocity increases just along the semicircle of radius r = ρ behind the harbor. But in front of the harbor, the w2 invariant function remains negative regardless of the velocity.

Our prediction of an exponential increase in the invariants due to the velocity is

not consistent with the invariants’ plots. A potential reason for this discrepancy is a

dominant term inside of each CM invariant that overcomes the exponential increase in

vs. The dominant term in the invariant functions must either not depend on vs or the values for σ are the dominant factor. The research in this dissertation may be extended

either greater values of vs or lower values of the other variables to further

68 investigate this anomaly. If the CM invariants can be manipulated by diﬀerent choices of vs, then a theoretical ship could control its navigation.

69 m m (a) v = 0.0 s (b) v = 0.01 s

m m (c) v = 0.1 s (d) v = 1 s

m m (e) v = 10 s (f) v = 100 s

Figure 4.11: The velocity evolution of R, the Ricci scalar for the Natário warp drive at a constant velocity. It is understood that vs is multiplied by c. The other variables 1 are set to σ = 50,000 m and ρ = 100 m.

70 m m (a) v = 0.0 s (b) v = 0.01 s

m m (c) v = 0.1 s (d) v = 1 s

m m (e) v = 10 s (f) v = 100 s

Figure 4.12: The velocity evolution of the r1 invariant for the Natário warp drive at a constant velocity. It is understood that vs is multiplied by c. The other variables are 1 set to σ = 50,000 m and ρ = 100 m.

71 m m (a) v = 0.0 s (b) v = 0.01 s

m m (c) v = 0.1 s (d) v = 1 s

m m (e) v = 10 s (f) v = 100 s

Figure 4.13: The velocity Evolution of the r2 invariant for the Natário warp drive at a constant velocity. It is understood that vs is multiplied by c. The other variables are 1 set to σ = 50,000 m and ρ = 100 m.

72 m m (a) v = 0.0 s (b) v = 0.01 s

m m (c) v = 0.1 s (d) v = 1 s

m m (e) v = 10 s (f) v = 100 s

Figure 4.14: The velocity evolution of the w2 invariant for the Natário warp drive at 1 a constant velocity. The other variables are set to σ = 50,000 m and ρ = 100 m.

73 4.3.2 Invariant Plots of Skin Depth for Natário

Figs. 4.15 and 4.16, plot the Natário invariants while changing the skin depth

from σ = 500000 to σ = 100000. Unlike the shape expected from inspecting the in-

variant functions, the shape of the invariants remains the same. Since sech(r −ρ) → 1 as (r−ρ → 0), the spike in the invariant functions match the values of the limit of the sech and Eq. (4.14). The dominant term(s) in the invariant must then be proportional to sech(r − ρ). The σ plots add further evidence that the shape of the CM invariants is a consequence of the “top-hat” shape function.

1 1 (a) The invariant R with σ = 50,000 m (b) The invariant R with σ = 100,000 m

1 1 (c) The invariant r1 with σ = 50,000 m (d) The invariant r1 with σ = 100,000 m

Figure 4.15: The warp bubble skin depth for the Ricci scalar and r1 for the Natário m warp drive at a constant velocity. The other variables were chosen to be v = 1 s , and ρ = 100 m in natural units.

74 1 1 (a) The invariant r2 with σ = 50,000 m (b) The invariant r2 with σ = 100,000 m

1 1 (c) The invariant w2 with σ = 50,000 m (d) The invariant w2 with σ = 100,000 m

Figure 4.16: The warp bubble skin depth for r2 and w2 for the Natário warp drive at m a constant velocity. The other variables were chosen to be v = 1 s , and ρ = 100 m in natural units.

75 4.3.3 Invariant Plots of Radius for Natário

Like the Alcubierre plots in Section 4.2.3, the main effect of changing ρ is to increase the size of the Natário safe harbor. As ρ increases from ρ = 50 to ρ = 100 in Figures 4.17 through 4.18, the sizes of the safe harbor and the bubble double. The plots confirm that the radius ρ moderates the size of the bubble in the same fashion as Section 4.2.3. The spacing between the fringes in r1, r2, and w2 is not affected by changing ρ. The nature of the internal structure in the warp bubble itself remains a topic for further research.

(a) The invariant w2 with ρ = 50 m (b) The invariant R with ρ = 100 m

Figure 4.17: The warp bubble radius for the Ricci scalar and r1 for the Natário warp m drive at a constant velocity. The other variables were chosen to be v = 1 s , and 1 σ = 50, 000 m in natural units.

76 (a) The invariant r2 with ρ = 50 m (b) The invariant r2 with ρ = 100 m

Figure 4.18: The warp bubble radius for the r2, and w2 for the Natário warp drive at m 1 a constant velocity. The other variables were chosen to be v = 1 s , and σ = 50, 000 m in natural units.

77 CHAPTER FIVE

Warp Drives Moving at a Constant Acceleration

The final set of spacetimes to be investigated in this dissertation are Natário

warp drive spacetimes moving at a constant acceleration. The earliest papers acknowl-

edged the need for an accelerating bubble to transport a ship from a state of rest in its

dock to its desired FTL velocity. However, no papers initially considered accelerating

solutions of the ADM metric in Eq. (4.1). Loup considered non-unity values for the

lapse function N and showed that these choices led to a warp bubble transporting at

a constant acceleration [36]. Later, he expanded his work to six line elements for the

Natário spacetime with constant acceleration for contravariant, covariant, and

mixed spacetimes [44]. Deriving the appropriate lapse functions for the other

potential warp drives remains an ongoing area of research.

In this chapter, the work of Loup will be presented to expand the warp drive line solutions to ADM spacetimes moving at a constant acceleration. Then, the CM invariants for the covariant form of the accelerating Natário metric will be plotted.

The size of the invariant functions for these spacetimes is prohibitively large, so they will be included as a link to a .pdf document. However, the invariant functions’ main

features such as singularities will be discussed. The diﬀerent free variables vs, t, σ, and

ρ will be varied individually to see their eﬀect on the CM invariant plots. Finally, the

eﬀect of each of these free variables will be discussed.

78 Figure 5.1: Foliation of spacetime into 3D spacelike hypersurfaces at three diﬀerent times from [46].

5.1 The Accelerating Natário Spacetime

Consider the three spacelike hypersurfaces in Fig. 5.1 for a Natário warp drive

undergoing a constant acceleration. Then, the lapse of time tf = t3 − t2 for the warp to evolve to the hypersurface Σ3 from the hypersurface Σ2 must be less than the time lapse ti = t2 − t1 since the constant acceleration requires that vf > vi. The lapse function N in Eq. (4.1) measures this lapse of proper time between diﬀerent

hypersurfaces. Choosing c = 1 and Gtt = 1, the lapse function will be

2 t N = (1 + Xt + XtX ), (5.1)

where Xt is the time component of a covariant vector ﬁeld for Natário’s warp drive.

An accelerating Natário warp drive spacetime must exhibit the same properties as the constant velocity Natário spacetime in Section 4.3. Namely, nX = 0 and

X = vs = 0 for the values of the radial coordinate r inside the warp bubble’s harbor and nX = vs(t)dx and X = vs = vs for the values of the radial coordinate r outside of the warp bubble. A natural choice for the variable velocity of the warp bubble is

vs = 2n(rs)at. (5.2)

79 where n(rs) is the Natário shape function, a is the constant acceleration of the warp bubble, and t is the time variable. Since time between the Cauchy surfaces decreases due to the lapse, the shift vector N i = X describing the relative velocity of Eulerian observers inside of each hypersurface must also be modiﬁed to nX = vs?(dx)+x?(dvs), where ? is the Hodge star product. For constant velocity, x ? (dvs) vanishes. But for a constant acceleration, the total diﬀerential is

dn(rs) dvs = 2 at drs + n(rs)adt (5.3) drs in spherical polar coordinates. Combining Eq. (5.1), Eq. (5.2) and Eq. (4.13), the

Natário vector ﬁeld nX for variable velocities is

t rs θ nX = X dt + X drs + X rsdθ (5.4) with the contravariant shift vector components given by

Xt = 2n(rs)ars cos θ, (5.5)

2 0 Xrs = 2 2n(rs) + rsn (rs) at cos θ, (5.6)

0 Xθ = −2n(rs)at [2n(rs) + rsn (rs)] sin θ. (5.7)

The speciﬁc equation for the Natário warp drive line element in the parallel covariant 3 + 1 ADM is

2 2 2 2 2 2 2 2 2 2 2 ds = (1−2Xt+(Xt) −(Xrs ) −(Xθ) )dt +2(Xrs drs+Xθrsdθ)dt−drs −rs dθ −rs sin θ dϕ (5.8) in spherical coordinates∗ and where a is the constant acceleration. The covariant shift vector components can be obtained by raising Eq. (5.5) through Eq. (5.7) by

∗ (0 ≤ rs < ∞; 0 ≤ θ ≤ π; 0 ≤ ϕ ≤ 2π), and (−∞ < t < ∞)

80 the metric to get

Xt = 2n(rs)ars cos θ, (5.9)

2 0 Xrs = 2 2n(rs) + rsn (rs) at cos θ, (5.10)

0 2 Xθ = −2n(rs)at [2n(rs) + rsn (rs)] rs sin θ. (5.11)

The Natário warp drive continuous shape function is Eq. (4.14). The comoving null

tetrad for Eq. (5.3) is     1 − Xt + Xrs 1 − Xt − Xrs          −1   1      li =   , ki =   ,  0   0          0 0     Xθ Xθ          0   0      mi =   , m¯ i =   . (5.12)  −r   −r          ir sin θ −ir sin θ

The comoving null tetrad describes light rays traveling parallel with the warp bubble.

Substituting Eqs. (5.8) and (5.12) into the procedure from Chapter 2, one may derive the four CM invariants. The Ricci scalar is included in Appendix C as Eq. (C.1) as a demonstrative example. Due to the remaining three invariants’ massive size, a link to a .pdf ﬁle for each is included in Appendix C. Each of the four invariants has two singularities as can be seen in the Ricci scalar in Appendix C. The ﬁrst is at r = 0 and the second is at

0 = ar cos(θ) + ar tanh((r − ρ)σ) cos(θ) − 2. (5.13)

Each invariant has the second singularity to increasing powers with w2 maxing out at n = 12. The eﬀect of these singularities will be analyzed in the plots in the next section.

81 5.2 Invariant Plots of Time for Natário

The invariants evolve dynamically over time. Figure 5.2 shows how the Ricci

1 scalar evolves from t = 0 s to t = 100 s, while setting ρ = 100 m, σ = 50,000 m , m and a = 1.0 s2 . The ﬁgures provide rich details of the features in and around the warp bubble. Each plot has a safe harbor within r ≤ 100 where the curvature plots

are ﬂat. Consequently, the singularity identiﬁed in the invariant functions must not

be an intrinsic singularity. A spaceship riding in the interior of the harbor would

experience only ﬂat space throughout the entire time evolution. The warp bubble

π 3π travels perpendicular to the wake along the θ = ( 2 , 2 ) line. The wake is a volume of large curvature as shown on the plots and is the primary eﬀect of the singularity

noted in the previous section. While the form of the wake quickly reaches a constant shape, the Ricci scalar’s magnitude increases approximately proportional to time. The

linear increase in the warp bubble’s curvature suggests an engineering constraint on a

maximum achievable global velocity. The wake’s shape also shows that there is

internal structure to the warp bubble. Finally, it can be seen that far in front and behind the warp bubble that the invariants are zero; thus, the space is asymptotically flat.

Choosing the same values for ρ, σ, and a as the Ricci scalar and varying the time,

Fig. 5.3 shows the time evolution for the r1 invariant. It has many similar features to the Ricci scalar. It contains the safe harbor, a wake running perpendicular to the direction of motion, is asymptotically flat in front and behind the bubble, and increases approximately linearly with time. The first apparent diﬀerence is the positive magnitude and lack of internal structure in the wakes. Subtly, the wake increases in angular size as time increases.

The invariant r2 shares the same basic properties of the Ricci scalar and r1. It contains the safe harbor, a wake running perpendicular to the direction of motion,

82 is asymptotically flat in front and behind the bubble, and increases approximately

linearly with time. It is similar in shape to r1 and has the same internal structure as the Ricci scalar, but it increases in magnitude more drastically.

The invariant w2 shares the same basic properties of the Ricci scalar, r1, and r2. It contains the safe harbor, a wake running perpendicular to the direction of motion, is asymptotically flat in front and behind the bubble, and increases approximately linearly with time. The invariant’s wake contains a small amount of internal structure at lower time values, but as time reaches 100 s, the internal structure begins to form crenellations. As time continues to evolve, the crenellations travel out parallel to the length of the wake from the center of the warp bubble. It can be speculated that this would cause an erratic flight path of the bubble since the crenellations are not symmetric. It can also be speculated that the Ricci scalar and r2 would exhibit crenellations at higher time values.

π 3π The wake along the θ = ( 2 , 2 ) line appears to propagate to inﬁnity instanta- neously. A distant Eulerian observer along that axis should be able to detect it the

split second the warp drive accelerates. Due to the size of the invariant functions, the

wake would also impact the observer such as with a gravitational wave. Consequently,

the wake would causally eﬀect an observer at inﬁnity and violate the laws of causality.

Moreover, the location of the wake matches the singularity identiﬁed in the invariant

functions; thus, it must be an intrinsic curvature singularity. If the wake is an inherent

feature of any accelerating warp drive spacetime and not due to the speciﬁc choices

in the design of the warp drive such as the shape function, then the wake would be

evidence that an accelerating warp drive is not possible in our universe. The wake

and the identiﬁed intrinsic curvature singularity is under continued investigation for

its eﬀect on accelerating warp drive spacetimes.

83 (a) t = 0.0 s (b) t = 1.0 s

Figure 5.2: The time evolution of R, the Ricci scalar. The other variables were chosen m 1 to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

84 (a) r1 and t = 0.0 s (b) r1 and t = 1.0 s

Figure 5.3: The time evolution of the invariant r1. The other variables were chosen to m 1 be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

85 (a) r2 and t = 0.0 s (b) r2 and t = 1.0 s

Figure 5.4: The time evolution of the invariant r2. The other variables were chosen to m 1 be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

86 (a) w2 and t = 0.0 s (b) w2 and t = 1.0 s

(e) w2 and t = 200.0 s (f) w2 and t = 300.0 s

Figure 5.5: The time evolution for the invariant w2. The other variables were chosen m 1 to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

87 5.3 Invariant Plots of Acceleration for Natário

Varying the acceleration of the invariants speeds up or slows down the time

1 evolution of the previous section. Setting ρ = 100 m, σ = 50,000 m , and t = 1.0 s, Fig. 5.6 shows the acceleration’s variation for the Ricci scalar. The first plot of a = 0 is consistent with the lapse function being zero for all time. The distance between hypersurfaces remains constant. The space is flat and no warp bubble forms. The second plot corresponds to a time slice between Figs. 5.2a and 5.2b. Similarly, the third plot is identical to Fig. 5.2b, and the fourth plot is also identical to Fig. 5.2c. It can be concluded that modifying the acceleration parameter corresponds with modifying the rate of change of the hypersurfaces. The analysis in the previous section holds for this case as well.

The plots of the invariants r1, r2 and w2 follow a similar process to the Ricci scalar. They are plotted in Figs. 5.7 and 5.8. When a = 0, their plots are identical to Fig. 5.6a. Their next plots can be seen as additional time slices between the plots shown in Figs. 5.3a and 5.3d for r1 and Figs. 5.4a and 5.5d for r2. Some additional features are present in the plots. The curvature invariants warp much more significantly and non-symmetrically than their counterparts for the Ricci scalar, as can be seen in Fig. 5.8c. The nonzero invariant inside the spaceship found in the previous time slices is also present, but due to the low magnitude of the r1, r2, and w2 invariants, this can be safely ignored.

88 m m (a) a = 0.0 s2 (b) a = 0.1 s2

m m (c) a = 1.0 s2 (d) a = 10.0 s2

Figure 5.6: Varying the acceleration of R, the Ricci scalar. The other variables were m 1 chosen to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

89 m (a) r1 and a = 0.1 s2

m (b) r1 and a = 1.0 s2

m (c) r1 and a = 10.0 s2

Figure 5.7: Varying acceleration for the invariant r1. The other variables were chosen m 1 to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

90 m (a) r2 and a = 0.1 s2

m (b) r2 and a = 1.0 s2

m (c) r2 and a = 10.0 s2

Figure 5.8: Varying acceleration for the invariant r2. The other variables were chosen m 1 to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

91 m (a) w2 and a = 0.1 s2

m (b) w2 and a = 1.0 s2

m (c) w2 and a = 10.0 s2

Figure 5.9: Varying acceleration for the invariant w2. The other variables were chosen m 1 to be a = 1 s2 , σ = 50,000 m , and ρ = 100 m in natural units.

92 5.4 Invariant Plots of Skin Depth for Natário

Varying the skin depth of the warp bubble, σ, does not noticeably aﬀect the

invariant plots reaching the same conclusion as Section 4.3.2. Figures 5.10 and 5.11

present the plots for each of the four invariants while doubling the skin depth and

m setting σ = 100,000 , t1 = 1.0 s, and a = 1.0 2 . The shape of the invariant does not m s change in the ﬁgures after the doubling. It can be speculated that either the invariants

are independent of skin depth or that the impact of skin depth is minimal compared

to the other variables.

1 1 (a) The invariant R with σ = 50,000 m (b) The invariant R with σ = 100,000 m

1 1 (c) The invariant r1 with σ = 50,000 m (d) The invariant r1 with σ = 100,000 m

Figure 5.10: The warp bubble skin depth for the Ricci scalar and r1. The other variables m were chosen to be t = 1 s, a = 1 s2 , and ρ = 100 m in natural units.

93 1 1 (a) The invariant r2 with σ = 50,000 m (b) The invariant r2 with σ = 100,000 m

1 1 (c) The invariant w2 with σ = 50,000 m (d) The invariant w2 with σ = 100,000 m

Figure 5.11: The warp bubble skin depth for r2 and w2. The other variables were m chosen to be t = 1 s, a = 1 s2 , and ρ = 100 m in natural units.

5.5 Invariant Plots of Radius for Natário

Varying the radius of the warp bubble, ρ, increases the size of the safe harbor inside the invariants, reaching the same conclusion as Section 4.3.3. Setting ρ = 100 m,

m t = 1.0 s, and a = 1.0 2 , the following plots for the invariants result in Figs. 5.12 s and 5.13. In the ﬁgure, the radial coordinate clearly doubles in each invariant without aﬀecting the shape of the plots. The safe harbor of ρ ≤ 100 in the left hand column also doubles in size to ρ ≤ 200. The only other pertinent feature is in the internal structure of w2. The structures are reduced implying that they cluster near the center.

94 (a) Ricci scalar with ρ = 100 m (b) Ricci scalar with ρ = 200 m

Figure 5.12: The warp bubble radius for the Ricci scalar and r1. The other variables m 1 were chosen to be t = 1 s, a = 1 s2 , and σ = 50,000 m in natural units.

95 (a) The invariant r2 with ρ = 100 m (b) The invariant r2 with ρ = 200 m

Figure 5.13: The warp bubble radius for r2 and w2. The other variables were chosen m 1 to be t = 1 s, a = 1 s2 , and σ = 50,000 m in natural units.

96 CHAPTER SIX Conclusion

The spacetime elements considered in this research can at best be described as speculative and at worst as physical impossibilities. No experimental evidence exists of either a warp drive or a wormhole. It is well known that wormholes and warp drives violate the null energy conditions (NEC) and require exotic matter to stabilize [1, 4].

Moreover, it appears excessively diﬃcult to engineer a local spacetime in a way that a human could use for travel. Wormholes either appear at galactic scales as a possibility lying beyond a black hole’s event horizon or as a solution to how spacetime warps on the Planck scale. Dark energy seems to be the only possibility for constructing the

Spacetime expansion and contraction required to propel a warp drive. Dark energy −30___ g acts on Hubble volume scales and its energy density is on the scale of 7 × 10 3 . cm Exponentially increasing its energy density to propel a spacecraft capable of carrying

a human being is an engineering impossibility for the foreseeable future. The most

dangerous potential consequence is how easy it is to create a CTC by adjusting the

relative velocities of the wormhole throats or the warp-drive bubbles. CTCs create time machines, which implies that both spacetimes violate causality. Causality is

fundamental to our understanding of physics, so both types of spacetime should be prohibited from existing in the known and possible universe by a short proof of contradiction.

Then, the important question is why should these spacetimes be researched at all. First, all spacetimes considered herein are solutions to Einstein’s equations.

They appear to be fully possible inside the known frameworks or semi-classical ap- proximations of GR. If causality prohibits these spacetimes, then how and why they

97 cannot exist needs to be understood. So by researching them, additional constraints

on possible spacetimes inside GR can be deduced. Second, simple extensions of GR

containing QM and even classical scalar field theories show physical violations of the

NECs [10]. The energy conditions in GR provide a precise definition of local energy

densities. They lay the basis for the amount a light cone may be tipped over as discussed in Section 1.1.2. Probing mathematically realistic but physically unlikely theories of wormholes and warp drives will help our understanding of the breadth of

NEC violations. Finally, assume for a moment that wormholes and warp drives are both mathematically possible and physically realistic. Then, methods and techniques must be developed to identify them when they occur in nature. For example, the jump discontinuity found in Fig. 2.2 can lead to a redshift that would distinguish a wormhole from a black hole. Second, the wakes observed for each warp drive spacetime should generate gravitational waves. Potentially, these waves could be detected by significantly more sensitive LIGO-like experiments. If either of these were detected, this would provide significant evidence for their physical existence.

The research contained in this dissertation and in the program in Appendix

A forms a foundation for the computation of any set of invariants. The program computes everything needed to find any SP invariant from a given line element.

It computes each tensor and all NP indices in Section 2.1. While not discussed in this dissertation, the program may also solve the Einstein equations and obtain other tensors such as Eijkl. The real power of the program is its modality. It can be easily adapted for any given set of invariants in addition to the CM invariants discussed herein. In the next section, how the program may be adapted to other sets of invari-ants or GR calculations will be presented.

98 6.1 Invariant Research

The program in Appendix A forms the basis for this research, and it can be used

to find the invariants of any given basis for any given metric. It takes a metric and a

null tetrad as inputs and outputs the needed set of invariants. In this research, the

question was whether any curvature singularities existed in the considered spacetimes.

If a singularity was discovered, its nature would be revealed by computing and plotting

only the CM invariants for the considered spacetimes. However, sets of invariants can

also be used in studying the Segre and Petrov type of the line element and the

equivalence problem.∗ To a limited extent for the wormhole metrics, the research shows that the MT wormhole and the exponential metric may be equivalent with appropriate choices of shape function and redshift function. As it stands, the research and program can be adapted to study these problems for any given set of invariants or line element.

In this section, several of the ongoing uses of the program will be presented.

Computing and plotting the invariant functions has significant advantages for the inspection of these exotic spacetimes. As mentioned previously, the advantage of plotting the invariants is that they are free from coordinate mapping distortions and other artifacts of the chosen coordinates. The resulting invariants properly illustrate the entire underlying spacetime independent of the coordinate system chosen. Plotting the invariants exposes the presence of any artifacts, divergences, or discontinuities

anywhere on the manifold. Once the artifacts are revealed by the invariants, they can

be related mathematically to the textbook tensors. Any artifact’s eﬀect can then be

analyzed based on where the curvature invariant locates it, what type of artifact the

curvature invariant reveals it to be, and how the artifact may aﬀect the object’s

motion based on the plots of the curvature invariant at that location.

∗ The question of whether two spacetime metrics are equivalent.

99 A second advantage is the relative ease with which the invariants can be plotted.

Software packages exist or can be developed to calculate the textbook tensors such as the one provided in Appendix A. Then, the CM invariants can be derived from the textbook using minimal edits to the software packages. The CM invariants were chosen to be computed and plotted in this paper because they had general independence, were of lowest possible degree, and a minimal independent set for any Petrov Type and for any specific choice of the Ricci tensor [41]. It is of research interest that other choices for the set of invariants exist, such as the Cartan invariants previously discussed, as wells as the Witten and Petrov invariants [21, 47]. These additional sets of invariants may answer several, unconsidered questions, such as whether the underlying spacetime is unique. These sets may also be computed and plotted without diﬃculty. They are related to the CM invariants by polynomial functions. Since the invariants are either scalars or pseudoscalars, they can be straightforwardly plotted and visually interpreted.

The first extension of this research in this paper is to include the non-independent

CM invariants presented in Eqs. (2.37) through (2.46). While the syzygies for Class B spacetimes reduced this group to the four in Eq. (2.67), other classes of spacetimes will have diﬀerent sets of syzygies and diﬀerent sets of independent invariants. The remaining invariants may then be computed. The program in Appendix A currently computes the tetrad and the tetrad components of the Ricci tensor and the Weyl tensor. The functional relationships with the non-independent CM invariants are the next expansion to the program’s code.

Another extension of this work is to expand the program to other sets of invariants such as the Cartan invariants. An alternate question to the one of completeness for a set of invariants is how to construct a set of invariants from the Riemann tensor, and/or its derivatives that will suﬃciently cover a spacetime [20]. Such a set may be

100 constructed from the SP invariants; however, the SP invariants will not uniquely cover the spacetime. A better alternative considers the Cartan set of invariants, which provide a unique coordinate-independent characterization. The coordinate components of diﬀerent metrics may then be compared given the coordinate components for curvature and the derivatives of the Riemann tensor.

The Cartan invariants are the non-zero components of the Riemann tensor Rαβγδ and its covariant derivatives [20, 47]. Invariants that are constructed from or are equal

to the Cartan invariants of any order are called extended invariants. The algorithm to

compute the Cartan invariants in any dimension is as follows:

(1) Set the order of diﬀerentiation q to 0.

(2) Calculate the derivatives of the Riemann tensor up to the qth order.

(3) Find the canonical form of the Riemann tensor and its derivatives.

(4) Fix the frame as far as possible by this canonical form and note the residual

frame freedom (the group of allowed transformations is the linear isotropy

group Hq). The dimension Hq is the dimension of the remaining vertical freedom of the frame bundle.

(5) Find the number tq of independent functions of spacetime position in the components of the Riemann tensor and its covariant derivatives, in the canonical

form. This tells the remaining horizontal freedom.

(6) If the isotropy group and number of independent functions are the same as in

the previous step, let p + 1 = q, and the algorithm terminates, if they diﬀer

(or if q = 0), increase q by 1 and return to step 2.

As the algorithm follows a simple set of ﬁnite steps, it can be automated and added to

the computer program in Appendix A. Upon completing this algorithm, suﬃciently

smooth metrics may be compared as a test of the equivalence.

101 The research in this dissertation has already hinted that two of the metrics, the

MT wormhole and the exponential metric, rely on the same underlying spacetime. A

possibility exists that every type of wormhole is a specific choice of either the shape

function b(r) and/or the redshift function Φ(r) of the MT metric. Even more tanta-

lizing is that wormholes and warp drives have the same topological structure. If true,

then the two will be connected by a suitable choice of shape functions b(r), f(rs), and/

or n(rs). Computing the Cartan invariants may confirm or deny this hypothesis.

Research is underway to adapt the program in this manner to construct the Cartan

Invariants based on the procedure outlined in this section. Its primary purpose is to answer the question of equivalence among the previous wormhole and warp drive metrics among others to be considered below.

6.2 Wormhole Research

Chapter 3 demonstrates how to compute and plot the curvature invariants of various wormhole line elements. The CM curvature invariants reveal the entire wormhole spacetime manifold and whether the wormhole is traversable or not. As examples, plotting the curvature invariants of the (i) spherically symmetric MT, (ii) TS

Schwarzschild and (iii) Exponential metric wormholes showed they are traversable in agreement with [4, 18, 11, 12]. The invariants of the MT wormhole were found to be non-zero and are plotted in Figs. 3.1a–3.1d. A divergence is found in all four. The divergence does not aﬀect the wormhole’s traversability as it is outside the physical range of the radial coordinate, r ∈ (r0, ∞). For the TS Schwarzschild wormhole, w2 is found to be the single non-zero invariant. As plotted in Fig. 3.4, it has a divergence at the center and a ring discontinuity. The divergence is outside the physical radial coordinate and can be safely ignored. The ring discontinuity represents a jump due

102 to the δ-function from the TS formalism. It is shown to be inversely proportional to a

−14; thus, it will not aﬀect any transport through the wormhole. The SP invariants of

the exponential metric were found to be non-zero and were plotted in Figs. 3.5a–3.5d.

The plots are continuous across the entire manifold and traversable.

Potentially, the ring discontinuity in the TS Schwarzschild wormhole may lead

to a redshift of light rays that pass through the wormhole. The redshift could be used

to distinguish a TS Schwarzschild wormhole from the famous Schwarzschild black

holes. The TS Schwarzschild wormhole is the most common example of a large class of

wormholes. The class includes wormholes with diﬀerent radii of curvature, R, masses,

M and/or diﬀerent charge, Q, on either side of their throat, and time-dependent wormholes. For charged wormholes, a second ring artifact at r = Q is likely to exist since the metric has a singularity at that point. Similar shifts could be expected also from their more complicated line elements such as the rotating traversable wormholes

[17]. Significant research using the methods and programs in this dissertation is underway to compute the invariants of the rotating traversable wormholes, which will aid in their potential identification.

One future application of studying curvature invariants of wormholes is an investigation of the rotating traversable wormhole in [17]. The rotating wormhole is the most general extension of the MT wormhole discussed in Section 3.2. Its metric is

ds2 = −N 2dt2 + eµdr2 + r2K2 dθ2 + sin2 θ(dφ − ωdt)2 , (6.1)

where N is the redshift function, µ is the shape function, K determines proper radial

distance, and ω is the wormhole’s angular velocity. It should be expected that the

polar symmetry observed in the MT plots will be broken by the angular velocity.

Several interesting features of rotating wormholes include speciﬁc geodesics that do

not encounter exotic matter and an ergoregion surrounding the throat. Consequently,

a hypothetical interstellar traveler would choose the rotating traversable wormhole for 103 transport over the ones discussed previously. Research is ongoing into this wormhole metric.

Another prospective future application of this work is an investigation of wormholes with throats that change dynamically over time as in [4]. The invariants for these wormholes should have the size of the central discontinuity change over time. The ring discontinuity in the invariant functions will change as a function of time as a primary consequence. Hence, dynamic wormholes are more technically demanding to study as compared to static wormholes. Consequently, it can be expected that the compu-tation of a dynamic wormhole’s invariants and their plots increase in diﬃculty and computational runtime. The program in Appendix A should use the

NP indices to compute the invariants for this class of wormhole. Like rotating wormholes, dynamic wormholes are a more physically realistic class of wormholes, and they can be related to primordial black holes.

6.3 Warp Drive Research

The research in Chapters Four and Five demonstrate how computing and plotting the curvature invariants for various variables of warp drive spacetimes can reveal their underlying curvature. While the individual functions are mammoth in size and may take days, weeks, or even months to calculate, their plots can be quickly scanned and understood. The plots give the magnitude of curvature at each point around the ship. Where the curvature invariant’s magnitudes are large, space is greatly warped, and vice versa. Also by observing the changes in slopes on the plots, the rate at which spacetime is being folded can be analyzed. Using this information can help map the spacetime around the ship and aid potential navigation.

104 The curvature invariant functions and plots were displayed for the Alcubierre

warp drive at constant velocity, the Natário warp drive at constant velocity, and the

accelerating Natário warp drive metric. Diﬀerent choices of the free variables were

varied to see the individual eﬀect on each invariant. The curvature invariants reveal a

safe harbor for a ship to travel inside the warp bubble and an asymptotically ﬂat space

outside the bubble in all cases. At the radial position r = ρ of the warp bubble(s),

the curvature invariants have local maxima implying that ρ is the location where

spacetime is warped the most. For the constant velocity Alcubierre warp drive, the

warp bubbles resemble two troughs with simple internal structures. For the constant

velocity Natário warp drive, the warp bubbles peak around r = ρ and display rich

internal structure. For the accelerating Natário warp drive, two wakes radiate out from

π 3π the warp bubble along the θ = ( 2 , 2 ) axis. The wakes have rich internal structures that ripple and bubble as time advances. The internal structures of the warp bubble

found in this paper are novel and require more diligent research to discover their eﬀects

on the warp drive’s flight.

For the accelerating Natário warp drive, a variation of time shows that each invariant’s plots experience a sudden jump from positive curvature in the direction of motion to negative curvature. As time progresses, the shapes of the R, r1, and r2 invariants remain constant, but the magnitude of the invariants increases roughly linearly in time and the angular arc of the wake subtlety along the polar axis. The w2 invariant begins to exhibit crenellations in the interior of the warp bubble after 100s.

By varying the acceleration, the invariant plots skip through the time slices and internal structures become more prominent. Changing the skin depth did not change either the shape or magnitude of the invariant plots. Doubling the radius did double the size of the warp bubble and safe harbor without aﬀecting the shape of the invariants. The invariant plots give a rich and detailed understanding of the curvature of spacetime surrounding a warp drive.

105 In addition to the research presented in this dissertation on inspecting the diﬀer- ent invariants, further work can be done in mapping warp drive spacetimes. The work in

Chapter 5 can be further expanded by considering time slices greater than 300 s.

Potentially, crenellations like the ones observed for the w2 invariant exist within the warp bubbles for the R, r1 and r2 invariants. The crenellations would make controlling the direction of the warp bubble challenging. As the crenellations bubble out along the polar axis, the warp bubble would be pulled chaotically by the crenellation’s large curvature. The crenellations imply that high frequency gravitational waves would be produced by an accelerating warp drive. Potentially, these waves could be detected by an extremely sensitive detector. A detector like the one proposed recently would be suitable [48]. Mapping the eﬀect on the warp bubble’s path is of critical importance to plotting a complete journey to a distant star.

The linear increase in the magnitude of the curvature implies that the warp drive requires a linear increase in the total amount of energy needed to accelerate the warp bubble. The invariant plots in Figs. 5.1 to 5.4 show that there is a linear increase in the magnitude of the curvature the longer the warp drive accelerates. The invariant plots in Figs. 5.5 to 5.8 reveal how a greater magnitude of the acceleration causes the magnitude of the invariants to increase. From the stress-energy tensor for the warp drive, these observations indicate that the energy requirements for an accelerating warp drive increase proportionally over time [32, 35, 44]. Therefore, a realistic warp drive will be able to accelerate to some finite vs that is potentially greater than c. The superluminal censorship theorem should be revisited in light of this new evidence

[10]. Further research is needed to establish the maximum achievable vs for a warp drive.

The next logical extension of the research on warp drives is to combine the constant velocity Natário and the constant acceleration Natário warp drives into a

106 complete trip to a distant star such as our closest star, Alpha Centauri. It would consist of four stages. The first stage would be under traditional rocket propulsion as the spaceship flies a safe distance away from its origin. Second, the warp bubble would be activated for a period of time to accelerate it at vs = 2n(rs)at until a maximum velocity of vs = 200c is reached. The third stage would be a period of constant velocity at vs = 200c as the ship travels between the stars. Fourth, the warp bubble would decelerate at vs = −2n(rs)at as the warp bubble is slowly deactivated and the ship returned to standard propulsion. It should be noted that Eqs. (5.8) through (5.11)

would need to be modified for infinitesimal hypersurface lapses increasing in proper

time for the deceleration. Finally, the ship would continue under rocket propulsion as it

begins its orbit of the destination star or planet.

The flight could answer several questions about warp drive propulsion. The

eﬀect of the wake and crenellations of the accelerating Natário warp drive should be identified and studied to see their eﬀect on the origin and destination. If the ef-fect is disastrous, the distance the ship would have to travel under traditional rocket propulsion before the warp drive is turned on would be known. The duration of each

individual stage of the FTL trip could then be found. Once the duration is known, the

amount of “exotic matter” and negative energy could be calculated using the methods

in [32, 35, 44]. Finally if the crenellations do cause gravitational waves, methods to

detect approaching or departing warp bubbles could be developed.

The technique of plotting the invariants can be applied to the other warp drive

space times such as Alcubierre’s at a constant acceleration, Krasnikov’s at either

constant velocity or constant acceleration, or Van Den Broeck’s at either constant

velocity or constant acceleration [33, 34, 44]. In addition, the lapse functions for the

Krasnikov and Van Den Broeck’s warp drives would need to be identified and then

107 their accelerating line elements could be derived. After plotting their line elements for the invariants, each proposed warp drive could be compared and contrasted to their corresponding invariants at a constant velocity contained in Chapter 4. The Cartan invariants from Section 6.1 computed for each warp drive element. The warp drive’s Cartan invariants may then be compared with the wormholes and other spacetimes for equivalence.

6.4 Closing Thoughts

The study of wormholes and warp drives is a rich vein that contains many nuggets to be mined. The research in this dissertation adds significantly to their study.

We have investigated the curvature invariants and confirmed that wormholes may be traversable. We discovered that there is a significant redshift in the TS Schwarzschild wormhole that may be used to distinguish it from the classic Schwarzschild black hole.

We compared the exponential metric to the MT metric and saw that their shape functions are potentially identical. The curvature invariants contained no singularities for the Alcubierre and Natário warp drives moving at a constant velocity. Their plots show how warp bubbles evolve over the diﬀerent variables vs, σ and ρ. They reveal the never-before-seen internal structures of the warp bubbles. Under every condition, a safe harbor existed in the warp bubbles that allows a ship to travel peacefully. The

Natário warp drive at a constant acceleration shared many of these properties. It also contains a sizable wake with rich internal structures. Unfortunately, it also contained a curvature singularity along the polar angle, which requires additional study.

None of these findings would have been possible without developing and vigor- ously testing the Mathematica R program contained in Appendix A. It is extensive in its applications as it contains every tensor in GR, each NP index, and the independent

108 CM curvature invariants. While this research focused on wormholes and warp

drives, the program may compute the CM invariants for any known spacetime. The

program takes as its input a metric and a tetrad, then outputs the CM curvature

invariants. Including these inputs is standard practice for any research paper on

novel space-times in GR. The program can then easily identify any curvature singularities in the spacetimes by following the same procedure as in this research.

The program may be easily adapted to any basis set of invariants. The basis for

computing any element in a set of invariants is already contained in Chapter 2 and included in the program. A future researcher only needs to modify the “Computing the Invariants” section of the program for the particular set of invariants of interest such as the Cartan invariants discussed in Section 6.1. The program lays the foundation for many branches of future research in Numerical GR.

While wormholes and warp drives capture our imaginations about what is possible, a realistic engineering of one is many years beyond what humanity can currently achieve. Taking an aﬀect that is as visible as Dark energy on a multi-galactic scale and shrinking it to the scale of a human is beyond any construction of GR currently known.

However, these spacetimes are prime examples of the boundary between what is possible in the mathematics of GR and what is observed experimentally. They pro-vide an opportunity to test and analyze where our understanding of GR ends. The intention of the research contained in this dissertation is to explore this boundary intensely. It plays with our notions of possibility. The aim is to ask what is possible and why is the possibility not observed. This research does just that and may act as a stepping stone to further advances in our knowledge and our imagination.

109 APPENDICES

110 APPENDIX A

Mathematica Program

The individual Mathematica programs for each spacetime metric contained in this dissertation may be found at the following website locations:

(1) Morris-Thorne metric —

https://baylor.box.com/s/fj3bhd8trjnbg6tc1zqt867oaqzmqld1 (2) Thin-Shell Schwarzschild metric —

https://baylor.box.com/s/fm1bt2asy3ulyyaqv9ro0f3o2p3k4fc1 (3) Exponential metric —

https://baylor.box.com/s/rfgfmlajr637g3aetd3zv28n8o70b1n5 (4) Alcubierre metric at a constant velocity —

https://baylor.box.com/s/3rllwrve15ut31kj5cruwcbr66slta9s (5) Natário Metric at a constant velocity —

https://baylor.box.com/s/lba8r6cr2uukfxveji9foejh5a9su8fc (6) The invariants, R, for the Natário metric at a constant acceleration —

https://baylor.box.com/s/lyod1zkjlmfc961ny5vosm0rxwbruxbb

(7) The invariants, r1, for the Natário metric at a constant acceleration — https://baylor.box.com/s/7y9nkx4pq2q4qnzxesyz0knxai3qoj9y

(8) The invariants, r2, for the Natário metric at a constant acceleration — https://baylor.box.com/s/tbxgl1gwsxeup2y27d8icqde737f3lv3

(9) The invariants, w2, for the Natário metric at a constant acceleration — https://baylor.box.com/s/b794s0enxthvaxt97he48f2wv0xm446p

111 Plotting the Invariants for the Natario Metric

Clear[v, G, c, n, t, r, θ, ϕ, Τ, Φ, zo, R, vs, n, rs, f, df, ρ, σ, vs, ns, a, Τ, Ρ, Θ1, Θ2, Ρ, P, Θ1, g, gi, Γ, RieDijkl, RieUiDjkl, RicDij, RicUiDj, RicUij, WeylDijUkl, WeylDijkl, WeylUijDkl, WeylUijkl, WeylUjDikl, DualWeylUijDkl, DualWeylUjDikl, ϵ, r1, r2, w2, lDi, kDi, mDi, mbDi, lUi, kUi, mUi, mbUi, Φ00, Φ10, Φ01, Φ11, Φ21, Φ12, Φ22, Ψ0, Ψ1, Ψ2, Ψ3, Ψ4, npr1, npr2, npw2, PltPoints, size]

Defining the Natario Metric

First, we need to define an array to hold the variables, t, r, θ, and ϕ, that we will be using. This step will allow us to take the appropriate derivatives later. Please remember to reevaluate each cell in order whenever you reopen this notebook. If you do not, you will just get a ton of errors. In advance, we are going to work in natural units: G=c=1.

v = {t, r, θ, ϕ}; n = Length[v] MatrixForm[v]

Next, we need to define the parallel covariant Natario Metric, gi j , using the defined variables.

g = SparseArray{1, 1} → 1 -(Ρ[t, r, θ])2 - Θ1[t, r, θ]2, {1, 2} → Ρ[t, r, θ], {2, 1} → Ρ[t, r, θ], {2, 2} → -1, {1, 3} → r Θ1[t, r, θ], {3, 1} → r Θ1[t, r, θ], {3, 3} → -r2, {4, 4} → -r2 (Sin[θ])2; MatrixForm[ %]

i j i j Next, we need the inverse metric, g . Its definition comes from the fact that gi j g = 1.

gi = Simplify[Inverse[g]]; MatrixForm[gi]

i i i i Finally, we need the null tetrad (li, ki, mi mi) and their dual l , k , m m . I have hand derived the following tetrad for the Natario Metric at a constant velocity. 1 li = ((1 + f vs) dt - dx);W 2 1 ki = ((1 - f vs) dt + dx); 2 1 li = (dy + idz); 2 1 li = (dy - idz). 2

Printed by Wolfram Mathematica Student Edition

112 2 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb

1 + Ρ[t, r, θ] 1 - lDi = 1 ; 0 2 0 1 - Θ1[t, r, θ] 1 kDi = 1 ; 0 2 0 Θ1[t, r, θ] 1 mDi = 0 ; -r 2 I r Sin[θ] Θ1[t, r, θ] 1 mbDi = 0 ; -r 2 -I r Sin[θ]

n lUi = SimplifyTable gi[[i, j]] lDi[[j]], {i, 1, n}; j=1 n kUi = SimplifyTable gi[[i, j]] kDi[[j]], {i, 1, n}; j=1 n mUi = SimplifyTable gi[[i, j]] mDi[[j]], {i, 1, n}; j=1 n mbUi = SimplifyTable gi[[i, j]] mbDi[[j]], {i, 1, n}; j=1

Finding the Connection

We will use the Table Function to find the different elements of the each tensor starting with the connec- i tion, Γ j l. From D’Inverno, we will use the formula i 1 gi l g g g Γ j k = 2 (∂j l k + ∂k l j - ∂l j k) To get Γ type  Gamma  without the interior spaces. To read the output, it goes i → row of vectors, j → column of vectors, and k → row of individual vectors. 1 n Γ = Table  (gi[[i, l]] (∂v[[j]] g[[l, k]] + ∂v[[k]] g[[l, j]] - ∂v[[l]] g[[j, k]])), 2 l=1 {i, n}, {j, n}, {k, n}; MatrixForm[ %];

Finding the Riemann Tensor

Next we need the Riemann Tensor with mixed indices. We will use the Formula: i i i m i m i R j k l = ∂k Γ j l - ∂l Γ j k + (Γ )j l Γ m k -(Γ )j k Γ m l I have named the tables with a U indicated a raised indice and a D to indicate a lowered indice. This i convention will be held for the entirity of the Mathematica Notebook. For example, RieUiDjkl=R j k l. Printed by Wolfram Mathematica Student Edition

113 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb 3

To read the output, it goes i → row of matrices, j → column of matrices, k → row of individual matrix, l → column of individual matrix.

RieUiDjkl = 4 Table∂v[[k]] Γ[[i, j, l]] - ∂v[[l]] Γ[[i, j, k]] +  (Γ[[m, j, l]] * Γ[[i, m, k]]) - m=1 4  (Γ[[m, j, k]] * Γ[[i, m, l]]), {i, 4}, {j, 4}, {k, 4}, {l, 4}; m=1 MatrixForm[ %]; In addition, we will need the Riemann Tensor with its indices all lowered for our computation of the m Weyl Tensor. For brevity’s sake, we will compute it here as Ri j k l = gi m (R )j ,k,l.

RieDijkl = 4 Table (g[[i, m]] * RieUiDjkl[[m, j, k, l]]), {i, 4}, {j, 4}, {k, 4}, {l, 4}; m=1 MatrixForm[ %];

Finding the Ricci Tensor

Next we need the Riemann Tensor. We will use the k k m k m k Formula : Ri j = ∂k Γ i j - ∂j Γ i k + Γ i j Γ m k - Γ i k Γ m j

RicDij = 4 4 Table ∂v[[k]] Γ[[k, i, j]] - ∂v[[j]] Γ[[k, i, k]] +  (Γ[[m, i, j]] * Γ[[k, m, k]]) - k=1 m=1 4  (Γ[[m, i, k]] * Γ[[k, m, j]]) , {i, 4}, {j, 4}; m=1

MatrixForm[%]; In addition, we will need the Ricci Tensor with all of its indices raised to compute all invariants. We will i j i k j l define it as R = g g Rk l.

4 4 RicUij = Table  (gi[[i, k]] * gi[[j, l]] * RicDij[[k, l]]), {i, 4}, {j, 4}; k=1 l=1 MatrixForm[%];

i i k Finally, we will need the mixed Ricci Tensor. It is defined as R j = g Rk j

4 RicUiDj = Table (gi[[i, k]] * RicDij[[k, j]]), {i, 4}, {j, 4}; k=1 MatrixForm[%];

Printed by Wolfram Mathematica Student Edition

114 4 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb

Finding the Ricci Scalar and Trace-Free Ricci Tensors

i j Following, we need the Ricci Scalar. We will use the Formula: R = g Ri j .

4 4 R = Simplify  (gi[[i, j]] RicDij[[i, j]]) i=1 j=1 R Next , we need the lowered trace - free Ricci Tensor. We will use the Formula : Si j = Ri j - gi j . 4 R SDij = TableRicDij[[i, j]] - g[[i, j]], {i, 4}, {j, 4}; 4

MatrixForm[%];

j j k In addition, we will need the mixed trace-free Ricci Tensor. We will define it as Si = g Si k.

4 SDiUj = Table (gi[[j, k]] * SDij[[i, k]]), {i, 4}, {j, 4}; k=1 MatrixForm[%]

i j i k j Finally, we will need the raised trace-free Ricci Tensor. It is defined as S = g Sk

4 SUij = Table (gi[[i, k]] * SDiUj[[k, j]]), {i, 4}, {j, 4}; k=1 MatrixForm[%];

Finding the Weyl Tensor

Now that we have the Riemann Tensor, the Ricci Tensor, and the Ricci Scalar. We can find the Weyl Tensor. We will use the formula: C R 1 g R g R g R g R 1 g g g g R. i j k l = i j k l + 2 ( i l j k + j k i l - i k j l - j l i k) + 6 ( i k j l - i l j k) The format of the naming conventions and output follows the same pattern as that of the Riemann Tensor. Avoid using the indice “n” as we have labeled that as the length of our vector.

WeylDijkl = 1 TableRieDijkl[[i, j, k, l]] + (g[[i, l]] RicDij[[j, k]] + g[[j, k]] RicDij[[i, l]] - 2 g[[i, k]] RicDij[[j, l]] - g[[j, l]] RicDij[[i, k]]) + 1 (g[[i, k]] g[[j, l]] - g[[i, l]] g[[j, k]]) R, {i, 4}, {j, 4}, {k, 4}, {l, 4}; 6 MatrixForm[ %];

Printed by Wolfram Mathematica Student Edition

115 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb 5

WeylUjDikl = 4 Table (gi[[j, m]] * WeylDijkl[[i, m, k, l]]), {i, 4}, {j, 4}, {k, 4}, {l, 4}; m=1 MatrixForm[ %];

j j j Checking the Weyl Tensor for tracelessness i.e. that Cj i l = Ci j l = Ci l j = 0.

4 (*MatrixFormSimplifyTable∑j=1WeylUjDikl[[j,j,i,l]],{i,4},{l,4} 4 MatrixFormSimplifyTable∑j=1WeylUjDikl[[i,j,j,l]],{i,4},{l,4} 4 MatrixFormSimplifyTable∑j=1WeylUjDikl[[i,j,l,j]],{i,4},{l,4}*) i j i m j Second, C k l = g Ci k l

WeylUijDkl = 4 Table (gi[[i, m]] * WeylUjDikl[[m, j, k, l]]), {i, 4}, {j, 4}, {k, 4}, {l, 4}; m=1 MatrixForm[ %]

k l k m l o Third, Ci j = g g Ci j m o.

4 4 WeylDijUkl = Table  (gi[[k, m]] gi[[l, o]] WeylDijkl[[i, j, m, o]]), m=1 o=1 {i, 4}, {j, 4}, {k, 4}, {l, 4}; MatrixForm[ %]

i j k l i m j o k l Fourth, C = g g Cm o

4 4 WeylUijkl = Table  (gi[[i, m]] gi[[j, o]] * WeylDijUkl[[m, o, k, l]]), m=1 o=1 {i, 4}, {j, 4}, {k, 4}, {l, 4}; MatrixForm[ %]; Next we will need the duals of several of the Weyl tensors. The dual of a Weyl Tensor is defined as C * 1 C m n where is the Levi-Civita tensor Density. i j k l ≡ 2 ϵi j k l k l ϵi j k l It is hard set to its form in 3+1D. If you wish for a different mension change the 4 below.

MatrixForm[LeviCivitaTensor[4]];

Printed by Wolfram Mathematica Student Edition

116 6 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 ϵ = ; 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The first Dual we need is C * ij 1 i j m o C m o. ( )kl = 2 ϵ k l

1 4 4 DualWeylUijDkl = Table *   (ϵ[[i, j, m, o]] * WeylUijDkl[[m, o, k, l]]), 2 m=1 o=1 {i, 4}, {j, 4}, {k, 4}, {l, 4}; MatrixForm[ %];

The second dual we need is C * j 1 i j m o C j ( )i k l = 2 ϵ i m o

1 4 4 DualWeylUjDikl = Table *   (ϵ[[i, j, m, n]] * WeylUjDikl[[i, j, m, o]]), 2 m=1 o=1 {i, 4}, {j, 4}, {k, 4}, {l, 4}; MatrixForm[ %];

Finding the Invariants

Clear[ρ, σ, vs, ns, vs, t, r, θ, ϕ, Ρ, Θ1, Θ2, Ρ,P]; The second is simply the Ricci Scalar, R. I will dispaly it here for reference.

R The third one is r 1 S b S a. 1 ≡ 4 a b

1 4 4 r1 =   (SDiUj[[a, b]] SDiUj[[b, a]]) 4 a=1 b=1 The fourth one is r 1 S b S c S a. 2 ≡ - 8 a b c

Printed by Wolfram Mathematica Student Edition

117 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb 7

1 4 4 4 r2 = -    (SDiUj[[a, b]] SDiUj[[b, c]] SDiUj[[c, a]]) 8 a=1 b=1 c=1

1 c d e f a b The first one we will find is w2 ≡ - Ca b c d C e f C . Assuming 8 M = M and r = r so they are purely real.

1 4 4 4 4 4 4 w2 = -       (WeylDijkl[[a, b, c, d]] * 8 a=1 b=1 c=1 d=1 e=1 f=1 WeylUijDkl[[c, d, e, f]] WeylUijkl[[e, f, a, b]])

Finding the Invariants using NP Formalism

We now have everything we need to calculate the invariants as outlined in the paper “A new way to See inside Black Holes” by Henry, Overduin and Wilcomb using the Carminati and McLeneghan (CM) invariants. Assuming M = M and r = r. We are going to use the NP formalism and Null Tetrad components to find the CM invariants and compare with the ones derived above. We will be using equations (7.10) through(7.20) in Stephani et. al. The first one we need is 1 S ka kb . Φ00 = 2 a b = Φ00

Clear[Φ, b] 1 n n Φ00 = Simplify   SDij[[i, j]] kUi[[i]] kUi[[j]]; 2 i=1 j=1 Next, 1 S ka mb . Φ01 = 2 a b = Φ10 1 n n Φ01 = Simplify   SDij[[i, j]] kUi[[i]] mUi[[j]]; 2 i=1 j=1 Φ10 = FullSimplify[Conjugate[Φ01]];

Next, 1 S ma mb . Φ02 = 2 a b = Φ20 1 n n Φ02 = Simplify   SDij[[i, j]] mUi[[i]] mUi[[j]]; 2 i=1 j=1 Φ20 = FullSimplify[Conjugate[Φ02]];

Next, 1 S ka lb ma mb . Φ11 = 4 a b +  = Φ11 1 n n Φ11 = Simplify   SDij[[i, j]] (kUi[[i]] lUi[[j]] + mUi[[i]] mbUi[[j]]); 4 i=1 j=1 Next, 1 S la mb . Φ12 = 2 a b = Φ21 1 n n Φ12 = Simplify   SDij[[i, j]] lUi[[i]] mUi[[j]]; 2 i=1 j=1 Φ21 = FullSimplify[Conjugate[Φ12]]; Printed by Wolfram Mathematica Student Edition

118 8 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb

Next, 1 S la lb . Φ22 = 2 a b = Φ22 1 n n Φ22 = Simplify   SDij[[i, j]] lUi[[i]] lUi[[j]]; 2 i=1 j=1 Now, we need the Tetrad Components to find w2. a b c d First, Ψ0 = Ca b c d k m k m .

n n n n Ψ0 = Simplify    WeylDijkl[[i, j, k, l]] kUi[[i]] mUi[[j]] kUi[[k]] mUi[[l]]; i=1 j=1 k=1 l=1 a b c d Next, Ψ1 = Ca b c d k l k m .

n n n n Ψ1 = Simplify    WeylDijkl[[i, j, k, l]] kUi[[i]] lUi[[j]] kUi[[k]] mUi[[l]]; i=1 j=1 k=1 l=1 a b c d Next, Ψ1 = Ca b c d k l k m .

n n n n Ψ1 = Simplify    WeylDijkl[[i, j, k, l]] kUi[[i]] lUi[[j]] kUi[[k]] mUi[[l]]; i=1 j=1 k=1 l=1 a b c d Next, Ψ2 = -Ca b c d k m l m .

Ψ2 = n n n n Simplify-     WeylDijkl[[i, j, k, l]] kUi[[i]] mUi[[j]] lUi[[k]] mbUi[[l]]; i=1 j=1 k=1 l=1 a b c d Next, Ψ3 = Ca b c d l k l m .

n n n n Ψ3 = Simplify    WeylDijkl[[i, j, k, l]] lUi[[i]] kUi[[j]] lUi[[k]] mbUi[[l]]; i=1 j=1 k=1 l=1 a b c d Finally, Ψ4 = Ca b c d l m l m .

Ψ4 = n n n n Simplify    WeylDijkl[[i, j, k, l]] lUi[[i]] mbUi[[j]] lUi[[k]] mbUi[[l]]; i=1 j=1 k=1 l=1 The first one is simply the Ricci Scalar, R. I will display it here for reference.

(*R*) 2 The second one is r1 ≡ 2 Φ20 Φ02 + 2 Φ22 Φ00 - 4 Φ12 Φ10 - 4 Φ21 Φ01 + 4 Φ11 .

npr1 = Simplify2 Φ20 Φ02 + 2 Φ22 Φ00 - 4 Φ12 Φ10 - 4 Φ21 Φ01 + 4 Φ112 The third one is

r2 ≡ 6 Φ02 Φ21 Φ10 - 6 Φ11 Φ02 Φ20 + 6 Φ01 Φ12 Φ20 - 6 Φ12 Φ00 Φ21 - 6 Φ22 Φ01 Φ10 + 6 Φ22 Φ11 Φ00.

npr2 = Simplify[ 6 Φ02 Φ21 Φ10 - 6 Φ11 Φ02 Φ20 + 6 Φ01 Φ12 Φ20 - 6 Φ12 Φ00 Φ21 - 6 Φ22 Φ01 Φ10 + 6 Φ22 Φ11 Φ00] 3 2 2 2 The fourth one one is w2 ≡ 6 Ψ4 Ψ0 Ψ2 - 6 Ψ2 - 6 Ψ1 Ψ4 - 6 Ψ3 Ψ4 - 6 Ψ3 Ψ0 + 12 Ψ2 Ψ1 Ψ3.

npw2 = 6 Ψ4 Ψ0 Ψ2 - 6 Ψ23 - 6 Ψ12 Ψ4 - 6 Ψ32 Ψ0 + 12 Ψ2 Ψ1 Ψ3

Printed by Wolfram Mathematica Student Edition

119 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb 9

Plotting the Ricci Scalar

Now that the program has calculated each one of the identified Invariants, we need to plot each one. Thanks to Cooper Watson’s program “Kerr_newman_skeleton_file.nb” for the Mathematica code lines to accomplish this. First, we need to make some choice for what n[ρ[t]] and rs[t,x,y,z] are. I will make the standard choice for the Natario Drive, r[t_, x_, y_, z_] := (x - xs[t])2 + y2 + z2 ;. n : 1 1 1 1 Tanh r [ρ] = 2  -  2 ( - [σ ( - ρ)]) So ρ is the radius of the warp bubble and σ is the skin depth of it. Now, these plots will contain a large amount of information due to changes in a and time. So, I will break the invariant plots up into different notebooks.

Clear[ρ, σ, vs, ns, vs, t, r, θ, ϕ, Ρ, Θ1, Θ2, Ρ, P, m]; Ρ[t_, r_, θ_] = 2 vs ns[r] Cos[θ];

Θ1[t_, r_, θ_] = -vs 2 ns[r] + r ∂r ns[r] Sin[θ]; 1 1 ns[r_] = 1 - 1 - Tanh[σ (r - ρ)] ; 2 2 P = Simplify[R] PltPoints = 250; m = 2;

Plotting σ=50000 and ρ=100 as in Loup

σ = 50 000; ρ = 100;

Velocity=0 Velocity=0.01 Velocity=0.1 Velocity=1

Clear[a, t, vs, r, θ] vs = 1 * ns[r];

t = 1; RevolutionPlot3D[{P}, {r, 0, m ρ}, {θ, 0, 2 π}, PlotStyle → Directive[Orange, Opacity[0.7], Specularity[White, 10]], PlotPoints → PltPoints, AxesLabel → {"r: radius", "r:radius", "Invariant"}, AxesStyle → Bold]

Printed by Wolfram Mathematica Student Edition

120 10 Displaying the code for the Santosuosso Invariants for the non-accelerating Natario Metric.nb

Velocity=10 Velocity=100

Plotting σ=75000 and ρ=100 as in Loup Plotting σ=100000 and ρ=100 as in Loup Plotting σ=50000 and ρ=50 as in Loup Plotting σ=50000 and ρ=150 as in Loup Plotting σ=50000 and ρ=200 as in Loup

Plotting the Invariant r1 Plotting the Invariant r2 Plotting the Invariant w2

Printed by Wolfram Mathematica Student Edition

121 APPENDIX B

Invariants for the Natário Metric at Constant Velocity

1 θ θ r = v 2σ2 cos8 ( )sech4((r − ρ)σ)((32r4v 2σ4(sin( ) 1 1024r2 s 2 s 2 3θ θ − sin( ))2 tanh6 ((r − ρ)σ) + 64r3v 2σ3(rσ − 1)(sin( ) 2 s 2 3θ − sin( ))2 tanh5 ((r − ρ)σ) 2 2 2 2 2 2 2 2 2 2 2 2 2 − 4r σ (−r σ vs + 4rσvs + (r σ − 4rσ − 4) cos(4θ)vs − 12vs + 16r σ θ − 8(v 2 + 2r2σ2) cos(2θ)) sec2 ( ) tanh4 ((r − ρ)σ) + 2rσ(−4r2σ2v 2 + 48rσv 2 s 2 s s 2 2 2 2 2 2 + (4r σ + 16rσ + 3) cos(4θ)vs − 7vs + 192r σ θ + 4(v 2(8rσ + 1) − 48r2σ2) cos(2θ)) sec2 ( ) tanh3 ((r − ρ)σ) s 2 2 2 2 2 2 2 2 2 2 2 + (48r σ vs − 28rσvs + (16r σ + 12rσ − 3) cos(4θ)vs + 7vs − 768r σ θ + 4((8r2σ2 + 4rσ − 1)v 2 + 160r2σ2) cos(2θ)) sec2 ( ) tanh2 ((r − ρ)σ) s 2 2 2 2 + 2(−7rσvs + 3(rσ − 1) cos(4θ)vs + 7vs + 320rσ θ + 4(v 2(rσ − 1) − 16rσ) cos(2θ)) sec2 ( ) tanh((r − ρ)σ) s 2 θ θ − 2(3 cos(2θ) + 5)(cos(2θ)v 2 − v 2 + 32) sec2 ( )) sec6 ( ) s s 2 2 θ − 16rσsech2((r − ρ)σ) tan2 ( )(2r3v 2σ3(3 cos(2θ) − 1) tanh4 ((r − ρ)σ) 2 s 2 2 2 3 + r vs σ (2rσ + (10rσ − 9) cos(2θ) + 9) tanh ((r − ρ)σ)

2 2 2 + rσ((4r σ + 5rσ − 9)vs

2 2 2 2 2 2 + (4r σ − 13rσ + 9) cos(2θ)vs − 32r σ ) tanh ((r − ρ)σ)

2 2 2 2 2 2 2 2 + ((−4r σ − 5rσ + 9)vs − (4r σ − 13rσ − 7) cos(2θ)vs + 96r σ ) tanh((r − ρ)σ)

2 2 2 2 + 9vs + 4rvs σ − 32rσ + 7vs cos(2θ) + 4rvs σ cos(2θ))

122 θ 1 sec4 ( ) + r2σ2sech4((r − ρ)σ)((4r2σ2v 2 + 16rσv 2 − (4r2σ2 + 16rσ − 3) cos(4θ)v 2 2 4 s s s θ + 109v 2 − 64r2σ2 − 4(v 2 − 16r2σ2) cos(2θ)) sec8 ( ) s s 2 θ θ − 416r2v 2σ2(cos(2θ) − 2) tan2 ( ) tanh2 ((r − ρ)σ) sec4 ( ) s 2 2 θ θ − 64rv 2σ(2rσ + 2r cos(2θ)σ + 17) tan2 ( ) tanh((r − ρ)σ) sec4 ( ) s 2 2 θ θ + 704r4v 2σ4 tan4 ( ) tanh4 ((r − ρ)σ) − 2816r3v 2σ3 tan4 ( ) tanh3 ((r − ρ)σ))), s 2 s 2 (B.1)

123 1 θ 1 r = − 3v 4σ3 cos12 ( )sech6((r − ρ)σ)( (tanh((r − ρ)σ) + 1) 2 32768r3 s 2 2 θ × (512r5v 2σ5 cos4 (θ) tan2 ( ) tanh7 ((r − ρ)σ) + 32r4v 2σ4(4rσ+ s 2 s θ θ 3θ × (4rσ − 3) cos(2θ) − 5) sec2 ( )(sin( ) − sin( ))2 tanh6 ((r − ρ)σ) 2 2 2 3 3 2 2 2 2 2 2 2 2 2 2 + 4r σ (2r σ vs − 10rσvs − 2(r σ − 5rσ − 4) cos(4θ)vs − r σ cos(6θ)vs

2 2 2 2 2 + 3rσ cos(6θ)vs + 4 cos(6θ)vs + 24vs − 64r σ θ + ((r2σ2 − 3rσ + 28)v 2 + 64r2σ2) cos(2θ)) sec4 ( ) tanh5 ((r − ρ)σ) s 2 2 2 2 2 2 2 2 2 2 2 + 2r σ (−10r σ vs + 96rσvs + 3r σ cos(6θ)vs + 16rσ cos(6θ)vs

2 2 2 2 2 2 2 2 2 + 4 cos(6θ)vs − 20vs + 816r σ + (vs (−3r σ + 112rσ + 12) − 832r σ ) cos(2θ) θ + 2((5r2σ2 + 16rσ + 2)v 2 + 8r2σ2) cos(4θ)) sec4 ( ) tanh4 ((r − ρ)σ) s 2 2 2 2 2 2 2 2 2 2 + rσ(96r σ vs − 80rσvs + 16r σ cos(6θ)vs + 16rσ cos(6θ)vs − 3 cos(6θ)vs

2 2 2 2 2 2 2 2 + 22vs − 3968r σ + ((112r σ + 48rσ − 13)vs + 3072r σ ) cos(2θ) θ + 2(v 2(16r2σ2 + 8rσ − 3) − 64r2σ2) cos(4θ)) sec4 ( ) tanh3 ((r − ρ)σ) s 2 2 2 2 2 2 2 2 2 2 + 2(−20r σ vs + 22rσvs + 4r σ cos(6θ)vs − 3rσ cos(6θ)vs − cos(6θ)vs

2 2 2 2 2 2 2 2 − 2vs + 2448r σ + (vs (12r σ − 13rσ + 1) − 448r σ ) cos(2θ) θ + 2((2r2σ2 − 3rσ + 1)v 2 + 24r2σ2) cos(4θ)) sec4 ( ) tanh2 ((r − ρ)σ) s 2 2 2 2 2 2 − (−22rσvs + 3rσ cos(6θ)vs + 4 cos(6θ)vs + 8vs + 3136rσ + ((13rσ − 4)vs θ + 1024rσ) cos(2θ) + (v 2(6rσ − 8) − 64rσ) cos(4θ)) sec4 ( ) tanh((r − ρ)σ) s 2 θ θ + 32(cos(4θ)v 2 − v 2 + 112 cos(2θ) + 144) tan2 ( )) sec8 ( ) s s 2 2 θ θ + 2r2σ2sech4((r − ρ)σ) tan2 ( )(−32r5v 2σ5(cos(2θ) − 3) tan2 ( ) tanh6 ((r − ρ)σ) 2 s 2 θ − 16r4v 2σ4(−2rσ + (6rσ − 7) cos(2θ) + 31) tan2 ( ) tanh5 ((r − ρ)σ) s 2 3 3 2 2 2 2 2 2 2 2 2 2 + r σ (−4r σ vs − 57rσvs + (4r σ − 19rσ − 9) cos(4θ)vs + 141vs + 64r σ θ + 4(v 2(19rσ − 37) − 16r2σ2) cos(2θ)) sec4 ( ) tanh4 ((r − ρ)σ) s 2

124 2 2 2 2 2 2 2 2 2 2 2 2 − r σ (−12r σ vs − 69rσvs + (12r σ + 17rσ − 15) cos(4θ)vs + 215vs + 320r σ θ + 4(3v 2(19rσ − 6) − 80r2σ2) cos(2θ)) sec4 ( ) tanh3 ((r − ρ)σ) s 2 2 2 2 2 2 2 2 2 2 2 − rσ(60r σ vs + 119rσvs + (4r σ − 31rσ + 29) cos(4θ)vs − 157vs − 512r σ θ + 8rσ((8rσ − 23)v 2 + 64rσ) cos(2θ)) sec4 ( ) tanh2 ((r − ρ)σ) s 2 3 3 3 2 3 2 2 2 2 2 2 2 + (−128r σ + 12r vs σ − 416r σ + 94r vs σ + 113rvs σ − 98vs

3 3 2 2 2 2 2 + 4((4r σ + 28r σ − 12rσ − 19)vs + 136r σ ) cos(2θ) θ + v 2(4r3σ3 + 18r2σ2 − 33rσ − 18) cos(4θ)) sec4 ( ) tanh((r − ρ)σ) s 2 2 2 2 2 2 2 2 2 2 2 + 2(−r σ vs − 22rσvs + (r σ − 2rσ − 9) cos(4θ)vs − 49vs + 144r σ + 224rσ θ θ + (16rσ(7rσ + 6) − 2v 2(12rσ + 19)) cos(2θ)) sec4 ( )) sec4 ( ) s 2 2 1 + rσsech2((r − ρ)σ)(r2σ2(72r2σ2v 2 − 664rσv 2 + 44r2σ2 cos(6θ)v 2 8 s s s 2 2 2 2 2 + 44rσ cos(6θ)vs + 69 cos(6θ)vs + 750vs − 3904r σ

2 2 2 2 2 + ((−108r σ + 532rσ + 139)vs + 5376r σ ) cos(2θ) θ − 2((4r2σ2 − 44rσ − 97)v 2 + 736r2σ2) cos(4θ)) tanh4 ((r − ρ)σ) sec8 ( ) s 2 3 3 3 2 3 3 2 3 2 2 2 2 2 + 2rσ(512r σ − 16r vs σ + 8r vs cos(6θ)σ + 2240r σ − 130r vs σ

2 2 2 2 2 2 + 5r vs cos(6θ)σ + 726rvs σ + 73rvs cos(6θ)σ − 186vs

2 3 3 2 2 2 2 + (vs (−8r σ + 139r σ + 135rσ + 137) − 256r σ (2rσ + 13)) cos(2θ)

3 3 2 2 2 2 2 + 2((8r σ − 7r σ + 109rσ + 13)vs + 544r σ ) cos(4θ) θ + 23v 2 cos(6θ)) tanh3 ((r − ρ)σ) sec8 ( ) s 2 3 3 3 2 3 3 2 3 2 2 2 2 2 + (−3968r σ + 48r vs σ − 8r vs cos(6θ)σ − 3328r σ + 654r vs σ

2 2 2 2 2 2 + 85r vs cos(6θ)σ − 728rvs σ + 100rvs cos(6θ)σ + 112vs

3 3 2 2 2 2 2 + ((8r σ + 123r σ + 540rσ − 82)vs + 3584r σ (rσ + 2)) cos(2θ)

3 3 2 2 2 2 2 + ((−48r σ + 290r σ + 88rσ − 32)vs + 384r σ (rσ − 6)) cos(4θ)

2 2 8 θ 2 2 + 2vs cos(6θ)) tanh ((r − ρ)σ) sec ( ) + 2(8rσvs + 4rσ cos(6θ)vs 2

125 2 2 2 + cos(6θ)vs + 56vs − 704rσ + (256(rσ + 7) − vs (4rσ + 41)) cos(2θ) θ − 8(v 2(rσ + 2) − 4(14rσ + 3)) cos(4θ) + 1184) sec8 ( ) s 2 2 2 2 2 2 2 2 + 2(−24r σ vs − 170rσvs + 4r σ cos(6θ)vs

2 2 2 2 2 + 31rσ cos(6θ)vs + 2 cos(6θ)vs + 112vs + 2112r σ − 192rσ

2 2 2 2 2 2 − ((4r σ − 129rσ + 82)vs + 256rσ(5rσ + 12)) cos(2θ) + 2((12r σ + 5rσ − 16)vs θ + 32rσ(3 − 13rσ)) cos(4θ)) tanh((r − ρ)σ) sec8 ( ) 2 4 4 2 2 2 2 2 2 2 2 2 2 − 64r σ (−r σ vs + 8rσvs + (r σ − 8rσ + 3) cos(4θ)vs − 23vs + 16r σ θ θ − 4(3v 2 + 4r2σ2) cos(2θ)) tan2 ( ) tanh6 ((r − ρ)σ) sec4 ( ) s 2 2 3 3 2 2 2 2 2 2 2 2 2 2 + 32r σ (−8r σ vs + 68rσvs + (8r σ − 20rσ − 13) cos(4θ)vs − 119vs + 256r σ θ θ + 4(v 2(4rσ − 15) − 64r2σ2) cos(2θ)) tan2 ( ) tanh5 ((r − ρ)σ) sec4 ( ) s 2 2 θ + 2048r6v 2σ6 cos2 (θ) tan4 ( ) tanh8 ((r − ρ)σ) s 2 θ θ + 4096r5v 2σ5(rσ − 2) cos2 (θ) tan4 ( ) tanh7 ((r − ρ)σ)) sec4 ( ) s 2 2 1 + r3σ3sech6((r − ρ)σ)(− (−(3(2r2σ2 + 8rσ − 69)v 2 + 256r2σ2) cos(2θ) 16 s 2 2 2 2 2 2 2 2 + 4(3(r σ + 4rσ + 11)vs + 40r σ ) cos(4θ) + 3(vs (2r σ + 8rσ + 3) cos(6θ) θ 1 − 4(v 2(r2σ2 + 4rσ − 7) − 8r2σ2))) sec12 ( ) − r2σ2(−4r2σ2v 2 − 32rσv 2 s 2 2 s s 2 2 2 2 2 2 + (4r σ + 32rσ + 45) cos(4θ)vs − 201vs + 64r σ θ θ + 4(69v 2 − 16r2σ2) cos(2θ)) tan2 ( ) tanh2 ((r − ρ)σ) sec8 ( ) s 2 2 2 2 2 2 2 2 2 2 2 2 + rσ(−4r σ vs − 52rσvs + (4r σ + 4rσ + 33) cos(4θ)vs − 61vs + 64r σ θ θ + (v 2(52 − 48rσ) − 64r2σ2) cos(2θ)) tan2 ( ) tanh((r − ρ)σ) sec8 ( ) s 2 2 θ θ − 16r4v 2σ4(20 cos(2θ) − 27) tan4 ( ) tanh4 ((r − ρ)σ) sec4 ( ) s 2 2 θ θ − 64r3v 2σ3(rσ + (rσ − 4) cos(2θ) + 13) tan4 ( ) tanh3 ((r − ρ)σ) sec4 ( ) s 2 2 θ θ + 192r6v 2σ6 tan6 ( ) tanh6 ((r − ρ)σ) − 1152r5v 2σ5 tan6 ( ) tanh5 ((r − ρ)σ))), s 2 s 2 (B.2)

126 1 w = v 4σ3sech6((r − ρ)σ)(−32v 2σ3(rv σ sin(θ)sech2((r − ρ)σ) 2 2415919104 s s s 3 + 2(vs cos(θ) + vs sin(θ) + vs(cos(θ) + sin(θ)) tanh((r − ρ)σ) + 4))

× (sech2((r − ρ)σ)(6 cos2 (θ) + 7 sin2 (θ) + 4r2σ2 sin2 (θ) tanh2 ((r − ρ)σ)

− 8rσ sin2 (θ) tanh((r − ρ)σ))

− 3(3 cos(2θ) + 1) tanh((r − ρ)σ)(tanh((r − ρ)σ) + 1))3 27i + sech8((r − ρ)σ) sin4 (θ)(cosh(2(r − ρ)σ)(4 + 4i) + (4 + 4i) + (1 + i)v cos(θ) r3 s

+ ivs cos(θ + 2i(r − ρ)σ) + vs cos(θ − 2i(r − ρ)σ) + (1 + i)vs sin(θ)

+ (1 + i)rvsσ sin(θ) + vs sin(θ + 2i(r − ρ)σ)

2 2 2 2 2 2 2 2 + ivs sin(θ − 2i(r − ρ)σ)) (−r σ vs − 2rσvs + r σ cos(2θ)vs

2 2 2 + 2rσ cos(2θ)vs − 2i cos(2θ + 2i(r − ρ)σ)vs + (1 − i)rσ cos(2θ + 2i(r − ρ)σ)vs

2 2 − i cos(2θ + 4i(r − ρ)σ)vs + 2i cos(2θ − 2i(r − ρ)σ)vs

2 2 2 + (1 + i)rσ cos(2θ − 2i(r − ρ)σ)vs + i cos(2θ − 4i(r − ρ)σ)vs − 2rσ sin(2θ)vs

2 2 2 − 2 sin(2θ)vs − (1 + i)rσ sin(2θ + 2i(r − ρ)σ)vs − 2 sin(2θ + 2i(r − ρ)σ)vs

2 2 − sin(2θ + 4i(r − ρ)σ)vs − (1 − i)rσ sin(2θ − 2i(r − ρ)σ)vs

2 2 2 − 2 sin(2θ − 2i(r − ρ)σ)vs − sin(2θ − 4i(r − ρ)σ)vs − 2rσ sinh(2(r − ρ)σ)vs

2 2 2 − 4 sinh(2(r − ρ)σ)vs − 2 sinh(4(r − ρ)σ)vs − 2vs − 12 cos(θ)vs

− (8 + 4i) cos(θ + 2i(r − ρ)σ)vs − (2 + 2i) cos(θ + 4i(r − ρ)σ)vs

− (8 − 4i) cos(θ − 2i(r − ρ)σ)vs − (2 − 2i) cos(θ − 4i(r − ρ)σ)vs − 8rσ sin(θ)vs

− 12 sin(θ)vs − (8 − 4i) sin(θ + 2i(r − ρ)σ)vs − 4rσ sin(θ + 2i(r − ρ)σ)vs

− (2 − 2i) sin(θ + 4i(r − ρ)σ)vs − (8 + 4i) sin(θ − 2i(r − ρ)σ)vs

− 4rσ sin(θ − 2i(r − ρ)σ)vs − (2 + 2i) sin(θ − 4i(r − ρ)σ)vs

2 − 2((rσ + 2)vs + 16) cosh(2(r − ρ)σ)

2 − 2(vs + 4) cosh(4(r − ρ)σ) − 24)(tanh((r − ρ)σ) + 1)

127 2 2 3 × (rσ tanh((r − ρ)σ) − 1)((4rσ − vs(r σ − rσ + 1) cos(θ)) tanh ((r − ρ)σ)

2 2 2 2 2 − 3(2r σ + vs(r σ − rσ + 1) cos(θ) − 2) tanh ((r − ρ)σ)

2 2 2 2 2 2 − 3(vs(r σ − rσ + 1) cos(θ) − 4rσ) tanh((r − ρ)σ) − 2r σ − r vsσ cos(θ)

2 2 2 2 2 − vs cos(θ) + rvsσ cos(θ) + sech ((r − ρ)σ)(6r σ + vs(3r σ + rσ − 3) cos(θ)

2 2 2 + (4rσ + vs(3r σ + rσ − 1) cos(θ)) tanh((r − ρ)σ) + 6) + 2) 2304 + v 2(σ sin4 (θ)(tanh((r − ρ)σ) + 1)2(rσ tanh((r − ρ)σ) − 1)2 r2 s 2 × (rvsσ sin(θ)sech ((r − ρ)σ)

+ 2(vs cos(θ) + vs sin(θ) + vs(cos(θ) + sin(θ)) tanh((r − ρ)σ) + 4))

× (sech2((r − ρ)σ)(6 cos2 (θ) + 7 sin2 (θ) + 4r2σ2 sin2 (θ) tanh2 ((r − ρ)σ)

− 8rσ sin2 (θ) tanh((r − ρ)σ))

− 3(3 cos(2θ) + 1) tanh((r − ρ)σ)(tanh((r − ρ)σ) + 1))

2 2 2 2 4 × (r vs σ sin (θ)sech ((r − ρ)σ)

2 + 2rvsσ(2 sin(θ)(vs cos(θ) + vs sin(θ) + 2) + vs(2 sin (θ) + sin(2θ))

× tanh((r − ρ)σ))sech2((r − ρ)σ)

2 2 2 2 2 2 + 2(−sech ((r − ρ)σ)(cos(θ) + sin(θ)) vs + (cos(θ) + sin(θ)) tanh ((r − ρ)σ)vs

2 2 + 6 cos(θ) sin(θ)vs + 3vs + 8 cos(θ)vs + 8 sin(θ)vs

+ 4(cos(θ) + sin(θ))(vs cos(θ) + vs sin(θ) + 2) tanh((r − ρ)σ)vs + 16))) 144σ sin2 (θ) − (rv σ sin(θ)sech2((r − ρ)σ) + 2(v cos(θ) + v sin(θ) r2 s s s 3 2 2 2 + vs(cos(θ) + sin(θ)) tanh((r − ρ)σ) + 4)) (sech ((r − ρ)σ)(6 cos (θ) + 7 sin (θ)

+ 4r2σ2 sin2 (θ) tanh2 ((r − ρ)σ) − 8rσ sin2 (θ) tanh((r − ρ)σ))

− 3(3 cos(2θ) + 1) tanh((r − ρ)σ)(tanh((r − ρ)σ) + 1))

2 × (rσ(2rσ + vs(rσ + 1) cos(θ))sech ((r − ρ)σ)

2 2 3 2 − 2(r vsσ cos(θ) tanh ((r − ρ)σ) + rσ(2rσ + vs(rσ − 1) cos(θ)) tanh ((r − ρ)σ)

128 + ((vs − rvsσ) cos(θ) − 4rσ) tanh((r − ρ)σ) + vs cos(θ) − 2)) 1 × ( r2v 2σ2(rσ + 1) sin(2θ)sech4((r − ρ)σ) + rσ(−r2v 2σ2 sin(2θ) tanh3 ((r − ρ)σ) 2 s s 2 2 2 − rvs σ(rσ − 1) sin(2θ) tanh ((r − ρ)σ) + 2vs cos(θ)(rσ cos(θ) + cos(θ)

2 2 + 2rσ sin(θ)) tanh((r − ρ)σ) + 2vs (rσ + 1) cos (θ) + 4vs(rσ + 1) cos(θ)

2 2 + rσ(vs sin(2θ) − 8))sech ((r − ρ)σ)

2 2 2 4 − 2(r vs σ (cos(2θ) + sin(2θ) + 1) tanh ((r − ρ)σ)

3 + 2rvsσ cos(θ)(2rσ + vs(2rσ − 1) cos(θ) + vs(2rσ − 1) sin(θ)) tanh ((r − ρ)σ)

2 2 2 2 2 + (−8r σ + 4rvs(rσ − 1) cos(θ)σ + 2vs (rσ − 1) cos (θ)

2 2 2 2 2 + vs (rσ − 1) sin(2θ)) tanh ((r − ρ)σ) + (−2(rσ − 2) cos (θ)vs

2 2 2 − (rσ − 2) sin(2θ)vs − 4(rσ − 1) cos(θ)vs + 16rσ) tanh((r − ρ)σ) + 2vs cos (θ)

2 + 4vs cos(θ) + vs sin(2θ) + 8)) 864 + sin4 (θ)(tanh((r − ρ)σ) + 1)(rσ tanh((r − ρ)σ) − 1)(rv σ sin(θ)sech2((r − ρ)σ) r3 s 2 + 2(vs cos(θ) + vs sin(θ) + vs(cos(θ) + sin(θ)) tanh((r − ρ)σ) + 4)) 1 × ( r2v 2σ2(rσ + 1) sin(2θ)sech4((r − ρ)σ) + rσ(−r2v 2σ2 sin(2θ) tanh3 ((r − ρ)σ) 2 s s 2 2 2 − rvs σ(rσ − 1) sin(2θ) tanh ((r − ρ)σ) + 2vs cos(θ)(rσ cos(θ) + cos(θ) + 2rσ sin(θ))

2 2 × tanh((r − ρ)σ) + 2vs (rσ + 1) cos (θ) + 4vs(rσ + 1) cos(θ)

2 2 + rσ(vs sin(2θ) − 8))sech ((r − ρ)σ)

2 2 2 4 − 2(r vs σ (cos(2θ) + sin(2θ) + 1) tanh ((r − ρ)σ)

3 + 2rvsσ cos(θ)(2rσ + vs(2rσ − 1) cos(θ) + vs(2rσ − 1) sin(θ)) tanh ((r − ρ)σ)

2 2 2 2 2 + (−8r σ + 4rvs(rσ − 1) cos(θ)σ + 2vs (rσ − 1) cos (θ)

2 2 2 + vs (rσ − 1) sin(2θ)) tanh ((r − ρ)σ)

2 2 2 + (−2(rσ − 2) cos (θ)vs − (rσ − 2) sin(2θ)vs − 4(rσ − 1) cos(θ)vs

2 2 2 2 + 16rσ) tanh((r − ρ)σ) + 2vs cos (θ) + 4vs cos(θ) + vs sin(2θ) + 8)) ). (B.3)

129 APPENDIX C

Invariants for the Natário Metric at Constant Acceleration

a R = − 32r2(ar cos(θ) + ar tanh((r − ρ)σ) cos(θ) − 2)3 × (ar8t2σ4 sin2 (θ)(ar cos(θ) + ar tanh((r − ρ)σ) cos(θ) − 2)sech8((r − ρ)σ)

− 4ar3t2σ3(aσ cos(θ) sin2 (θ) tanh3 ((r − ρ)σ)r6

+ (a(2rσ − 3) cos(θ) − 2σ) sin2 (θ) tanh2 ((r − ρ)σ)r5

− a(3r2 + 2) cos(θ) sin2 (θ)r3 + (a(σr3 − 6r2 − 2) cos(θ)

− 2r(rσ − 3)) sin2 (θ) tanh((r − ρ)σ)r3 + 2(3r2 + 2) sin2 (θ)r2

+ 16a(2rσ + 3) cos3 (θ)r − 16(r2 + 4σr − 2) cos2 (θ))sech6((r − ρ)σ)

+ 4ar2t2σ2(aσ2 cos(θ) sin2 (θ) tanh5 ((r − ρ)σ)r7

+ σ(3a(rσ − 2) cos(θ) − 2σ) sin2 (θ) tanh4 ((r − ρ)σ)r6 + 3a cos3 (θ)r5

+ 5a sin(θ) sin(2θ)r5 − 6 cos2 (θ)r4 − 16 sin2 (θ)r4 − 64a cos3 (θ)r3

+ 13a cos(θ) sin2 (θ)r3 − 16aσ cos3 (θ)r2

+ 184 cos2 (θ)r2 − 58 sin2 (θ)r2 + 104a cos3 (θ)r + (−4σ(rσ − 3) sin2 (θ)r4

+ a(3σ2r4 − 18σr3 + 10r2 − 4σr + 1) cos(θ) sin2 (θ)r2

+ a(3r4 + 32σr3 + 8(16σ2 − 1)r2 − 96σr + 16) cos3 (θ)) tanh3 ((r − ρ)σ)r

+ 32σ cos2 (θ)r + 4a cos(θ) sin2 (θ)r − 432 cos2 (θ) − 8 sin2 (θ)

+ (a(σ2r4 − 18σr3 + 30r2 − 8σr + 15) cos(θ) sin2 (θ)r3

− 2(σ2r4 − 12σr3 + 8r2 − 12σr + 5) sin2 (θ)r2

+ a(9r4 + 64σr3 + 16(8σ2 − 5)r2 − 432σr + 88) cos3 (θ)r

130 − 2(3r4 + 32σr3 + 4(32σ2 − 3)r2 − 112σr + 24) cos2 (θ)) tanh2 ((r − ρ)σ)

+ (ar(9r4 + 32σr3 − 136r2 − 352σr + 176) cos3 (θ)

− 4(3r4 + 16σr3 − 52r2 − 192σr + 72) cos2 (θ)

− ar(6σr5 − 30r4 + 4σr3 − 27r2 − 4) sin2 (θ) cos(θ)

+ 4r2(3σr3 − 8r2 + 6σr − 17) sin2 (θ)) tanh((r − ρ)σ))sech4((r − ρ)σ)

− 8rσ(−6a2t2 cos3 (θ)r5 − 3a2t2 cos(θ) sin2 (θ)r5 + 12at2 cos2 (θ)r4 + 8a cos2 (θ)r4

− 2at2 sin2 (θ)r4 + 32a2 cos3 (θ)r3 + 52a2t2 cos3 (θ)r3 − 5a2t2 cos(θ) sin2 (θ)r3 1 − a2t2 tanh5 ((r − ρ)σ)r2σ cos(θ) 2 × (−r4 + 3r2 + 16σr + (r4 + 5r2 + 16σr − 8) cos(2θ) − 8)

− 16 cos(θ)r3 − 132at2 cos2 (θ)r2 − 192a cos2 (θ)r2 + 46at2 sin2 (θ)r2 1 − at2(4aσ cos(3θ)r5 + 4σr4 + 3a cos(3θ)r4 + 36aσ cos(3θ)r3 − 12σr2 4 + 48aσ2 cos(3θ)r2 − 23a cos(3θ)r2 − 64σ2r − 72aσ cos(3θ)r + 32σ

− a(4σr5 − 21r4 − 92σr3 + (57 − 144σ2)r2 + 216σr − 24) cos(θ)

− 4σ(r4 + 5r2 + 16σr − 8) cos(2θ) + 8a cos(3θ)) tanh4 ((r − ρ)σ)r

− 12a2t2 cos3 (θ)r − 2a2t2 cos(θ) sin2 (θ)r + 256 cos(θ)r

− 2a(2ar(4σr3 + t2(6r4 + 18σr3 + 4(3σ2 − 7)r2 − 42σr + 9)) cos3 (θ)

− 2t2(3r4 + 14σr3 + (16σ2 − 13)r2 − 36σr + 8) cos2 (θ)

− ar(r2 + 1)t2(3σr3 − 6r2 − 1) sin2 (θ) cos(θ)

+ r2t2(3σr3 + r2 + 11σr − 9) sin2 (θ)) tanh3 ((r − ρ)σ) + 80at2 cos2 (θ) + 4at2 sin2 (θ)

+ 2a(−2ar(9t2r4 + 8(2σt2 + σ)r3 + (t2(4σ2 − 54) − 8)r2 − 38t2σr + 15t2) cos3 (θ)

+ 2((9t2 + 2)r4 + 2(11t2 + 12)σr3 + (t2(8σ2 − 59) − 4)r2 − 60t2σr + 36t2) cos2 (θ)

+ ar(r2 + 1)t2(2σr3 − 9r2 − 3) sin2 (θ) cos(θ)

+ t2(−3σr5 − 3r4 − 19σr3 + 41r2 + 2) sin2 (θ)) tanh2 ((r − ρ)σ)

+ (−4a2r(6t2r4 + (5t2 + 4)σr3 − 4(11t2 + 4)r2 − 12t2σr + 11t2) cos3 (θ)

131 + 4a((9t2 + 4)r4 + 2(5t2 + 12)σr3 − (79t2 + 52)r2 − 28t2σr + 48t2) cos2 (θ)

+ r(a2(r2 + 1)t2(σr3 − 12r2 − 6) sin2 (θ) − 16(r2 + 8σr − 2)) cos(θ)

− 2at2(σr5 + 3r4 + 9σr3 − 55r2 − 4) sin2 (θ)) tanh((r − ρ)σ))sech2((r − ρ)σ)

+ (tanh((r − ρ)σ) + 1)2(2a2rt2 cos(θ)(13r4 − 22r2

+ (11r4 − 26r2 − 5) cos(2θ) − 3) tanh3 ((r − ρ)σ)

+ at2(33a cos(3θ)r5 − 20r4 − 78a cos(3θ)r3 + 88r2 + 3a(37r4 − 70r2 − 11) cos(θ)r

− 15a cos(3θ)r − 4(19r4 − 42r2 + 3) cos(2θ) − 20) tanh2 ((r − ρ)σ)

+ a(33at2 cos(3θ)r5 − 40t2r4 − 64r4 − 78at2 cos(3θ)r3 + 176t2r2 + 64r2

+ 3a(37r4 − 70r2 − 11)t2 cos(θ)r − 15at2 cos(3θ)r − 40t2 − 8((19t2 + 8)r4

− 2(21t2 + 4)r2 + 3t2) cos(2θ)) tanh((r − ρ)σ) + 4(4a2r(3r4 − 6r2 − 1)t2 cos3 (θ)

− 8a((3t2 + 4)r4 − 4(2t2 + 1)r2 + t2) cos2 (θ) + r(a2(r2 + 1)2t2 sin2 (θ)

+ 64(r2 − 1)) cos(θ) + 2a(7r4 − 10r2 − 1)t2 sin2 (θ)))). (C.1)

The remaining invariants may be found at

(1) r1 — https://drive.google.com/file/d/1Lkww1oZizm-C7lifv-TfWIdJqQEHEn-e/view?

usp=sharing

(2) r2 — https://drive.google.com/file/d/1Rd7sXTKFE6F2v6p-uLUx-V3VFJ_A9r9o/view?

usp=sharing

(3) w2 — https://drive.google.com/file/d/17WnjSULu6PxcnHtk_FBlO1ZkFIkVcdMr/view?

usp=sharing

132 REFERENCES

[1] F. S. N. Lobo, (2017), “Wormhole Basics,” in Wormholes, Warp Drives and Energy Conditions, Fund. Theor. Phys. 189, ed. F. S. N. Lobo, Springer, Cham, CH, pp. 11-33. doi:10.1007/978-3-319-55182-1

[2] L. Flamm, (1916), “Comments on Einstein’s Theory of Gravity,” Physikalische Zeitschrift 17:448-454

[3] A. Einstein, and N. Rosen, (1935), "The Particle Problem in the General Theory of Relativity," Phys. Rev., 48:73-77, doi:10.1103/PhysRev.48.73.

[4] M. Visser, (1995), Lorentzian Wormholes: From Einstein to Hawking (New York: AIP Press).

[5] D. C. D’Inverno, (1992), Introducing Einstein’s Relativity (Oxford: University Press)

[6] C. W. Misner, K. S. Thorne, and J. A. Wheeler, (1973),“Gravitation,” San Francisco (Princeton: Princeton University Press), 1279p

[7] W. J. van Stockum, (1937) “The gravitational ﬁeld of a distribution of particles rotating about an axis of symmetry,” Proc. Roy. Soc. Edinburgh 57, 135.

[8] F. S. N. Lobo, (2008) “Closed timelike curves and causality violation,” Submitted to: Class.Quant.Grav. [arXiv:1008.1127 [gr-qc]].

[9] M. Visser, B. Bassett and S. Liberati, (2000) “Superluminal censorship,” Nucl. Phys. Proc. Suppl. 88, 267 doi:10.1016/S0920-5632(00)00782-9 [gr-qc/9810026].

[10] C. Barcelo and M. Visser, (2002) “Twilight for the energy conditions?,” Int. J. Mod. Phys. D 11, 1553 doi:10.1142/S0218271802002888 [gr-qc/0205066].

[11] M. S. Morris and K. S. Thorne, (1988) “Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity,” Am. J. Phys. 56, 395. doi:10.1119/1.15620

[12] M. S. Morris,K. S. Thorne and U. Yurtsever, (1988), “Wormholes, time machines, and the weak energy conditions,” Phys. Rev. Lett., Vol. 61, pp. 1446 − 1449.

[13] H. G. Ellis (1973), “Ether ﬂow through a drainhole: A particle model in general relativity,” J. Math. Phys., Vol. 14, pp. 104 − 118.

133 [14] K. A. Bronnikov,(1973), “Scalar-tensor theory and scalar charge,” Acta Physica Polonica, Vol. B4, pp. 251 − 266.

[15] J. A. Wheeler, (1955) “Geons,” Phys. Rev. 97, 511. doi:10.1103/PhysRev.97.511

[16] J. A. Wheeler, (1957) “On the Nature of quantum geometrodynamics,” Annals Phys. 2, 604. doi:10.1016/0003-4916(57)90050-7

[17] E. Teo, (1998) “Rotating traversable wormholes,” Phys. Rev. D 58, 024014 doi:10.1103/PhysRevD.58.024014 [gr-qc/9803098].

[18] P. Boonserm, T. Ngampitipan, A. Simpson and M. Visser, (2018) “Exponential metric represents a traversable wormhole,” Phys. Rev. D 98, No. 8, 084048 doi:10.1103/PhysRevD.98.084048 [arXiv:1805.03781 [gr-qc]].

[19] R. Penrose and W. Rindler, (1986),“Spinors and Space-Time,” Cambridge Monographs on Mathematical Physics, (Cambridge: Cambridge University Press), doi:10.1017/CBO9780511564048

[20] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers,and E. Herlt, (2003), “Exact solutions of Einstein’s ﬁeld equations,” Cambridge Monographs on Mathematical Physics, (Cambridge: Cambridge University Press). doi:10.1017/CBO9780511535185

[21] E. Zakhary and C. B. G. McIntosh, (1997), “A Complete Set of Riemann Invariants,” Gen. Relativ. Gravit., Vol. 29, pp. 539 − 581.

[22] J. Overduin, M. Coplan, K. Wilcomb and R. C. Henry, (2020 “Curvature Invariants for Charged and Rotating Black Holes,” Universe 6, no. 2, 22. doi:10.3390/universe6020022

[23] Baker, J. G. and Campanelli, M. (2000) “Making use of geometrical invariants in black hole collisions,” Phys. Rev. D 62, 127501 doi:10.1103/PhysRevD.62.127501 [gr-qc/0003031].

[24] C. Cherubini, D. Bini, S. Capozziello, and D. Ruﬃni, (2002) “Second order scalar invariants of the Riemann tensor: Applications to black hole space-times,” Int. J. Mod. Phys. D 11, 827 doi:10.1142/S0218271802002037 [gr-qc/0302095].

[25] M. A. H. MacCallum, (2006) “On singularities, horizons, invariants, and the results of Antoci, Liebscher and Mihich (GRG 38, 15 (2006) and earlier),” Gen. Rel. Grav. 38, pp.1887 doi:10.1007/s10714-006-0346-6 [gr-qc/0608033].

[26] A. Coley, S. Hervik, and N. Pelavas, (2009) “Spacetimes characterized by their scalar curvature invariants,” Class. Quant. Grav. 26, 025013 doi:10.1088/0264-9381/26/2/025013 [arXiv:0901.0791 [gr-qc]].

134 [27] M. Abdelqader and K. Lake, (2015) “Invariant characterization of the Kerr spacetime: Locating the horizon and measuring the mass and spin of rotating black holes using curvature invariants,” Phys. Rev. D 91, No. 8, 084017 doi:10.1103/PhysRevD.91.084017 [arXiv:1412.8757 [gr-qc]].

[28] M. A. H, MacCallum, (2015), “Spacetime invariants and their uses,” arXiv:1504.06857v1 [gr-qc].

[29] D. N. Page and A. A. Shoom, (2015) “Local Invariants Vanishing on Stationary Horizons: A Diagnostic for Locating Black Holes,” Phys. Rev. Lett. 114, No. 14, 141102 doi:10.1103/PhysRevLett.114.141102 [arXiv:1501.03510 [gr-qc]].

[30] A. Coley, and D. McNutt, (2018) “Identiﬁcation of black hole horizons using scalar curvature invariants,” Class. Quant. Grav. 35, no. 2, 025013 doi:10.1088/1361-6382/aa9804 [arXiv:1710.08773 [gr-qc]].

[31] B. Mattingly et al., “Curvature Invariants for Lorentzian Traversable Wormholes,” Universe 6, no. 1, 11 (2020) doi:10.3390/universe6010011 [arXiv:1806.10985 [gr-qc]].

[32] M. Alcubierre, (1994) “The Warp drive: Hyperfast travel within general relativity,” Class. Quant. Grav. 11, L73 doi:10.1088/0264-9381/11/5/001 [gr-qc/0009013].

[33] S. V. Krasnikov, (1998) “Hyperfast travel in general relativity,” Phys. Rev. D 57, 4760 doi:10.1103/PhysRevD.57.4760 [gr-qc/9511068].

[34] C. Van Den Broeck, (1999)“A ’Warp drive’ with reasonable total energy requirements,” Class. Quant. Grav. 16, 3973 doi:10.1088/0264-9381/16/12/314 [gr-qc/9905084].

[35] J. Natário, (2002) “Warp drive with zero expansion”, Class. Quant. Grav. 19, 1157 doi:10.1088/0264-9381/19/6/308 [gr-qc/0110086].

[36] F. Loup. (2017) “An extended version of the Natário warp drive equation based in the original 3 + 1 ADM formalism which encompasses accelerations and variable velocities.” [Research Report] Residencia de Estudantes Universitas. 2017. < hal − 01655423 >

[37] E. W. Davis, (2009) “Chapter15: Faster-Than-Light Approaches in General Relativity,” Frontiers of Propulsion Science, Progress in Astronautics & Aeronautics Series, Vol 227, American Inst. of Aeronautics & Astronautics Press, Reston, VA, (2nd printing with corrections, 2012), pp. 473 − 509.

[38] O. Groen and S. Hervik, (2007), “Einstein’s general theory of relativity: With modern applications in cosmology,” New York, (USA: Springer) pp. 538

135 [39] S. Carroll, (2014) “Spacetime and Geometry: An Introduction to General Relativity,” (Pearson Education Limited)

[40] E. B. Christoﬀel (1869), “Ueber die Transformation der homogenen Diﬀerentialausdrücke zweiten Grades,” Journal für die reine und angewandte Mathematik, Vol. 70, pp. 46 − 70.

[41] J. Carminati and R. G. McLenaghan, (1991), “Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space,” Journal of Mathematical Physics, Vol. 32, No. 11, pp. 3135 − 3140, doi:10.1063/1.529470.

[42] K. Santosuosso, D. Pollney, N. Pelavas, P. Musgrave, and K. Lake, (1998) “Invariants of the Riemann tensor for class B warped product spacetimes,” Comput. Phys. Commun. No. 115, pp. 381, doi:10.1016/S0010-4655(98)00134-9 [gr-qc/9809012].

[43] P. Marquet, (2009) “The Generalized Warp Drive Concept in the EGR Theory.” The Abraham Zelmanov Journal. No. 2.

[44] F. Loup. (2018) "Six diﬀerent Natario warp drive spacetime metric equations." [Research Report] Residencia de Estudantes Universitas. < ffhal − 01862911f >

[45] R. L. Arnowitt, S. Deser and C. W. Misner, (1959) “Dynamical Structure and Deﬁnition of Energy in General Relativity,” Phys. Rev. 116, 1322 . doi:10.1103/PhysRev.116.1322

[46] J. Frauendiener, (2011) “Miguel Alcubierre: Introduction to 3 + 1 numerical relativity,” Gen. Rel. Grav. 43, 2931. doi:10.1007/s10714-011-1195-5

[47] D. Brooks, M. A. H. MacCallum, D. Gregoris, A. Forget, A. Coley, P. c. Chavy-Waddy, and D. D. McNutt, (2018) “Cartan Invariants and Event Horizon Detection, Extended Version,” Gen. Relativ. Grav. 50, no. 4, 37 [arXiv:1709.03362 [gr-qc]].

[48] R. C. Woods, R. M. L. Baker, F. Li, G. V. Stephenson, E. W. Davis, and A. W. Beckwith, (2011) "A new theoretical technique for the measurement of high frequency relic gravitational waves." Journal of Modern Physics 2, pp. 498 − 518 doi:10.4236/jmp.2011.26060

[49] A. G. Agnese, and M. La Camera, (2002) “Traceless stress energy and traversable wormholes,” Nuovo Cim. B 117, 647 [gr-qc/0203067].

[50] R. V. Korolev, and S. V. Sushkov, (2014) “Exact wormhole solutions with nonminimal kinetic coupling,” Phys. Rev. D 90, 124025 doi:10.1103/PhysRevD.90.124025 [arXiv:1408.1235 [gr-qc]].

136 [51] J. Carot and J. D. Costa, (1993) “On the Geometry of Warped Spacetimes,”Classical and Quantum Gravity 10 no. 3 pp. 461 − 482 doi:10.1088/0264-9381/10/3/007

137