archived as http://www.stealthskater.com/Documents/WarpDrive_02.doc
read more at http://www.stealthskater.com/Science.htm#Halerewicz
note: because important
web-sites are frequently "here today but gone tomorrow", the
following was archived from http://da_theoretical1.tripod.com/wdphysics.html
on April 22, 2004 . This is NOT
an attempt to divert readers from the aforementioned web-site. Indeed, the reader should only read this
back-up copy if it cannot be found at the original author's site.
Hyperbolic Geometrodynamic
Warp Drives
& Beyond
Draft Copy June 19, 2003
Paul Hoiland 1
Edward Halerewicz, Jr. 2
----------------------------------------------
This e-book was originally intended as a reference source for the E Somino Ad Astra (ESAA) online Group. Actually the group name was somewhat of a misnomer as ESAA was really the motto of the group, with the range of discussions evolving within the group the more dignified name Advanced Theoretical Physics Group (ATPG) was christened or ESAA-ATPG. While the original ESAA-ATPG has disbanded and allowed for the founding of the Journal of Advanced theoretical Propulsion Methods (http://www.joatp.org ), the ESAA spirit continues on with the authors with the public availability of this document. It is the hopes of the authors that this document may go beyond a simple reference source and inspire new knowledge and frontiers in science. As such a single copy of this E-book may be saved and printed for private purposes but it is not meant to be commercially distributed and otherwise used without the explicit written permission of the authors © 2003.
About the Authors
Paul Hoiland is a theoretical physicist and engineer who has a background in General Relativity as well as string theory. He has published a number of popular scientific articles in the New York Post, as well as in scientific journals. He is a self proclaimed educator of science, he is also the founder of the Transtator Industries research group and the Journal of Advanced theoretical Propulsion Methods (http://www.joatp.org).
Edward Halerewicz, Jr. is an independent
researcher, with an interest in exploring the limits of modern science as well
as popularizing such concepts to the public. He was one of the founding members
of the ESAA group, and has worked to bring the warp drive into acceptable
limits of existing theory. http://da_theoretical1.tripod.com/index.htm
PREFACE
The initial intent of this E-book was to give a
general survey over what has become to be known as the “
The broader scope of this E-Book is to also explore aspects of theoretical physics which go beyond epistemology purposes and to investigate there practical uses for the benefit of society as a whole. The specific practical use of General Relativity which this E-Book attempts to cover is how it may be possible to use warp drive metrics as a means of allowing interstellar communication and transportation on human time scales. This could have several benefits for the collective world society, such as the stimulation of economic growth, the potential for practical human colonizations of space, the ability to protect our planet and home from celestial hazards, and the possibility of unprecedented peaceful international cooperation amongst all nations. While these benefits may not be realizable at present given time they should begin to reveal themselves and this the authors feel should be the direction in which the physical sciences should follow. It is hoped that the practical uses of the theoretical possibilities known today may lead to this direction and it is hoped that this document serves as a good start for this vision of the future.
Contents
1 Underlying Physics
1.1 Special Relativity . . .
. . . . . . . . . . . . . . . . . . . . .
1.2 General Relativity . . .
. . . . . . . . . . . . . . . . . . . . .
1.2.1 A Metric Defined . . .
. . . . . . . . . . . . . . . . .
1.2.2 Minkowski Metric Tensor
. . . . . . . . . . . . . . . .
1.2.3 Wormholes . . . . . .
. . . . . . . . . . . . . . . . . .
1.2.4 Inflationary Universe
. . . . . . . . . . . . . . . . . . .
1.3 What is the
2 Warp Drive
2.1 Alcubierre’s
2.1.1 Flaws of the Warp
Drive . . . . . . . . . . . . . . . . .
2.2 The Energy Conditions .
. . . . . . . . . . . . . . . . . . . . .
2.3 Wormholes . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
2.4 GR FTL Ambiguity . . . .
. . . . . . . . . . . . . . . . . . . .
2.4.1 Accelerated Observers
. . . . . . . . . . . . . . . . . .
2.4.2 tetrads . . . . . . .
. . . . . . . . . . . . . . . . . . . .
2.5 Cosmology . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
3 After Shocks
3.1 Van Den Broeck . . . . .
. . . . . . . . . . . . . . . . . . . . .
3.1.1 Quantum Fields . . . .
. . . . . . . . . . . . . . . . . .
3.2 Warp Drives &
Wormholes . . . . . . . . . . . . . . . . . . . .
3.3 Time Travel &
Causality . . . . . . . . . . . . . . . . . . . . .
3.3.1 The Two-Dimensional
ESAA Metric . . . . . . . . . .
3.4 Photon-ed to Death . . .
. . . . . . . . . . . . . . . . . . . . .
4 Null Manifolds
4.1 The Causal Photon Problem
. . . . . . . . . . . . . . . . . . .
4.1.1 The Cauchy Problem . .
. . . . . . . . . . . . . . . . .
4.2 The Shield Effect . . .
. . . . . . . . . . . . . . . . . . . . .
4.2.1 Cauchy Geodesics . . .
. . . . . . . . . . . . . . . . .
5 Negative Energy Problems 45
5.1 Quantum Theory in Brief
. . . . . . . . . . . . . . . . . . .
5.2 Virtual Particles . . .
. . . . . . . . . . . . . . . . . . . . .
5.2.1 The Casimir Effect . .
. . . . . . . . . . . . . . . . . .
5.2.2 Casimir Tensor . . . .
. . . . . . . . . . . . . . . . . .
5.3 Quantum Inequalities . .
. . . . . . . . . . . . . . . . . . . . .
5.4 Exotic Energies
Nullified . . . . . . . . . . . . . . . . . . . . .
6 The Mysterious Vacuum
6.1 EM Stress-Energy . . . .
. . . . . . . . . . . . . . . . . . . . .
6.2 Vacuum Squeezing . . . .
. . . . . . . . . . . . . . . . . . . .
6.3 PV Warp Motion . . . . .
. . . . . . . . . . . . . . . . . . . .
7 The Affects of Scalar Fields
7.1 The Anti-Gravity Non
Sense . . . . . . . . . . . . . . . . . . .
7.2 Scalar Fields . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
7.3 Maxwell’s Equations . .
. . . . . . . . . . . . . . . . . . . . .
8 EM ZPF Force
8.1 EM Field in gauge theory
. . . . . . . . . . . . . . . . . . . .
8.2 Ghost Scalars . . . . .
. . . . . . . . . . . . . . . . . . . . . .
9 Quantum Entanglement
9.1 Bell States . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
9.2 Quantum Teleportation .
. . . . . . . . . . . . . . . . . . . . .
9.3 Boost Entanglement . . .
. . . . . . . . . . . . . . . . . . . .
10 Vacuum Engineering
10.1 Non-gravitational
Effects . . . . . . . . . . . . . . . . . . . . .
10.2 The GSL Issue . . . . .
. . . . . . . . . . . . . . . . . . . . .
10.2.1 Dealing With Entropy
Issues . . . . . . . . . . . . . .
10.3 Quintessence Scalars .
. . . . . . . . . . . . . . . . . . . . . .
11 Beyond the Fourth Dimension
11.1 Superstring Theory in
Brief . . . . . . . . . . . . . . . . . . .
11.2 Type I: Bosonic strings
. . . . . . . . . . . . . . . . . . . . . .
11.3 Type IIA: Fermionic
strings . . . . . . . . . . . . . . . . . . .
11.4 Heterotic strings . . .
. . . . . . . . . . . . . . . . . . . . .
11.5 Type IIB: Compactified
Topologies . . . . . . . . . . . . . .
11.6 Shortcuts in Hyperspace
. . . . . . . . . . . . . . . . . . . . .
12 Navigation Issues
12.1 Measurement . . . . . .
. . . . . . . . . . . . . . . . . . . .
13 Ship Structure Requirements
13.1 Initial Thoughts . . .
. . . . . . . . . . . . . . . . . . . . . . .
13.2 Structural Requirements
. . . . . . . . . . . . . . . . . . . . .
13.2.1 Shielding
13.2.2
13.3 Oxygen Generation
13.4 Cooling / Heating /
Humidity
13.5 Electrical Systems
13.6 Sensor Equipment
13.7 Computer Systems
Bibliography
Chapter 1 - Underlying Physics
“Basic Warp
Design is requirement at the Academy. Actually the first Chapter is called
Zephram Cochrane1 .” – Lavar Burton from the 1996 film Star Trek:
First Contact
The
Using a more general approach for a potential “FTL” interpretation of a moving body, one can easily see how the curvature associated with General Relativity could allow for “warp” propulsion. As one can say that if space-time is not completely at (i.e. having a non-trivial curvature) as seen by a preferred observer that observer may record different transit times for a single distant vector at different displaced locations in spacetime (this discrepancy of the local behavior of physics in spacetime is the very background from which warp drives originate). It is within this chapter where we will investigate the ‘strict’ rules of Special Relativity and later modify them in accordance to the bizarreness of General Relativity. This modification then allows us to discuss the possibility of apparent FTL travel, although it will become clear that this pursuit comes at a great cost.
This book is written for those who are familiar with
both the special and general theories of relativity. The brief introduction
which this chapter covers on the two theories are only meant to be a refresher
to the reader. This is done so that you the reader will have a conceptual
source of the physics behind the
1.1 Special Relativity on Speed Limits
Let’s start by considering the well-known velocity-of-light speed limit, as viewed by Special Relativity (SR) and by General Relativity (GR). In the context of SR, the speed-of-light is the absolute speed limit within the Universe for any object having a real mass, there are two reasons for this. First there is the limit imposed by what we in physics term a Lorentz invariant frame. Simply explained, it's giving a fast material object additional kinetic energy and has the main affect of causing an apparent
-----------------------------------------------------------------------------
1The father of warp drive in the ‘Star Trek Universe.’
increase in mass-energy of the system rather than speed, with mass-energy going infinite as the speed snuggles up to the velocity-of-light. This occurs mathematically because we can attribute kinetic energy to a system by the following formula
KE = ˝ mv2 (1.1)
Now the reason why a mass (in reality momentum)
increase occurs is not presently understood (that is beyond mathematical
computation), it is simply an observed phenomena. For example the acceleration
rate of a bullet can be calculated with the aid of a mathematical function
known as the beta function 
, but why this
function is accurate gets into metaphysics.
A similar phenomenon was noted in a series of
experiments conducted by Antoon Hendrik Lorentz, as the acceleration rates of
electrons also seemed to be strangely governed by the beta function. Thus for
all intents an purposes acting to increase the mass of an electron. However the
mass-energy increase only exist relative to an external observer, the
mass-energy increase is actually a consequence of the dynamics of the a system
in relation to momentum. Which is given
through the equality p = m0v, thereby leading to an
unknown coupling constant which we call gamma, so that the momentum of the
system is governed by p = γ(m0v) where 
. Therefore it is clearly seen that it is the
momentum of the system which increases and not the rest mass m0, as a result
the total kinetic energy of a system can be calculated from 
(1.2)
By this mechanism, relativistic mass increase limits bodies with mass to sub-light velocities, this result comes from the fact that as v → c the “accelerated mass” m0 increases asymptotically. This then requires a massive body to obtain an infinite kinetic energy to maintain the equality v = c. In theory, with “light masses” (relative to the gamma function) you could get close to c, which is intentionally done with the world’s particle accelerators at will. But, you can never reach c as long as we have to deal with what is called inertia (the resistance of matter to accelerate) in the classical sense.
There is also a second FTL prohibition supplied by Special Relativity. Suppose a device like the “ansible” of LeGuin and Card were discovered that permitted a faster-than-light or instantaneous communication, then an absolute frame would form. However, since SR is based on the treatment of all reference frames (i.e., coordinate systems moving at some constant velocity) with perfect even-handedness and democracy [gravitation is often said to be the most democratic force in the Universe, we will explore this in later chapters]. Therefore, FTL communication is implicitly ruled out by Special Relativity because it could be used to perform “simultaneity tests” of the readings of separated clocks which would reveal a preferred or “true” (universal) reference frame for the Universe as a whole. The existence of such a preferred frame is in conflict with the building blocks of Special Relativity. However, in more recent times that second prohibition has begun to be looked at more closely. Recent articles with the Los Alamos National Laboratories (LANL) e-print server and other publication mediums have begun to ask further questions about the preferred reference frame issue. There are modifications to Einsteins original theory of relativity that do allow for such.
Secondly, experiments with quantum entanglement and quantum tunneling have also added to the reason this secondary prohibition has been brought into question. It is from these contradictions with other existing physical theories which allows us to seriously explore the question: “Is FTL travel really possible?”
1.2 General Relativity
General Relativity treats Special Relativity as a restricted sub-theory that applies locally to any region of space sufficiently small that its curvature can be neglected. General Relativity does not forbid apparent FTL travel (as seen by a preferred topological observer) or communication, but it does require that the local restrictions of Special Relativity must apply (that is Special Relativity applies to ones local geometry). Light speed is the local speed limit, but the broader considerations of General Relativity may provide an end-run way of circumventing this local statute. It is possible to change the metrics of a local region of space-time to one in which the local limit on c has been increased. It is also possible to alter them so that c is reduced. Neither of these situations would violate anything Special Relativity dictates (although it may require a recalculation for the value of c). They would simply be using a special frame of reference different from the ideal one (which we associate with the vacuum state of the Universe). For example within those modified regions Lorentz invariance would still apply. You could not out run light within that region. But, you could out run light within our normal space-time reference.
So you may ask why does General Relativity allow for apparent FTL travel, and not Special Relativity? In the earlier section we explored Lorentz invariance and found why c ≤ v travel can not be overcomed by massive bodies (or for that matter light-like propagations). But what is so special about General Relativity you ask? First General Relativity is a physical theory based upon a differential geometry along with the second postulate of Special Relativity; that c = constant. Now one has to ask how does this geometry affect the apparent value for c? The question is not really valid as Lorentz invariance still applies (see Section 1.1). However the affects of geometry on an observer modifies the way in which that observer records events. An altered geometry can affect the rate at which clocks tick for example. If the rate at which a clock ticks slows down locally an outside observer will measure, this time lapse. However for the occupant in the curved region there is no time displacement. In plain English this tells us that since time slows, vector speeds may appear to decelerate in certain regions as well. This is a consequence of the time dilation affects of Special Relativity, mathematically represented as:
(1.3)
Meaning that outside these regions time will appear to move faster to an observer with a slow clock. V’ola this leaves us with “shortcuts” in spacetime, thus the speed-of-light locally depends on the rate at which your clock ticks (often refereed to as coordinate time). However, this can not do justice to the grandiose of that remarkable theory. We will be gradually pulling out tricks from GR as well as other theories to explain how the warp drive can work throughout this book.
Before closing on this section let us review the foundations of GR as they will be discussed throughout this book. One should keep in mind that within the frame work of General Relativity geometry and energy are connected in a most fundamental way. The exact relation is given through the seemingly simple formula
Gij
= Tij (1.4)
G is known as the Einstein tensor which represents the geometry of space and time. T is known as the stress-energy tensor and acts to measurement the energy within spacetime. The subscripts i,j represent a chosen coordinate system, or measurement system (this is an imaginary metric to keep a chosen tensor positive for conceptual reasons, although physically this is not the case). If you noticed we called Eq. 1.4 seemingly simple, the reason for this is because G actually represents a complicated set of differential geometry known as Riemannian geometry. The differential geometry is affected by classical physics through partial derivatives known as Christoffel symbols. Although Riemannian geometry is very complex we can generally write a spacetime geometry in the form of several tensors such as Rabjk . Although solving the equations is a bit overwhelming to new comers as the geometry reduces (or contracts) under special operations by something we call a Ricci tensor Rab . This means that for computations involving GR you will often see Eq. 1.4 expressed in some variant of
(1.5)
The new term gab is called
a metric tensor because it acts as a tool to measure space. It is from the
concept of a metric tensor where we can take measurements of spacetime as
mentioned earlier in the section to make judgments about the curvature of
spacetime. This very abstract geometry G
intern affects T and T can later affect G
(also note that this does not represent
1.2.1 A Metric Defined
A metric is a distance on some surface. For example, in at Cartesian space the metric is simply the straight line distance between two points. On a sphere it would be the distance between two points (given by Gallian coordinates), as confined to lie along the surface of the sphere. In the space-time of Special Relativity the metric is defined as
ds2
= c2 dt2 − dx2 − dy2
− dz2 (1.6)
You can think of the metric as the distance between points in space-time. Related to the metric is the very important idea of a metric tensor, which is usually represented by the symbol g which is called a determinant, sometimes with the indices explicitly shown (e.g., gab). The metric tensor is one of the most important concepts in relativity, since it is the metric which determines the distance between nearby points, and hence it specifies the geometry of a space-time situation.
One can represent the metric tensor in terms of a matrix as we show below. In four-dimensional spacetime (whether at or curved by the presence of masses) the metric tensor will have 16 elements. However, like all other tensors which you will experience in this book, the metric tensor is symmetric, so for example g12 = g21. This means that there are 10 independent terms in the metric tensor, these are also known as scalar components.
(1.7)
Misner, Thorne and Wheeler have a unique way of picturing the metric tensor g as a two slot machine g(_,_). The precise numerical operation of the g machine depends on where you are on a surface (i.e., what the local curvature is). When one puts in the same vector, say u, into both slots of the g machine, one gets the square of the length of the vector u as an output. If one puts two different vectors u, v then the g machine gives the scalar (or dot) product of the two vectors u and v.
You might say so what is the big deal – it’s just something to calculate the dot product of two vectors. The subtle difference is that in curved space-time the “angle” between vectors depends on where you are in space-time, and therefore g is a machine which operates differently at different space-time points. Viewed backwards, if you know how to calculate the dot products of vectors everywhere, you must understand the curvature of the space (and therefore the g machine specifies the geometry). For this reason, the metric tensor enters into virtually every operation in General Relativity (and some in Special Relativity).
1.2.2 Minkowski Metric
Tensor
In the at spacetime of Special Relativity, g is represented by the Minkowski Metric. The Minkowski metric tensor is that of at space-time (Special Relativity) is written as follows:
(1.8)
It is more conventional to express tensor relationships in terms of summations, rather than as matrices. We can express the tensor relationship for the space-time interval in the following way:
(1.9)
which yields, if one uses the terms dx0 = cdt,dx1 = dx,dx2 = dy,dx3 = dz
one receives ds2 = cdt2 − dx2 − dy2 − dz2 (1.10)
and writing Eq. 1.10 in matrix form gives us
(1.11)
Note that the Minkowski metric tensor has only diagonal terms. When we go to the curved space of General Relativity in the presence of masses we will employ tensors which have off-axis components. This is yet another example of why Eq. 1.4 in Section 1.2 is deceptively simple. Since our topic again is not so much GR, but Warp Drives, we bring up these important background issues as needed.
1.2.3 Wormholes
To explore what we mean by a topological (or geometrical) shortcut in GR, we start with the very popularized concept of a wormhole. A wormhole is a gravitational/topological shortcut capable of connecting two widely separated locations in space, say 26 light-years apart. A body might take a few minutes to move through this shortcut trough space at velocities v << c in a region known as the neck of a wormhole. However, by transiting through the wormhole the body has traveled 26 light-years in global (larger) spacetime in just a few minutes, producing an effective speed of a million times the conventional velocity-of-light to an outward observer. The body simply moved through a region that has a shorter distance to its destination. By taking this short cut an external observer will record a faster than light transient time. Yet, in reality it never exceeded c in the local metric of the global manifold, its simply a topological trick!
However these wormholes come at a great cost, although they are mathematically viable they are not necessarily physically viable. To elaborate, wormholes are generally considered as nothing more than a mathematical curiosity (however there is a minority in the physics community that believe they may be physically possible). The reason for the debate over their physical nature is that they were accidentally discovered by a thought experiment of Einstein and Rosen, initiated through a symmetry operation in the mathematics of GR (as a result a wormhole is sometimes refereed to as an “Einstein-Rosen Bridge”). As a black hole forms the null geodesics (the path of light) of the dying star converge to a singularity. At the singularity the known number system completely fails and all meaningful information about the dead star is lost. However because GR is based upon symmetries, mathematically there should exist an imaginary copy of the collapsing black hole. If the imaginary (mirror) copy somehow converged, the singularity is adverted and a transversable hole (wormhole) is formed in space. There have been a number of topological studies performed such that one does not need an imaginary black hole to collide with a real one for a wormhole to form. However the gravitational forces acting on the throat of the wormhole are enormous and strongly relativistic in nature, which causes the “neck” to collapse. Thus the only way left to hold open the wormhole’s neck is the mysterious acceptance of a form of energy called negative energy or exotic energy, which just happens to violate the laws of thermodynamics (such that the T = 0 term in Section 1.2 becomes T < 0).
1.2.4 Inflationary
Universe
Another example of FTL behavior in General Relativity is the expansion of the Universe itself. As the Universe expands, new space is seemingly being created between any two separated objects. The objects may be at rest with respect to their local environment and with respect to the Cosmic Microwave Background (CMB) radiation, but the distance between them may grow at a rate greater than the velocity-of-light. According to the standard model of cosmology, parts of the Universe are receding from us at FTL speeds, and therefore are completely isolated from us (resulting from our corner of the Universe forming a local cosmological horizon). The recently observed accelerated expansion of the Universe by S. Perlmutter [1] and others even shows the presence of this affect more than the original standard model. Adding in variances in the velocity-of-light over time we find that at one time the Universe had a higher velocity-of-light than it now does. So, the idea of altering the local velocity-of-light, or of traveling FTL in a normal space-time region is not new to physics. It is simply determined by finding methods based upon General Relativity to do an end-run around the local restrictions of Special Relativity.
The present “accelerating” (in quotations as this is a non Lorentzian expansion) Universe gives clear evidenced that seemingly apparent FTL travel is possible and is presently at work within our Universe. The concept of an accelerating Universe was first conceived by ad hoc adaptation Einstein made to his famous field equation (see Eq. 1.5). He added a term to the right hand side of his equation called the Cosmological Constant, represented by the Greek symbol Λ. Einstein invented this concept to maintain the idea of a static Universe, until Edwin Hubble showed that the Universe is in fact expanding. Einstein later withdrew this term calling it “the biggest blunder of my life”. What is even more strange is that if you put the term to the left side of the Einstein Field Equation (Eq. 1.5), it acts to remove energy, thus the Cosmogonical Constant defies the law of conservation of energy as T mentioned in Section 1.2 can takes values which are T ≤ 0. It is rather disconcerting that something in the large-scale of the Universe is violating the very nature of GR, nothing is sacred anymore. However, this does not necessarily mean that GR is wrong, and that the energy conditions as far as GR is concerned are not violated. GR for example cannot take quantum action into affect because quantum theory has non locality as its basic building blocks, which violate the principles upon which relativity is built. There is also the possibility that we are missing something and the Universe’s apparent acceleration may exist because we are unaware of something else that may exist. So that said even the “accelerating Universe,” does not provide evidence that objects such as wormholes do in fact exist.
1.3 What is the
Now to the heart of the matter, what exactly is a
Figure
1.1: The
functions which create a
very peculiar distortion of space at the edges of the at-space volume. The
hyperbolic structure was chosen so that such a spacetime would not produce a
Causal Time-like Curve (CTC), or what the general public calls a time machine.
In affect, new space is rapidly being created (like an expanding Universe) at
the backside of the moving volume, and existing space is being annihilated in front
of the metric. The effect is what we term a dipolar region of space. It causes
space-time to flow, think of a snow plow, it eats the snow and then spits it
out. This is in affect how Alcuiberre’s 
. This finding is not really to surprising in
the present, for example Olum has shown that FTL travel requires negative
energy [3].
So the relevant question now is that since Warp Drives violate the law of
conservation of energy can they exist in reality. Unfortunately there are no
easy answers to this question, but throughout this book we try to examine this
possibility. Welcome to the wonderful
world of differential travel schemes using c in GR, you now have
the background to tackle the bigger questions.
Chapter 2 Fundamental Warp Drive Metrics
"Captain’ I cannota’ change the laws of physics". –Chief Engineer “Scotty” from the Star Trek series
We now wish to introduce the reader to the heart of
the matter, what is a
In this chapter we explore in relative detail the geometrical requirements of a warp drive spacetime. From here it is learned how Warp Drives use geometry to satisfy a remote observers local definition of FTL travel. When we explore the geometry in detail we also become more aware of the violations of GR explored in Chapter 1. Generally from this chapter we learn of the specific nature of warp drives, and why there Nature defies General Relativity.
2.1 Dr. Alcubierres Original Metric
To explore how The Alcubierre Spacetime works we
will explore the mathematics of his paper [2] throughout this section. To do so let us
imagine that you are in space ship who’s position is given in Cartesian
coordinates by an arbitrary function xs(0, 0). Now
imagine that your space ship is coasting so that its future position is
determined by an arbitrary function of time xs(t),
this function yields the velocity of your space ship by the standard relation vs(t)
= dxs(t)/dt. Therefore the
One would then have to form a bubble around oneself, lets say that three spatial coordinates are given with x = y = z = 0. With the ship’s position xs = 0, the invariant lengths of this space are then given by:
(2.1)
One can now write the arbitrary function of the warp drive spacetime as a function of the ship’s positions rs.
(2.2)
When the velocity of the warp drive is set to v = 1 then the Warp Drive Spacetime can be described by the metric:
ds2
= − dt2 + (dx − vs f(rs)dt)2
+ dy2 + dz2
. (2.3)
2.1.1
Problems with the Alcubierre metric
The initial problem of the warp drive was recognized by Alcubierre himself in his breakthrough paper. It was even more focused upon by Dr.s Ford and Pfenning [4] in their works. The problem being a weak conservation principle in General Relativity (GR) is broken, simply known as the Weak Energy Condition (WEC). The formula for a non-trivial space is given by TabVa Vb ≥0 for all time like vectors Va . The problem which arises is that it causes the flat spatial metric within the zero Christoffel space to obtain a positive curvature. Thus violating relativity by a local expansion of space-time (that is the creation of an exotic energy region). Further, there is a restraint placed upon any negative energy condition that an equal or greater rebound pulse of positive energy must follow the initial negative energy. We know from Quantum Mechanics that such violations do occur at the sub-atomic level. The fundamental question is can they exist at the Macroscopic level. Recent work with what has become known as the Casimir Effect, which demonstrates that they can. This has now brought up a new energy condition referred to as the Average Weak Energy Condition (AWEC). Basically, under it negative energy can exist in a local region of space-time as long as globally the composite energy condition remains positive.
The idea of space-time expanding and contracting is not at all new, it should be noted that Einstein predicted such an affect. From the local metric gab acting on a Riemannian surface geometry, the curvature connection of space-time is derived directly from a metric with a signature of rank 2 (referring to the indices of the metric). Thus the curvature connection would be given in a linear form do to the presence of the metric, hence space would be required to expand and contract from the metrics signature. This expansion and contraction of spacetime is called gravitational radiation, which is assumed to propagate at the speed-of-light c (although there is some recent in direct evidence that points to this). The key between this idea and Alcubierre’s proposal is that his paper requires two separate sets of gravitational waves. However, do to the spatial metric proposed by Alcubierre one set of the gravitational waves would be required to defy the WEC since his original metric required a large negative energy it would also violate the AWEC. Both of these issues