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DefenseIntelligence
Reference
Document
Acquisition Threat Support
2 April 2010
ICOD: 1 December 2009
DIA-08-1004-001

Warp Drive, Dark Energy, andthe Manipulation of Extra
Dimensions

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U N C LA SSI FI E D/ MaikaaarsicakatiffEenfrS

Warp Drive, Dark Energy, and the Manipulation of Extra
Dimensions

Prepared by:(b)(3):10 USC 424

Defense Intelligence Agency
Authors:(b)(6)

Administrative Note
COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized.

This product is one in a series of advanced technology reports produced in FY 2009
under the Defense Intelligence Agency, (b)(3):10 USC 424
Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments or questions pertaining tothis document should be addressed to (b)(3):10 USC 424;(b)(6)I, AAWSA Program
Manager, Defense Intelligence Agency, ATTN: (b)(1) 10 USC 424 Bldg 6000, Washington,
DC 20340-5100.

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UNCLASSIFIED/rem nc-c*CIn "r"Contents
Introduction
2. General Relativistic Warp Drives

1

2.1 Warp Drive Requirements

2

3. The Cosmological Constant
3.1 Einstein's Equation and the Introduction of A
4. Casimir Energy and the Quantum Vacuum

445

4.1 The Casimir Effect6

5. Extra Space Dimensions

8

5.1 Kaluza-Klein Theory

8

5.2 Large Extra Dimensions

10

5.3 Randall Sundrum Brane Models

11

5.4 Extra Dimension Summary

12

6. Dark Energy as a Higher Dimensional Artifact

12

7. Warp Drive and Higher Dimensional Manipulation

15

7.1 Adjusting Higher Dimensions for Propulsions

16

7.2 The Geometry of Extra Dimensions

17

7.3 Higher Dimensions and Stabilization

17

7.4 Elementary Warp Drive Calculations

20

7.5 Future Experiments

22

7.6 The Development of the Technology

23

8. Summary

24

FiguresFigure 1. York Extrinsic Time (9) Plot
Figure 2. The Interior Region of Parallel Conducting Plates
Figure 3. Internal Structure of a Seemingly One-Dimensional Object
Figure 4. Manipulated Extra Dimensioniii

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Figure 5. Artist's Conception of a Futuristic Warp Drive Spacecraft
Figure 6. A Toroidal Higher Dimension
Figure 7. A Combination of Phenomenologically Viable Fields
Figure 8. False Vacuum Minima
Figure 9. Thick and Thin Shell Warp Bubble

16
17
19
19
21

TablesTable 1. Transit Times to Various Exotic Destinations at 100 Times the Speed ofLightvi
Table 2. Negative Energy Required for Warp Bubble (Larger Negative Energy)
3
Table 3. Negative Energy Required for Warp Bubble
22

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Warp Drive, Dark Energy, and the Manipulation of Extra
Dimensions
Introduction
If one is to realistically entertain the notion of interstellar exploration in
timeframes of a human lifespan, a dramatic shift in the traditional approach tospacecraft propulsion is necessary. It has been known and well tested since
the time of Einstein that all matter is restricted to motion at sublight velocities(« 3 x 108 m/s, the speed of light, or c), and that as matter approaches the
speed of light, its mass asymptotically approaches infinity. This mass increase
ensures that an infinite amount of energy would be necessary to travel at the
speed of light, and, thus, this speed is impossible to reach and represents anabsolute speed limit to all matter traveling through spacetime.
Even if an engine were designed that could propel a spacecraft to an
appreciable fraction of light speed, travel to even the closest stars would takemany decades in the frame of reference of an observer on Earth. Althoughthese lengthy transit times would not make interstellar exploration impossible,
they would certainly dampen the enthusiasm of governments or private
individuals funding these missions. After all, a mission whose success isperhaps a century away would be difficult to justify. In recent years, however,physicists have discovered two loopholes to Einstein's ultimate speed limit:
the Einstein-Rosen bridge (commonly referred to as a "wormhole") and thewarp drive. Fundamentally, both ideas involve manipulation of spacetime itself
in some exotic way that allows for faster-than-light (FTL) travel.
Essentially, the wormhole involves connecting two potentially distant regionsof space by a topological shortcut. Theoretically, one would enter thewormhole and instantaneously be transported to the exit located in a distantregion of space. Although no observational evidence of wormholes exists,
theoretically they can exist as a valid solution to general relativity.
The warp drive—the main focus this paper—involves local manipulation of thefabric of space in the immediate vicinity of a spacecraft. The basic idea is tocreate an asymmetric bubble of space that is contracting in front of thespacecraft while expanding behind it. Using this form of locomotion, thespacecraft remains stationary inside this "warp bubble," and the movement of
space itself facilitates the relative motion of the spacecraft. The most
attractive feature of the warp drive is that the theory of relativity places noknown restrictions on the motion of space itself, thus allowing for aconvenient circumvention of the speed of light barrier.
An advanced aerospace platform incorporating warp drive technology would
profoundly alter the capacity to explore—and potentially to colonize—theuniverse. Because a warp drive is not limited by the speed of light, one canonly guess the top speeds such a technology might be capable of achieving.
For the sake of argument, let's consider the duration of trips taken by a

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spacecraft capable of 1000 for an array of exotic destinations of possible
interest. As Table 1 shows, trips to the planets within our own solar system
would take hours rather than years, and journeys to local star system would
be measured in weeks rather than hundreds of thousands of years.
Table 1. Transit Times to Various Exotic Destinations
at 100 Times the Speed of Light
Destination
Mars
Jupiter
NeptuneAlpha Centauri
Epsilon Eridani
The Orion Nebula

Transit Time
193 seconds
36 minutes
4 hours
15 days
38 days
1.3 years

Until recently, the warp drive was a concept reserved for science fiction.
However, a 1994 paper by Miguel Alcubierre placed the idea on a more solid
theoretical footing. Alcubierre (Reference 1) demonstrated that a specific
Lorentzian manifold could be chosen that exhibited bubble-like features
reminiscent of the warp drive from the popular Star Trek television series. The
bubble allowed for the surrounding spacetime to move at FTL speeds, and the
inhabitants of the bubble would feel no acceleration effects because spacetime
itself would be in motion instead of the spacecraft and its inhabitants.
A number of papers have emerged in recent years that build on this originalidea. However, these papers do not typically address how one might actuallycreate the necessary spacetime bubble. Our own research directly addresses
this question from a new and unique perspective and introduces a novel
paradigm shift in the field of warp drive study (Reference 2). More formally,
our work approaches the physics of warp drive from the perspective ofquantum field theory; this diverges from the more traditional approach towarp drives, which utilizes the physics of general relativity. One of the
improvements the model introduces is a dramatic reduction in the overall
energy required to create such a phenomenon.
The roadmap to this new idea was the observation that spacetime is currently
known to be in a state of accelerated expansion, as demonstrated by theredshifting of galaxies, and the belief that if the mechanism for this expansion
could be understood, then it might ultimately be controlled. A popular term
used by cosmologists today is "dark energy," an exotic and ubiquitous form ofenergy that is believed to constitute over 70 percent of the matter-energy
content of the universe (Reference 3-6). One salient feature of dark energy is
its intrinsic ability to generate negative pressure, causing the fabric of space
to expand in the way that is currently observed (Reference 7).

' This speed, while somewhat arbitrary, highlights the fact that our galaxy would become far more accessible if or
when one discovers how to surpass the speed of light barrier.

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Although we know what dark energy does, we do not yet fully understand its
nature. We do not understand why it exists or how it is created; we simply
know it provides an ever-present force on spacetime, causing the universe toexpand. Indeed, recent high-precision experimental observations indicate darkenergy may be a cosmological vacuum energy (Reference 8-10). Theseobservations are based on the magnitudes of high-redshift supernova andhave been a source of high research activity of late owing to the unexpecteddiscovery that the rate of expansion of the universe is increasing (commonlyreferred to as accelerated expansion).
One tantalizing aspect of dark energy is that if it were fully understood, and if
a technology were developed that could generate and harness the exotic
effects of dark energy on the fabric of space, then a warp drive would be one
step closer to technological reality. While a full understanding of the true
nature of dark energy may be many years away, it is entirely feasible thatexperimental breakthroughs at the Large Hadron Collider or developments inthe field of M-theory could lead to a quantum leap in our understanding of thisunusual form of energy and perhaps help to direct technological innovations.
Our own research focuses on gaining an understanding of the physical originof dark energy. By exploring novel ideas at the forefront of theoretical physics,
one is able to propose a physically viable model incorporating some of the
cutting-edge ideas emerging from string theory and quantum field theory. This
leads to a deeper understanding of the possible origin of dark energy andallows consideration of a mechanism that would allow a sufficiently advanced
technology to control the dark energy density in any region of space, and thus
the expansion of space. This work has clear implications for the advancementof warp drive research.
This paper is structured as follows: Section 2 reviews the more traditional
general relativistic warp drives, the energy required to create them, and thephysics required to understand them. Section 3 discusses the cosmologicalconstant, a term featured in Einstein's equation that regulates the contractionand expansion of the spacetime. Section 4 introduces the Casimir energy,
which, under certain conditions, may be the phenomenon that physicallygenerates the cosmological constant. Section 5 discusses higher dimensions inphysics and their importance in the context of Casimir energy calculations.
Section 6 introduces the formulas that demonstrate that the Casimir energies
in higher dimensions may in fact be the dark energy that is responsible for theaccelerated expansion of the universe. Section 7 relates all the previous
concepts together and introduces the novel warp drive paradigm. Section 8
performs original calculations of the energy required to create a superluminal
warp drive. Finally, the paper speculates about the technological progress thatwould be necessary to turn this model into a reality.

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2. General Relativistic Warp Drives
Alcubierre (Reference 1) derived a spacetime metric motivated by cosmological inflationthat would allow arbitrarily short travel times between two distant points in space. The"warp drive" metric uses coordinates (t, x, y, z) and curve (or worldline) x = xsh(t), y =0, z = 0, lying in the t-x plane passing through the origin. Note that Xsh is the x-axis
coordinate position of the moving spaceship (or warp bubble) frame. The metric'
specifying this particular spacetime geometry is (Reference 1):
ds 2 =+[dx — v(01.(r,„(tildir + dy 2 + dz 2

(2.1)

where c is the speed of light, vsh(t) is the speed associated with the curve (or warp
bubble speed), and rsh(t) is the Euclidean distance from the curve. The warp bubble
shape function f (rsh) is any smooth positive function that satisfies f (0) = 1 anddecreases away from the origin to vanish when rth > R for some distance R. The
geometry of each spatial slice is flat, and spacetime is flat where f (r9,) vanishes but iscurved where it does not vanish.
The driving mechanism of Equation (2.1) is the York extrinsic time, a. This quantity isdefined as (Reference 1):(2.2)

The a behavior of the warp drive bubble
provides for the simultaneous expansionof space behind the spacecraft and acorresponding contraction of space infront of the spacecraft. Figure 1 illustrates
the 9 behavior of the warp drive bubble
geometry. Thus the spacecraft isenveloped within a warp bubble and canbe made to exhibit an arbitrarily large
faster-than-light (FTL) speed (vsh » c)
as viewed by external coordinate
observers. Even though the worldlines
inside the warp bubble region are
spacelike for all external observers, themoving spaceship (warp bubble) frame
itself never travels outside of its local
comoving light cone and thus does not
violate special relativity.

Figure 1. York Extrinsic Time (9) Plot

A spacetime metric (ds2 ), or line element, is a Lorentz-invariant distance function between any two points inspacetime that is defined by ds2 =fix', where g„, is the metric tensor which is a 4x4 matrix that encodes thegeometry of spacetime and dx- is the infinitesimal coordinate separation between two points. The Greek indices (y,v = 0...3) denote spacetime coordinates, x°...x3, such that x1 ..x3 = space coordinates and x° = time coordinate.

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2.1 WARP DRIVE REQUIREMENTS
Implementation of Fit interstellar travel via warp drives requires engineering ofspacetime into very specialized local geometries as shown by Equation (2.1). The
analysis of these via the general relativistic field equation plus the resultant source
matter equations of state demonstrates that such geometries require the use of"exotic" matter in order to produce the requisite FTL spacetime modification. Exotic
matter is generally defined by general relativity (GR) physics to be matter thatpossesses (renormalized) negative energy density and/or negative stress-tension (=positive outward pressure, aka gravitational repulsion). The term is widely
misunderstood and misapplied by the non-GR community. Also, it has been claimed
that FTL spacetimes are not plausible because exotic matter violates the generalrelativistic energy conditions.2 However, this has been shown to be a spurious issue(Reference 11).
The energy density for the Alcubierre (Reference 1) warp drive that is derived from thegeneral relativistic field equation is complex, so we instead use a more simple formula
to express the net energy required, E q,, to build a warp bubble around a spaceship(Reference 12):v2(.4 R 2 a

(2.3)

Ewar p= -(1.21 x

)2.
)

,
u

where G is Newton's universal gravitation constant (6.673 X 10-11 N.m2M-2
y N,) v‘‘arp is thedimensionless speed of the warp bubble, R (> 0) is the radius of the warp bubble, and CS(> 0) is proportional to the inverse of the warp bubble wall thickness A (i.e., a - 1/A).
Equation (2.3) characterizes the amount of negative energy that one needs to localizein the walls of the warp bubble. Table 2 presents a tabulation of the required negative
energy as a function of the "warp factor," v,,,,,p. One can compare the values of amp in
the table with the (positive) rest-energy contained in the Sun (1.79 X 1047 3). The
consequence of Equation (2.3) and Table 2 is that if one wants to travel at hyperlight
speeds, then the warp bubble energy requirement will be an enormous negative
number. And this remains true even if one engineers an arbitrarily low sublight speed
warp bubble. Engineering a warp drive bubble is quite daunting given these results.

2 The condition for ordinary, classical (non-exotic) forms of matter that we are familiar with in nature is that pE > p
and/or pi_ e 0, where id is the energy density and p is the pressure/stress-tension of some source of matter. These
conditions represent two examples of what are variously called the "standard" energy conditions: Weak Energy
Condition (WEC: pE a 0, pE + pa 0), Null Energy Condition (NEC: pEp 0), Dominant Energy Condition (DEC),and Strong Energy Condition (SEC). These energy conditions forbid negative energy density between material
objects to occur in nature, but they are mere hypotheses. The energy conditions were developed to establish aseries of mathematical hypotheses governing the behavior of collapsed-matter singularities in the study ofcosmology and black holes.

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Table 2. Negative Energy Required for Warp Bubble(Larger Negative Energy)
Warp Factor, vw,,,p

Eva rp (I )10 5 (= 3 km/s)—3.03 x 1040
1094 (= 30 km/s)—3.03 x 1042
0.01 (= 3,000 km/s)—3.03 x 104°
0.5 (= 150,000 km/s)—7.59 x 1049
1 (= light speed)—3.03 x 1050
51
2 (= 600,000 km/s)—1.21 x 10
10 (= 3.0 x 106 km/s)—3.03 x 1052
54
100 (= 3.0 x 102 km/s)—3.03 x 10
Assume: R = 50 in. = 10 m
Lobo and Visser (Reference 12) constructed an improved model of the warp drive
spacetime by applying linearized gravity to the weak-field warp drive case and testing
the energy conditions to first and second orders of v„.wp. The fundamental basis of theirmodel is that it specifically includes a finite mass spaceship that interacts with the warp
bubble. Their results verified that all warp drive spacetimes violate the energy
conditions and will continue to do so for arbitrarily low warp bubble speed. They also
found that the energy condition violations in this class of spacetimes is generic to theform of the geometry under consideration and is not a side effect of the superluminal
properties. Based on these facts plus Equation (2.3) and Table 2, it appears that for allconceivable laboratory experiments in which negative energy can be created in minuteamounts, the warp bubble speed will be absurdly low.
Coupling of the finite spaceship mass with the warp bubble leads to the (quite
reasonable) condition that the net total energy stored in the warp bubble be less than
the total rest-energy of the spaceship itself, which places a strong constraint upon the(dimensionless) speed of the warp bubble (Reference 3):
R A
R2(7A1xt0 -8(

(2.4)
M 1

R-

where frisk!, and /2,h,i, are the mass and size of the spaceship, respectively, and R is theradius of the warp bubble. Equation (2.4) indicates that for any reasonable values ofthe engineering parameters inside the brackets, v,„,p will be absurdly low. This result isdue to the intrinsic nonlinearity of the general relativistic field equation. To illustrate
this point, the example starship parameters from Table 2 (R = 50 m, A (9 1/ri = 10-3 m)
are inserted into Equation (2.4) and assume Alsillp = 106 kg to find that v„.,„p1.72 X
10 14 (or 5.16 x 10 6 m/s). Garden snails can crawl faster than this. And if R and Aiship

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are kept constant, then A = 3.37 x 1024 m (or 3.57 x 108 light-years) in order for Vv, „rp
1, which is an unrealistic requirement on the warp bubble design.
Because this energy requirement is so phenomenally high one finds it of paramountimportance to explore new ideas in the field of warp drive technology. What now follows
is a pedagogically rich review of the novel warp drive concept that we have beendeveloping since 2005.

3. The Cosmological Constant
Einstein is famous for a multitude of achievements in the field of physics. Arguably his
most notable contribution is the General Theory of Relativity, a geometric description ofgravitation whose fundamental idea relates the matter and the energy content of theuniverse to the geometry of spacetime. Simply put, the presence of matter and energy
causes spacetime to curve, and this curvature controls how matter and energy movethrough spacetime. General relativity has been the prevailing theory of gravitation since
1915 and thus far has unambiguously passed observational and experimental tests. It
remains an active area of research and technology is still being developed to test
certain features of the theory. Gravitational waves, for example, are one prediction
from GR; however, technology is only now reaching the stage of maturity to allow for
the detection of these waves.

3.1 EINSTEIN'S EQUATION AND THE INTRODUCTION OF A
Upon completion of GR, Einstein applied his theory to the entire universe. He firmly
believed in Mach's principle, and the only way to satisfy this was to assume that spaceis globally closed and that the metric tensor should be determined uniquely from theenergy-momentum tensor (Reference 13). He also assumed that the universe wasstatic, which was a reasonable assumption at the time because observational
astronomy had not advanced to a level that contradicted this paradigm. In 1917, whena static solution to his equations could not be found, he introduced the cosmologicalconstant A (Reference 14): 3

1,

2

greG
= 4 +

(3.1)

In this equation R„, is the Ricci curvature tensor, R is the Ricci curvature scalar, T,„ isthe stress-energy-momentum tensor,4 and gl „ is the spacetime metric. The left-handside of Equation (3.1) encodes the curvature in the geometry of spacetime, and theright-hand side encodes the source of matter-energy that curves spacetime.
The addition of A can be understood as a term in the equation which allows one toadjust theory to match observation. In Einstein's case, he chose to add A to ensure thatthe universe was static and unchanging. In later years, he often referred to thisamendment to his equations as his "biggest blunder." Several years after GR had beenformulated, the astronomer Edwin Hubble discovered the phenomenon of galactic
redshifting, which strongly indicated that the universe was indeed expanding. This
3 Pronounced "lambda."" T„, encodes the density and flux of a matter source's energy and momentum.

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UNCLASSIFIED//Pentheoretical prediction from GR was ignored by Einstein because of his belief in a static
universe.
Even though Einstein retracted the addition of A into his equations, it is now known thatit does indeed play a role and is typically included in GR equations. Data from precise
astronomical observations strongly suggest that an extremely small, yet non-zero A is anecessary feature of GR and is responsible for the expansion of the universe that isobserved.
From a physical perspective, A represents an inherent energy density associated withempty space. One way to envision this is to take a perfectly insulating box into deep
space, and then to remove all matter and all energy from this box so that it encloses a
perfect void. Even in this emptiness a residual energy field would remain. According to
GR, the effect of this energy would be to cause the region of space to expand, albeit atan extremely small rate. To summarize, A is a ubiquitous, ever present feature ofspace, and its presence causes space to expand.
In the late 1990s it emerged that not only is the universe expanding, but the rate of
expansion is, in fact, increasing. Since then, it has become more popular to refer to A
as dark energy, and the remainder of this paper will follow this convention.
Although the role of dark energy is extremely well understood mathematically, and in
the context of its effects on spacetime, its physical nature is still a mystery. One knowsthat it is homogeneous, not particularly dense, and that it does not interact with any of
the fundamental forces of nature. One also knows that it exerts negative pressure on
spacetime, which explains the observed accelerated expansion (Reference 15, 16). As
there is yet to be a reasonable explanation for the fundamental origin of dark energy,
the problem is considered serious and has been tackled by a large number of eminentand respected physicists, including previous Nobel prize winners (Reference 17).
Because dark energy is intimately related to the expansion of space, and because thisexpansion is exactly the feature that would allow for a warp drive to function, an
understanding of this mysterious energy is of paramount importance in thedevelopment of this novel propulsion technology.

4. Casimir Energy and the Quantum Vacuum
A central theme in this paper is the notion of the quantum vacuum. To a particle
physicist, the term "vacuum" means the ground state of a quantum field in somequantum theory for matter. In general, this ground state must obey Lorentz invariance,at least with regards to three spatial dimensions, meaning that the vacuum must lookidentical to all observers.
At all energies probed by experiments to date, the universe is accurately described as aset of quantum fields. To a non-physicist a quantum field may, at first, be a strange
concept to grasp. This is because one generally likes to visualize the things one thinks
about; for example, an electron and even a photon provides something one can, on
some level, picture in one's minds. Simply put, a quantum field is an intangible
mathematical object whose properties are ideal in explaining nature. Theories havereached such an advanced level that the familiar physical images that one appreciates

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must be abandoned for more erudite mathematical constructions which are better
suited at describing the building blocks of nature (Reference 18-20).
If one takes the Fourier transform of a free quantum field,5 each mode of a fixedwavelength behaves like a simple harmonic oscillator. A quantum mechanical property
of a simple harmonic oscillator is that the ground state exhibits zero-point fluctuations
as a consequence of the Heisenberg Uncertainty Principle. One way to understand thesezero-point fluctuations is to imagine releasing a pendulum and watching as dissipative
forces slowly try to bring the pendulum to a stop. The uncertainty principle would
ensure that the pendulum was never able to come to a complete rest, but insteadwould exhibit microscopic oscillations around the equilibrium position indefinitely. Of
course, for a real macroscopic pendulum, these fluctuations would be miniscule and all
but impossible to detect; however, the analogy with a quantum harmonic oscillator
holds well. The expectation value of the energy associated with the ground state energyof a quantum oscillator is:(E)=Tik„2(4.1)

In this formula c and h are the speed of light and Planck's reduced constant (1.055 x
10-34 1s), respectively, and k is the wave-vector related to the momentum of thequantum field. One of the features of this ground state energy is that the wave vector
has an infinite degree of freedom. Clearly this sum is divergent; however, this is acommon feature of quantum field theory, and an array of mathematical techniques
known as renormalization exists to deal with the infinities that arise.

4.1 THE CASIMIR EFFECT
The quantum fluctuations of the vacuum fields give rise to a number of phenomena;
however, one is particularly striking. The Casimir Effect, which will be explored in moredetail in this paper, is arguably the most salient manifestation of the quantum vacuum.
In 1948, H. Casimir published a profound paper where he explained the van der Waals
interaction in terms of the zero-point energy of a quantized field (Reference 19). In its
most basic form, the Casimir Effect it is realized through the interaction of a pair ofneutral parallel conducting plates (with separation distance d). The presence of theplates modifies the quantum vacuum, and this modification causes the plates to bepulled toward each other with a force:

F—

240d4

(4.2)

This is a profound result in the sense that the origin of this force cannot be traced backto one of the four fundamental forces of nature (gravity, electromagnetism, and the twonuclear forces), but is a force that is entirely due to a modification of the quantum
vacuum.

5 By "free" we mean that the field does not interact with other fields.

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Figure 2. The Interior Region of Parallel Conducting Plates. The region experiences a reduced quantum
vacuum energy density owing to the boundary condition the plates impose on the fields. This generates ameasurable attractive force that pushes the plates together.

For many years, the paper remained unknown (Reference 22), but from the 1970s
onward the Casimir effect received increasing attention, and over the last decade it has
become very popular (Reference 23). The Casimir effect is a purely quantum effect. In
classical electrodynamics the force between the plates is zero. The ideal scenario occurs
at zero temperature when there are no real photons (only virtual photons) between theplates; thus, it is the ground state of the quantum electrodynamic vacuum whichcauses the attraction. The most important feature of the Casimir effect is that eventhough it is purely quantum in nature, it manifests itself macroscopically. For example,
for two parallel plates of area A = 1 cm2 separated by a distance of d = 1 pm the force
of attraction is F z 1.3 x 10 7 N. This force is certainly within the range of laboratory
force-measuring techniques.
Typically, the calculations of the expectation value of the vacuum are divergent,6 sosome form of renormalization must be performed. A full review of the experimental
verifications of the Casimir effect are beyond the scope of this paper, but it is certainly
worth mentioning that experiments at Washington University using ultra-sensitive
Atomic Force Microscopes have experimentally verified the theoretical predictions of the
Casimir force to within 1 percent accuracy (Reference 24, 25). Needless to say, many
physicists consider this to be a real and well established phenomenon.

6 Divergent meaning the equation predicts an infinite result. Divergences are typical in many calculations using
quantum field theory, and an array of ingenious tools is used by physicists to extract finite and meaningful results.

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In summary, quantum field theory predicts that the vacuum is an interlaced cobweb ofquantum fields which are never strictly at rest, and which exhibit zero-point
fluctuations. These fluctuations give rise to real and measurable phenomenon, with the
Casimir effect being the most poignant. It seems only natural to attempt to relate theideas from the previous section regarding a ubiquitous dark energy field to thisquantum vacuum energy. If a relationship can be established, one would be a step
closer to the technological realization of warp drive.

5. Extra Space Dimensions
In connection with the Casimir effect, extra dimensions provide a rich arena for one togenerate models that explain the origin of dark energy. Technically speaking, theCasimir effect is a direct consequence of the non-trivial boundary conditions that thepresence of the conducting plates imposes upon the quantum vacuum. The quantum
modes on the interior region of the plates are restricted, and there is a pressure
difference when compared to the quantum vacuum on the exterior region of the plates.
It is this pressure difference that causes the plates to attract.7
A very similar phenomenon to the Casimir effect can occur when the quantum vacuum
energy in extra space dimensions are considered. The exploration of this idea has
important ramifications in the context of explaining dark energy. Before one canaddress these ideas it is necessary to review the role of higher space dimensions inphysics.
It was Riemann, with his development of differential geometry in the 19th century, whoprovided the necessary tools to study higher dimensional descriptions of the world(Reference 26). Riemann held the belief that 3-dimensional space was not enough toprovide an adequate description of nature. Improvements in physics led to Maxwell'sunified theory of electricity and magnetism, and then GR, which unified space and timewith Special Relativity (SR). Inspired by these unifications, physicists of the early 20th
century wanted to unify gravity and electromagnetism. The first attempt was by
Nordstrom in 1914, who used a scalar potential for the gravitational field. Later Weyl
and Kaluza, using Einstein's tensor potential, followed two separate paths. Weyl's
attempt involved an alteration of the geometry of spacetime in four dimensions. His
early attempts had physical consequences which did not match experimental data.
However, Weyl's work was extended by Einstein and Schrodinger independently in the
Einstein-Schrodinger non-symmetric field theory, which is widely regarded as the mostadvanced unified field theory based on classical physics.

5.1 KALUZA-KLEIN THEORYIn 1919 Kaluza (Reference 27) offered a unique approach to unifying gravity andelectromagnetism which involved adding an additional spatial dimension to GR, andpopulating this extra dimension with two mathematical objects called a vector potentialA,„ and a scalar potential 0. The line element in this theory is given by:' This analogy is not strictly true as different geometries can, in fact create repulsive Casimir forces and so thepressure analogy breaks down. It is, however, a useful visualization tool,

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UNCLASSIFIED//UNCLASSIFIED/MasalaWafrablaF0( g 'iv+ A .A.0)dx"dx r + 20 - &cif dy + dy 2

(5.1)

where the Greek indices run from 0 to 3 (0 represents the time coordinate and 1...3 thespace coordinates), and where the higher dimension is expressed using y. The powersof 0 are selected for later convenience. Upon solving Einstein's equation in this higher
dimensional space using this metric, Kaluza was able to accurately reproduce both GR
and electromagnetic theory. However, the theory made an additional prediction which
was not consistent with observation and was somewhat of an embarrassment for
Kaluza.
The main failure of Kaluza's idea, and anissue that caused Einstein to delay his
endorsement of the paper by two years,
was the very obvious fact that we clearly
exist in three spatial dimensions and not
four. This issue was tackled in 1926 byOskar Klein (Reference 28), whosuggested that the 581 dimension
compactifies, so as to have the geometryof a circle of extremely small radius. One
way to envisage this additional dimension
is to imagine a garden hose. It is only
when one magnifies the image thetoroidal structure is visible. From a long
distance it looks like a 1-dimensional line,
but a closer inspection reveals that every
point on the line is, in fact, a circle (seeFigure 3 for an illustration).

Figure 3. Internal Structure of a Seemingly OneDimensional Object

While Kaluza-Klein (KK) theory is considered elegant in its simplicity, it is not without
problems. One obvious criticism is that the theory is non-predictive, in that it does notextend Einstein's or Maxwell's theories, but merely synthesizes the formalism within a
new mathematical framework. A more serious criticism regards the introduction of the
5th dimension, which was, and still is, seen as an artificial construct since our universe
is apparently 4-dimensional.8
KK theory remained largely ignored and was considered somewhat obscure for the firsthalf of the twentieth century, as were the speculations regarding additional spatial
dimensions. However, the birth of string theory generated a renewed interest in theidea, largely due to string theory's promise of being a quantum theory of gravity.8
Unification is one of the main themes in the history of science and is a guiding principle
in theoretical physics. Countless examples exist where diverse and seemingly unrelated
phenomena have been understood in terms of a small number of underlying principles.
In the 1940s, it was demonstrated that quantum mechanics and electromagnetism
could be accurately described by quantum field theory, and by the 1970s the weak and89Including time, of course.
Quantum gravity is considered somewhat of a "holy grail" within the theoretical physics community.

9

UNCLASSIFIEDUNCLASSIFIED//lW
strong nuclear forces could also be described using QFT. The full theory, called theStandard Model of particle physics, is arguably the most successful physical theory todate.
Despite the successes of the Standard Model, incorporating gravitational interactions
using the methods of QFT has proven to be one of the most challenging problems facing
theoretical physics today. String theory is currently the best candidate for a quantum
theory of gravity (Reference 29). The divergences associated with point particle
gravitational interactions are removed in string theory via the extended nature of thestring. A review of string theory falls far outside the scope of this paper; however, thetheory contains numerous compelling features that deserve mention here. One of themost discussed aspects of string theory is its prediction that additional dimensions ofspace exist, and are indeed required for string theory to work. One of the predictions ofstring theory is that these additional dimensions are extremely small, on the order of
10-15 m. However, there are also popular models which attempt to explain certain
aspects of nature that involve large extra dimensions which we will briefly introduce.

5.2 LARGE EXTRA DIMENSIONS
Models with large extra dimensions have enjoyed a revived interest in physics. This
began with the Arkani-Hamed, Dimopoulos, and Dvali (ADD) proposal to lower thequantum gravity scale to the 10 to 100 TeV scalel° (accessible to the next generationof particle accelerators) by embedding the Standard Model fields in a 3+1 dimensionalbrane existing in a higher dimensional bulk spacetime (Reference 30, 31). Gravity is
free to propagate in the bulk, which effectively dilutes its strength. This idea wasinspired by M-theory, where it was recognized that the scale of quantum gravity could
be lowered from the Planck energy scale to the Grand Unification Theory (GUT) energy
scale (Reference 32-35).11 The assumptions underlying the ADD model are:•n-extra dimensions compactified on a torus with volume V. = (bin".

•Standard Model fields are localized to the brane.

•

Gravity can propagate in the bulk.

•

There is no cosmological constant in the bulk or on the boundary.

•

The brane is stiff.

The bulk action for this model can be given by:

Sh.rIA

2

.

"x

(5.2)

where g is the matrix determinant of the metric and the two tilde-quantities are the4+n dimensional Planck mass and Ricci scalar, respectively. By integrating out theextra dimensions, it is simple to show that:19 1 TeV = 1012 eV (1 eV = 1.602 k 10-19 3) is a mass-energy scale used in elementary particle physics" Planck energy (hc9/G) 29 > 1099 eV; GUT energy > 10'4 GeV (1 GeV = 109 eV).

10

UNCLASSIFIED/PPOWIIK#141444WISFSMIrY

UNCLASSIFIED/MeffeBMW&Wie*MtaF
M= (2.2)/1)4 "I(5.3)

Mr, is the Planck mass12 (in our usual 4-dimensional spacetime) and r is the size of theextra space dimensions. This remarkable result indicates that the Planck scale is, infact, a quantity that is derived from a more fundamental quantum gravity scale andalso the volume of the extra dimensions. Physically, this implies that the graviton isdiluted across the bulk with a diminished intersection with the familiar 3+1 dimensionalbrane.
In this type of model the size of the internal space is of order 1/M and the effective
cosmological constant is on the order Aril — 8/77M,

X 104 (Reference(Reference 36, 37). Setting

the value of the corresponding energy density equal to the known density of darkenergy, one finds that the extra-dimensional radius is r
10-3cm. Similar results arefound in models with more complicated internal spaces. This is an important result in
the context of dimensional manipulation.
Note that extra dimensions that are accessible to all the Standard Model fields can also
be realized. These models are known as Universal Extra Dimensions (UED). In the caseof recent experimental constraints, a compactification scale as low as 1 TeV is allowed.

5.3 RANDALL SUNDRUM BRAME MODELSThe idea that the universe can be modeled as a (mem)brane existing in a higher
dimensional bulk spacetime has received a huge amount of attention in recent years(Reference 38-46). It is possible that the brane energy density affects the spacetime
curvature, and an approximation can be achieved by first considering a model wherebranes are located at the two ends of a periodic 5th dimension. To ensure stability of themodel two branes are required to balance the bulk energy. To get a stable metric, theeffects of the brane on the spacetime must be compensated by a negative cosmologicalconstant in the bulk. Thus, the 5th dimension can be considered a slice of Anti-deSitter(Ads)13 space bounded by flat branes, and the price of keeping the branes flat is tointroduce curvature into the 5th dimension. Such models are termed warped extra
dimensions.
The Randall-Sundrum (RS1) model (Reference 47, 48) proposes a novel geometricalsolution to the hierarchy problem. The hierarchy problem questions why gravity is somuch weaker than the weak force (which is 1032 times stronger), and why the Higgs
boson is so much lighter than the Planck mass. In the RS1 setup, the Standard Model
fields are now confined to one of two 3-branes which lie at the endpoints (i.e., fixed
points) of an S1/Z2 orbifold,14 except for the Higgs field. One of the branes physicallycorresponds to "our" universe and is sometimes referred to as the IR or "visible" brane.
The closer a Standard Model field is to the visible brane, the greater its coupling to the2.18 x 10 8 kg.
Anti-deSitter space is a Lorentzian manifold with a constant negative scalar curvature. In terms of General
Relativity, this is a solution to Einstein's field equation with an attractive cosmological constant.
'4 An S1/Z, corresponds to a circular extra dimension with an additional symmetry. This type of projection is
popular in a number of higher dimensional models due to its ability to mathematically "project' out certain
phenomenologically undesirable fields.
12

13

11

UNCLASSIFIED/re" Q"I^In "Cc ^1'Y

UNCLASSIFIED/PreffeenatetAt

ais

Higgs, and therefore the greater the mass. The second brane is the UV or "hidden"
brane. The line element in F151 is described by the metric:— r 2c10 2 .

ds° =e° 21° ° ri p,dx"(5.4)

wr is the metric tensor for the D-dimensional Minkowski spacetime, the AdS curvatureradius is given by l/k and the interbrane separation is given by r. The 5th dimension iscompactified on the orbifold of length a, where a << a. The orbifold fixed points

located at cp= 0 and 9= a correspond to the location of the visible and the hidden
brane.
The exponential factor is referred to as the warp factor and is an appealing feature in
the RS1 model, as it can both generate a TeV mass scale from the Planck scale in thehigher dimensional theory and reduce the effective gravitational strength on the visible
brane through the suppression factor ewhile retaining a bulk width that is only acouple of orders of magnitude above the Planck scale. In addition to the stabilization of
the interbrane separation, the quantum effects from the bulk fields (aka the Casimirenergy) can also provide a mechanism for the generation of dark energy on the visible
brane.

5.4 EXTRA DIMENSION SUMMARY
Although there is still no direct evidence of extra spatial dimensions, there is thepossibility that the experiments planned at the Large Hadron Collider could detect
particle decay signatures that would indicate the presence of higher dimensions. In fact,
all the theoretical groundwork has been performed by theorists, and as soon as the LHC
is running smoothly it could, in principle, demonstrate the existence of higher
dimensions in a relatively short period of time. Any such discovery would represent atruly radical alteration of our understanding of nature, and many new questions will
emerge regarding the potential role extra dimensions could play in advanced
technologies (which will be discussed in more detail later in this paper).

6. Dark Energy as a Higher Dimensional Artifact
As discussed earlier, the vacuum of spacetime can be visualized as a sea of quantum
fields never fully at rest due to the Heisenberg Uncertainty Principle. The oscillations of
the vacuum radiate energy over a range of frequencies in much the same way that an
oscillating electron emits electromagnetic energy, and in this way, there exists a groundstate energy associated with space itself. The vacuum potential for a periodic scalar
field in the ADD model described above is given by (Reference 48-50):
.y2
in
„ log

V+

2

" t2iry

k2

n'T
R

in°

(6.1)

where k is the momentum modes of the quantum fields, R is the radius of the 5th
dimension, and in is the mass of the field. The prime on the summation indicates then = 0 term is excluded. The integral over the continuous momentum modes of the
12

UNCLASSIFIED/jaRmeaiallilmliSidiNieF

UNCLASSIFIED/PPefrePPI

Pter

quantum fields is divergent, as is the infinite summation over the extra-dimensional KK
modes; however, as mentioned earlier, dimensional regularization (a form ofrenormalization) can be used to extract a finite result. This equation differs from
Equation (4.1) in that the quantum field that we consider has both mass and a degreeof freedom in the higher dimension. Working with a scalar field, the result can later be
extended simply for the case of more phenomenologically viable fields (fermions, forexample).
Our own research focuses upon exploring a new way to handle the infinities arisingfrom Equation (6.1), and after performing a novel regularization it was discovered that:
(5 )
2

V' —

2

V

(6.2)

(2m rn)3' yr - r"-I

where Cis the Riemann zeta function, and K50(2mrn) is the modified Bessel function of
the second kind. Although the summation is infinite, the function converges rapidly andso a good approximation is obtained by performing the sum up to n = 10. Since
discovering this formula, the result agrees with derivations of this energy based on
different regularization methods and so one is confident in the validity of Equation(6.2).
It is relatively straightforward to calculate the contributions to the vacuum energy
density coming from each field in the Standard Model of particle physics. For example,
the electron is a fundamental quantum field whose ubiquitous ground state energy
contributes to the vacuum energy density. Similarly, the photon is a fundamental
quantum field whose ubiquitous ground state energy also contributes to the vacuum
energy density. In fact, all Standard Model fields contribute a finite and calculable
component to the overall energy density of space.
Equation (5.1) expresses the vacuum energy density for a periodic massive scalar field.
Using knowledge of supersymmetry multiplets it is possible to enumerate this energy
for all fields occurring in the Standard Model (Reference 49):(r) = —4V - (r ) ,15(r) = — V + (r)

V

4
JIFg

(6.3)

= 2V (r) .

Also, knowledge of the vacuum energy density for a massless field:

K(5)

(6.4)

mg2r"
will allow one to fully articulate the energy density of the vacuum in terms of thebuilding blocks of nature. This is expressed algebraically as:

13

UNCLASSIFIED/Elle

UNCLASSIFIED//(6.5)

where the index i runs over the spectrum of Standard Model fermions. Using Equation(6.5), and slight variations, we are able to computationally build a model which
demonstrates how the vacuum energy density varies as a function of higher
dimensional radius. It cannot be overstated that a full understanding of the vacuum
structure is of critical importance when attempting to understand the nature of darkenergy and to investigate its possible manipulation.
It is important to appreciate that, for example, a physical electron does not actually
have to be present at a specific point in space for it to contribute to the vacuum energy
density. But that the vacuum always has the potential to allow an electron to exist at
any point in space. Thus, at all points in space a virtual electron exists. This virtualelectron is a basic and fundamental feature of the intrinsic makeup of spacetime itself.
In this way, empty space has a ground state energy that is due to the virtual
contributions of all fields that occur in nature: electrons, quarks, photons, and indeed,
the entire particle zoo.
Many attempts have been made to relate this vacuum energy to dark energy; however,
because these quantum fields are free to oscillate over a wide range of possible
frequencies, when one calculates the sum15 of all the contributions from all possible
frequencies of the vacuum, an energy density far in excess of that seen in nature isrecovered (Reference 50).
Our own work (Reference 51, 53) has demonstrated that when the contribution due tothe extra-dimensional quantum vacuum fields is included, it is possible to "tune" thetheoretical energy density of the universe to agree with experimental observations
using extensions of Equation (6.5), provided allowance for certain exotic fields to exist
within the higher dimension. Although this may at first appear counterintuitive, one
novel feature of the quantum vacuum energy is that it can contribute both positive andnegative energy to the vacuum. The sign of the contribution is fundamentally due tothe nature of the underlying virtual quantum field. For example, virtual fermionic fields(e.g., electrons) contribute an overall positive energy to the vacuum, whereas virtual
bosonic fields contribute an overall negative energy. In this way, certain fieldcombinations allow for energy cancellations. The additional freedom encountered inhigher dimensional theories means that it is a fairly straightforward matter to adjust
the overall vacuum energy density to agree with the experimentally measured value forthe cosmological constant.
Essentially, this means that one is immediately presented with a natural explanation for
the existence of dark energy. Previous attempts to link dark energy to the vacuum
energy had yielded grossly high theoretical predictions far in excess of that observed innature; however, by including the contributions from higher dimensional fields we haveshown that the taming of this dark energy density is entirely possible. The significanceof this result is that it provides a foundation upon which to explore possibilities relating
to warp drive propulsion. More simply, once one knows why space expands, it becomes
possible to explore technological possibilities to potentially make space expand.
15

Technically we integrate over all the possible frequencies.

14

UNCLASSIFIED/jaGSWiralsklrkkrellef

UNCLASSIFIED/ / r011 ornamt-r
To summarize, the existence of dark energy may be attributed to the combined effectsof vacuum fluctuations in "normal" 3+1 dimensional spacetime plus higher dimensional
contributions. If this model proves to be correct, then ideas extending from thisparadigm could provide one with intriguing opportunities for technological intervention.

7. Warp Drive and Higher Dimensional Manipulation
Both our own research and previous work in higher dimensional Casimir energy16
demonstrate that the magnitude of the vacuum energy is intimately related to the sizeof the extra dimension. More precisely, the smaller the extra dimension, the greater the
Casimir energy (and vice versa: the bigger an extra dimension, the smaller the Casimirenergy). In fact, the energy is related to the radius of the higher dimension raised to
the fourth power, which means that very small changes in the radius of the extra
dimension generate dramatic changes to the vacuum energy density.

Figure 4. Manipulated Extra Dimension. A sufficiently advanced technology with the capacity to directly interact
with and manipulate an extra dimension would be able to locally adjust the dark energy density in a given region of
spacetime.

What this means is that if an advanced technology was able to influence the radius ofan extra dimension, then it would acquire direct control over dark energy, and hence
the expansion and contraction of space itself. As tremendous a feat as this may sound,at this early stage in the research it is one of the only viable mechanisms to generate awarp drive.

IS

Quantum vacuum energy and Casimir energy are often used interchangeably in the literature.

15

UNCLASSIFIED/ laRditaalre+PrWiteeentail

UNCLASSIFIED
It is worthwhile to expand more on the concept of what we mean by adjusting theradius of the extra dimension.

7.1 ADJUSTING HIGHER DIMENSIONS FOR PROPULSIONS
Figure 4 should assist in the visualization of what a higher dimensional space mightlook like. In this 1-dimensional example one can see that all points have an associatedhigher dimension. One assumes that generally the radius of the extra dimension isfixed. It is this fixed radius which generates the observed dark energy density and is
responsible for the homogeneity in the observed expansion of the universe.
If one were to locally adjust the higher dimensional radius then the dark energy density
would also change locally. More specifically, if one were to adjust the radius of the extra
dimension in the direct vicinity of a spacecraft, then the dark energy density would alsochange only in the vicinity of the spacecraft, as would the expansion of space. It is
important at this point to appreciate that globally, the universe would continue toexpand at the rate we observe today, but that only in the proximity of the spacecraft
would space be "stimulated" to expand at some modified rate.

Figure 5. Artist's Conception of a Futuristic Warp Drive Spacecraft. By locally adjusting the size of the extra
space dimension, the spacecraft is able to generate the necessary warp bubble required to surpass the light speed
barrier.

Of course, obvious questions present themselves. Can one really assume that a higher
dimension is circular? Also, if dark energy is responsible for the expansion of space, can
we assume that it can somehow be used to contract space, and not just expand it?

16

UNCLASSIFIEDHalefarakfrOrnmentaF

UNCLASSIFIED// POWSIFF)6+1.6freillfr*
7.2 THE GEOMETRY OF EXTRA DIMENSIONSTo answer the first question, our most developed higher dimensional theory (M-theory)
works in seven additional spatial dimensions. The shape is believed to be whatmathematicians call a Calabi-Yau manifold - a complex object that is notoriously
challenging to work with. Physicists often like to work with simpler models (a single
extra dimension, for example) with an uncomplicated shape like the circle. Although
this may at first appear to be a gross simplification, often these simplistic higher
dimensional models both reflect the flavor of the physics involved, and give accurate
predictions that are believed to deviate from nature only at extremely high energies.17
For this reason, many of the research
papers investigating higher dimensions
choose to work in the simpler circularhigher dimensional space. A smaller
fraction of papers explore two additional
higher dimensions, which are commonly
toroidal (see Figure 6), and an evensmaller fraction of papers work in the fullM-theoretic Calabi-Yau manifold. At theseearly stages of investigation, theadditional circular dimension represents
an adequate approximation. Should theresearch progress to a more highly
developed phase, then it may become
necessary to work within the Calabi-Yau
manifold.
Figure 6-A Toroidal Higher Dimension. This is one

With regards to the question of whetherof the many possible topologies explored in higher
dimensional theories.
space can be made to contract, it seems
possible if one can make the energy
density of a given region of space negative instead of positive. This type of space has
been well explored by physicists, and is known as anti-deSitter space. One of theunique features of Casimir energy is that under many conditions it is known to be
negative, and thus with a careful manipulation of the higher dimensional fields it is, inprinciple, possible to generate the required contraction of space.

7.3 HIGHER DIMENSIONS AND STABILIZATIONOur goal in this section is to explore the possibilities of manipulating a higher
dimension, which will influence the local dark energy density and thus the expansion
and contraction of spacetime in the vicinity of a spacecraft.
Before the issue of how to manipulate a higher dimension can be addressed, first one
must understand why an additional spatial dimension holds some fixed radius. This is awell know problem in higher dimensional physics and is commonly called the problem of"modulus stabilization." Broadly stated, the question is as follows: if there are
additional spatial dimensions, why do they not perpetually expand, like our familiar
dimensions of space, or alternatively, why do they not perpetually contract? What
17 For example, in the vicinity of a black hole, or in the first moments of the big bang.

17

UNCLASSIFIED/haL

UNCLASSIFIED//r91 er.1r1. "C'"mechanism is it that allows for this higher space to remain compact and stable? Of thehandful of theories that attempt to answer this problem, one is particularly appealing
due to its naturalness.
A recurring theme throughout this paper is Casimir energy. This energy is compelling
due to the fact that it is a natural feature intrinsic to the fabric of space itself. Casimirenergy in higher dimensions not only offers the promise of explaining the nature of darkenergy, as discussed in previous sections, but can also be utilized as a mechanism tostabilize the compact extra dimension.
We have discovered in our research that with a natural combination of fields, and asingle exotic component, we are able to generate a stable extra dimension (Reference
51, 53). The way to understand this is as follows: as mentioned earlier, each field innature contributes a component to the ground state of the vacuum (that is, the state ofminimum energy of space) due to the virtual particle contributions. When this energy is
calculated for the case of three large and one compact spatial dimension, one discovers
that this energy is a strong function of the size of the 5tI dimension.
In our research, we summed the contributions from all the known fields in nature using
extensions of Equation (6.5). We discovered that with the addition of one additional
exotic field confined to exist in only the 5th dimension, a stable higher dimensional
configuration was found. Our motivations for the addition of extra exotic fields have astrong phenomenological foundation, and these exotic fields been studied successfullyin the context of explaining solar neutrino oscillations (Reference 54-56). A full
discussion, however, is beyond the scope of this paper.
To understand what is meant by a stable higher dimensional configuration we refer toFigure 7, which is a plot illustrating the vacuum energy density as a function of higher
dimensional radius.
The x-axis represents the radius of the higher dimension and the y-axis the energy
density of spacetime.18 We have shown the contributions to the energy density comingfrom the different fields of the Standard Model, and also the additional exotic field. The
overall energy density is illustrated as a thick black line. The most important feature ofthis graph is the mathematical minimum. A good physical analogy here is to imagine
releasing a ball from the far left of the black line: the ball would roll down the line andbecome stuck in the minimum. The ball becoming stuck in this minimum is an excellent
analogy to the dynamics of the higher dimension, which begins at some unstable state,
but evolves into a stable configuration.

IS

We work in "normalized" units.

18

UNCLASSIFIED/

sis

UNCLASSIFIED//

6

Figure 7. A Combination of Phenomenologically Viable Fields Generates a Stable Minimum of the
Vacuum Energy Density at a Fixed Higher Dimensional Radius. This stable minimum, located at a positiveenergy, indicates a stable deSitter type spacetime and reflects a realistic dark energy model. Note in this plot thatthe radius of the extra dimensions is shown on the x-axis, and the enemy density on the y-axis (normalized units
are used).

- - Ailitical minimum

Natural muumunt

Anifical minimum

Figure 8. False Vacuum Minima are Created Around a Spacecraft. The different minima create an
asymmetric dark energy pressure on 3-space which generates the warp bubble. In this plot the x-axis represents
the radius of the extra dimension, and the y-axis represents the energy density (normalized units are used).

The key to creating a warp drive is to create a false vacuum minimum, i.e., to modify
the vacuum spectrum and inject some field which creates a deSitter minimum at therear of the craft and an anti-deSitter minimum at the front of the craft. What thisrequires is a technology that would allow us to artificially manipulate the field content
illustrated in Figure 7, shifting the location of the minimum. In this basic
representation, the spacecraft would sit in a stable region of space corresponding to thenatural minimum of the extra dimension. At the front and rear of the craft, regions of
19

UNCLASSIFIED/MIRS•f44+1

U N C LA SSI FI E D / frPORsOPFIreshir&eifffrOML
false minima would be artificially created via the adjustment of the extra dimension(see Figure 8). These modified regions would correspond to increased and negative
dark energy densities, thus creating the warp bubble previously discussed.

7.4 ELEMENTARY WARP DRIVE CALCULATIONSThis section will explore the calculations relevant to propulsion. Specifically, it will
determine the energy required to accelerate a spacecraft to the speed of light. By
associating the cosmological constant with the higher dimensional Casimir energy, astraightforward relation between A and the radius of the extra dimensions can be
determined:
1
R4

(7.1)

A simpler way of developing the relationship between the energy density of space andthe expansion of space is to express A as a function of Hubble's constant, H:
H cc Al A ,(7.2)

which is a standard result obtained from GR. From these two formula, a relationshipbetween the expansion of space and the radius of the extra dimension can be shown:(7.3)

This formula really expresses the foundation of this novel warp drive concept: that asufficiently advanced technology with the ability to adjust the radius of the extra
dimension locally would be able to locally adjust the expansion and contraction of
spacetime around a spacecraft. This asymmetric expansion creates the warp bubble
illustrated in Figure 1. The spacecraft would always move within its own light cone andthus would not contradict any law of special relativity. The possibility that the higher
dimensional radius might vary from place to place has been explored in the context ofstring theory (Reference 57), and so is a valid academic pursuit. However, it has neverbefore been suggested that this might facilitate a new and exotic form of propulsion.
Equations (7.1) and (7.2) used together express a relationship between the expansionof space and the energy density of space. A sufficient increase in the energy density ofspace would generate a proportional increase in the expansion of space. Thus calculate
the energy density of space that would be necessary to generate a local expansion of
space at the speed of light. A local expansion at the speed of light, Hubble's constantmust be increased by a factor of:
1-1, =10 26 H ,(7.4)

where H, represents the modified Hubble's constant, and the subscript c indicates that itis the Hubble's constant for the case of light speed expansion. H must be increased bya factor of 1026 to achieve a local expansion of space equivalent to the speed of light.
20

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Another relevant calculation regards the local energy density required to generate thisexpansion. A similar calculation to the one just performed indicates that thecosmological constant, an expression which characterizes the energy density of space,must be increased by a factor of 1052. This implies an energy density of:
Pe 1042jim3(7.5)

This is, indeed, an incredible number. However, the total energy requirement would be
reduced if we assumed a "thin shell" of modified spacetime. Figure 9 illustrates the casewhere the warp bubble thickness is reduced. More detailed research is necessary before
we could accurately predict the minimum shell thickness that would be necessary tosupport a stable warp field. However, if a shell could be produced that was merely asingle Planck length in thickness, then the energy requirements would be reduced
immensely.

U N
10

10

lc

5

Figure 9. Two Images Illustrating the Contrast Between a Thick (left) and Thin (right) Shell Warp
Bubble. A thin shell warp drive model has the capacity to dramatically reduce the energy requirements of a warp
drive.

Under this new paradigm of warp drive, where the necessary contracting and expansionof spacetime is generated by a manipulation of the radius of the extra dimension, one
can clearly see that the energy requirements are reduced immensely when compared tothe calculations of Lobo and Visser (Reference 12). Table 3 illustrates the energy
requirements for a range of multiple of the speed of light. All calculations are order-ofmagnitude.

21

UNCLASSIFIED/SEIMPILAUNCLASSIFIED/MNIffeffilatetAteTable 3. Negative Energy Required for Warp Bubble
Warp Factor, vwarp

Ewarp M

i0-5 (= 3 km/s)—1.00 x l0"

10-4 (= 30 km/s)
0.01 (= 3,000 km/s)
0.5 (= 150,000 km/s)—1.00 x 1034

1 (= light speed)—1.00 x 1042

2 (= 600,000 km/s)—4.00 x 1042

10 (= 3.0 x 106 km/s)
100 (= 3.0 x 107 km/s)—1.00 x 1044
—3.03 x 1046

—1.00 x 1038
—2.50 x 104l

Upon comparing Tables 2 and 3, immediately one can see the drastic energy reductions
that are apparent when one uses the dimensional warp drive paradigm. The energy
requirements are reduced by a factor of 108. This energy is based on a warp bubble
that encompasses 100 m3 of space. As discussed earlier, this is a "worst case" scenario,
and the energy requirements could be further reduced, perhaps by many orders ofmagnitude, by utilizing the thin shell model illustrated in Figure 9.
These are, perhaps, the most important numerical results of this paper, as they set an
upper limit on the energy requirements necessary to generate a warp bubble, and also
on the energy requirements necessary to surpass the speed of light. Even though thisenergy requirement is a vast improvement on the calculations of Visser and Lobo, theenergies are still far in excess of those available in the foreseeable future.
One interesting prospect to test this theory is to consider the maximum energy densitythat is achievable by modern technology, and to calculate the expansion of space that
this energy would generate. One could then conceivably contemplate a table-top
experiment if the numbers allowed. What follows is a description of how this mightwork.

7.5 FUTURE EXPERIMENTS
At this point we take a diversion into pure speculation as to what technologicaladvancements may be necessary to build a device that might test our dark energy
theory. One knows from the Friedmann equations of cosmology that normal matter andenergy generate:iia

4RG(p -43P),
3

(7.6)

where a is the so-called cosmological scale factor, p is the energy density and p is thepressure. If one defines the quantity "= pip", one sees immediately that an accelerated
expansion of spacetime requires negative pressure. Currently, all known matter andenergy generate w > 1; however, dark energy has w = -I. If it were possible totechnologically create dark energy in the lab, then this would assist with our

22

UNCLASSIFIED/PPEN

EPMeirUNCLASSIFIED

iseer

understanding and experimentation with warp drive technology. To understand how,
first discuss an analogy with energy w > 1.
In a handful of labs scattered around the country, petawatt lasers are being built andtested. These are lasers of profound capability, able to generate short laser pulses ofintensity » 1015 W/m6, with peak focused power densities of >> 1067 W/m3, andenergy densities of >> 1016 3/m3. If it were possible to construct an exotic device
whose energy output had w < 1, then using calculations identical to those to generate
Table 3, it would seem straightforward to show that the energy density produced by ananalogous dark energy laser would translate into a local expansion of spacecorresponding to v,„ip= 10-13 or — 10-5 m/s for every meter of space that the laser
travels. It may be possible to construct an experiment that could measure the modified
expansion of space along the length of such a laser to test the predictions of this paper.

7.6 THE DEVELOPMENT OF THE TECHNOLOGY
Although a practical warp drive could be many years away from realization, there are anumber of technological developments that may, in fact, be necessary in order to allow
prototype experiments to begin. First, a more complete understanding of dark energy isof paramount importance. As mentioned earlier, dark energy contributes approximately
70 percent of the overall energy density of the universe, and is responsible for the
expansion of space. As one acquires deeper understanding of this energy, attempts togenerate dark energy in the lab would no doubt be a critical component to a working
warp drive.
Other crucial developments include determining whether extra dimension are, in fact,
real. Once this has been determined with absolute certainty, the role of extra
dimensions and their relationship with the universe that we are more familiar with will
become a more serious focus of scientific research and attention.
As has been discussed, the stabilization of the extra dimensions and the accelerated
expansion of the remaining four dimensions can be realized via the Casimir energy. The
acceleration of the 3-dimensional subspace is a naturally occurring phenomenon which
occurs when the extra dimensions are stabilized.
An important feature of the higher dimensional model is the dependence of the darkenergy density on the size of the extra dimensions. In models with large extra
dimensions, the interaction of the graviton KK tower with the Standard Model fields aresuppressed by the higher dimensional Planck scale and the corresponding couplings areinverse TeV in strength. This can be seen more clearly when we consider the expansionof the metric tensor in models with large extra dimensions computed within linearized
gravity models:
1
I)

where the capital letter indices indicate summation over the higher dimensions, M is
the modified Planck mass, 174,,,, corresponds to flat (Minkowski) spacetime and it„,corresponds to the bulk graviton fluctuations. The graviton interaction term in theaction is expressed by:
23

UNCLASSIFIED//aPara6446400E6INtsis

7.7

UNCLASSIFIED/;re7.8

where TAIN is the higher dimensional energy-momentum tensor. The interaction of thegraviton with the graviton KK states and with the SM fields are obtained by integrating
the action over the extra coordinates. Because all these states are coupled with theuniversal strength IIMpi this leads to the compelling possibility of the control of the sizeof the extra dimensions by processes at energies that will be accessible via the particle
accelerators of the near future. Although the coupling is extremely small, the effective
coupling is enhanced by the large number of KK states.
Referring to Figure 7, additional energy in the form of matter or radiation with the TeV
energy scale can alter the shape of the effective potential. In particular, the extrema
determining the size of the extra dimensions are modified with the change of the
Casimir energy density and hence, the dark energy density, in the models underconsideration.
In the ADD model discussed in Section 5.2, the Standard Model fields are confined to a
4-dimensional brane and a new gravity scale M„=G 2-"»TeV is introduced in the D= 4 + d dimensions, where d is the number of extra dimensions. This is determined
quite easily by forming the higher dimensional action and integrating out the extra
dimension.
The Atlas experiment at CERN's Large Hadron Collider will have the capability to probe
the ADD type extra dimensions up to MD - 8 TeV. In the Randall Sundrum scenario, thehierarchy is explained by the warp factor in the AdS5 bulk geometry. A lower bound canbe placed on the lowest KK mass by electroweak precision tests (masses on the orderof 1 TeV are allowed). In the Universal Extra Dimension scenario, Tevatron results
constrain the compactification scale to Mc > 400 GeV. Because the Atlas experimentwill be sensitive to Mc - 3 TeV , the estimates in this section indicate that if nature is infact described by one of these higher dimensional scenarios, then the additional
dimensions can be probed.
Even more exciting is the possibility that their size may be controlled at the energies
accessible to the Large Hadron Collider. When the Casimir energy to play the role ofdark energy is involved, one can see that the possibility for the direct control of thelocal dark energy density by controlling the size of the additional dimensions is possible.

8. Summary
The idea that a sufficiently advanced technology may interact with, and acquire directcontrol over, the higher dimensions is a tantalizing possibility, and one that is mostcertainly worthy of deeper investigation. Control of this higher dimensional space may
be a source of technological control over the dark energy density and could ultimately
play a role in the development of exotic propulsion technologies; specifically, a warp
drive.
Of course, this may not be actualized until many years in the future, but consider the
many spectacular physical phenomena that are believed to be true at this early point in
24

UNCLASSIFIED//12F1 SIIiWLL USE SJLVUNCLASSIFIED
the 215t century. One believes that an energy field called the Higgs boson permeates
spacetime and that the interaction of matter with this field is what is responsible forparticles acquiring mass. One believes that an exotic ubiquitous energy source,
unimaginatively named dark energy, is responsible for the current accelerated
expansion of the universe based on observation of supernova in galaxies billions of lightyears from Earth. One also believes that the universe may not consist of the threespatial dimension of length, breadth, width, and one of time, but that, in fact, there
may be as many as seven additional compactified dimensions assuming the topology ofa Calabi-Yau manifold, and that the fundamental building blocks of the universe are, infact, extended string-like entities.
Modern physics is full of many exciting and marvelously imaginative creations. Because
one understands these curiosities, one could potentially harness these elements ofnature for one's own technological ends. This is by no means a certainty, but if we may
make predictions based on the innovative history of scientific pioneers of the past, thenit seems entirely possible that the creative minds of the future may indeed find ways toaccomplish what, to us, may seem like magic.

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