Warp Drive paper done for DIA AATIP program

2 Apгil 2010
СОО: 1 December 2009
DiA-08-1004-001
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Defense
Intelligence
Reference
Document
Acquisition Threat Support

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

2. General Relativistic Warp Drives
2.1 Warp Drive Requirements
З .. The Cosmological Coi1stant
З.1 Einstein’s Equation and the Introduction of J
4. Casimir Energy an·d the Quantum Vacuum
4 .1 The Casimir Effect ”
5. Extra Space Dimensions
5;2 Large Extra Dimensions
5.3 Randall Sundrum вrane models.
5.4 Extra Dimension Summary ”
б. Daгk Eneгgy as а Higheг Dimensional Artifact
7. Warp Drive and Higher Dimensional Manipulation
7.1 Adjustlng Higher Dimensions for Propulsions
7.2 The Geometry of Extra Dimensions
7.З Higher Dimensions and Stabllization
7.4 Elementary Warp Dгive Calculations
7.5 Future Experiments
7.6 The Development of the Technology
8 .. Summary
Figures
Figuгe 1. York Extrinsic Time (9-) Pfot.”” •••.••• ” •••••.••• “” •• ” ••• ” .••• ” ••. ” ••• “” •••..• “.;.” .•. 1
ri.guгe 2. The Interior Region of Parallel Conducting Plates .••.•..•.•..•.•…•…•.•… i •••••• 7
Figure З. Internal Structure of а Seemingly One-Dimensional Object “” •••••••.• 1 •••••• 9
F{gure 4. Manipulated Extra Dimension.” .•• ” •..•.••• ” •• “” …••• ” ..•.•..•. ” .••.• ” ..•.•.•.. .! …. ·15
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Warp Drive, Dark Energy, and the Manipulation of Extra
Dimensions
Introduction
If one is to realistically enteгtain the notion of interstellar exploration in ·
timeframes of а human lifespan, а dramatic shift in the traditional approach to
spacecraft propulsion is necessary. It has been known and well tested since
the time of Einstein that all matter is restricted to motion at suЬlight velocities
( << З х 108 m/s, the speed of light, or с), 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 Ье necessary to travel at the
speed of fight, and, thus, this speed is impossiЫe to reach and represents an
absolute speed limit to all matter traveling through spacetime.
Even if an engine were designed that could propel а spacecraft to ап
appreciaЫe fraction of light speed, travel to even the closest stars would take
many decades in the frame of reference of an observer оп Eaгth. Although
these lengthy transit times would not make interstellar exploration impossiЬle,
they would ceгtainly dampen the enthusiasm of governments or private
individuals funding these missions. After all, а mission whose success is
perhaps а century away would Ье 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 а “wormhole”) and the
warp drive. Fundamentally, both ideas involve manipulation of spacetime itself
in some exotic way that allows for faster-than-light (FTL) travel.
Essentially, the wormhole jnvolves connecting two potentially distant regions
of space Ьу а topological shortcut. Theoretically, one would enter the
wormhole and instantaneously Ье transported to the exit located in а distant
region of space. Although no observational evidence of wormholes exists:
theoretically they сап exist as а valid solution to general relativity.
The warp drive-the main focus this paper-involves local manipulation of the
fabric of space in the immediate vicinity of а spacecraft. The basic idea is to
create an asymmetric ЬuЬЫе of space that is contracting in front of the
spacecraft while expanding behind it. Using this form of locomotion, the
spacecraft remains stationary inside this “warp bubЫe,” 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 no
known restrictions оп the motion of space itself, thus allowing for а
convenient circumvention of the speed of light barrier.
j
An advanced aerospace platform incorporating warp drive techno1ogy would
profoundly alter the capacity to explore-and potentially to colonize the
universe. Because а warp drive is not limited Ьу the speed of light, one can
only guess the top speeds such а technology might Ье сараЫе of achievirig.
For the sake of argument, let’s consider the duration of trips taken Ьу а
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spacecraft сараьtе of 100с 1 for ап array of exotic destinations of possiЫe
int~rest. As ТаЫе 1 shows, trips to the planets within ouг own solar system
would take hours rather than years, and journeys to local star system would
Ье measured in weeks rather than hundreds of thousands of·years. 1 \ .. ·. . .
ТаЫе 1. Transit Times to Various Exotic .Destinations
at 100 Times the Speed of Light
1
Destination 1 Transit Time
Магs 1 193 seconds
Jupiter 1 36 minutes
Neptune 1 4 hours
Alpha Centauri … 15 days -·-·-·····-·
Epsilon Eridani 38 days
The Orion Nebula i.з years
Until гecently, the warp drive was а concept reseгv~d for science fiction. !
However, а 1994 рарег Ьу Miguel AlcuЬierre placed the idea on а more solid
theoretical footing. Alcublerre (Reference 1) demonstrated that а specific
Lorentzian manifold could Ье chosen that exhiblted bubЬle-like featuгes
reminiscent of the warp drive from the popular Star Trek television series. The
ЬuЬЫе allowed for the surrounding spacetime to move at FTL speeds, and the
inhaЬitants of .the ЬuЬЫе would feel no acceleration effects because spacetime
itself would Ье in motion instead of the spacecraft and its inhabltants.
А number of papers have emerged in recent years that build on this original
idea. However, these papers do not typically address how one might actually
create the necessary spacetime ЬuЬЫе. Our own research directly addresses
this question from а new and unique perspective and introduces а novel
paradigm shift in the field of warp drive study (Reference 2). More formarly,
our work approaches the physics of warp drive from the perspective of
quantum field theory; this diverges from the more traditional approach to
wаф dгives, which utilizes the physics of general relativity. One of the
improvements the model introduces is а dramatic reduction in the overali
energy required to create such а phenomenon.
The roadmap to this new idea was the observation that spacetime is currently known to Ье in а state of accelerated expansion, as demonstrated Ьу the
redshifting of galaxies, and the belief that if the mechanism for this expansion
could Ье understood, then it might ultimately Ье controlled. А populaг teгm
used Ьу cosmologists today is “dark energy,” an exot1c and ub1quitous form of
energy that is believed to constitute over 70 percent of the matteг-energy
content of the universe (Reference 3-6). Опе salient feature of dark energy is
its intrinsic abllity 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, high1ights the fact that our galaxy woulld become fаг mоге accessible if ог wheen one discoveгs how to surpass the speed of light batтier. J
~
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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 to
expand. Indeed, recent high-precision experimental observations indicate dark
energy may Ье а cosmological vacuum energy (Reference 8-10). These ‘
obseгvations are ьased оп the magnitudes of high-redshift supernova and
have been а source of high research activity of late owing to the unexpected
discovery that the rate of expansion of the universe is increasing ( commonly
refeгred to as accelerated expansion). ]
One tantalizing aspect of dark energy is that if it were fully understood, and if
а technology were developed that could generate and harness the exotic
effects of dark energy on the fabric of space, then а warp drive would Ье one
step closer to .technological reality. While а full understanding of the true
nature of dark energy may Ье many уеагs away, it is entirely feasiЬle that
experimental breakthroughs at the Large Hadron Collider ог developments in
the field of M-theory could lead to а quantum leap in our understanding qf this
unusual form of energy and perhaps help to direct technological innovations.
Our own reseaгch focuses on gaining an understanding of the physical origin
of dark energy. Ву exploring novel ideas at the forefront of theoretical physics,
one is аЫе to propose а physically viaЬle model incorporating some of the
cutting-edge ideas emerging from string theory and quantum field theory.  This
leads to а deeper understanding of the possiЫe origin of dark energy and
allows consideration of а mechanism that would allow а 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 advancement
of warp drive research. ‘
This рарег is structured as follows: Section 2 reviews the mоге traditional
general relativistic warp drives, the energy required to create them, and the
physics гequired to understand them. Section З discusses the cosmological
constant, а term featured in Einstein’s equation that regulates the contraction
and expansion of the spacetime. Section 4 introduces the Casimir energy,
which, under certain conditions, may Ье the phenomenon that physically
generates the cosmological constant. Section 5 discusses higher dimensions in
physics and their importance in the context of Casimir energy calculation.
Section 6 introduces the formulas that demonstrate that the Casimir energies
in higheг dimensions may in fact Ье the dark energy that is responsiЫe for the
accelerated expansion of the universe. Section 7 relates all the previous
concepts together and introduces the novel warp drive paгadigm. Section 8
performs original calculations of the energy required to сгеаtе а superluminal
warp drive. Finally, the paper speculates about the technological progress that
would Ье necessary to turn this model into а reality. ;
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2. General Relativistic Warp Drives
AlcuЬierre (Reference 1) derived а spacetime metric motivated Ьу cosmological l inflation
that would allow arbltrarily short travel times between two distant points in space. The
“warp drive” metric uses coordinates (t, х, у, z) and curve (or worldline) х = Xsh(t), у =
, z = , lying in the t-x plane passing through the origin. Note that Xsn is the x -axis
coordinate position of the moving spaceship (or warp ЬuЬЫе) frame. The metric
specifying this particular spacetime geometry is (Reference 1): ·
(2.1)
where С is the speed Of light, V5t1(t) is the speed associated wjth the CUГVe (ОГ warp
ЬuЬЫе speed), and sh(t) is the Euclidean distance from the curve. The warp buЫble
shape function f (rsh ) is any smooth positive function that satisfies f (О) = 1 and j
decreases away from the origin to vanisli when sh > R for some distance R. The i
geometry of each spatial slice is flat, and spacetime is flat where f (rsh) vanishes put is
curved where it does not vanish.
The driving mechanism of EqLration ( 2.1) is the York extrinsic time, Э. This quantity is
defined as (Reference 1): :
The Э behavior of the warp drive ЬuЬЫ е
provides for the simultaneous expansion
of space behind the spacecraft and а
corresponding contraction of space in
fгont of the spacecraft. Figure 1 illustrates
the .Э. behavior of the warp drive ЬuЬЫе
geometry. Thus the spacecraft is
enveloped within а warp ЬuЬЫе and сап
Ье made to exhiblt an arbltrarily large
faster-than-light (FГL) speed ( vsh > > с)
as viewed Ьу external coordinate
observers. Even though the worldlines
inside the warp ЬuЬЫе region are
spacelike fог all external observers, the
moving spaceship (warp ЬuЬЫе) frame
itself never travels outside of its local
comoving light cone and thus does not
violate special relativity. Figuгe 1 . Уогk Extгinsic Time ) Plot
(2.2)
1 А spacetime metric (ds1 ), or line eternent, is а Lorentz-invariant distance function between any two points iin
spacetime that is defined Ьу ds2 .= g ,,dx” dx’·, where g,., is the metric tensor which is а 4,4 matrix tt1at епс 4еs the
geometry of spacetime and dx” is the infinitesimal coordinate separation between two points . The Greek indices (µ,
v =· 0 … 3) denote spacetime coordinates, х” … , such that “.х3 “‘ space coordinates and х” .,, time coordinates .
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2.1 WARP DRIVE REQUIREMENTS
Implementation of FТL interstellar travel via warp drives requires engineering of
spacetime iпto very specialized local geometries as shown Ьу Equatioп (2.1). The
analysis оf these via the general relativistic field equation plus the resultant source
for latter equations of state demonstrates that such geometries require the use of :
“exotic” matter in order to produce the requisite FТL spacetime modification. Exotic
matter is generally defined Ьу general relativity (GR) physics to Ье matter that
possesses (renormalized) negative eпergy density aпd/or negative stress tension
positive outward pressure, aka gravitational repulsion). The term is widely
misunderstood and misapplied bу the non-GR community. Also, it has been claimed
that FТL spacetimes аге not plausiЬle because exotic matter violates the general
relativistic energy conditions. However, this has been shown to Ье а spurious issue
(Refereпce 11).
The епегgу density for the Alcublerre (Reference 1) warp drive that is derived from the
general relativistic field equation is complex, so we instead use а more simple formula
to express the net energy required, E.· rтJJ to build а warp ЬuЬЫе around а spaceship
(Reference 12):
‘ <1- R 2 v;..,rp с с;
G (2.3)
= -(I 21 10 ) ,rr R2
cr ,
where G is Newton’s universal avitat oп constant (6.673 х 10-н N·m2
/kg2
), v\\arr\is the
dimensionless speed of the warp ЬuЬЫе, R (> О) is the radius of the warp bubЬlej and v
(> О) is proportional to the inverse of the warp ЬuЬЫе wall thickness д (i.e., cr – 1/д) .
Equation (2. 3) characterizes the amount of negative епегgу that one needs to localize
in the walls of the warp ЬuЬЫе . ТаЫе 2 presents а tabulation of the required negative
energy as а function of the “warp factor,” v”·arp · One саn compare the values of в.. rp in
the tаЫе with the (positive) est-eпergy contained in the Sun (1.79 х 1047 J). The:
consequence of Equation (2. 3) and ТаЫе 2 is that if one wants to traveJ at hyper/(ght
speeds, then the warp ЬuЬЫе energy requirement will Ье an enormous negative пumber. And this remains true even if one engiпeers ап arЬitrarily !ow suЬlight speed
warp ЬuЬЫе. Engineering а warp drive ЬuЬЫе is quite daunting given these results.
The condition for ordinary, classical (non exotic) forms of matter that we аге familiar with in nature is tha\ ре > р
and/ or Р< , where РЕ is the energy density and р is the pressure/stress-tension of some source of matter.  These
conditions гергеsепt two examples of what аге variously called the “standard” energy conditions: Weak Energy
Condition (WEC: р, :<: , о; + р <:О), Null Energy Condition (NEC: р, + р 2: О), Dominant Епегgу Condition (D,EC),
апd Strong Eпегgу Condition (SEC). Тhese energy conditions forbld negative energy density between material
objects to occur in nature, but they аге mere hypotheses. The energy conditions were developed to estaЬlish а
series of mathematical hypotheses governing the behavior of collapsed-matter singularities in the study of cosmology and Ыасk holes. ·
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ТаЫе 2. Negative Energy Required for Warp ВuЬЫе
(Larger Negative Energy)
\Vагр Fa toг, V,varp E “‘r’.rp (J)
10-
5 (= 3 km/s) -3.03 х 1040
10-: (= 30 km/s) – 3.03 х ·1042
0.0 1 (= 3,000 km/s) -3.03 х 1046
0.5 (= 150,000 km/s) -7.59 х 1049
1 (= !ight speed) -3.03 х 1050
2 (= 600,000 km/s) -1.21 10 ; 1
l о (= 3.0 х 106 km/s) .ОЗ х 1052
100 (= 3.0 х J 07 krp/s) – 3.03 х 1054
‘ Ass ne: R= 50 m, с;= 10’ 1
Lobo and Visser (Rеfегеnсе 12) constructed an improved model of the wагр drive
spacetime Ьу applying linearized gravity to the weak-field wагр drive case and testing
the energy conditions to first аnd second orders of v,,1l!’p· The fundamental basis of their
model is that it specifically includes а finite mass spaceship that interacts with the warp
ЬuЬЫе . Their results verified that all warp drive spacetimes violate the eneгgy
conditions and will continue to do so fог arbitrarily low warp ЬuЬЫе speed. They also
found that the energy condition violations in this class of spacetimes is generic to the
form of the geometry under consideгation and is not а side effect of the superluminal
properties. Based on these facts plus Equation (2.3) and tаЫе 2, it appears that for all
conceivaЫe laboratory experiments in which negative energy саn Ье created in minute
amounts, the warp ЬuЬ Ые speed will Ье absurdly low.
Coupling of the finite spaceship mass with the warp uЬЫе leads to the (quite
reasonaЫe) condition that the net total energy stored in the warp ЬuЬЫе Ье less than
the total rest-energy of the spaceship itself, which places а strong constraint upon the
( dimensionless) speed of the warp ЬuЬЫе (Reference 3) : ‘
(2.4)
where Л;fsh ip and R s11ip are the mass and size of the spaceship, respectively, and R ~ the
radiOs of the .warp ЬuЬЫе. Equation (2.4) indicates that for any reasonaЫe values of
the engineering parameters inside the brackets, warp will Ье absurdly low. This result is
due to the intrinsic nonlinearity of the general relativistic field equation. То illustrate
this point, the example starship parameters from tаЫе 2 (R = 50 m, Л ,…, 1/cr = 16-3 m) 1 .
аге inserted into Equation (2.4) and assume А1;; ;р = 106 kg to find that 1\,-arp :::; 1.7 t х
10 –
14 (or 5. 16 х 10-
5 m/s). Garden snails can crawl faster than this. And if R and Nfs111p аге kept constant, then л = 3.37 х 1024 m (or 3.57 х 108 light-years) in order for’ v, warp
1, which is an unrealistic requirement оп the warp ЬuЬЫе design. ·
Because this energy requirement is so phenomenally high one finds it of paramount
importance to explore new ideas in the field of warp drive technology. What now follows
is а pedagogically rich review of the novel warp drive concept that we have been
developing since 2005 .
З. The Cosmological Constant
Einstein is famous for а multitude of achievements in the field of physics. ArguaЬly his
most notaЫe contribution is the General Theory of Relativity, а geometric description of
gravitation whose fundamental idea relates the matter and the energy content of the
universe 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 move
through spacetime. General relativity has been the prevailing theory of gravity since
1915 and thus far has unambiguously passed observational and experimental tests. It
remains an active агеа 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 ТНЕ INTRODUCTION OF Л
Upon completion of GR, Einstein applied his theory to the entire universe. Не firm!y believed in Mach’s principle, and the only way to satisfy this was to assume that space
is globally closed and that the metric tensor should Ье determined. uniquely from the
energy-momentum tensor (Reference 13). Не also assumed that the universe was
static, which was а reasonaЫe assumption at the time because observational
astronomy had not advanced to а level that contradicted this paradigm. In 1917, when
а static solution to his equations could not Ье found, he introduced the cosmological
constant Л (Reference 14): 3 ·
R 1 R SnG Т \ )11′ -? g)(V = – ,,-. /11’ + j gJIV ‘
:… с
(3.1)
In this equation is the Ricci curvature tensor, R is the Ricci curvature scalar, TJ,. is
the stress-energy-momentum tensor,
4 and gµ•· is the spacetime metric. The left-h~n d
side of Eq uation (3.1) encodes the curvature in the geometry of spacetime, and the
right-hand side encodes the source of matter-energy that curves spacetime. ·
The addition of [?] can Ье understood as а term in the equation v~hfch allows one to
adjust theory to match observation. In Einstein’s case, he chose to add л to ensure that
the universe was static and unchanging. In later years, he often referred to this
amendment to his equations as his “Ьiggest Ыunder.” Several years after GR had been
formulated, the astronomer Edwin НuЬ Ые discovered the phenomenon of galactic
redshifting, which strongly indicated that the universe was indeed expanding. This encodes the density and flux of· а matter source’s energy and momentum.
This theoretical prediction from GR was ignored Ьу Einstein because of his belief in а static
universe.
Even though Einstein retracted the addition of л into his equations, it is now known that
it does indeed play а role and is typically included in GR equations. Data from precise
astronomical observations strongly suggest that ап extremely small, yet пon-zer<? л is а
пecessary feature of GR and is responsiЫe for the ехрапsiоп of the universe that is
observed .
From а physical perspective, А represents an inherent energy density associated with
empty space . One way to envision this is to take а perfectly insulating Ьох into deep
space, аnd then to remove all matter аnd all energy from this Ьох so that it encloses а
perfect void . Even in this emptiпess, а residual energy field would remain. According to
GR, the effect of this energy would Ье to cause the region of space to ехраnd albeit at
an extremely small гаtе. То summarize, л is а ublquitous, ever present feature of
space, and its presence causes space to expand.
In the late 1990s it enierged 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 tо л
as dark eпergy, and the remainder of this рарег wШ fol/ow this convention.
Although the гоlе of dark energy is extremely well understood mathematically, afld in
tl1e context of its effects оп spacetime, its physical nature is still а mystery. One knows
that it is homogeneous, поt particularly dense, апd 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 Ье а reasonaЫe explanation for the fundamental origin of dark energy ,
the proЫem is considered serious and has Ьееп tackled Ьу а large number of em фent
and respected physicists, including previous Nobel prize winners (Refer·ence 17),
Because dark energy is intimately related to the expansion of space, and because’ this
expansion is exactly the feйture that would allow for а warp drive to function, any
understanding of this mysterious energy is of paramount importance in the
development of this novel propulsion technology.
4. Casimir Energy and the Quantum Vacuum
А central theme in this рарег is the notion of the quantum vacuum. То а particle
physicist, the term “vacuum” means the ground state of а quantLim field in some
quantum theory for matter. In geпeral, this ground state must оЬеу Lorentz invariance,
at least with regaгds to three spatial dimensions, meaning that the vacuum must look
identical to all observers.
At all energies ргоЬеd Ьу experiments to date, the universe is accurately described as а
set of quantum fields. То а non-physicist а quantum field may, at first, Ье а strange
concept to grasp. This is because one generally likes to visualize the things one thinks
about; for example, an electron and even а photon provides something one can, on
some level, picture in one’s minds. Simply put, а quantum field is an intangiЫe j
mathematical object whose properties are ideal in explaining nature. Theories have
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must Ье abandoned for more .erudite mathematical constructions which are bett~r
suited at describlng the build ing Ыocks of nature (Reference 18-20).
‘ If one takes the Fourier transform of а fгее quantum field,5 each mode of а fixea
wavelength behaves like а simple harmoлic oscillator. А quantum mechanical prqperty
of а simple harmonic oscillator is that the ground state exhiblts zero-poi.nt fluctu~tions
as а consequence of the Heisenberg Uncertainty Principle. Опе way to understan ф these
zero-point fluctuations is to imag ine releasing а pendulum and watching as dissiPiative
forces slowly try to bring the pendulum to а stop. The uncertainty principle would
ensure that the pendulum was never аЫе to соте to а complete rest, bt.Jt insteac,I
woul.d exhiblt microscopic oscillations around the equilibrium position indefinitely.i Of
course, for а геаl macroscopic pendulum, these fluctuations would Ье miлiscu le and all
but impossiЫe to detect; however, the analogy with а quantum harmonic oscШat r
holds well. The expectation value of the energy associated with the ground state erg y
of а quantum oscillator is: ‘
с “” (E)=-“f/ik
2 n=I · ”
(4.1)
1
In this formula с and h are the speed of light and Planck’s reduced constant (1.05!5 х
10-
34 J-s), respectively, ar:id k is the wave-vector rela ted to the momeritum of the •
quantum field. One of the features of this grou nd state energy is that the wave vector
h.as an iпfinite degree of freedom . Clearly t his sum is divergent; however, this is ijl
common feature of quantum field theory, and an array of mathematical techniques
known as renormalization exjsts to deal v’Vith tt1e infinities that arise.
4 .1 ТНЕ CASIMI R EFFECT
The quantum fluctuations of the vacuum fields give rise to а number of pherюme ;
however, one is part1Clilarly striking. The Casimir Effect, which wil l Ье explored in jmore
detail in this раре г, is arguaЬly the most sa!ient manifestation of the quantum vaquum.
In 1948, Н. Casimir puЫ ished а profound paper where he explained the van der \fi/aals
interaction in terms of the zero-point energy of а quantized field (Reference 19). ~n its
most basic form, the Casimir Effect it is realized through the interaction of а pair qf
neutral parallel conducting plates (with sepa ration distance с!). The presence of th ~
plates modifies the quantum vacuum, and this modification causes the plates to ь!=
pulled toward each other with а force: ‘
ticn2
F=–.- 240d 4
j
\(4. 2)
This is а profound result in the sense that the origin of this force caлnot Ье traced [back
to one of the four fuпdamental forces of nature (gravity, electromagnetism, and ~e two
nuclear forces), but is а force that is entirely due to а modification of the quantum. ‘ vacuum.
5 Ву “free” we rnean that the field does not iпteгact with ott1er fields.
6
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