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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