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arXiv:0712.1649v1

[gr-qc]11Dec2007

Warp Drive: A New Approach

Richard K Obousy and Gerald Cleaver

Baylor University, Waco, Texas, 76706, USA

(Dated: February 29, 2008)

Certain classes of higher dimensional models suggest that the Casimir effect is a candidate forthe cosmological constant. In this paper we demonstrate that a sufficiently advanced civilizationcould, in principal, manipulate the radius of the extra dimension to locally adjust the value of the

cosmological constant. This adjustment could be tuned to generate an expansion/contraction ofspacetime around a spacecraft creating an exotic form of field-propulsion. Due to the fact thatspacetime expansion itself is not restricted by relativity, a faster-than-light warp drive could becreated. Calculations of the energy requirements of such a drive are performed and an ultimatespeed limit, based on the Planckian limits on the size of the extra dimensions is found.

I. INTRODUCTION

A 1994 paper written by M. Alcubierre [2] demon-strated that, within the framework of general relativity,a modification of spacetime could be created that wouldallow a spacecraft to travel with arbitrarily large speeds.In a manner identical to the inflationary stage of the uni-verse, the spacecraft would have a relative speed, definedas change of proper spatial distance over proper spatialtime, faster than the speed of light. Since the originalpaper, numerous authors have investigated and built onthe original work of Alcubierre.

Warp drives provide an unique and inspiring opportu-nity to ask the question what constraints do the laws ofphysics place on the abilities of an arbitrarily advancedcivilization[21]. They also serve as a tool for teachinggeneral relativity, as well as an exciting way to attractyoung students to the field by stimulating their imagina-tion. In this paper an original mechanism to generate thenecessary warp bubble is proposed. The main focus of

the paper is to demonstrate that the manipulation of theradius of one, or more, of the extra dimensions found inhigher dimensional quantum gravity theories, especiallythose that are based on or inspired by string/M-theory,creates a local asymmetry in the cosmological constantwhich could be used to propel a space vehicle.

Warp drives have not been the sole interest of theo-retical physicists as was demonstrated by the formationof the NASA Breakthrough Propulsion Program and theBritish Aerospace Project Greenglow, both of whose pur-pose was to investigate and expand on these ideas regard-ing exotic field propulsion.

At such an early stage in the theoretical development

of the ideas presented in this paper, it is challenging tomake predictions on how this warp drive might func-tion. Naively one could envision a spacecraft with anexotic power generator that could create the necessaryenergies to locally manipulate the extra dimension(s). Inthis way, an advanced spacecraft would expand/contract

Electronic address: [email protected] ; Electronic

address: [email protected]

the compactified spacetime around it, thereby creatingthe propulsion effect.

The first four sections of this paper review the neces-sary physics required to appreciate the new warp drivemodel. The remainder of the paper will introduce thepropulsion concept. Calculations regarding speed limitsand energy requirements will also be presented.

II. THE COSMOLOGICAL CONSTANT AND

QUANTUM FIELD THEORY

Current observations of distant supernova [12] indi-cate that the universe is expanding. This expansion isrealized in Einsteins equations .

R 12Rg = 8GT g (1)

through the cosmological constant term, ThisLambda term, as it is known, provides a necessary ad-

dition to the gravitational field to correlate with what isobserved in nature.

The fundamental origin of Lambda is still a mysterynearly a century after its introduction into cosmology.Physicists are not certain what generates the cosmologi-cal constant term in Einsteins equations; we simply knowthat it is there. Several ideas exist as to the nature ofthis field. Dark energy, for example, is a popular contem-porary phrase for the Lambda term. Efforts have beenmade to explain the cosmological constant using the moremodern quantum field theory, created after Einstein andhis gravitational equations [6].

Quantum field theory (QFT) is widely regarded as one

of the most, if not the most, successful physical theoriesof all time. Its predictions have been verified in particleaccelerators; no experiment has ever shown a result thatcontradicts QFT. It would thus seem only natural to tryto account for the cosmological constant using QFT. Thecalculations [7] are based on summing all the zero pointoscillations with a Planck scale cut-off which would giveus an estimate to the overall vacuum energy densityEvac.

Evac = 12

M0

d3k

(2)3

k2 + m2 M

4

162(2)
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The energy density predicted using this equation isof order 1071(GeV)4, which conflicts with the observedvalue of 1048(GeV)4. Indeed, this is genererally viewedas, by far, the worst predication of theoretical physics-being off by a factor of 10119! This failure of quantumtheory has recently been reexamined using brane worldscenarios born of string theory [10], [4], [13], [1], [9], [23].

One (partial) fix to the vacuum energy calculations is

the introduction of supersymmetry (SUSY). The basicidea is that all known particles have an associated su-perparticle whose spin differs by exactly one half (in units). In the case of unbroken SUSY, i.e., when particleand supersymmetric partner have exactly equal masses,the superparticle additions to the vacuum energy per-fectly cancel the particle contributions and the result-ing vacuum energy is reduced to zero. However, whenSUSY is broken and the difference of the particle massand the supersymmetric partner mass is on a scale ofMS 104GeV, then the resulting vacuum energy ison the order (MS)

4 1016GeV and the discrepancy is(only) off by a factor of 1060! In this paper we assumethat even the SUSY-breaking contribution is somehowcancelled and that all contributions to the vacuum en-ergy density are higher dimensional.

III. KALUZA KLEIN MODELS

Kaluza-Klein (KK) theory was an early attempt tounify gravitation with electromagnetism. Kaluza initiallypostulated that a fifth spatial dimension could be intro-duced into Einsteins equations [15]. When the equationswere solved the additional dimension would generate notonly the standard metric, but also a vector field A thatcould be associated with the electromagnetic field and a

scalar field . Kaluzas starting point was the metric:

g =

g AA A

A

(3)

Where and run over 0,1,2,3 and the tilde quantitesrun over 0,1,2,3,5. A and A are vector fields and is ascalar field. Now start with a source free spacetime (pure-gravity) in five dimensions. The corresponding action ofthe system is

S5 = d5x

gR (4)

Where g is the five dimensional determinant of the metricand R is the five dimensional Ricci Scalar. After integrat-ing out the extra dimension and performing a conformalrescaling of the metric g

g, it is trivial to show

that the four dimensional action can be written as:

S=

d4x

g(R + 1

2

14e

3FF

) (5)

From this action we retrieve the spin-2 graviton, thespin-1 photon and discover a spin-0 scalar field. This in-deed was a success in unifying gravity with electromag-netism. However the unexpected existence of the scalarfield was an embarrassment at the time. This stimulatedlater work on Brans-Dicke theories of gravity where thescalar field has the physical effect of changing the effec-tive gravitational constant from place to place.

Kaluzas idea suffered from a very obvious drawback.If there is a fifth dimension, where is it? In 1926 OskarKlein suggested that the fifth dimension compactifies soas to have the geometry of a circle of extremely smallradius [17]. Thus, the space has topology R4 S1. Oneway to envisage this spacetime is to imagine a hosepipe.From a long distance it looks like a one dimensional linebut a closer inspection reveals that evey point on the lineis in fact a circle.

IV. EXTRA DIMENSIONS AND THE CASIMIR

EFFECT

The Casimir effect is one of the most salient manifes-tations of the vacuum fluctuations. In its most elemen-tary form, it is the interaction of a pair of neutral, paral-lel conducting planes whose existence modify the groundstate of the quantum vacuum in the interior portion ofthe plates creating a force which attracts the plates toeach other. For the electromagnetic field, this force ofattraction is:

F = A480

hc

a4(6)

where A is the area of the plates, h is Plancks constant,c is the speed of light and a is the plate separation. Fora review on the Casimir effect see [20].

The Casimir effect can be extended to regions of non-trivial topology. For example, on S1, a circular manifold,one can associate 0 and 2 with the location of the platesand the Casimir energy can be calculated. This becomesrelevant when we consider models with additional spatialdimensions.

Two models with extra dimensions have become par-ticularly popular in recent years. These are the Arkani-Hamed-Dimopoulos-Dvali (ADD) [3] and the Randall-Sundrum (RS) models [22]. Both are attempts to ex-plain what has become known as a heirachy problem inphysics, which questions why the gravitational force is somuch weaker than the other forces of nature. In the ADDmodel it is proposed that the force carriers of the stan-dard model (the photon, W+, W, Z0 and the gluons)are constrained to exist on the usual four dimensionalspacetime, which we will call the bulk. Gravity however,is free to move both in the bulk and in the extra dimen-sions. These dimensions can be as large as m [16]. //


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Because only gravity can propagate in the extra dimen-sions, we cannot see them as we see in the bulk throughthe electromagnetic field (light), nor can we observe themthrough weak or strong force interactions, which are alsorestricted to the bulk. It is the freedom of the gravitonto propagate off of the bulk that dilutes the field strengthaccounting for the apparent weakness of gravity. In theRS models, it is the warping of the extra dimension that

is the root cause of the weakness of gravity.In both the ADD and RS models one can account forthe cosmological constant by calculating the contributionof extra dimensional graviton fields to the Casimir energyand then associating this energy with the cosmologicalconstant [10]. As an example we will demonstrate a cal-culation of the vacuum energy due to compactification inthe ADD scenario in which the spacetime orbifold will beM4 T2 . Matter fields will reside on the 3-brane (ouruniverse) and gravity is free to propagate in the bulk.The calculations closely follow [14].

The four dimensional vacuum energy is given by:

Evac =

n1,n2=1

1

2 d3k

(2)3k2 +

42(n21

+ n22

)

R2 (7)

Where R is the radius of the extra dimensions and n1and n2 are the KK (Kaluza Klein) graviton modes. Wewill now perform dimensional regularization on the kintegral by first rewriting it as:

I(mn) = 2

d3k

(2)3

k2 + m2n

1

2

(8)

Where we have made the substitution, m2n =42(n2

1+n2

2)

R2 . The term is inserted for dimensional con-

sistency and will be taken to zero. The integral can beexpressed as a beta function which can be expanded to:

I(mn) = m4n

322

1

+ 2log2 1

2 log(m

2n

2)

(9)

The vacuum energy can thus be expressed as:

Evac = 1322

n1,n2=1

m4n

1

+ 2log2 1

2 log(m

2n

2)

(10)

The summation is clearly divergent so we next use azeta function regularization.

n1,n2=1

m4n =

4

R2

2 n1,n2=1

(n41 + n42 + n

21n

22) (11)

=

4

R2

22(0)(4) + 2(2 (12)

Using the fact that (2m) = 0, for m and naturalnumber, we obtain the vacuum energy:

Evac4+2 =1

322

n1,n2=1

m4nlog(m2n2

) (13)

= 2R4

n1,n2=1

(n21 + n22)2log(n21 + n22) (14)

= 2

R4(0)(4) (15)

This result can be generalized to 4+n dimensions

Evac = 2

R4

(2 + n)(3 + n)

2 1

[(0)]

n1 (4)

(16)where the graviton degrees of freedom is expressed in

the brackets. This vacuum energy due to the massiveKK modes is associated with the cosmological constant.

V. WARP DRIVES

Numerous papers discussing the idea of warp driveshave emerged in the literature in recent years. See forexample [19]. The basic idea is to formulate a solutionto Einsteins equations whereby a warp bubble is drivenby a local expansion of spacetime behind the bubble anda contraction ahead of the bubble. One common featureof these papers is that their physical foundation is the

General Theory of Relativity. An element missing fromall the papers is that there is little or no suggestion as tohow such a warp bubble may be created.

The aim of this paper is not to discuss the plausibil-ity of warp drive, the questions associated with violationof the null energy condition, or issues regarding causal-ity. The aim of this paper is to suggest that a warpbubble could be generated using ideas and mathematicsfrom quantum field theory, and to hypothesize how sucha bubble could be created by a sufficiently advanced tech-nology.

By associating the cosmological constant with theCasimir Energy due to the Kaluza Klein modes of gravi-tons in higher dimensions, especially in the context of M-theory derived or inspired models, it is possible to forma relationship between and the radius of the compactextra dimension. We know from eqn.(16)

Evac = 1R4

(17)

An easier way of developing this relationship is to putthings in terms of Hubbles constant H which describesthe rate of expansion of space per unit distance of space.


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H

, (18)

or in terms of the radius of the extra dimension we have

H 1R2

. (19)

This result indicates that a sufficiently advanced tech-nology with the ability to locally increase or decrease theradius of the extra dimension would be able to locallyadjust the expansion and contraction of spacetime cre-ating the hypothetical warp bubble discussed earlier. Aspacecraft with the ability to create such a bubble willalways move inside its own local light-cone. However theship can utilize the expansion of space-time behind theship to move away from some object at any desired speedor equivalently to contract the space-time in front of theship to approach any object.

In the context of general relativity a similar phe-nomenology is produced for the case of anisotropic cos-mological models, in which it is the contraction of theextra dimension that has the effect of expanding another[18]. For example consider a toy universe with one ad-ditional spatial dimension with the following metric

ds2 = dt2 a2(t)dx2 b2(t)dy2 (20)In this toy universe we will assume spacetime is empty,

that there is no cosmological constant and that all spatialdimensions are locally flat.

T = g = 0 (21)

The action of the Einstein theory of gravity generalizedto five dimensions will be

S(5) =

d4xdy

g(5)

M2516

R(5)

(22)

Solving the vacuum Einstein equations

G = 0, (23)

we obtain for the G11 component

G11 =3a(ba + ab)

a2b(24)

Rewriting aa

= Ha andbb

= Hb where Ha and Hbcorresponds to the Hubble constant in three space andthe Hubble constant in the extra dimension respectively,we find that solving for G11 = 0 yields

Ha = Hb. (25)

This remarkable result indicates that in a vacuum,the shear of a contracting dimension is able to inflate theremaining dimensions. In other words the expansion ofthe 3-volume is associated with the contraction of theone-volume.

Even in the limit of flat spacetime with zero cosmo-logical constant, general relativity shows that the physicsof the compactified space effects the expansion rate of

the non-compact space.The main difference to note hereis that the quantum field theoretic result demonstratesthat a fixed compactification radius can also result in ex-pansion of the three-volume as is shown in eqn.(16) dueto the Casimir effect, whereas the general relativistic ap-proach suggests that a changing compactifification radiusresults in expansion. Both add credibility to the warpdrive concept presented here.

VI. ENERGY REQUIREMENTS

In this section we perform some elementary calcula-

tions to determine how much energy would be requiredto reach superluminal speeds. We also determine an ab-solute speed limit based on fundamental physical limita-tions.

The currently accepted value for the Hubble constantis 70 km/sec/Mpsc. A straightforward conversion intoSI units gives H = 2.17 1018(m/s)/m. This tells usthat one meter of space expands to two meters of spaceif one were prepared to wait two billion billion seconds orsixty five billion years. The fundamental idea behind thewarp drive presented in this paper is to increase Hubblesconstant such that space no longer expands at such asedentary rate, but locally expands at an arbitrarily fastvelocity. For example, if we want space to locally expand

at the speed of light, a simple calculation shows us bywhat factor we would need to increase H.

HcH

108

1018= 1026 (26)

Where Hc is the modified Hubble constant (subscriptc for speed of light). This results implies that H wouldhave to be increased by a factor of 1026 for space to ex-pand at the speed of light. Since we know that H 1

R2,

we can naively form the relation

HcH

= R2R2c

= 1026 (27)

or,

Rc = 1013R (28)

Where Rc is the modified radius of the extra dimen-sion. This indicates that the extra dimensional radiusmust be locally reduced by a factor of 1013 to stimulate


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space to expand at the speed of light. In the ADD modelthe size of the extra dimension can be as large as 106m.If we use this number as a prototype extra-dimensionalradius, this would have to be shrunk to 1019m forlightspeed expansion.

An interesting calculation is the energy required tocreate the necessary warp bubble. The accepted value of

the cosmological constant is 1047

(GeV)4

. Convert-ing again into SI units gives 1010J/m3 . Now, fora warp bubble expanding at the speed of light we wouldneed to increase this again by a factor of 1052 as we haveH

. We can say

c = 1052 = 1042J/m3 (29)

where c is the local value of the cosmological con-stant when space is expanding at c. Let us consider aspacecraft of dimensions

Vcraft = 10m 10m 10m = 1000m3

. (30)

If we postulate that the warp bubble must, at least,encompass the volume of the craft, the total amount ofenergy injected locally would equal

Ec = c Vcraft = 1045J. (31)Assuming some arbitrarily advanced civilization were

able to create such an effect we might postulate thatthis civilization were able to utilize the most efficientmethod of energy production - matter antimatter anni-hilation. Using E= mc2 this warp bubble would require

around 1028Kg of antimatter to generate, roughly themass-energy of the planet Jupiter.

This energy requirement would drop dramatically ifwe assumed a thin-shell of modified spacetime instead ofbubble encompassing the volume of the craft.

VII. ULTIMATE SPEED LIMIT

It is known from string theory that the absolute min-imum size for an extra dimension is the Planck length,1035m. This places an ultimate speed limit on the ex-

pansion of space based on the idea that there is a limitto the minimum radius of the extra dimension. From theabove arguments it is straightforward to form the relation

HmaxH

=R2

R2min=

1012

1070= 1058. (32)

Here Hmax is the maximum rate of spacetime expan-sion based on the minimum radius of the extra dimensionRmin. In this formula we have again used a prototype

extra-dimensional radius of 106m which is the upperbound based on current experimental limits. Using thesevalues and the known value of H in SI units we obtain

Hmax = 1058H 1040(m/s)/m (33)

A quick conversion into multiples of the speed of lightreveals

Vmax = 1032c, (34)

which would require on the order of 1099Kg of anti-matter, more mass energy than is contained within theuniverse. At this velocity it would be possible to crossthe known universe in a little over 1015 seconds.

The authors wish to reiterate that the calculations inthe previous two sections are extremely back of the en-velope and merely serve as interesting figures to contem-plate, based on the formula

Evac

=

1

R4. (35)

Note that it is not really possible to travel faster thanlight in a local sense. One can however, make a roundtrip between two points in an arbitrarily short time asmeasured by an observer who remained at rest at thestarting point. See [24] for details on violations of thenull energy conditions and causality.

VIII. CONCLUSIONS

In this paper we have calculated the vacuum energydue to extra dimensional graviton contributions to the

Casimir energy and associated this energy with the cos-mological constant. It has been shown that this energy isintimately related to the size of the extra dimension. Wehave picked the ADD model, however similar approachescan be used for alternative models, for example the RSmodel of warped extra dimension where a similar relationcan be found.

We have proposed that a sufficiently advanced civi-lization could utilize this relation to generate a localizedexpansion/contraction of spacetime creating a warp bub-ble in which to travel at arbitrarily high velocities. Onevital aspect of future research would be how to locallymanipulate an extra dimension. String theory suggeststhat dimensions are globally held compact by stringswrapping around them [5], [8]. If this is indeed the case,then it may be possible to even locally increase or de-crease the string tension, or even locally counter the ef-fects of some string winding modes. This would achievethe desired effect of changing the size of the extra dimen-sions which would lead to propulsion under this model.It would thus be prudent to research this area further andperform calculations as to the energies required to affectan extra dimension and to try and relate this energy tothe acceleration a spacecraft might experience.


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