Pathology Free Warp Drive2A

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Non-Pathological Warp Drive within Randall-Sundram Brane World Relativity

By: Doctor Paul Karl Hoiland

Abstract: In the original article by M. Alcubierre[1] there was an attempt made toconstruct a metric within the confines of General Relativity whereby a spacecraft couldtraverse an arbitrarily long distance in an arbitrarily short time period. This concept ispopularly referred to as Warp Drive in Science Fictore lore. While many Authors,including those in ESAA, have made attempts from time to time to place this work byAlcubierre on a solid physical footing, several physical manifestation shortcomings havebeen identified. Of these shortcomings, two of the strongest prohibitions have remained:1.) those raised by Authors like Ford[2], dealing with quantum energy conditionviolations, and (2)those originally raised by Jose Natario[3] when it comes to objects inthe path of a warp driven craft and navigation in general. Without a proper physical

solution to these problems the prospect of ever engineering a superluminal version ofpropulsion remains problematic at best. It is our contention that there is a Brane modelsolution to these problems. If and when certain Brane world models becomeexperimentally verified such a solution could be implemented and that such a solutioninvolves no pathologies such as no blueshifted photons, no local violation of normalquantum energy conditions, no violation locally of the speed of light, and no generalproblems as far as navigation goes.

PACS numbers 04.50. + h, 04.20. Jb, 04.20.CvMSC: 83E15, 83E30

INTRODUCTIONIn this Model that we have employed our 3D + 1 dimensional universe is but part of alarger bulk where the zero point field of Quantum mechanics becomes the brane ofpopular M-Theory. In the case of the extra dimension under the Randal-Sundrum Model,the length or volume of space-time is determined by the bulk cosmological constant[4] onboth a local and global level. Gravity becomes higher dimensional at scales where[7]

1/r 1/r1+N. (1)

Basically showing us that as scale changes, so does the manifestion of gravity.

Thus, as a starting point we will consider a five-dimensional Einstein gravity coupled toan unspecified as yet, energy-momentum tensor TAB which is independent of the fourspace-time coordinates x. We will for the sake of argument here assume that TAB resultsfrom a certain combination of branes, bulk cosmological constant and classical fieldconfigurations which preserve Poincare invariance. Einsteins equations can be stated as:

GAB = 8TAB + gAB (2)


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gives rise to a metric of the following form:

ds2 = A(z) (ds42 dz2) (3)

where z is an extra coordinate, and we assume that the four-dimensional metric ds42

isflat. The volume of the extra space in this construction is determined by an integral

-+ A5/2(z)dz . (4)

In our specific model A(z) = (1 + H|z|)-2, where H is proportional to the square root of thecosmological constant. This warp factor gives rise to a finite expression[5]. An infinitevolume arises if the value of A(z) tends to a nonzero constant when z goes to infinity. Inall cases where the volume is non-infinite there is a normalizable zero-mode graviton inthe spectrum fluctuations of this background[5]. Parametrizing the four-dimensionalgraviton fluctuations we find:

ds2 = A(z)[( + h(x,z))dxdx dz2]. (5)

The linearized Schrdinger equation for the excitation h(x,z) =A-3/4(z)(z)h(0)(x) takesthe form:

( -z2 + {(/A) (2/4A2)})(z) = m2 (z) (6)

where h(0) = -m2h(0) and the prime denotes differentiation with respect to z. Thespectrum remains normalizable as long as the volume does not go infinite and z is just aconstant.

Since in our model A(z) = (1 + H|z|)-2, where H is proportional to the square root of thecosmological constant any modification on the local level of the cosmological constantwill result in a modification to the extra volume and indirectly in the value of Einsteinsgravity via the equation:

GAB = 8TAB + gAB. (7)

An example is seen in the normal expected flat Minkowski condition metric

ds2=2(z)dxdx-dz2 (8)

since is the normal four-dimensional Minkowski metric with

(z)=e-k|z| (9)

being the warp factor[see Authors Notes] where

k=(4/3)G(5) (10)


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=e (2kt)+Yt [13a]1/=e (2kt) +Yt [13b]

and as such

d/dt=de(2kt) +Yt

/dt [14a]= e (2kt) +Yt(-1)((2k)+Y). [14b]

Hince,

(1/ )(da/dt)=e(2kt) +Yte-(sqrt(2kt) +Yt)(-1)(2k) +Y [15a](1/)(d/dt)=(-1)(2k)+Y [15b]

combining equations[12] and [15], it follows

T00=-6k2 + 3[(-1)(2k) +Y]2 [16a]

T

00

=-6k

2

+ 3(Y

2

+ 2(2Y)k +2k

2

[16b]T00=-6k2 + 3Y2 + 6(2Y)k + 6k2 [16c]T00=3Y2 + 6(2Y)k [16d]

If we maintain the condition where 3Y2>-6(2Y)k, then T00 will remain positive up to thelimit of T00=0.

We can now observe that k is large for brane lensing. In turn, H does not need to be largeand dH/dt


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Since Frame A assumes optically that frame B is the same volume any Photon sent inframe B would be measured by frame A to have traveled superluminally even thoughthere has been no change in the local value of c between frame A and frame B. It is onlythe volume that has been altered by brane lensing which cannot be measured directly byframe A.

Going further if we consider equation (37) in [8], we have

T00 = -6k2 + 3(1/(d/dt))2 (17a)T00 = -6k2 + 3(d ln()/dt)2 (17b)T00 = (g00)2T00, (17c)

where g00 = 1 since c = 1.

We find that:

= e-(2kt + Yt) (18a)1/ = e2kt +Yt (18b)

and so it follows:

d/dt = de-(2k + Yt)/dt = e (2kt + Yt)(-1)(2k +Y). (19)

Hence, it can be said:


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(1/a)(d/dt) = e2kt +Yte-(kt +Yt)(-1)(2k + Y) (20a)(1/a)(d/dt) = (-1)(2k + Y) (20b)

Combining equations it follows

T00

= -6k2

+ 3[(-1)(2k + Y)]2

(21a)T00 = -6k2 +3(Y2 + 22Y k + 2k2) (21b)T00 = -6k2 + 3Y2 + 62Y k + 2k2) (21c)T00 = 3Y2 + 62Y k (21d)

When 3Y2 > -62Y k, T00 is always positive. From this we can now show that the stressenergy tensor for this type of warp drive remains positive and low. The limit case iswhen T00 = 0. The total energy need to achieve such a warp drive condition is given by:

E = T00dV (22)

From this we can determine that for dV we must have:

3((1/)(d/dt)2 > 6k2 (23)

for the normal RS Model condition at the present time. The expression for used in thiswork is an ideal one and further work would be required to find a more realistic one. It isthe Authors suggestion that a collider could be used to generate the local energyrequirement to modify brane tension, T00, to further refine the value of and eventually ahybrid version of a collider based multiple particle generator could be used to generate aworking warp field.

The idea in general is to use such a warp field at some future date after development of aproper conventional propulsion system capable of sustained acceleration at 1G or betterup to a terminal velocity of .5 to .75C. The Warp Field generator system would form abrane lensed conduit where the volume of space-time was decreased within which thecraft would then travel under this future to be developed propulsion. In effect, even witha terminal velocity of .5C if the volume was reduced in half then an effective travel timeequal to 1C or better becomes possible. The control factor on maximum travel timebecomes limited by the energy requirement of the warp field generator for a given warpfactor and by the maximum acceleration of the conventional propulsion system itself.

Travel times from point to point within such an augmented system would be acombination of conventional relativistic time dilation plus the brane lensing volumedecrease effect upon those travel times. However, given the vast distances and traveltimes involved from the perspective of earth any fractional decrease would be of anadvantage when it comes to interstellar flight in general. Since actual velocity evenwithin the warp conduit remains subluminal normal navigational instrumentation andcommunication becomes possible once the severe time expanse of two-waycommunication at C limited means is accepted. Within short range of the craft since the


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actual velocity remains less than C radar and ladar systems should function to warn theoccupants of such a powered craft of any objects in their direct path.

CONCLUSIONS

The title of this work was Non-Pathological Warp Drive within Randall-Sundram BraneWorld Relativity. This title was chosen aptly as an extension of the original work byMiguel Alcubierre on Warp Drive within Einsteins General Relativity. The theory andconcepts proposed in this work are based upon eventually Brane World theory beingexperimentally proven as valid. At the current time neither String Theory, nor itsextension Branes or M-Theory have been able to be supported by experimental evidencedue mostly to the extreme energy levels involved where their effects become detectable.But such may change in the future since models like the Randall- Sundrum one offermany avenues within reach of experimentally testing in the near future. A lot of thespeculation and theory that has arisen over the last decade or so dealing with possiblesuperluminal modes of travel has arisen because of the legacy of Einstein and of

Quantum mechanics. We know from experimental evidence that on quantum scalesparticles can tunnel and seemingly cross distances along shorter paths. The object ofsuch theories as this one we have outlined are to find such short cuts for larger macroscale objects like spacecraft and at the same time to better understand the universe welive in. But it must be remembered that all of this is theory designed by the mind of manin his quest for knowledge of the unknown.

REFERENCES

[1] Miguel Alcubierre, Classical and Quantum Gravity, vol. 11, L73-L77, (1994)

[2] Michael J. Pfenning & L.H. Ford, The unphysical nature of "Warp Drive",gr-qc/9702026and

[3] J. Natario, Warp Drive with Zero Expansion, Class. Quant. Grav 19 (2002) ] gr-qc/0110086

[4] L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83,4690 (1999)

[5] M. Gogberashvili, hep-ph/9812296 and hep-ph/9812365

[6] Chung, D. Freese, Can Geodesics in Extra Dimensions Solve the CosmologicalHorizon Problem, Phys. Rev. D62 (2000) 063513[7] N. Arkani-Hamed, S. Dimopoulos, D. Davli, Phys. Lett. B429, 263 (1998) andPhys. Rev. D59, 0860 (1999)[8] Marco Cavaglia, Black hole and brane production in TeV gravity: A review,Int.J.Mod.Phys. A18 (2003) 1843-1882


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AUTHORS NOTES

1.) Any variance induces an (anti-)de Sitter rather than flat geometry on the brane.

(See V. Rubakov, Large and Infinite Extra Dimensions, hep-ph/ 0104152) Thisresults in a brane lensing effect which depending upon how you vary the branetension will either make objects on the bulk appear closer or further away fromthe point of brane tension change with resulting volume changes on and off thebrane. (See Jonathan Bagger, Michele Redi, Radion effective theory in thedetuned Randall-Sundrum model, JHEP class, http://arxiv.org/PS_cache/hep-th/pdf/0312/0312220.pdf)
http://arxiv.org/PS_cache/hep-th/pdf/0312/0312220.pdfhttp://arxiv.org/PS_cache/hep-th/pdf/0312/0312220.pdfhttp://arxiv.org/PS_cache/hep-th/pdf/0312/0312220.pdfhttp://arxiv.org/PS_cache/hep-th/pdf/0312/0312220.pdf

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