UNCL ASSIF IED//F OR OF F ICIAL USE ONL Y Defense Intelligence Reference Docum ent Acquisition Threat Support April 2010 : 1 D ecember 2009 .-08-1004-001 W arp Drive, Dark Energy, and the M anipulation of Extra Dim ensions UNCL ASSI F IED //F OR OF F ICIAL USE ONI=¥ U N CL ASSI F I ED //F OR OF F ICIAL USE ONL Y W arp Drive, Dark Energy, and the M anipulation of Extra Dim ensions Prepared by: A cquisition Support Division (DWO-3) Defense Warning Office Directorate for A nalysis Defense Intelligence A gency A uthors: AAP Person 74, AAP Person 58 A dministrative Note C O PYR IG H T W AR N IN G : Further dissem ination of the photographs in this publication is not authorized. This product is one in a series of a dva nced technology reports produced in FY 2009 under the D efense Intelligence Agency, D efense W a rning Office's Adva nced Aerospa ce W ea pon S ystem Applica tions (AAW S A) Progra m. Comments or questions perta ining to this document should be a ddressed to|AAP Person 1 | AAW S A Progra m Ma na ger, D efense Intelligence Agency, ATTN: CLAR/D W O-3, Bldg 6000, W a shington, D C 20340-5100. UNCL ASSIF IED//BQB QEEICIAL USE ONL Y UNCL ASSIF IED//F OR OF F ICIAL UDE ONL Y Contents Introduction...........................................................................................................v 2. General Relativistic Warp Drives..................................... 1 2.1 Warp Drive Requirements............................... 2 3. The Cosmological Constant................................. 4 3.1 Einstein's Equation and the Introduction of A ............................. 4 4. Casimir Energy and the Quantum Vacuum....................................... 5 4.1 The Casimir Effect................. 6 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 H igher Dimensional A rtifact....................... 12 7. Warp Drive and H igher Dimensional Manipulation.............. 15 7.1 A djusting H igher Dimensions for Propulsions............. 16 7.2 The Geometry of Extra Dimensions............................................ 17 7.3 H igher 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 F igures Figure 1. York Extrinsic Time (J) Plot.....................................................................1 Figure 2. The Interior Region of Parallel Conducting Plates.................. 7 Figure 3. Internal Structure of a Seemingly One-Dimensional Object....................9 Figure 4. Manipulated Extra Dimension................................... 15 iii UNCL ASSIF IED//F OR OF F ICIAL UGE ONL Y UNCL AS5IF IED//F OR OF F ICIAL UOC ONL Y Figure 5. A rtist's Conception of a Futuristic Warp Drive Spacecraft.................. 16 Figure 6. A Toroidal H igher Dimension............................................ 17 Figure 7. A Combination of Phenomenologically Viable Fields........................... 19 Figure 8. False Vacuum Minima................................. 19 Figure 9. Thick and Thin Shell Warp Bubble............................................ 21 Tables Table 1. Transit Times to Various Exotic Destinations at 100 Times the Speed of Light.............................................................................. ....vi Table 2. Negative Energy Required for Warp Bubble (Larger Negative Energy)....3 Table 3. Negative Energy Required for Warp Bubble ...................... 22 iv UNCL ASSIF IED//F OR OF F ICIAL USE ONL Y UNCL AS5IF IED//EQn OK iCIfth USE ONL Y W arp Drive, Dark Energy, and the M anipulation of Extra Dim ensions 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 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 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 an absolute 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 take many decades in the frame of reference of an observer on Earth. A lthough these lengthy transit times would not make interstellar exploration impossible, they would certainly dampen the enthusiasm of governments or private individuals funding these missions. A fter all, a mission whose success is perhaps 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 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 involves connecting two potentially distant regions of space by a topological shortcut. Theoretically, one would enter the wormhole and instantaneously be transported to the exit located in a distant region of space. A lthough 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 the fabric of space in the immediate vicinity of a spacecraft. The basic idea is to create an asymmetric bubble 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 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 no known restrictions on the motion of space itself, thus allowing for a convenient circumvention of the speed of light barrier. A n advanced aerospace platform incorporating warp drive technology would profoundly alter the capacity to explore—and potentially to colonize—the universe. Because a warp drive is not limited by the speed of light, one can only 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 v UNCL ASSIF IED//rOR OF F ICIAL USE ONL Y uncla ssifiED//ron official use only spacecraft capable of 100c1 for an array of exotic destinations of possible interest. A s 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. 1 This speed, while somewha t a rbitra ry, highlights the fa ct tha t our ga la xy would become fa r more a ccessible if or when one discovers how to surpa ss the speed of light ba rrier. Table 1. Transit Times to Various Exotic Destinations at 100 Times the Speed of Light Destination Transit Time Ma rs 193 seconds Jupiter 36 minutes Neptune 4 hours Alpha Centa uri 15 da ys Epsilon Erida ni 38 da ys The Orion Nebula 1.3 yea rs Until recently, the warp drive was a concept reserved for science fiction. H owever, a 1994 paper by Miguel A lcubierre placed the idea on a more solid theoretical footing. A lcubierre (Reference 1) demonstrated that a specific Lorentzian manifold could be chosen that exhibited bubble-like features reminiscent of the warp drive from the popular S tar 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 original idea. H owever, these papers do not typically address how one might actually create 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 of quantum field theory; this diverges from the more traditional approach to warp 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 the redshifting 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 of energy 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). vi UNCL ASSIF IED//F OR OF F ICIAL UOE ONL Y UNCL ASSIF IED//F OR OF F ICIAL USE ONL Y A lthough 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 be a cosmological vacuum energy (Reference 8-10). These observations are based on the magnitudes of high-redshift supernova and have been a source of high research activity of late owing to the unexpected discovery that the rate of expansion of the universe is increasing (commonly referred 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 that experimental breakthroughs at the Large H adron Collider or developments in the field of M-theory could lead to a quantum leap in our understanding of this unusual form of energy and perhaps help to direct technological innovations. Our own research focuses on gaining an understanding of the physical origin of 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 and allows 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 advancement of 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 the physics required to understand them. Section 3 discusses the cosmological constant, a 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 be the phenomenon that physically generates the cosmological constant. Section 5 discusses higher dimensions in physics 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 the accelerated 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 that would be necessary to turn this model into a reality. vii UNCL ASSIF IED//F OR OF F ICIAL UOE ONL Y UNCL AS5IF IED//F OR OF F ICIAL UGC ONL Y 2. General Relativistic W arp Drives Alcubierre (Reference 1) derived a spa cetime metric motiva ted by cosmologica l infla tion tha t would a llow a rbitra rily short tra vel times between two dista nt points in spa ce. The "wa rp drive" metric uses coordina tes (t, x, y, z) a nd curve (or worldline) x = xSh(t), y = 0, z = 0, lying in the t-x pla ne pa ssing through the origin. Note tha t x sk is the x-a xis coordina te position of the moving spa ceship (or wa rp bubble) fra me. The metric1 specifying this pa rticula r spa cetime geometry is (Reference 1): 1 A spa cetime metric (ds2), or line element, is a Lorentz-inva ria nt dista nce function between a ny two points in spa cetime tha t is defined by ds2 = g^dx^dx', where g^,. is the metric tensor which is a 4x4 ma trix tha t encodes the geometry of spa cetime a nd dx" is the infinitesima l coordina te sepa ra tion between two points. The Greek indices (n, v = 0...3) denote spa cetime coordina tes, x°...x3, such tha t xL-.x3 - spa ce coordina tes a nd x° = time coordina te. ds2 = —c2dt2 +[rfr- vsh(/)/(^(Z))^]* + dy2 + dz2, (2.1) where c is the speed of light, Vsh(t) is the speed a ssocia ted with the curve (or wa rp bubble speed), a nd rSh(t) is the Euclidea n dista nce from the curve. The wa rp bubble sha pe function f (rSh) is a ny smooth positive function tha t sa tisfies f (0) = 1 a nd decrea ses a wa y from the origin to va nish when rS h > R for some dista nce R. The geometry of ea ch spa tia l slice is fla t, a nd spa cetime is fla t where f (rSh) va nishes but is curved where it does not va nish. The driving mecha nism of Equa tion (2.1) is the York extrinsic time, 9. This qua ntity is defined a s (Reference 1): 9 = ^ c xh ^f (2.2) The 9 beha vior of the wa rp drive bubble provides for the simulta neous expa nsion of spa ce behind the spa cecra ft a nd a corresponding contra ction of spa ce in front of the spa cecra ft. Figure 1 illustra tes the 9 beha vior of the wa rp drive bubble geometry. Thus the spa cecra ft is enveloped within a wa rp bubble a nd ca n be ma de to exhibit a n a rbitra rily la rge fa ster-tha n-light (FTL) speed (vS h >> c) a s viewed by externa l coordina te observers. Even though the worldlines inside the wa rp bubble region a re spa celike for a ll externa l observers, the moving spa ceship (wa rp bubble) fra me itself never tra vels outside of its loca l comoving light cone a nd thus does not viola te specia l rela tivity. F igure 1. York Extrinsic Tim e (9) Plot 1 UNCL ASSI F IED/ / F OR OF F ICIAL USE ONL Y UNCL AS5IF IED//rOR OF F ICIAL USE ONL Y 2.1 W ARP DRIVE REQUIREM ENTS Implementa tion of FTL interstella r tra vel via wa rp drives requires engineering of spa cetime into very specia lized loca l geometries a s shown by Equa tion (2.1). The a na lysis of these via the genera l rela tivistic field equa tion plus the resulta nt source ma tter equa tions of sta te demonstra tes tha t such geometries require the use of "exotic" ma tter in order to produce the requisite FTL spa cetime modifica tion. Exotic ma tter is genera lly defined by genera l rela tivity (GR) physics to be ma tter tha t possesses (renorma lized) nega tive energy density a nd/or nega tive stress-tension (= positive outwa rd pressure, a ka gra vita tiona l repulsion). The term is widely misunderstood a nd misa pplied by the non-GR community. Also, it ha s been cla imed tha t FTL spa cetimes a re not pla usible beca use exotic ma tter viola tes the genera l rela tivistic energy conditions.2 However, this ha s been shown to be a spurious issue (Reference 11). 2 The condition for ordina ry, cla ssica l (non-exotic) forms of ma tter tha t we a re fa milia r with in na ture is tha t pg > p a nd/or pt > 0, where pE is the energy density a nd p is the pressure/stress-tension of some source of ma tter. These conditions represent two exa mples of wha t a re va riously ca lled the "sta nda rd" energy conditions: W ea k Energy Condition (W EC: pg > 0, pe + p > 0), Null Energy Condition (NEC: pE + p > 0), D omina nt Energy Condition (D EC), a nd S trong Energy Condition (S EC). These energy conditions forbid nega tive energy density between ma teria l objects to occur in na ture, but they a re mere hypotheses. The energy conditions were developed to esta blish a series of ma thema tica l hypotheses governing the beha vior of colla psed-ma tter singula rities in the study of cosmology a nd bla ck holes. The energy density for the Alcubierre (Reference 1) wa rp drive tha t is derived from the genera l rela tivistic field equa tion is complex, so we instea d use a more simple formula to express the net energy required, Twp, to build a wa rp bubble a round a spa ceship (Reference 12): £wuip G (2.3) ^-^IxlO44)^/?2o, \ / warp * where G is Newton's universa l gra vita tion consta nt (6.673 x 10 11 Nm2/kg2), vW a ip is the dimensionless speed of the wa rp bubble, W (> 0) is the ra dius of the wa rp bubble, a nd o (> 0) is proportiona l to the inverse of the wa rp bubble wa ll thickness A (i.e., a ~ 1/A). Equa tion (2.3) cha ra cterizes the a mount of nega tive energy tha t one needs to loca lize in the wa lls of the wa rp bubble. Ta ble 2 presents a ta bula tion of the required nega tive energy a s a function of the "wa rp fa ctor," vwa rp. One ca n compa re the va lues of £W iup in the ta ble with the (positive) rest-energy conta ined in the S un (1.79 x 1047 J). The consequence of Equa tion (2.3) a nd Ta ble 2 is tha t if one wa nts to tra vel a t hyperlight speeds, then the wa rp bubble energy requirement will be a n enormous nega tive number. And this rema ins true even if one engineers a n a rbitra rily low sublight speed wa rp bubble. Engineering a wa rp drive bubble is quite da unting given these results. 2 U N CL ASSI F I E D/ /f«R-em «M M »M M4^ UNCL ASSIF IED//F OR OF F ICIAL UOC ONL Y Table 2. Negative Energy Required for Warp Bubble (Larger Negative Energy) W a rp Fa ctor, vwa rp £varp (J) I0’5 (= 3 km/s) -3.03 x 1040 10"4 (= 30 km/s) -3.03 x 1042 0.01 (= 3,000 km/s) -3.03 x 1046 0.5 (= 150,000 km/s) -7.59 x 1049 1 (= light speed) -3.03 x 1050 2 (= 600,000 km/s) -1.21 x 1051 10 (= 3.0 x IO6 km/s) -3.03 x 1052 100 (= 3.0 x 107 km/s) -3.03 x 1054 A ssum e: 7? = 50 m . o = 10’m 1 Lobo a nd Visser (Reference 12) constructed a n improved model of the wa rp drive spa cetime by a pplying linea rized gra vity to the wea k-field wa rp drive ca se a nd testing the energy conditions to first a nd second orders of vwa (p. The funda menta l ba sis of their model is tha t it specifica lly includes a finite ma ss spa ceship tha t intera cts with the wa rp bubble. Their results verified tha t a ll wa rp drive spa cetimes viola te the energy conditions a nd will continue to do so for a rbitra rily low wa rp bubble speed. They a lso found tha t the energy condition viola tions in this cla ss of spa cetimes is generic to the form of the geometry under considera tion a nd is not a side effect of the superlumina l properties. Ba sed on these fa cts plus Equa tion (2.3) a nd Ta ble 2, it a ppea rs tha t for a ll conceiva ble la bora tory experiments in which nega tive energy ca n be crea ted in minute a mounts, the wa rp bubble speed will be a bsurdly low. Coupling of the finite spa ceship ma ss with the wa rp bubble lea ds to the (quite rea sona ble) condition tha t the net tota l energy stored in the wa rp bubble be less tha n the tota l rest-energy of the spa ceship itself, which pla ces a strong constra int upon the (dimensionless) speed of the wa rp bubble (Reference 3): £ W^M' (2-4) ( M'"ship where MShiP a nd /?ShiP a re the ma ss a nd size of the spa ceship, respectively, a nd R is the ra dius of the wa rp bubble. Equa tion (2.4) indica tes tha t for a ny rea sona ble va lues of the engineering pa ra meters inside the bra ckets, vwa rp will be a bsurdly low. This result is due to the intrinsic nonlinea rity of the genera l rela tivistic field equa tion. To illustra te this point, the exa mple sta rship pa ra meters from Ta ble 2 (/? = 50 m, A ~ 1/a = IO-3 m) a re inserted into Equa tion (2.4) a nd a ssume MShiP = 106 kg to find tha t vwa rp < 1.72 x 10 14 (or 5.16 x 10 6 m/s). Ga rden sna ils ca n cra wl fa ster tha n this. And if R a nd MShiP 3 U N CL ASSI F I E D/ /f^ft^H«Afe-W S»N4¥ UNCL AS5IF IED//rOR OF F ICIAL UOC ONL Y a re kept consta nt, then A = 3.37 x 1024 m (or 3.57 x 108 light-yea rs) in order for vwa rp < 1, which is a n unrea listic requirement on the wa rp bubble design. Beca use this energy requirement is so phenomena lly high one finds it of pa ra mount importa nce to explore new idea s in the field of wa rp drive technology. W ha t now follows is a peda gogica lly rich review of the novel wa rp drive concept tha t we ha ve been developing since 2005. 3. The Cosm ological Constant Einstein is fa mous for a multitude of a chievements in the field of physics. Argua bly his most nota ble contribution is the Genera l Theory of Rela tivity, a geometric description of gra vita tion whose funda menta l idea rela tes the ma tter a nd the energy content of the universe to the geometry of spa cetime. S imply put, the presence of ma tter a nd energy ca uses spa cetime to curve, a nd this curva ture controls how ma tter a nd energy move through spa cetime. Genera l rela tivity ha s been the preva iling theory of gra vita tion since 1915 a nd thus fa r ha s una mbiguously pa ssed observa tiona l a nd experimenta l tests. It rema ins a n a ctive a rea of resea rch a nd technology is still being developed to test certa in fea tures of the theory. Gra vita tiona l wa ves, for exa mple, a re one prediction from GR; however, technology is only now rea ching the sta ge of ma turity to a llow for the detection of these wa ves. 3.1 EINSTEIN'S EQUATION AND THE INTRODUCTION OF A Upon completion of GR, Einstein a pplied his theory to the entire universe. He firmly believed in Ma ch's principle, a nd the only wa y to sa tisfy this wa s to a ssume tha t spa ce is globa lly closed a nd tha t the metric tensor should be determined uniquely from the energy-momentum tensor (Reference 13). He a lso a ssumed tha t the universe wa s sta tic, which wa s a rea sona ble a ssumption a t the time beca use observa tiona l a stronomy ha d not a dva nced to a level tha t contra dicted this pa ra digm. In 1917, when a sta tic solution to his equa tions could not be found, he introduced the cosmologica l consta nt A (Reference 14): 3 3 Pronounced "la mbda ." 4 T^ encodes the density a nd flux of a ma tter source's energy a nd momentum. I SttCj Rfiv ~ ^8#? ~ ~^fiv ^ ^8(3.1) 2 c In this equa tion R/IV is the Ricci curva ture tensor, R is the Ricci curva ture sca la r, T^ is the stress-energy-momentum tensor,4 a nd gfiv is the spa cetime metric. The left-ha nd side of Equa tion (3.1) encodes the curva ture in the geometry of spa cetime, a nd the right-ha nd side encodes the source of ma tter-energy tha t curves spa cetime. The a ddition of A ca n be understood a s a term in the equa tion which a llows one to a djust theory to ma tch observa tion. In Einstein's ca se, he chose to a dd A to ensure tha t the universe wa s sta tic a nd uncha nging. In la ter yea rs, he often referred to this a mendment to his equa tions a s his "biggest blunder." S evera l yea rs a fter GR ha d been formula ted, the a stronomer Edwin Hubble discovered the phenomenon of ga la ctic redshifting, which strongly indica ted tha t the universe wa s indeed expa nding. This 4 UNCL ASSI F I ED//F OR OF F ICIAL USE QNhY UNCL ASSIF IED//F OR OF F ICIAL USE ONL Y theoretica l prediction from GR wa s ignored by Einstein beca use of his belief in a sta tic universe. Even though Einstein retra cted the a ddition of A into his equa tions, it is now known tha t it does indeed pla y a role a nd is typica lly included in GR equa tions. D a ta from precise a stronomica l observa tions strongly suggest tha t a n extremely sma ll, yet non-zero A is a necessa ry fea ture of GR a nd is responsible for the expa nsion of the universe tha t is observed. From a physica l perspective, A represents a n inherent energy density a ssocia ted with empty spa ce. One wa y to envision this is to ta ke a perfectly insula ting box into deep spa ce, a nd then to remove a ll ma tter a nd a ll energy from this box so tha t it encloses a perfect void. Even in this emptiness, a residua l energy field would rema in. According to GR, the effect of this energy would be to ca use the region of spa ce to expa nd, a lbeit a t a n extremely sma ll ra te. To summa rize, A is a ubiquitous, ever present fea ture of spa ce, a nd its presence ca uses spa ce to expa nd. In the la te 1990s it emerged tha t not only is the universe expa nding, but the ra te of expa nsion is, in fa ct, increa sing. S ince then, it ha s become more popula r to refer to A a s da rk energy, a nd the rema inder of this pa per will follow this convention. Although the role of da rk energy is extremely well understood ma thema tica lly, a nd in the context of its effects on spa cetime, its physica l na ture is still a mystery. One knows tha t it is homogeneous, not pa rticula rly dense, a nd tha t it does not intera ct with a ny of the funda menta l forces of na ture. One a lso knows tha t it exerts nega tive pressure on spa cetime, which expla ins the observed a ccelera ted expa nsion (Reference 15, 16). As there is yet to be a rea sona ble expla na tion for the funda menta l origin of da rk energy, the problem is considered serious a nd ha s been ta ckled by a la rge number of eminent a nd respected physicists, including previous Nobel prize winners (Reference 17). Beca use da rk energy is intima tely rela ted to the expa nsion of spa ce, a nd beca use this expa nsion is exa ctly the fea ture tha t would a llow for a wa rp drive to function, a n understa nding of this mysterious energy is of pa ra mount importa nce in the development of this novel propulsion technology. 4. Casim ir Energy and the Quantum Vacuum A centra l theme in this pa per is the notion of the qua ntum va cuum. To a pa rticle physicist, the term "va cuum" mea ns the ground sta te of a qua ntum field in some qua ntum theory for ma tter. In genera l, this ground sta te must obey Lorentz inva ria nce, a t lea st with rega rds to three spa tia l dimensions, mea ning tha t the va cuum must look identica l to a ll observers. At a ll energies probed by experiments to da te, the universe is a ccura tely described a s a set of qua ntum fields. To a non-physicist a qua ntum field ma y, a t first, be a stra nge concept to gra sp. This is beca use one genera lly likes to visua lize the things one thinks a bout; for exa mple, a n electron a nd even a photon provides something one ca n, on some level, picture in one's minds. S imply put, a qua ntum field is a n inta ngible ma thema tica l object whose properties a re idea l in expla ining na ture. Theories ha ve rea ched such a n a dva nced level tha t the fa milia r physica l ima ges tha t one a pprecia tes 5 UNCL ASSI F IED//TOR OF F ICIAL USE ONL Y UNCL ASSIF IED//F OR OF F ICIAL UOC ONL ¥ must be a ba ndoned for more erudite ma thema tica l constructions which a re better suited a t describing the building blocks of na ture (Reference 18-20). If one ta kes the Fourier tra nsform of a free qua ntum field,5 ea ch mode of a fixed wa velength beha ves like a simple ha rmonic oscilla tor. A qua ntum mecha nica l property of a simple ha rmonic oscilla tor is tha t the ground sta te exhibits zero-point fluctua tions a s a consequence of the Heisenberg Uncerta inty Principle. One wa y to understa nd these zero-point fluctua tions is to ima gine relea sing a pendulum a nd wa tching a s dissipa tive forces slowly try to bring the pendulum to a stop. The uncerta inty principle would ensure tha t the pendulum wa s never a ble to come to a complete rest, but instea d would exhibit microscopic oscilla tions a round the equilibrium position indefinitely. Of course, for a rea l ma croscopic pendulum, these fluctua tions would be miniscule a nd a ll but impossible to detect; however, the a na logy with a qua ntum ha rmonic oscilla tor holds well. The expecta tion va lue of the energy a ssocia ted with the ground sta te energy of a qua ntum oscilla tor is: (£) = nlM (4-1) In this formula c a nd h a re the speed of light a nd Pla nck's reduced consta nt (1.055 x 10’34 Is), respectively, a nd Zeis the wa ve-vector rela ted to the momentum of the qua ntum field. One of the fea tures of this ground sta te energy is tha t the wa ve vector ha s a n infinite degree of freedom. Clea rly this sum is divergent; however, this is a common fea ture of qua ntum field theory, a nd a n a rra y of ma thema tica l techniques known a s renorma liza tion exists to dea l with the infinities tha t a rise. 4.1 THE CASIMIR EF F ECT The qua ntum fluctua tions of the va cuum fields give rise to a number of phenomena ; however, one is pa rticula rly striking. The Ca simir Effect, which will be explored in more deta il in this pa per, is a rgua bly the most sa lient ma nifesta tion of the qua ntum va cuum. In 1948, H. Ca simir published a profound pa per where he expla ined the va n der W a a ls intera ction in terms of the zero-point energy of a qua ntized field (Reference 19). In its most ba sic form, the Ca simir Effect it is rea lized through the intera ction of a pa ir of neutra l pa ra llel conducting pla tes (with sepa ra tion dista nce d). The presence of the pla tes modifies the qua ntum va cuum, a nd this modifica tion ca uses the pla tes to be pulled towa rd ea ch other with a force: qc^2 24(W 2 (4.2) This is a profound result in the sense tha t the origin of this force ca nnot be tra ced ba ck to one of the four funda menta l forces of na ture (gra vity, electroma gnetism, a nd the two nuclea r forces), but is a force tha t is entirely due to a modifica tion of the qua ntum va cuum. 5 By "free" we mea n tha t the field does not intera ct with other fields. 6 U N CL ASSI F I E D/ /f^R^F ««Ate-W ® & ^Nfe* UNCL ASSIF IED//reH OF F ICIAL USE ONL Y F igure 2. The Interior Region of Parallel Conducting Plates. The region experiences a reduced qua ntum va cuum energy density owing to the bounda ry condition the pla tes impose on the fields. This genera tes a mea sura ble a ttra ctive force tha t pushes the pla tes together. For ma ny yea rs, the pa per rema ined unknown (Reference 22), but from the 1970s onwa rd the Ca simir effect received increa sing a ttention, a nd over the la st deca de it ha s become very popula r (Reference 23). The Ca simir effect is a purely qua ntum effect. In cla ssica l electrodyna mics the force between the pla tes is zero. The idea l scena rio occurs a t zero tempera ture when there a re no rea l photons (only virtua l photons) between the pla tes; thus, it is the ground sta te of the qua ntum electrodyna mic va cuum which ca uses the a ttra ction. The most importa nt fea ture of the Ca simir effect is tha t even though it is purely quantum in na ture, it ma nifests itself ma croscopica lly. For exa mple, for two pa ra llel pla tes of a rea A = 1 cm2 sepa ra ted by a dista nce of <7 = 1 //m the force of a ttra ction is F « 1.3 x IO-7 N. This force is certa inly within the ra nge of la bora tory force-mea suring techniques. Typica lly, the ca lcula tions of the expecta tion va lue of the va cuum a re divergent,6 so some form of renorma liza tion must be performed. A full review of the experimenta l verifica tions of the Ca simir effect a re beyond the scope of this pa per, but it is certa inly worth mentioning tha t experiments a t W a shington University using ultra -sensitive Atomic Force Microscopes ha ve experimenta lly verified the theoretica l predictions of the Ca simir force to within 1 percent a ccura cy (Reference 24, 25). Needless to sa y, ma ny physicists consider this to be a rea l a nd well esta blished phenomenon. 6 D ivergent mea ning the equa tion predicts a n infinite result. D ivergences a re typica l in ma ny ca lcula tions using qua ntum field theory, a nd a n a rra y of ingenious tools is used by physicists to extra ct finite a nd mea ningful results. UNCL ASSI F I ED//rOR OF F ICIAL USE ONL Y UNCL AS5IF IED//F OR OF F ICIAL UBC ONL Y In summa ry, qua ntum field theory predicts tha t the va cuum is a n interla ced cobweb of qua ntum fields which a re never strictly a t rest, a nd which exhibit zero-point fluctua tions. These fluctua tions give rise to rea l a nd mea sura ble phenomenon, with the Ca simir effect being the most poigna nt. It seems only na tura l to a ttempt to rela te the idea s from the previous section rega rding a ubiquitous da rk energy field to this qua ntum va cuum energy. If a rela tionship ca n be esta blished, one would be a step closer to the technologica l rea liza tion of wa rp drive. 5. Extra Space Dim ensions In connection with the Ca simir effect, extra dimensions provide a rich a rena for one to genera te models tha t expla in the origin of da rk energy. Technica lly spea king, the Ca simir effect is a direct consequence of the non-trivia l bounda ry conditions tha t the presence of the conducting pla tes imposes upon the qua ntum va cuum. The qua ntum modes on the interior region of the pla tes a re restricted, a nd there is a pressure difference when compa red to the qua ntum va cuum on the exterior region of the pla tes. It is this pressure difference tha t ca uses the pla tes to a ttra ct.7 7 This a na logy is not strictly true a s different geometries ca n, in fa ct, crea te repulsive Ca simir forces a nd so the pressure a na logy brea ks down. It is, however, a useful visua liza tion tool. A very simila r phenomenon to the Ca simir effect ca n occur when the qua ntum va cuum energy in extra space dimensions a re considered. The explora tion of this idea ha s importa nt ra mifica tions in the context of expla ining da rk energy. Before one ca n a ddress these idea s it is necessa ry to review the role of higher spa ce dimensions in physics. It wa s Riema nn, with his development of differentia l geometry in the 19th century, who provided the necessa ry tools to study higher dimensiona l descriptions of the world (Reference 26). Riema nn held the belief tha t 3-dimensiona l spa ce wa s not enough to provide a n a dequa te description of na ture. Improvements in physics led to Ma xwell's unified theory of electricity a nd ma gnetism, a nd then GR, which unified spa ce a nd time with S pecia l Rela tivity (S R). Inspired by these unifica tions, physicists of the ea rly 20th century wa nted to unify gra vity a nd electroma gnetism. The first a ttempt wa s by Nordstrom in 1914, who used a sca la r potentia l for the gra vita tiona l field. La ter W eyl a nd Ka luza , using Einstein's tensor potentia l, followed two sepa ra te pa ths. W eyl's a ttempt involved a n a ltera tion of the geometry of spa cetime in four dimensions. His ea rly a ttempts ha d physica l consequences which did not ma tch experimenta l da ta . However, W eyl's work wa s extended by Einstein a nd S chrodinger independently in the Einstein-S chrodinger non-symmetric field theory, which is widely rega rded a s the most a dva nced unified field theory ba sed on cla ssica l physics. 5.1 K AL UZA-K L EIN THEORY In 1919 Ka luza (Reference 27) offered a unique a pproa ch to unifying gra vity a nd electroma gnetism which involved a dding a n a dditiona l spa tia l dimension to GR, a nd popula ting this extra dimension with two ma thema tica l objects ca lled a vector potentia l Afl a nd a sca la r potentia l . The line element in this theory is given by: 8 UNCL ASSI F IED/ / F OR OF F ICIAL USE ONL Y UNCL AS5IF IED//rOR OF F ICIAL UOB ONL Y els' = ^ ^(^,, + A^A^dx" dx' + 2^'^A^dx^dy + '-dy2 (5.1) where the Greek indices run from 0 to 3 (0 represents the time coordina te a nd 1...3 the spa ce coordina tes), a nd where the higher dimension is expressed using y. The powers of 10M GeV (1 GeV = 109 eV). u n cla ssi fi ED//rerem ettrts»Nt* 10 UN CL ASSI F I ED//F OR OF F ICIAL UOC ONL ¥ M2^^)'^^2 (5.3) Mpi is the Pla nck ma ss12 (in our usua l 4-dimensiona l spa cetime) a nd r is the size of the extra spa ce dimensions. This rema rka ble result indica tes tha t the Pla nck sca le is, in fa ct, a qua ntity tha t is derived from a more funda menta l qua ntum gra vity sca le a nd a lso the volume of the extra dimensions. Physica lly, this implies tha t the gra viton is diluted a cross the bulk with a diminished intersection with the fa milia r 3+1 dimensiona l bra ne. 12 2.18 x 10 a kg. 13 Anti-deS itter spa ce is a Lorentzia n ma nifold with a consta nt nega tive sca la r curva ture. In terms of Genera l Rela tivity, this is a solution to Einstein's field equa tion with a n a ttra ctive cosmologica l consta nt. 14 An S 7Z ? corresponds to a circula r extra dimension with a n a dditiona l symmetry. This type of projection is popula r in a number of higher dimensiona l models due to its a bility to ma thema tica lly "project" out certa in phenomenologica lly undesira ble fields. In this type of model the size of the interna l spa ce is of order 1/M a nd the effective , IO’2 cosmologica l consta nt is on the order A , ., - StiM ^ x—— (Reference 36, 37). S etting a the va lue of the corresponding energy density equa l to the known density of da rk energy, one finds tha t the extra -dimensiona l ra dius is r ~ 10 3cm. S imila r results a re found in models with more complica ted interna l spa ces. This is a n importa nt result in the context of dimensiona l ma nipula tion. Note tha t extra dimensions tha t a re a ccessible to all the S ta nda rd Model fields ca n a lso be rea lized. These models a re known a s Universa l Extra D imensions (UED ). In the ca se of recent experimenta l constra ints, a compa ctifica tion sca le a s low a s 1 TeV is a llowed. 5.3 RANDAL L SUNDRUM BRANE M ODEL S The idea tha t the universe ca n be modeled a s a (mem)bra ne existing in a higher dimensiona l bulk spa cetime ha s received a huge a mount of a ttention in recent yea rs (Reference 38-46). It is possible tha t the bra ne energy density a ffects the spa cetime curva ture, a nd a n a pproxima tion ca n be a chieved by first considering a model where bra nes a re loca ted a t the two ends of a periodic 5th dimension. To ensure sta bility of the model two bra nes a re required to ba la nce the bulk energy. To get a sta ble metric, the effects of the bra ne on the spa cetime must be compensa ted by a nega tive cosmologica l consta nt in the bulk. Thus, the 5th dimension ca n be considered a slice of Anti-deS itter (AdS )13 spa ce bounded by fla t bra nes, a nd the price of keeping the bra nes flat is to introduce curva ture into the 5th dimension. S uch models a re termed wa rped extra dimensions. The Ra nda ll-S undrum (RS I) model (Reference 47, 48) proposes a novel geometrica l solution to the hiera rchy problem. The hiera rchy problem questions why gra vity is so much wea ker tha n the wea k force (which is 1032 times stronger), a nd why the Higgs boson is so much lighter tha n the Pla nck ma ss. In the RS I setup, the S ta nda rd Model fields a re now confined to one of two 3-bra nes which lie a t the endpoints (i.e., fixed points) of a n S1/Zz orbifold,14 except for the Higgs field. One of the bra nes physica lly corresponds to "our" universe a nd is sometimes referred to a s the IR or "visible" bra ne. The closer a S ta nda rd Model field is to the visible bra ne, the grea ter its coupling to the 11 U N CL ASSI F I E D/ /f^R^F ««Ate-U® & ^Nfe¥ UNCL ASSIF IED//F OR OF F ICIAL UCE ONhY Higgs, a nd therefore the grea ter the ma ss. The second bra ne is the UV or "hidden bra ne. The line element in RS I is described by the metric: ds2 - e “*' ^ rjpvdd‘dxv - rd^r . (5.4) ^■is the metric tensor for the D -dimensiona l Minkowski spa cetime, the AdS curva ture ra dius is given by 1/^ a nd the interbra ne sepa ra tion is given by r. The 5th dimension is compa ctified on the orbifold of length a , where a < (p < a . The orbifold fixed points loca ted a t ip= 0 a nd qj= a correspond to the loca tion of the visible a nd the hidden bra ne. The exponentia l fa ctor is referred to a s the wa rp fa ctor a nd is a n a ppea ling fea ture in the RS I model, a s it ca n both genera te a TeV ma ss sca le from the Pla nck sca le in the higher dimensiona l theory a nd reduce the effective gra vita tiona l strength on the visible bra ne through the suppression fa ctor e-2*'^ while reta ining a bulk width tha t is only a couple of orders of ma gnitude a bove the Pla nck sca le. In a ddition to the sta biliza tion of the interbra ne sepa ra tion, the qua ntum effects from the bulk fields (a ka the Ca simir energy) ca n a lso provide a mecha nism for the genera tion of da rk energy on the visible bra ne. 5.4 EXTRA DIM ENSION SUM M ARY Although there is still no direct evidence of extra spa tia l dimensions, there is the possibility tha t the experiments pla nned a t the La rge Ha dron Collider could detect pa rticle deca y signa tures tha t would indica te the presence of higher dimensions. In fa ct, a ll the theoretica l groundwork ha s been performed by theorists, a nd a s soon a s the LHC is running smoothly it could, in principle, demonstra te the existence of higher dimensions in a rela tively short period of time. Any such discovery would represent a truly ra dica l a ltera tion of our understa nding of na ture, a nd ma ny new questions will emerge rega rding the potentia l role extra dimensions could pla y in a dva nced technologies (which will be discussed in more deta il la ter in this pa per). 6. Dark Energy as a Higher Dim ensional Artifact As discussed ea rlier, the va cuum of spa cetime ca n be visua lized a s a sea of qua ntum fields never fully a t rest due to the Heisenberg Uncerta inty Principle. The oscilla tions of the va cuum ra dia te energy over a ra nge of frequencies in much the sa me wa y tha t a n oscilla ting electron emits electroma gnetic energy, a nd in this wa y, there exists a ground sta te energy a ssocia ted with spa ce itself. The va cuum potentia l for a periodic sca la r field in the AD D model described a bove is given by (Reference 48-50): V+ IE y r —T^’ O ' (6-1) where k is the momentum modes of the qua ntum fields, R is the ra dius of the 5th dimension, a nd m is the ma ss of the field. The prime on the summa tion indica tes the n = 0 term is excluded. The integra l over the continuous momentum modes of the 12 UNCL ASSI F I ED//rOn OF F ICIAL USE ONL Y UNCL ASSIF IED//F OR OF F ICIAL UOE ONL ¥ qua ntum fields is divergent, a s is the infinite summa tion over the extra -dimensiona l KK modes; however, a s mentioned ea rlier, dimensiona l regula riza tion (a form of renorma liza tion) ca n be used to extra ct a finite result. This equa tion differs from Equa tion (4.1) in tha t the qua ntum field tha t we consider ha s both ma ss a nd a degree of freedom in the higher dimension. W orking with a sca la r field, the result ca n la ter be extended simply for the ca se of more phenomenologica lly via ble fields (fermions, for exa mple). Our own resea rch focuses upon exploring a new wa y to ha ndle the infinities a rising from Equa tion (6.1), a nd a fter performing a novel regula riza tion it wa s discovered tha t: U — \m~ co — ^-^XK d2m m ) 32^2r2 "= 1 2 (6.2) where 4 is the Riema nn zeta function, a nd Kw.(2mrn) is the modified Bessel function of the second kind. Although the summa tion is infinite, the function converges ra pidly a nd so a good a pproxima tion is obta ined by performing the sum up to n = 10. S ince discovering this formula , the result a grees with deriva tions of this energy ba sed on different regula riza tion methods a nd so one is confident in the va lidity of Equa tion (6.2). It is rela tively stra ightforwa rd to ca lcula te the contributions to the va cuum energy density coming from ea ch field in the S ta nda rd Model of pa rticle physics. For exa mple, the electron is a funda menta l qua ntum field whose ubiquitous ground sta te energy contributes to the va cuum energy density. S imila rly, the photon is a funda menta l qua ntum field whose ubiquitous ground sta te energy a lso contributes to the va cuum energy density. In fa ct, a ll S ta nda rd Model fields contribute a finite a nd ca lcula ble component to the overa ll energy density of spa ce. Equa tion (5.1) expresses the va cuum energy density for a periodic ma ssive sca la r field. Using knowledge of supersymmetry multiplets it is possible to enumera te this energy for a ll fields occurring in the S ta nda rd Model (Reference 49): r,.J<'-2V (n. Also, knowledge of the va cuum energy density for a massless field: 64 7" r will a llow one to fully a rticula te the energy density of the va cuum in terms of the building blocks of na ture. This is expressed a lgebra ica lly a s: U N CL ASSI F I ED//W R OF F ICIAL USE ONh¥ UNCL ASSIF IED//F OR OF F ICIAL USE ONL Y (6-5) where the index i runs over the spectrum of S ta nda rd Model fermions. Using Equa tion (6.5), a nd slight va ria tions, we a re a ble to computa tiona lly build a model which demonstra tes how the va cuum energy density va ries a s a function of higher dimensiona l ra dius. It ca nnot be oversta ted tha t a full understa nding of the va cuum structure is of critica l importa nce when a ttempting to understa nd the na ture of da rk energy a nd to investiga te its possible ma nipula tion. It is importa nt to a pprecia te tha t, for exa mple, a physica l electron does not a ctua lly ha ve to be present a t a specific point in spa ce for it to contribute to the va cuum energy density. But tha t the va cuum a lwa ys ha s the potentia l to a llow a n electron to exist a t a ny point in spa ce. Thus, a t a ll points in spa ce a virtual electron exists. This virtua l electron is a ba sic a nd funda menta l fea ture of the intrinsic ma keup of spa cetime itself. In this wa y, empty spa ce ha s a ground sta te energy tha t is due to the virtual contributions of a ll fields tha t occur in na ture: electrons, qua rks, photons, a nd indeed, the entire pa rticle zoo. Ma ny a ttempts ha ve been ma de to rela te this va cuum energy to da rk energy; however, beca use these qua ntum fields a re free to oscilla te over a wide ra nge of possible frequencies, when one ca lcula tes the sum15 of a ll the contributions from a ll possible frequencies of the va cuum, a n energy density fa r in excess of tha t seen in na ture is recovered (Reference 50). 15 Technica lly we integra te over a ll the possible frequencies. Our own work (Reference 51, 53) ha s demonstra ted tha t when the contribution due to the extra -dimensiona l qua ntum va cuum fields is included, it is possible to "tune" the theoretica l energy density of the universe to a gree with experimenta l observa tions using extensions of Equa tion (6.5), provided a llowa nce for certa in exotic fields to exist within the higher dimension. Although this ma y a t first a ppea r counterintuitive, one novel fea ture of the qua ntum va cuum energy is tha t it ca n contribute both positive and nega tive energy to the va cuum. The sign of the contribution is funda menta lly due to the na ture of the underlying virtua l qua ntum field. For exa mple, virtua l fermionic fields (e.g., electrons) contribute a n overa ll positive energy to the va cuum, wherea s virtua l bosonic fields contribute a n overa ll negative energy. In this wa y, certa in field combina tions a llow for energy cancellations. The a dditiona l freedom encountered in higher dimensiona l theories mea ns tha t it is a fa irly stra ightforwa rd ma tter to a djust the overa ll va cuum energy density to a gree with the experimenta lly mea sured va lue for the cosmologica l consta nt. Essentia lly, this mea ns tha t one is immedia tely presented with a na tura l expla na tion for the existence of da rk energy. Previous a ttempts to link da rk energy to the va cuum energy ha d yielded grossly high theoretica l predictions fa r in excess of tha t observed in na ture; however, by including the contributions from higher dimensiona l fields we ha ve shown tha t the ta ming of this da rk energy density is entirely possible. The significa nce of this result is tha t it provides a founda tion upon which to explore possibilities rela ting to wa rp drive propulsion. More simply, once one knows why spa ce expa nds, it becomes possible to explore technologica l possibilities to potentia lly ma ke spa ce expa nd. 14 UNCL ASSI F I ED//rOn OF F ICIAL USE ONL Y UNCL AS5IF IED//rOn OF F ICIAL UOC ONL Y To summa rize, the existence of da rk energy ma y be a ttributed to the combined effects of va cuum fluctua tions in "norma l" 3+1 dimensiona l spa cetime plus higher dimensiona l contributions. If this model proves to be correct, then idea s extending from this pa ra digm could provide one with intriguing opportunities for technologica l intervention. 7. W arp Drive and Higher Dim ensional M anipulation Both our own resea rch a nd previous work in higher dimensiona l Ca simir energy16 demonstra te tha t the ma gnitude of the va cuum energy is intima tely rela ted to the size of the extra dimension. More precisely, the sma ller the extra dimension, the grea ter the Ca simir energy (a nd vice versa : the bigger a n extra dimension, the sma ller the Ca simir energy). In fa ct, the energy is rela ted to the ra dius of the higher dimension ra ised to the fourth power, which mea ns tha t very sma ll cha nges in the ra dius of the extra dimension genera te dra ma tic cha nges to the va cuum energy density. 16 Qua ntum va cuum energy a nd Ca simir energy a re often used intercha ngea bly in the litera ture. F igure 4. M anipulated Extra Dim ension. A sufficiently a dva nced technology with the ca pa city to directly intera ct with a nd ma nipula te a n extra dimension would be a ble to loca lly a djust the da rk energy density in a given region of spa cetime. W ha t this mea ns is tha t if a n a dva nced technology wa s a ble to influence the ra dius of a n extra dimension, then it would a cquire direct control over da rk energy, a nd hence the expa nsion a nd contra ction of spa ce itself. As tremendous a fea t a s this ma y sound, a t this ea rly sta ge in the resea rch it is one of the only via ble mecha nisms to genera te a wa rp drive. UNCL ASSI F IED/ / F OR OF F ICIAL USE ONL Y UNCL AS5IF IED//F OR OF F ICIAL USE ONL Y It is worthwhile to expa nd more on the concept of wha t we mea n by a djusting the ra dius of the extra dimension. 7.1 ADJUSTING HIGHER DIM ENSIONS F OR PROPUL SIONS Figure 4 should a ssist in the visua liza tion of wha t a higher dimensiona l spa ce might look like. In this 1-dimensiona l exa mple one ca n see tha t all points ha ve a n a ssocia ted higher dimension. One a ssumes tha t genera lly the ra dius of the extra dimension is fixed. It is this fixed ra dius which genera tes the observed da rk energy density a nd is responsible for the homogeneity in the observed expa nsion of the universe. If one were to locally a djust the higher dimensiona l ra dius then the da rk energy density would a lso cha nge loca lly. More specifica lly, if one were to a djust the ra dius of the extra dimension in the direct vicinity of a spa cecra ft, then the da rk energy density would a lso cha nge only in the vicinity of the spa cecra ft, a s would the expa nsion of spa ce. It is importa nt a t this point to a pprecia te tha t globally, the universe would continue to expa nd a t the ra te we observe toda y, but tha t only in the proximity of the spa cecra ft would spa ce be "stimula ted" to expa nd a t some modified ra te. F igure 5. Artist's Conception of a F uturistic W arp Drive Spacecraft. By loca lly a djusting the size of the extra spa ce dimension, the spa cecra ft is a ble to genera te the necessa ry wa rp bubble required to surpa ss the light speed ba rrier. Of course, obvious questions present themselves. Ca n one rea lly a ssume tha t a higher dimension is circula r? Also, if da rk energy is responsible for the expa nsion of spa ce, ca n we a ssume tha t it ca n somehow be used to contract spa ce, a nd not just expa nd it? 16 UNCL ASSI F I ED//ron OF F ICIAL UGE ONL Y UNCL AS5IF IEDZ/W R OF F ICIAL UOC ONL Y 7.2 THE GEOM ETRY OF EXTRA DIM ENSIONS To a nswer the first question, our most developed higher dimensiona l theory (M-theory) works in seven a dditiona l spa tia l dimensions. The sha pe is believed to be wha t ma thema ticia ns ca ll a Ca la bi-Ya u ma nifold - a complex object tha t is notoriously cha llenging to work with. Physicists often like to work with simpler models (a single extra dimension, for exa mple) with a n uncomplica ted sha pe like the circle. Although this ma y a t first a ppea r to be a gross simplifica tion, often these simplistic higher dimensiona l models both reflect the fla vor of the physics involved, a nd give a ccura te predictions tha t a re believed to devia te from na ture only a t extremely high energies.17 17 For exa mple, in the vicinity of a bla ck hole, or in the first moments of the big ba ng. For this rea son, ma ny of the resea rch pa pers investiga ting higher dimensions choose to work in the simpler circula r higher dimensiona l spa ce. A sma ller fra ction of pa pers explore two a dditiona l higher dimensions, which a re commonly toroida l (see Figure 6), a nd a n even sma ller fra ction of pa pers work in the full M-theoretic Ca la bi-Ya u ma nifold. At these ea rly sta ges of investiga tion, the a dditiona l circula r dimension represents a n a dequa te a pproxima tion. S hould the resea rch progress to a more highly developed pha se, then it ma y become necessa ry to work within the Ca la bi-Ya u ma nifold. F igure 6. A Toroidal Higher Dim ension. This is one of the ma ny possible topologies explored in higher dimensiona l theories. W ith rega rds to the question of whether spa ce ca n be ma de to contract, it seems possible if one ca n ma ke the energy density of a given region of spa ce nega tive instea d of positive. This type of spa ce ha s been well explored by physicists, a nd is known a s a nti-deS itter spa ce. One of the unique fea tures of Ca simir energy is tha t under ma ny conditions it is known to be nega tive, a nd thus with a ca reful ma nipula tion of the higher dimensiona l fields it is, in principle, possible to genera te the required contra ction of spa ce. 7.3 HIGHER DIM ENSIONS AND STABIL IZATION Our goa l in this section is to explore the possibilities of ma nipula ting a higher dimension, which will influence the loca l da rk energy density a nd thus the expa nsion a nd contra ction of spa cetime in the vicinity of a spa cecra ft. Before the issue of how to manipulate a higher dimension ca n be a ddressed, first one must understa nd why a n a dditiona l spa tia l dimension holds some fixed ra dius. This is a well know problem in higher dimensiona l physics a nd is commonly ca lled the problem of "modulus sta biliza tion." Broa dly sta ted, the question is a s follows: if there a re a dditiona l spa tia l dimensions, why do they not perpetua lly expa nd, like our fa milia r dimensions of spa ce, or a lterna tively, why do they not perpetua lly contra ct? W ha t 17 UNCL ASSI F I ED//F OR OF F ICIAL USE ONhY UNCL ASSIF IED//F OR OF F ICIAL UDC ONL Y mecha nism is it tha t a llows for this higher spa ce to rema in compa ct a nd sta ble? Of the ha ndful of theories tha t a ttempt to a nswer this problem, one is pa rticula rly a ppea ling due to its na tura lness. A recurring theme throughout this pa per is Ca simir energy. This energy is compelling due to the fa ct tha t it is a na tura l fea ture intrinsic to the fa bric of spa ce itself. Ca simir energy in higher dimensions not only offers the promise of expla ining the na ture of da rk energy, a s discussed in previous sections, but ca n a lso be utilized a s a mecha nism to stabilize the compa ct extra dimension. W e ha ve discovered in our resea rch tha t with a na tura l combina tion of fields, a nd a single exotic component, we a re a ble to genera te a sta ble extra dimension (Reference 51, 53). The wa y to understa nd this is a s follows: a s mentioned ea rlier, ea ch field in na ture contributes a component to the ground sta te of the va cuum (tha t is, the sta te of minimum energy of spa ce) due to the virtua l pa rticle contributions. W hen this energy is ca lcula ted for the ca se of three la rge a nd one compact spa tia l dimension, one discovers tha t this energy is a strong function of the size of the 5th dimension. In our resea rch, we summed the contributions from a ll the known fields in na ture using extensions of Equa tion (6.5). W e discovered tha t with the a ddition of one a dditiona l exotic field confined to exist in only the 5th dimension, a sta ble higher dimensiona l configura tion wa s found. Our motiva tions for the a ddition of extra exotic fields ha ve a strong phenomenologica l founda tion, a nd these exotic fields been studied successfully in the context of expla ining sola r neutrino oscilla tions (Reference 54-56). A full discussion, however, is beyond the scope of this pa per. To understa nd wha t is mea nt by a sta ble higher dimensiona l configura tion we refer to Figure 7, which is a plot illustra ting the va cuum energy density a s a function of higher dimensiona l ra dius. The x-a xis represents the ra dius of the higher dimension a nd the y-a xis the energy density of spa cetime.18 W e ha ve shown the contributions to the energy density coming from the different fields of the S ta nda rd Model, a nd a lso the a dditiona l exotic field. The overa ll energy density is illustra ted a s a thick bla ck line. The most importa nt fea ture of this gra ph is the ma thema tica l minimum. A good physica l a na logy here is to ima gine relea sing a ba ll from the fa r left of the bla ck line: the ba ll would roll down the line a nd become stuck in the minimum. The ba ll becoming stuck in this minimum is a n excellent a na logy to the dyna mics of the higher dimension, which begins a t some unsta ble sta te, but evolves into a sta ble configura tion. 18 W e work in "norma lized" units. 18 UNCL ASSI F IED/ / F OR OF F ICIAL USE ONL Y UNCL AS5IF IED//F OR OF F ICIAL UOC ONL Y F igure 7. A Com bination of Phenom enologically Viable F ields Generates a Stable M inim um of the Vacuum Energy Density at a F ixed Higher Dim ensional Radius. This sta ble minimum, loca ted a t a positive energy, indica tes a sta ble deS itter type spa cetime a nd reflects a rea listic da rk energy model. Note in this plot tha t the ra dius of the extra dimensions is shown on the x-a xis, a nd the energy density on the y-a xis (norma lized units a re used). A rtifical m inim um N atural m inim um A rtifical m inim um F igure 8 . F alse Vacuum M inim a are Created Around a Spacecraft. The different minima crea te a n a symmetric da rk energy pressure on 3-spa ce which genera tes the wa rp bubble. In this plot the x-a xis represents the ra dius of the extra dimension, a nd the y-a xis represents the energy density (norma lized units a re used). The key to crea ting a wa rp drive is to crea te a fa lse va cuum minimum, i.e., to modify the va cuum spectrum a nd inject some field which crea tes a deS itter minimum a t the rea r of the cra ft a nd a n a nti-deS itter minimum a t the front of the cra ft. W ha t this requires is a technology tha t would a llow us to a rtificia lly ma nipula te the field content illustra ted in Figure 7, shifting the loca tion of the minimum. In this ba sic representa tion, the spa cecra ft would sit in a sta ble region of spa ce corresponding to the na tura l minimum of the extra dimension. At the front a nd rea r of the cra ft, regions of 19 UNCL ASSI F IED/ / F OR OF F ICIAL UOC ONL Y UNCL AS5IF IED//rOR OF F ICIAL UCE ONL Y fa lse minima would be a rtificia lly crea ted via the a djustment of the extra dimension (see Figure 8). These modified regions would correspond to increa sed a nd nega tive da rk energy densities, thus crea ting the wa rp bubble previously discussed. 7.4 EL EM ENTARY W ARP DRIVE CAL CUL ATIONS This section will explore the ca lcula tions releva nt to propulsion. S pecifica lly, it will determine the energy required to a ccelera te a spa cecra ft to the speed of light. By a ssocia ting the cosmologica l consta nt with the higher dimensiona l Ca simir energy, a stra ightforwa rd rela tion between A a nd the ra dius of the extra dimensions ca n be determined: ^ = a-f (7.1) A simpler wa y of developing the rela tionship between the energy density of spa ce a nd the expa nsion of spa ce is to express A a s a function of Hubble's consta nt, H: Hx^A, (7-2) which is a sta nda rd result obta ined from GR. From these two formula , a rela tionship between the expa nsion of spa ce a nd the ra dius of the extra dimension ca n be shown: (7.3) This formula rea lly expresses the founda tion of this novel wa rp drive concept: tha t a sufficiently a dva nced technology with the a bility to a djust the ra dius of the extra dimension locally would be a ble to loca lly a djust the expa nsion a nd contra ction of spa cetime a round a spa cecra ft. This a symmetric expa nsion crea tes the wa rp bubble illustra ted in Figure 1. The spa cecra ft would a lwa ys move within its own light cone a nd thus would not contra dict a ny la w of specia l rela tivity. The possibility tha t the higher dimensiona l ra dius might va ry from pla ce to pla ce ha s been explored in the context of string theory (Reference 57), a nd so is a va lid a ca demic pursuit. However, it ha s never before been suggested tha t this might fa cilita te a new a nd exotic form of propulsion. Equa tions (7.1) a nd (7.2) used together express a rela tionship between the expa nsion of spa ce a nd the energy density of spa ce. A sufficient increa se in the energy density of spa ce would genera te a proportiona l increa se in the expa nsion of spa ce. Thus ca lcula te the energy density of spa ce tha t would be necessa ry to genera te a loca l expa nsion of spa ce a t the speed of light. A loca l expa nsion a t the speed of light, Hubble's consta nt must be increa sed by a fa ctor of: H, = 102fW , (7.4) where W < represents the modified Hubble's consta nt, a nd the subscript c indica tes tha t it is the Hubble's consta nt for the ca se of light speed expa nsion. H must be increa sed by a fa ctor of 1026 to a chieve a loca l expa nsion of spa ce equiva lent to the speed of light. 20 UNCL ASSI F I ED//rOn OF F ICIAL USE ONL Y UNCL ASSIF IED//ron OF F ICIAL UOC ONL Y Another releva nt ca lcula tion rega rds the loca l energy density required to genera te this expa nsion. A simila r ca lcula tion to the one just performed indica tes tha t the cosmologica l consta nt, a n expression which cha ra cterizes the energy density of spa ce, must be increa sed by a fa ctor of 1052. This implies a n energy density of: pc = 1042J/m3 (7.5) This is, indeed, a n incredible number. However, the tota l energy requirement would be reduced if we a ssumed a "thin shell" of modified spa cetime. Figure 9 illustra tes the ca se where the wa rp bubble thickness is reduced. More deta iled resea rch is necessa ry before we could a ccura tely predict the minimum shell thickness tha t would be necessa ry to support a sta ble wa rp field. However, if a shell could be produced tha t wa s merely a single Pla nck length in thickness, then the energy requirements would be reduced immensely. F igure 9. Tw o Im ages Illustrating the Contrast Betw een a Thick (left) and Thin (right) Shell W arp Bubble. A thin shell wa rp drive model ha s the ca pa city to dra ma tica lly reduce the energy requirements of a wa rp drive. Under this new pa ra digm of wa rp drive, where the necessa ry contra cting a nd expa nsion of spa cetime is genera ted by a ma nipula tion of the ra dius of the extra dimension, one ca n clea rly see tha t the energy requirements a re reduced immensely when compa red to the ca lcula tions of Lobo a nd Visser (Reference 12). Ta ble 3 illustra tes the energy requirements for a ra nge of multiple of the speed of light. All ca lcula tions a re order-of- ma gnitude. 21 U N CL ASSI F I E D/ / F OR OF F ICIAL USE ONL Y UNCL ASSIF IED//F OR OF F ICIAL USE ONL Y Table 3. Negative Energy Required for Warp Bubble W a rp Fa ctor, vwa rp ^warp (J) 10-5 (= 3 km/s) -I.OOx I032 10 4 (= 30 km/s) -1.00 X IO34 0.01 (= 3,000 km/s) -I.OOx io38 0.5 (= 150,000 km/s) -2.50 x IO41 1 (= light speed) -I.OOx 1042 2 (= 600,000 km/s) -4.00 x 1042 10 (= 3.0 X IO6 km/s) -I.OOx 1044 100(= 3.0x 107km/s) -3.03 x 1046 Upon compa ring Ta bles 2 a nd 3, immedia tely one ca n see the dra stic energy reductions tha t a re a ppa rent when one uses the dimensiona l wa rp drive pa ra digm. The energy requirements a re reduced by a fa ctor of 108. This energy is ba sed on a wa rp bubble tha t encompa sses 100 m3 of spa ce. As discussed ea rlier, this is a "worst ca se" scena rio, a nd the energy requirements could be further reduced, perha ps by ma ny orders of ma gnitude, by utilizing the thin shell model illustra ted in Figure 9. These a re, perha ps, the most importa nt numerica l results of this pa per, a s they set a n upper limit on the energy requirements necessa ry to genera te a wa rp bubble, a nd a lso on the energy requirements necessa ry to surpa ss the speed of light. Even though this energy requirement is a va st improvement on the ca lcula tions of Visser a nd Lobo, the energies a re still fa r in excess of those a va ila ble in the foreseea ble future. One interesting prospect to test this theory is to consider the ma ximum energy density tha t is a chieva ble by modern technology, a nd to ca lcula te the expa nsion of spa ce tha t this energy would genera te. One could then conceiva bly contempla te a ta ble-top experiment if the numbers a llowed. W ha t follows is a description of how this might work. 7.5 F UTURE EXPERIM ENTS At this point we ta ke a diversion into pure specula tion a s to wha t technologica l a dva ncements ma y be necessa ry to build a device tha t might test our da rk energy theory. One knows from the Friedma nn equa tions of cosmology tha t norma l ma tter a nd energy genera te: - = ---—(p + 3/?), (7.6) a 3 where a is the so-ca lled cosmologica l sca le fa ctor, p is the energy density a nd p is the pressure. If one defines the qua ntity "= p/p", one sees immedia tely tha t a n a ccelera ted expa nsion of spa cetime requires nega tive pressure. Currently, a ll known ma tter a nd energy genera te w > 1; however, da rk energy ha s w = -1. If it were possible to technologica lly crea te da rk energy in the la b, then this would a ssist with our 22 UNCL ASSI F I ED//rOn OF F ICIAL USE ONL Y UNCL AS5IF IED//rOR official UOC ONL Y understa nding a nd experimenta tion with wa rp drive technology. To understa nd how, first discuss a n a na logy with energy w > 1. In a ha ndful of la bs sca ttered a round the country, peta wa tt la sers a re being built a nd tested. These a re la sers of profound ca pa bility, a ble to genera te short la ser pulses of intensity >> 1015 W /m2, with pea k focused power densities of >> 1027 W /m3, a nd energy densities of >> 1016 J/m3. If it were possible to construct a n exotic device whose energy output ha d w < 1, then using ca lcula tions identica l to those to genera te Ta ble 3, it would seem stra ightforwa rd to show tha t the energy density produced by a n a na logous da rk energy la ser would tra nsla te into a loca l expa nsion of spa ce corresponding to vW a rp= 1013 or ~ IO-5 m/sfor every meter of spa ce tha t the la ser tra vels. It ma y be possible to construct a n experiment tha t could mea sure the modified expa nsion of spa ce a long the length of such a la ser to test the predictions of this pa per. 7.6 THE DEVEL OPM ENT OF THE TECHNOL OGY Although a pra ctica l wa rp drive could be ma ny yea rs a wa y from rea liza tion, there a re a number of technologica l developments tha t ma y, in fa ct, be necessa ry in order to a llow prototype experiments to begin. First, a more complete understa nding of da rk energy is of pa ra mount importa nce. As mentioned ea rlier, da rk energy contributes a pproxima tely 70 percent of the overa ll energy density of the universe, a nd is responsible for the expa nsion of spa ce. As one a cquires deeper understa nding of this energy, a ttempts to genera te da rk energy in the la b would no doubt be a critica l component to a working wa rp drive. Other crucia l developments include determining whether extra dimension a re, in fa ct, rea l. Once this ha s been determined with a bsolute certa inty, the role of extra dimensions a nd their rela tionship with the universe tha t we a re more fa milia r with will become a more serious focus of scientific resea rch a nd a ttention. As ha s been discussed, the sta biliza tion of the extra dimensions a nd the a ccelera ted expa nsion of the rema ining four dimensions ca n be rea lized via the Ca simir energy. The a ccelera tion of the 3-dimensiona l subspa ce is a na tura lly occurring phenomenon which occurs when the extra dimensions a re sta bilized. An importa nt fea ture of the higher dimensiona l model is the dependence of the da rk energy density on the size of the extra dimensions. In models with la rge extra dimensions, the intera ction of the gra viton KK tower with the S ta nda rd Model fields a re suppressed by the higher dimensiona l Pla nck sca le a nd the corresponding couplings a re inverse TeV in strength. This ca n be seen more clea rly when we consider the expa nsion of the metric tensor in models with la rge extra dimensions computed within linea rized gra vity models: S M N ~ H M N + yD/2-l ^M N ’ ‘ where the ca pita l letter indices indica te summa tion over the higher dimensions, M is the modified Pla nck ma ss, t]MN corresponds to fla t (Minkowski) spa cetime a nd hMN corresponds to the bulk gra viton fluctua tions. The gra viton intera ction term in the a ction is expressed by: 23 UNCL ASSI F I ED//rOn ornciAL UOC ONL Y UNCL ASSIF IED//POR OF F ICIAL USE ONL Y 5 =—-—[d^xh Tm n 7.8 where TMN is the higher dimensiona l energy-momentum tensor. The intera ction of the gra viton with the gra viton KK sta tes a nd with the S M fields a re obta ined by integra ting the a ction over the extra coordina tes. Beca use a ll these sta tes a re coupled with the universa l strength 1/Ma this lea ds to the compelling possibility of the control of the size of the extra dimensions by processes a t energies tha t will be a ccessible via the pa rticle a ccelera tors of the nea r future. Although the coupling is extremely sma ll, the effective coupling is enha nced by the la rge number of KK sta tes. Referring to Figure 7, a dditiona l energy in the form of ma tter or ra dia tion with the TeV energy sca le ca n a lter the sha pe of the effective potentia l. In pa rticula r, the extrema determining the size of the extra dimensions a re modified with the cha nge of the Ca simir energy density a nd hence, the da rk energy density, in the models under considera tion. In the AD D model discussed in S ection 5.2, the S ta nda rd Model fields a re confined to a । 4-dimensiona l bra ne a nd a new gra vity sca le MD = G2~D » TeV is introduced in the D = 4 + d dimensions, where d is the number of extra dimensions. This is determined quite ea sily by forming the higher dimensiona l a ction a nd integra ting out the extra dimension. The Atla s experiment a t CERN's La rge Ha dron Collider will ha ve the ca pa bility to probe the AD D type extra dimensions up to M d ~ 8 TeV. In the Ra nda ll S undrum scena rio, the hiera rchy is expla ined by the wa rp fa ctor in the AdS 5 bulk geometry. A lower bound ca n be pla ced on the lowest KK ma ss by electrowea k precision tests (ma sses on the order of 1 TeV a re a llowed). In the Universa l Extra D imension scena rio, Teva tron results constra in the compa ctifica tion sca le to Me >400 GeV. Beca use the Atla s experiment will be sensitive to Me - 3 TeV , the estima tes in this section indica te tha t if na ture is in fa ct described by one of these higher dimensiona l scena rios, then the a dditiona l dimensions ca n be probed. Even more exciting is the possibility tha t their size ma y be controlled a t the energies a ccessible to the La rge Ha dron Collider. W hen the Ca simir energy to pla y the role of da rk energy is involved, one ca n see tha t the possibility for the direct control of the loca l da rk energy density by controlling the size of the a dditiona l dimensions is possible. 8 . Sum m ary The idea tha t a sufficiently a dva nced technology ma y intera ct with, a nd a cquire direct control over, the higher dimensions is a ta nta lizing possibility, a nd one tha t is most certa inly worthy of deeper investiga tion. Control of this higher dimensiona l spa ce ma y be a source of technologica l control over the da rk energy density a nd could ultima tely pla y a role in the development of exotic propulsion technologies; specifica lly, a wa rp drive. Of course, this ma y not be a ctua lized until ma ny yea rs in the future, but consider the ma ny specta cula r physica l phenomena tha t a re believed to be true a t this ea rly point in 24 U N CL ASSI F I E D/ / F OR OF F ICIAL UPC ONL Y UNCL ASSIF IED//ron OF F ICIAL W EE ONL Y the 21st century. One believes tha t a n energy field ca lled the Higgs boson permea tes spa cetime a nd tha t the intera ction of ma tter with this field is wha t is responsible for pa rticles a cquiring ma ss. One believes tha t a n exotic ubiquitous energy source, unima gina tively na med da rk energy, is responsible for the current a ccelera ted expa nsion of the universe ba sed on observa tion of supernova in ga la xies billions of light yea rs from Ea rth. One a lso believes tha t the universe ma y not consist of the three spa tia l dimension of length, brea dth, width, a nd one of time, but tha t, in fa ct, there ma y be a s ma ny a s seven a dditiona l compa ctified dimensions a ssuming the topology of a Ca la bi-Ya u ma nifold, a nd tha t the funda menta l building blocks of the universe a re, in fa ct, extended string-like entities. Modern physics is full of ma ny exciting a nd ma rvelously ima gina tive crea tions. Beca use one understa nds these curiosities, one could potentia lly ha rness these elements of na ture for one's own technologica l ends. 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