Warp Field Mechanics 101

Transcription

1 Warp Field Mechanic 101 Dr. Harold Sonny White NASA Johnon Space Center 2101 NASA Parkway, MC EP4 Houton, TX Abtract: Thi paper will begin with a hort review of the Alcubierre warp drive metric and decribe how the phenomenon might work baed on the original paper. The canonical form of the metric wa developed and publihed in [6] which provided key inight into the field potential and boot for the field which remedied a critical paradox in the original Alcubierre concept of operation. A modified concept of operation baed on the canonical form of the metric that remedie the paradox i preented and dicued. The idea of a warp drive in higher dimenional pace-time (manifold) will then be briefly conidered by comparing the null-like geodeic of the Alcubierre metric to the Chung-Freee metric to illutrate the mathematical role of hyperpace coordinate. The net effect of uing a warp drive technology coupled with conventional propulion ytem on an exploration miion will be dicued uing the nomenclature of early miion planning. Finally, an overview of the warp field interferometer tet bed being implemented in the Advanced Propulion Phyic Laboratory: Eaglework (APPL:E) at the Johnon Space Center will be detailed. While warp field mechanic ha not had a Chicago Pile moment, the tool neceary to detect a modet intance of the phenomenon are near at hand. Keyword: warp, boot, York Time, bulk, brane Introduction How hard i intertellar flight without ome form of a warp drive? Conider the Voyager 1 pacecraft [1], a mall mt pacecraft launched in 1977, it i currently out at ~116 Atronomical Unit (AU) after 33 year of flight with a cruie peed of 3.6 AU per year. Thi i the highet energy craft ever launched by mankind to date, yet it will take ~75000 year to reach Proxima Centauri, the nearet tar at 4.3 light year away in our neighboring trinary ytem, Alpha Centauri. Recent informal miion trade have been aeing the capabilitie of emerging high power EP ytem coupled to light nuclear reactor to accomplih the reference Thouand Atronomical Unit (TAU) [2] miion in ~15 year. Rough calculation ugget that uch a Nuclear Electric Propulion (NEP) robotic miion would pa Voyager 1 in jut a few year on it way to reaching 1000 AU in 15 year. While thi i a handy improvement over Voyager 1 tatitic - almot 2 order of magnitude, thi peedy robotic craft would till take thouand of year to cro the black ocean to Proxima Centauri. Clearly intertellar flight will not be an eay endeavor.

2 Background The tudy of intertellar flight i not a new puruit, and there have been numerou tudie publihed in the literature that conider how to approach robotic intertellar miion to ome of our cloet tellar neighbor, with the objective of having tranit time cloer to the 100 year mark rather than thouand of year. One of the mot familiar tudie i Project Daedelu [3] ponored by the Britih Interplanetary Society in The Daedelu tudy objective wa to conider a 50-year robotic miion to Barnard tar, which i ~6 light year away. The pacecraft detailed in the report wa quite maive weighing in at mt, 92% of which wa propellant for the fuion propulion ytem. For comparion, the International Space Station i a modet ~400 mt, thu the Daedelu pacecraft i nearly the equivalent of 150 International Space Station. Project Longhot [4], a joint NASA-NAVY tudy in the late 1980 to develop a robotic intertellar miion to Alpha Centaury, produced a 400 mt (67% propellant) robotic pacecraft that could reach Alpha Centaury in 100 year. At one ISS of ma, thi vehicle i eaier to viualize than it heftier older couin, Daedelu. There are many other tudie that have been performed over the year each having light permutation on the anwer, primarily depending on the integrated efficiency of converting propellant ma directly into pacecraft kinetic energy (matter-antimatter being among the bet). All reult are of coure bounded by the peed of light, meaning earth-bound oberver will likely perceive intertellar tranit time of outbound pacecraft in decade, centurie, or more. Alcubierre Metric I there a way within the framework of current phyic model uch that one could cro any given comic ditance in an arbitrarily hort period of time, while never breaking the peed of light? Thi i the quetion that motivated Miguel Alcubierre to develop and publih a poible mathematical olution to the quetion back in 1994 [5]. Since the expanion and contraction of pace doe not have a peed limit, Alcubierre developed a model (metric) within the domain of general relativity that ue thi phyic loop hole and ha almot all of the deired characteritic of a true intertellar pace drive, much like what i routinely depicted in cience fiction a a warp drive. The metric that i dicued in the paper i preented in equation 1. Thi ue the familiar coordinate, (t, x, y, z) and curve x = x (t), y = 0, z=0 where x i analogou to what i commonly referred to a a pacecraft trajectory. d 2 = c 2 dt 2 + [dx v (t)f(r )dt] 2 + dy 2 + dz 2 (1) In thi metric, v i defined a the velocity of the pacecraft moving frame, dx /dt, and r i the radial poition in the commoving pherical pace around the pacecraft origin. The f(r ) term i a top hat haping function that i defined a: f(r ) = tanh σ(r + R) tanh σ(r R) 2 tanh(σr)

3 The parameter σ and R when mapped into the metric given in equation 1 control the wall thickne and radiu of the warp bubble repectively. For very large σ, the wall thickne of the bubble become exceedingly thin, approaching zero thickne in the limit. The driving phenomenon that facilitate peedy travel to tellar neighbor i propoed to be the expanion and contraction of pace (York Time) hown in equation 2. Figure 1 how everal urface plot of the York Time urrounding the pacecraft. The region directly in front of the pacecraft experience the mot contraction of pace, while the region directly behind the pacecraft experience the mot expanion of pace. The phenomenon revere ign at the x = x ymmetry urface. A the warp bubble thickne i decreaed, the magnitude of the York Time increae. Thi behavior when mapped over to the energy denity requirement will be dicued in the next ection. θ = v x r df dr (2) Figure 1: York Time, θ, i depicted for everal different warp bubble wall thicknee, σ. The energy denity hown in equation 3 for the field ha a toroidal form that i axiymmetric about the x-axi, and ha a ymmetry urface at x = x. The energy denity i exactly zero along the x-axi. For a fixed target velocity v and warp bubble radiu R, varying the warp bubble thickne σ change the required peak energy denity for the field at a fixed velocity. Figure 2 how the relative change in energy denity for everal warp bubble wall thicknee. A i evident when comparing the magnitude, a the warp bubble i allowed to get thicker, the required denity i dratically greatly reduced, but the toroid grow from a thin equatorial belt to a diffue donut. The advantage of allowing a thicker warp bubble wall i that the integration of the total energy denity for the right-mot field i order of magnitude le that the left-mot field. The drawback i that the volume of the flat pace-time in the center of the bubble i reduced. Still, a minimal reduction in flat pace-time volume appear to yield a dratic reduction in total energy requirement that would likely outweigh reduced real-etate. Sloppy warp field would appear to be eaier to engineer than precie warp field. Some additional appealing characteritic of the metric i that the proper acceleration α i zero, meaning there i no acceleration felt in the flat pace-time volume inide the warp bubble when the field i turned on, and the coordinate

4 time t in the flat pace-time volume i the ame a proper time τ, meaning the clock on board the pacecraft proper beat at the ame rate a clock on earth. T 00 = 1 8π v 2 ρ 2 4r 2 df dr 2 (3) Figure 2: Energy denity, T 00, i depicted for everal different warp bubble wall thicknee, σ. The concept of operation a decribed by Alcubierre i that the pacecraft would depart the point of origin (e.g. earth) uing ome conventional propulion ytem and travel a ditance d, then bring the craft to a top relative to the departure point. The field would be turned on and the craft would zip off to it tellar detination, never locally breaking the peed of light, but covering the ditance in an arbitrarily hort time period of time jut the ame. The field would be turned off a imilar tandoff ditance from the detination, and the craft would finih the journey conventionally. Thi approach would allow a journey to ay Alpha Centauri a meaured by an earth bound oberver (and pacecraft clock) meaured in week or month, rather than decade or centurie. A paradox identified in [6] i an iue that arie due to the ymmetry of the energy denity about the x = x urface. When the energy denity i initiated, the choice in direction of the +x-axi i mathematically arbitrary, o how doe the pacecraft know which direction to go? Comparing Figure 1 to Figure 2 viually diplay the aymmetry of the York Time and the ymmetry of the energy denity. Both et of three frame were purpoely aligned to make direct comparion eaier. Thi aymmetry/ymmetry paradox iue can be potentially reolved when conidering the canonical form of the metric derived by uing a gauge tranformation in [6] a hown in equation 4. d 2 = (v 2 f(r ) 2 1) dt v f(r ) v 2 f(r ) 2 1 dx 2 dx 2 + dy 2 + dz 2 (4) Uing thi canonical form, the field potential φ and the boot γ can be determined uing the tandard identity γ = coh(φ). They are, repectively:

5 φ = 1 2 ln 1 v 2 f(r ) 2 and trivially, γ = coh 1 2 ln 1 v 2 f(r ) 2 Uing thi new information, a modified concept of operation i propoed that may reolve the aymmetry/ymmetry paradox. In thi modified concept of operation, the pacecraft depart earth and etablihe an initial ub-luminal velocity v i, then initiate the field. When active, the field boot act on the initial velocity a a calar multiplier reulting in a much higher apparent peed, <v eff >= γ v i a meaured by either an earth bound oberver or an oberver in the bubble. Within the hell thickne of the warp bubble region, the pacecraft never locally break the peed of light and the net effect a een by earth/hip oberver i analogou to watching a film in fat forward. Conider the following to help illutrate the point aume the pacecraft head out toward Alpha Centauri and ha a conventional propulion ytem capable of reaching 0.1c. The pacecraft initiate a boot field with a value of 100 which act on the initial velocity reulting in an apparent peed of 10c. The pacecraft will make it to Alpha Centauri in 0.43 year a meaured by an earth oberver and an oberver in the flat pace-time volume encapulated by the warp bubble. While thi line of reaon eem to reolve the paradox, it alo ugget that the York Time may not be the driving phenomenon, rather a econdary reult. In thi phyical explanation of the mathematic, the York Time might be thought of a perhap a Doppler train on pace a thi pherical region i propelled through pace. A pedetrian analog to ue to help enviion thi concept would be to conider the hydrodynamic preure gradient that form around a pherical body moving through a fluid the front hemiphere ha a high preure region while the rear hemiphere ha a low preure region. Analogouly, the warp bubble travelling through pace-time caue pace to pile up (contract) in front of the bubble, and tretch out (expand) behind the bubble. Figure 3 depict the boot field for the metric, and how that the toroidal ring of energy denity create pherical boot potential urrounding a flat pace-time volume. Alo note peudo-horizon at v 2 f(r ) 2 =1 where photon tranition from null-like to pace-like and back to null like upon exiting. Thi i not een unle the field meh i et fine enough. The coare meh on the right did not detect the horizon. Figure 3: Boot plot for the field

6 Chung-Freee Metric Additional work ha been publihed that expand the idea of a warp drive into higher dimenional pace-time. In 2000, Chung and Freee [7] publihed a higher dimenional pace-time model that i a modified Friedmann-Roberton-Walker (FRW) metric a hown in equation 5. The idea of a higher dimenional model i that the tandard 3+1 ubpace exit a a brane embedded in thi higher dimenional pace-time labeled the bulk. The ize and number of extra dimenion are not explored in thi paper; rather the dicuion will tick to the original form of the publihed metric. d 2 = c 2 dt 2 + a2 (t) e 2kU dx2 + du 2 (5) The dx 2 term repreent the 3+1 pace (on the brane), while the du 2 term repreent the bulk with the brane being located at U=0. The a(t) term i the cale factor, and k i a compactification factor for the extra pace dimenion. A conventional analogy to help viualize the brane-bulk relationhip, conider a 2D heet that exit in a 3D pace. The 2D inhabitant if the flat-land ubpace have a manifold that i mapped out with the imple metric, dx 2 + dy 2, where thi can be viewed a being analogou to the dx 2 term in equation 5. The remainder of the 3D bulk pace i mapped by the z-axi, and anything not on the heet would have a non-zero z-coordinate. Thi additional dz 2 term i, from the perpective of the 2D inhabitant, the du 2 term in equation 5. Anything not on the 2D heet would be labeled a being in the bulk with thi implified analogy. In order to illutrate the mathematical relationhip between a hyper drive and a warp drive, the nulllike geodeic for the Chung-Freee metric will be conidered and compared to the conjectured driving phenomenon in the Alcubierre metric, the boot. The equation for the null-like geodeic for equation 5 i (etting c=1): dx dt = eku a(t) 1 du2 dt 2 If du/dt i et to 1, then a tet photon that ha a velocity vector orthogonal to the brane would have a zero peed a meaured on the brane, dx/dt=0. If a tet photon ha du/dt=0, but arbitrarily large U coordinate, dx/dt will be large, poibly >>1. Remember that c wa et to 1, o dx/dt >1 i analogou to the hyper-fat travel character of the Alcubierre metric. The behavior of the null-like geodeic in the Chung-Freee metric become pace-like a U get large. The null-like geodeic in the Alcubierre metric become pace-like within the warp bubble, or where the boot get large. Thi ugget that the hyperpace coordinate erve the ame role a the boot, and the two can be informally related by the imple relationhip γ~e U. A large boot correpond to an object being further off the brane and into the bulk. Miion Planning with a Warp-enabled Sytem To thi point, the dicuion ha been centered on the intertellar capability of the model, but in the interet of addreing the crawl-walk-run paradigm that i a taple of the engineering and cientific

7 dicipline, a more dometic application within the earth gravitational well will be conidered. A a preamble, recall that the driving phenomenon for the Alcubierre metric wa peculated to be the boot acting on an initial velocity. Can thi peculation be hown to be conitent when uing the tool of early reference miion planning while conidering a warp-enabled ytem? Note that the energy denity for the metric i negative, o the proce of turning on a theoretical ytem with the ability to generate a negative energy denity, or a negative preure a wa hown in [8], will add an effective negative ma to the pacecraft overall ma budget. In the regime of reference miion development uing lowthrut electric propulion ytem for in-pace propulion, planner will cat part of the trade pace into a domain that compare a pacecraft pecific ma α to tranit time. While electric propulion ha excellent fuel economy due to high pecific impule that are meaured in thouand of econd, it require electric power meaured in 100 of kw to keep trip time manageable for human exploration cla payload. Figure 4 how a notional plot for a human exploration olar electric propulion tug ized to move payload up and down the earth well to L1 in thi cae. If time were of no conequence, then much of thi dicuion would be moot, but a experience how, time i a contraint that i traded with other miion contraint like delivered payload, power requirement, launch and aembly manifet, crew cycling frequency, miion objective, heliocentric tranfer date, and more. The pecific ma of an element for an exploration architecture or reference miion can be determined by dividing the pacecraft beginning of life wet ma by the power level. Specific ma can alo be calculated at the ubytem level if competing technologie are being compared for a particular function, but for thi exercie, the integrated vehicle pecific ma will be ued. The tranit time for a miion trajectory can then be calculated and plotted on a graph that compare pecific ma to tranit time. Thi can be done for a few dicrete vehicle configuration, and the curve that fit thee point will allow miion planner to extrapolate between the point when adding and ubtracting ma, either in the form of payload or ubytem, for a particular power level. Figure 4 how a imple plot of thi approach for two pecific impule/efficiency value repreenting notional engine choice. It i apparent from the graph that lower pecific impule yield reduced trip time, but thi alo reduce delivered payload. However, if negative ma i added to the pacecraft ma budget, then the effective pecific ma and tranit time are reduced without necearily reducing payload. A quetion to poe i what effect doe thi have mathematically? If energy i to be conerved, then ½ mv 2 would need to yield a higher effective velocity to compenate for apparent reduction in ma. Auming a point deign olution of 5000kg BOL ma coupled to a 100kW Hall thruter ytem (lower curve), the expected tranit time i ~70 day for a pecific ma of 50 kg/kw without the aid of a warp drive. If a very modet warp drive ytem i intalled that can generate a negative energy denity that integrate to ~2000kg of negative ma when active, the pecific ma i dropped from 50 to 30 which yield a reduced tranit time of ~40 day. A the amount of negative ma approache 5000 kg, the pecific ma of the pacecraft approache zero, and the tranit time become exceedingly mall, approaching zero in the limit. In thi implified context, the idea of a warp drive may have ome fruitful dometic application ubliminally, allowing it to be matured before it i engaged a a true intertellar drive ytem.

8 Figure 4: Trip time to L1 a a function of Beginning of Life (BOL) pecific ma. Advanced Propulion Phyic Lab: Eaglework A good quetion to ak at the end of thi dicuion i can an experiment be deigned to generate and meaure a very modet intantiation of a warp field? A briefly dicued by the author in [9], a Michelon-Morley interferometer may be a ueful tool for the detection of uch a phenomenon. Figure 5 depict a warp field interferometer experiment that ue a 633nm He-Ne laer to evaluate the effect of York Time perturbation within a mall (~1cm) pherical region. Acro 1cm, the experimental rig hould be able to meaure pace perturbation down to ~1 part in 10,000,000. A previouly dicued, the canonical form of the metric ugget that boot may be the driving phenomenon in the proce of phyically etablihing the phenomenon in a lab. Further, the energy denity character over a number of hell thicknee ugget that a toroidal donut of boot can etablih the pherical region. Baed on the expected enitivity of the rig, a 1cm diameter toroidal tet article (omething a imple a a very highvoltage capacitor ring) with a boot on the order of i neceary to generate an effect that can be effectively detected by the apparatu. The intenity and patial ditribution of the phenomenon can be quantified uing 2D analytic ignal technique comparing the detected interferometer fringe plot with the tet device off with the detected plot with the device energized. Figure 5 alo ha a numerical example of what the before and after fringe plot may look like with the preence of a pherical diturbance of the trength jut dicued. While thi would be a very modet intantiation of the phenomenon, it would likely be Chicago pile moment for thi area of reearch.

9 Figure 5: Warp Field Interferometer layout (here, φ i the phae angle). Concluion In thi paper, the mathematical characteritic of the Alcubierre metric were introduced and dicued, the canonical form wa preented and explored, and the idea of a warp drive wa even conidered within a higher dimenional manifold. The driving phenomenon wa conjectured to be the boot field a oppoed to purely the York Time which reolved the aymmetry/ymmetry paradox. An early idea of a warp drive wa briefly dicued within the context of miion planning to elucidate the impact uch a ubytem would have on the miion trade pace. Finally, a laboratory experiment that might produce a modet intantiation of the phenomenon wa dicued. While it would appear that the model ha nearly all the deirable mathematical characteritic of a true intertellar pace drive, the metric ha one le appealing characteritic it violate all three energy condition (trong, weak, and dominant [9]) becaue of the need for negative energy denity. Thi doe not necearily preclude the idea a the como i continually experiencing inflation a evidenced by obervation, but the alient quetion i can the idea be engineered to a point that it prove ueful for exploration. A ignificant finding from thi effort new to the literature i that for a target velocity and pacecraft ize, the peak energy denity

10 requirement can be greatly reduced by allowing the wall thickne of the warp bubble to increae. Analyi performed in upport of generating the plot hown in Figure 1 and 2 alo indicate a correponding reduction in total energy when converted from geometric unit (G=c=1) to SI unit, but till how that the idea will not be an eay tak. So it remain to be een if the evolution of the phrae penned by J. M. Barrie in the tory Peter Pan will ever be uttered on the bridge of ome majetic tarhip jut embarking on a daring miion of deep pace exploration taking humanity beyond the bound of thi olar ytem and boldly going out into the tar: 2 nd tar to the right, traight on till morning Godpeed Reference [1] Available at: [2] Nock, k. T., TAU A Miion to a Thouand Atronomical Unit, 19th AIAA/DGLR/JSASS International Electric Propulion Conference, Colorado Spring, (1987). [3] Bond, Martin, Project Daedalu: The Miion Profile, JBIS: Project Daedalu Final Report (1978). [4] Available at: [5] Alcubierre, M., The warp drive: hyper-fat travel within general relativity, Cla. Quant. Grav. 11, L73-L77 (1994). [6] White, H., A Dicuion on pace-time metric engineering, Gen. Rel. Grav. 35, (2003). [7] Chung, D. J. H., and Freee, K., Can geodeic in extra dimenion olve the comological horizon problem?, Phy. Rev. D 62, (2000). [8] White, H., Davi, E., The Alcubierre Warp Drive in Higher Dimenional Space-time, in proceeding of Space Technology and Application International Forum (STAIF 2006), edited by M. S. El-Genk, American Intitute of Phyic, Melville, New York, (2006). [9] S.W. Hawking and G.F.R. Elli, The Large Scale Structure of Spacetime, Cambridge Univerity Pre, (1973).

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