{{Found this partial paper under a deleted subfolder entitled Looking Glass, which was last updated Feb 4, 2017. File author was Erica Kravid. In spite of the directory being wiped several times, I was able to pull out this data.}}
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±•ññáîýmq󡶇oÌ`4hä>ïê.í{Û"ëô.$õFÍË´`ÿ,ó•ƒDÔöð%˽îh–0¹”¤¶ŸöëÂf}üp°løBª‘ñlþë{3¦ÕÙæ%‚‰X)ÉÊçÄBÓPygôÚ§¶o’û8˜ÅA¤ªž%s²‘Òecá
qò,y)±+©(âgoµ}ËÚç8à¤)Q‚¿Ë§˜-öñ}5‘Qgw¸‚;¼õ_ùÑ‚IHNS¦ü-Ïoµšã¡Z«ê²÷íñ“>ù”“Ø•:5ÒŠ.'¤™£&wç$KLc%éÝÞD2›©ç¼6W5AnpNï¤Ì€œ¾ÑC¬zVÇ|WÁŽK9Lthere a way within the framework of current physics models such that one could cross any given cosmic distance in an arbitrarily short period of time, while never breaking the speed of light? This is the question that motivated Miguel Alcubierre to develop and publish a possible mathematical solution to the question back in 1994. Since the expansion and contraction of space does not have a speed limit, Alcubierre developed a model (metric) within the domain of general relativity that uses this physics loophole and has almost all of the desired characteristics of a true interstellar space drive, much like what is routinely depicted in science fiction as a “warp drive”.The metric that is discussed in the paper is presented in equation 1. This uses the familiar coordinates,(t, x, y, z) and curve x = xs(t), y = 0, z=0 where x is analogous to what is commonly referred to as A spacecraft's trajectory.
The parameters σ and R when mapped into the metric given in equation 1 control the wall thickness and radius of the warp bubble respectively. For very large σ, the wall thickness of the bubble becomes exceedingly thin, approaching zero thickness in the limit. The driving phenomenon that facilitates speedy travel to stellar neighbors is proposed to be the expansion and contraction of space (York Time) shown in equation 2. Figure 1 shows several surface plots of the York Time surrounding the spacecraft.The region directly in front of the spacecraft experiences the most contraction of space, while the region directly behind the spacecraft experiences the most expansion of space. The phenomenon reverses sign at the x = xs symmetry surface. As the warp bubble thickness is decreased, the magnitude of the York Time increases. This behavior when mapped over to the energy density requirements will be discussed in the next section.
The energy density shown in equation 3 for the field has a toroidal form that is axisymmetric about the x-axis, and has a symmetry surface at x = xs. The energy density is exactly zero along the x-axis. For a fixed target velocity vs and warp bubble radius R, varying the warp bubble thickness σ changes the required peak energy density for the field at a fixed velocity. Figure 2 shows the relative change in energy density for several warp bubble wall thicknesses. As is evident when comparing the magnitudes,as the warp bubble is allowed to get thicker, the required density is drastically greatly reduced, but the toroid grows from a thin equatorial belt to a diffuse donut. The advantage of allowing a thicker warp bubble wall is that the integration of the total energy density for the right-most field is orders of magnitude less that the left-most field. The drawback is that the volume of the flat space-time in the center of the bubble is reduced. Still, a minimal reduction in flat space-time volume appears to yield a drastic reduction in total energy requirement that would likely outweigh reduced real-estate. Sloppy warp fields would appear to be “easier” to engineer than precise warp fields. Some additional appealing characteristics of the metric is that the proper acceleration α is zero, meaning there is no acceleration felt in the flat space-time volume inside the warp bubble when the field is turned on, and the coordinate time t in the flat space-time volume is the same as proper time τ, meaning the clock within the field of the Looking Glass beat at the same rate as clocks elsewhere on earth.
The concept of operations as described by Alcubierre in relation to space travel, is that the spacecraft would depart the point of origin (e.g. earth) using some conventional propulsion system and travel a distance d, then bring the craft to a stop relative to the departure point. The field would be turned on and the craft would zip off to its stellar destination, never locally breaking the speed of light, but covering the distance in an arbitrarily short time period of time just the same. The field would be turned off a similar standoff distance from the destination, and the craft would finish the journey conventionally. This approach would allow a journey to say, Alpha Centauri, as measured by an earth bound observer (and spacecraft clocks) measured in weeks or months, rather than decades or centuries.A paradox identified in [6] is an issue that arises due to the symmetry of the energy density about the x= xs surface. When the energy density is initiated, the choice in direction of the +x-axis is mathematically arbitrary, so how does the spacecraft “know” which direction to go? Comparing Figure 1 to Figure 2 visually displays the asymmetry of the York Time and the symmetry of the energy density. Both sets of three frames were purposely aligned to make direct comparison easier. This asymmetry/symmetry-paradox issue can be potentially resolved when considering the canonical form of the metric derived byusing a gauge transformation in [6] as shown in equation 4.
If we take this principle and apply it to the spacetime warping phenomenon found within the aperture of Looking Glass, we could reverse the properties of the Alcubierre field and potentially fold spacetime in such a way as to travel backwards or forwards—{€ú´5v=)h4Ð΢#ö‹æå¨Tþq;gOÒ`Vµ¢“A•³™®²uéVmÞ½¸‹½¹âٵŢñìða¦ÔZ“ªaˆR¢ãµN4²í~Ðq>ßF–ÙD2¢*ÑâXÅL[ ¯8IGO[ÁÒqæeebÛVÜi!Ó·‰×ÇcÎÜÙ|¡%åÌ-·®j”"@àJzs@ã€RBÒúNÏzÜ=z