Warp Drive Theory
The Warp Drive
The magic formula brought to us by Miguel Alcubierre is a metric of the form :
which is simply the velocity of the system, in classical mechanics this is given similarly through v=dx(distance)/dt(time). The d term arises through calculus, where one receives the geodesic relation for a curvature (i.e. an arc circle) and the line path. Also note that for consistency the terms dy2+dz2 are needed in the first equation, however, is not needed directly to understand the warp theory, and is removed to make the equation easier to handle. The rs term is given through
neglecting the y and z components it is the difference between the original coordinates and the warp drive coordinates. Where a localized region of space is propelled through the x direction (to the right in the figure below) by a velocity determined through the function f(rs) which resembles a "top hat" function (given through trigonometry):
This metric supposes a contraction of spacetime in front of a body, with a expansion behind it. The expansion and contraction can be seen through the coordinates x and r=(y2+z2)1/2 (which is shown in the figure below).
In fact what Alcubierre is proposing is using a form of bipolar (or "dual") gravitational waves as a method of propulsion. Gravitational waves in general relativity are planar and hence each wave expands and contracts, however, the Alcubierre metric in principle suggest that such an effect could be bipolar, possibly explaining the necessity for a "negative energy" requirement (however, this is purely speculation on my part). What this metric truly suggest is that such a manipulation of space would cause spacetime to propel a localized region of space (refereed to as a warp bubble) by expanding and contracting the metric field. Since gravitational radiation is believed to propagate at the speed of light the prolusion of this space is similar in principle as to how electric and magnetic field cause electromagnetic radiation to propagate.
Perhaps a work that is better suited in making the Alcubierre geometry a reality is the Van Den Broeck metric :
ds2=-dt2+B2(rs) [(dx-vsf(rs)dt)2 +dy2+dz2]
The basis of this model is to shrink a "warp bubble" (this refers to the flat spacetime within Alcubierre's warp metric) to microscopic proportions to negotiate around the negative energy conditions. This is beacuse the basis of the Alcubierre metric requires an enormous amount of negative energy, which according to classical conservation laws shouldn't exist. So the Broeck metric shows basically how to shrink the "Alcubierre warp bubble," so that it requires less "negative energy."
This however only effects the external properties of the warp bubble while internally the effects of the bubble could be as large as one wished (this deals with the construction of energy densites within the field). The main benefit of this theory is that it dramatically lowers the negative energy requirements, thereby making warp drive look as if it could have a real future.
Worm holes also allow for possible mediums of FTL travel. One of the models of interest to me is a model that corresponds to a metric of the form :
As an observer travels through this space only standard relativistic effects are noticed, however on the return trip, it produces effects characteristic of warp drive. The neat little trick using this models is that as you travel to a nearby star you travel nearly at the speed of light, so that it would only appear to take a few days (i.e. standard relativistic effects).
While making your trip you would simply drag space behind you, and to return you would just ride back on "tube" which you have been "towing" with you. Thereby on the return trip you end up transversing less space, which may appear to be a form of warp drive explored at the begging of this page. The downside is that this equation is only two-dimensional, and as in the case of the Alcubierre metric a tremendous amount of negative energy is required.
 Alcubierre M. The Warp Drive: Hyper-Fast Travel Within General Relativity. Class.Quant.Grav. 11 (1994), L73-77.
 Broeck C. A `warp drive' with more reasonable total energy requirements. Class.Quant.Grav. 16 (1999) 3973-79
 Krasnikov S. Hyperfast interstellar travel in general relativity. [ Building Warp Drives in a Lab? ]
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