Time Travel Research Center © 1998 Cetin BAL  GSM:+90 05366063183  Turkey / Denizli
One of the oldest transportation facilities are ships. They use wind or current for their movement. A spaceship could do the same provided it were possible to construct a spacetime containing a 'stream' that carries the object in question at a sufficiently high velocity with respect to some point outside the 'stream'. This is achieved by using the metric suggested by Alcubierre. In the coordinates (t, x, y, z) it can be written as follows:
For an observer in the middle of this 'stream' (i.e. y = z = 0) it is clear that gamma(t) = (t, v_{s} t) is a timelike path for dv_{s}/dt = 0. Here v_{s} is an arbitrary parameter, which can be interpreted as constant velocity. In particular v_{s} can be bigger than 1, thus exceeding the speed of light (in natural units). For v_{s} = 0 or f_{s} = 0 the Lorentz metric is recovered. The function f_{s}(r_{s}) controls the shape of the 'stream'. It depends on r_{s} only, the distance from the curve x = y = z = 0. For r_{s} > R we want f_{s} = 0, for r_{s} < R  sigma we want f_{s} = 1. So f_{s} may look like this:
The spatial part of this metric hence describes a 'bubble' with radius R and walls of the width s. Adopting the denotation from popular science this may be called a warp bubble. As fr is approximately constant in the interior, tidal forces will be very small thus an extended observer such as a spaceship can be placed in the center without danger.
Of course this spacetime causes a deviation of light. In the case of a warp bubble moving upwards, light rays coming from the exterior region will be deflected and blueshifted as viewed from the center. This is shown in the diagrams below (left: v_{s} ~ 2, right: v_{s} ~ 4). Photons coming directly from below are infinitely redshifted and are hence not observable. So there exists a past event horizon (and analogously a future event horizon in the opposite direction) indicated by the shaded region. The travellers inside the bubble will see a fixed star background like in the middle part of the diagram. From the Einstein equations the energymomentumtensor can be calculated. From this the energy density and matter distribution are known. Once calculated, it is obvious that a negative energy density is necessary. The diagrams below show the cases v_{s} ~ 1 (left) and v_{s} ~ 5 (right).
Parts of the bubble are causally disconnected from the center. Hence the crew of the spaceship cannot control its way  the route must be planned by an observer in the causal past of the complete path. Therefore the Alcubierre metric rather describes a 'trolley'. Another problem with this metric is the occurence of
closed timelike curves. This means that the flow of information is not
only future directed. Hence the principle of causality is not fulfilled
any more and the topology of the spacetime becomes non trivial. Thus the
question is raised: is it principally possible to modify spacetime as
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