Science Forums 2

[paultrr]

The Alcubierre metric refered to was that utilized by Doctor Alcubierre when he devised an interesting mathamatically example out of GR to something proposed from time to time by science fiction(ie Warp Drive). The metric is simple a normal GR metric which involved using gravity to move some point ahead of a craft closer to the craft and at the same time move a point behind the craft further away from the craft(requiring negative energy) while all the time the craft remains in a non-accelerated frame of reference.

As for what is termed superluminal paths or FTL paths there is also the idea proposed by Fernando Loup a bit back in which using a path through hyperspace from multidimensional theory might solve travel times to the stars. The problem faced by that alternative idea can be summed up as follows:

The Geometry of Space-time in Special Relativity with hyperspace added in

SR uses a 'flat' 4 dimensional Minkowski space, usually referred to as space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.

The differential of distance(ds) in cartesian 3D space is defined as: ds^2 = dx_1^2 + dx_2^2 + dx_3^2 where (dx_1,dx_2,dx_3) are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, with units of c, so that the equation for the differential of distance becomes: ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2 In many situations it may be convenient to treat time as imaginary (e.g. it may simplify equations), in which case t in the above equation is replaced by i.t', and the metric becomes

ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + c^2(dt')^2 If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space, ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2 We see that the null geodesics lie along a dual-cone: defined by the equation

ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2 , or

dx_1^2 + dx_2^2 = c^2 dt^2 Which is the equation of a circle with r=c*dt.

If we extend this to three spatial dimensions, the null geodesics are continuous concentric spheres, with radius = distance = c*(+ or -)time.

ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2 dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2 This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event d = \sqrt{x_1^2+x_2^2+x_3^2} meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".) The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.

The biggest difference with adding in extra dimensional hyperspace is that the cone spreads out and shortens in height for the hyperspace frame almost becoming an instanton where far larger distances are encompassed in relation to our normal space-time frame. However, when you try and compare events from one frame to another with the limits our frame imposes you find that while event rates in hyperspace yield a superluminal path that path in relation to our frame moves into the future. Thus, while for anyone using such a superluminal path(see Fernando Loup’s works on Cern) their journey will appear to be faster than light in our normal space-time they have simply journeyed into our far future. What has happened is the null geodesics which are continuous concentric spheres, with radius = distance = c*(+ or -)time, has in the hyperspace frame, our normal concentric sphere a vastly shrunken version. Yet, in our frame of reference the hyperspace sphere may be many spheres removed from our own.

When a careful comparison of frame to frame is done one finds that unless one cares to limit this future time travel through hyperspace to C or less paths that such travel does not actually get around some of the current problems we face when it comes to traveling to the stars. However, interesting enough, such dual frames do not only offer solutions to how entanglement might work, they do offer a possible solution to creating metrics simular to the one proposed by Alcubierre. I say simular to his simply because that original metric has many problems when it comes to trying to make such a physical reality. One such example is the large amount of exotic energy required which violates quantum energy conditions for our space-time.

The main problem with almost all of the theoretical original ideas on FTL travel is if one trys to construct such fields one is required by the velocity of such fields to construct the field faster than the path itself allows. Fernando's hyperspace idea does allow such in a fashion simply because the path is one into the future lightcone in relation to our space-time. But, one would still have to solve the other problems like creation of negative energy in the first place as well as how to cross over into hyperspace in the first place.

Do not get me wrong here. There are possible solutions to all these problems out there. For one the amount of exotic energy required can be lowered in theory. But, all of these ideas are based upon being able to engineer in a way we simply do not have the ability to do at this time. What few bother to realize is most of this off the wall thinking has added to our understanding of nature itself on many fronts. At one time it was almost taboo to suggest there could be any theoretical way around the speed of light. Now there are even mainline researchers out there sugessting that C itself may not always have been a constant. Thinking about such has actual value and is a worthy path of research. For one, if we ever do want to actually travel to the stars we will need to find a way to either reach C or exceed it. Current propulsion methods simply will not work for star travel when it comes to travel time, supplies, fuel, etc.