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I've been reading "Time Travel and Warp Drives" by Allen Everett and Thomas Roman, and the book had an interesting section on Tachyons. At one point it presented a system Leonard Parker (of the University of Wisconsin-Milwaukee) created whereby coordinate transformations for Tachyons could be done that would respect the "light barrier" and avoid paradoxes.

In the context of a "toy" two-dimensional spacetime, instead of having

[itex](ct')^{2}-x'^{2} = (ct)^{2}-x^{2}[/itex]

as regular Lorentzian transformations do, for Tachyons it had

[itex]-(ct')^{2}+x'^{2} = (ct)^{2}-x^{2}[/itex]

The system allowed for Tachyons that wouldn't be able to invoke paradoxes, since Tachyons could, in this model, only travel in the positive [itex]x[/itex] direction, just as regular matter can only travel in the positive [itex]t[/itex] direction, thus it couldn't return to its original point in spacetime and form a closed time-like curve.

This failed in higher dimensions, though, since it either required more dimensions of time so the other two dimensions of space would have something to interchange with, or it would introduce a preferred direction (the positive [itex]x[/itex] axis that acts as "forward in time" for the Tachyons).

My thought was that, "well, but why couldn't the arbitrary direction of its travel act as this [itex]x[/itex]-axis for any specific Tachyon, so it wouldn't be the same for every Tachyon, thus not introduce a preferred frame?"

Later in the text, though, it simply says; "Allen and his collaborators were able to demonstrate, beyond question, that these papers [claiming that it was possible to construct a theory of superluminal coordinate transformations that did not involve the introduction of a preferred direction] were mathematically inconsistent."

It gets my mind thinking on other tangents, and actually wondering about the significance of a preferred direction, and some cosmological models that might somehow be related to such a concept (at least locally, within the Schwarzschild radius of a black hole, such a thing could be said to exist: the direction towards the singularity. Also, forward in time is a preferred direction for regular matter, though time is different from space, and it might be a little "inconsiderate" (ie, uncalled for) to invoke a whole new "arrow of time" problem).

But most immediately, before going off on that tangent; what I'm really asking here is; what about my question, earlier? Why couldn't the direction of travel of any Tachyon be its "preferred" [itex]x[/itex]-axis?

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# Superluminal (Tachyon) Transformations

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