Many sci-fi questions here are derailed by complaints that the proposed version of FTL breaks causality. However, relativity-safe FTL concepts do exist. What range of conditions allow FTL (movement, teleportation, or communication) but not time travel?
Time paradoxes I've seen are based on FTL interchange between relativistic reference frames. (edit: see update below) Problems occur when the differential between time frames is larger than the travel time. Is that correct? If so, we can avoid paradox by requiring lower velocities (FTL and/or reference).
Also, those paradoxes use reference frames moving away from each other. If the reference frames are moving towards each other, does paradox still occur?
My goal is to establish parameters for physics-tolerant FTL, so that future answers don't need to nitpick about closed timelike curves.
Here is one example that I think should work:
- vessels can shift to & from "hyperspace", but travel still requires local time (at minimum, hours per light year plus some overhead even if you don't move).
- vessels can't enter or exit hyperspace at high real velocity (>1% c) relative to some local center of mass (e.g. galactic core). Technobabble about nonlinear fluidic space available if needed.
- separate vessels enter separate instances of hyperspace, cannot intercommunicate.
In the "Sharp Blue" article, the diagrams display Lorentz transformation as a slanting of the space-time axes, and FTL is assumed to be instantaneous. But non-instant FTL would also have a slope, and it seems like paradox could be avoided if the FTL is steeper than the dilation angle.
Mathematically: dilation slope as a function of (relative) frame velocity goes from f(0)=0 to f(c)=1, while the FTL slope as a function of travel velocity goes from f(c)=1 to f(infinity)=0. I'd need to refresh my analytic geometry to make the terms cancel, but such values are determinable. Why is this approach not valid?
Dan Smolinske's answer explains why my thinking is incorrect: even if the endpoints of the FTL don't experience paradox directly, a relativistic observer traveling near an endpoint does.
celtschk's answer provides a solution: require a primary reference frame, such as the ether in Lorentz Ether Theory. Lorentz's math is more complex than Einstein's (Occam's Razor FTW) but their results are indistinguishable for velocities below c. They only differ during FTL; the ether can prevent paradox.