DISCUSSION AND CONCLUSIONS
An analysis has been given showing how faster-than-light travel can result in closed-loop backward time travel. The analysis focused on an extremely simple scenario – an object going out and coming back – effectively extending the twin paradox scenario to superluminal speeds. Further, only a single relativistic formula was used – that for velocity addition. A surprising result is that, in this scenario, backward time travel appears only when the turnaround location is moving away from the launch location, and, further, is bound to the creation and annihilation of object pairs. The underlying mathematical reason is that the negative time duration for the return trip needed to create closed-loop backward time travel is tied to spaceship motion away from the launch site, not toward it, as shown in Eq. (\ref{tback0}). This behavior neatly describes an (Earth-observed) spaceship moving out toward the planet on the “return back” leg of the trip in addition to the (Earth observed) spaceship moving out toward the planet on the initial “outward” leg of the trip. One knows that the spaceship does return, and so the second member of the pair-created spaceships remains on the Landing Pad. Note that when superluminal spaceship speeds are invoked, the spaceship always travels at superluminal speeds relative to the Earth, and never accelerates through \(c\).
It is tempting to explain away these results as meaningless because the relativistic velocity addition formula, Eq. (\ref{weq}) was applied to a regime where it might not hold: where one speed is superluminal. However, the validity of this formula in the superluminal regime should be testable in a conventional physics lab where illumination fronts or sweeping spots move superluminally, in contrast to a detector that moves subluminally. \cite{Nemiroff2016} Further, to our knowledge, no other relativistic velocity addition formula has even been published.
Although not defined in the above equations, it is consistent to conjecture that the superluminal spaceship has negative energy. \citep{Chase1993} This may be pleasing from an energy conservation standpoint because both pair creation and annihilation events always involve a single positive energy and a single negative energy spaceship – never two positive energy or two negative energy spaceships. Therefore, neither the creation nor annihilation of a spaceship pair, by themselves, demand that new energy be created or destroyed.
It is not clear how “real” the negative energy spaceships are to observers in inertial frames other than the Earth, including frames moving superluminally. The negative energy ships are surely real in the Earth frame in the sense that they give those observers positions from which spaceship images can emerge. However, these negative energy ships may not exist in some other reference frames, which appears to raise some unexplored paradoxes. Also unresolved presently is whether observer in a superluminal positive-energy ship that left the Launch Pad would be able to see a negative-energy ship that left from the Landing Pad. Since it is not in the scope of the above work to analyze what happens in inertial frames other than the Earth, then, unfortunately, this and other intriguing questions will remain, for now, unanswered.
Finally, this analysis may give some unexpected insight to physical scenarios that seem to depend on superluminal behavior. For example, implied non-locality in quantum entanglement typically posits some sort of limited superluminal connection between entangled particles, although one that does not allow for explicit superluminal communication. To the best of our knowledge, never has such supposed superluminal connection been tied through the special relativity addition formula to pair events. Perhaps one reason for this is that so few seem to know about it. Yet, as implied here – it may well be expected for observers in some reference frames.
\acknowledgments
The authors thank Qi Zhong, Teresa Wilson, and Chad Brisbois, for helpful conversations.
\({}^{1}\) Electronic address: nemiroff@mtu.edu
\({}^{2}\) Electronic address: dmrussel@mtu.edu