PLANET-FREE SCENARIOS

One might consider that the pair creation and annihilation arguments above only arise because of the “trick” of involving a planet that has a non-zero and positive speed \(u\). In this view the planet’s speed, along with the relativistic speed addition formula, act as a spurious door to mathematical possibilities that are physically absurd. As evidence, one might take an example where a spaceship leaves with a speed \(v\) relative to the Earth and then returns at a speed \(u\), again relative to the Earth. The arbitrary turnaround location can be labeled \(x_{turn}\). Then, in the Earth frame, the time it takes for the ship to reach the turnaround location would be \(\Delta t_{out}=x_{turn}/v\), and the time it takes for the ship to return from the turnaround location would be the same: \(\Delta t_{back}=x_{turn}/u\). When \(u>c\), a pair of virtual images of the ship will again be seen, for a while, on Earth. However, there are no velocities \(v\) and \(u\), subluminal or superluminal, where either \(\Delta t_{out}\) or \(\Delta t_{back}\) is negative, and so no \(v\) and \(u\) values exist that create \(\Delta t_{tot}<0\). Therefore, in this scenario the spaceship will never arrive back on Earth before it left. Does this counter-example disprove the presented analysis?
No. The scenario of the previous paragraph does not create a situation where an object returns to the same location at an earlier time – a closed-loop backward time travel event. Therefore this scenario does not address the main query posed by this work – how faster-than-light travel enables backward time travel.
Scenarios do exist, however, where superluminal travel creates closed-loop backward time travel events, but where no intermediary planet is involved. Such scenarios, which some might consider simpler, have the spaceship just go out at one speed, turn around at an arbitrary location, and return at another speed. So long as the return speed is generated and hence specified relative to the outbound speed, then the relativistic velocity addition formula Eq. (\ref{weq}) may be used, and the same types of results arise. Note that is only presumed that Eq. (\ref{weq}) is valid when one or both speeds \(v\) and \(u\) are superluminal – this presumption has never been verified.
A simple planet-free scenario is as follows. A spaceship leaves Earth at speed \(v\). At an arbitrary turnaround location, the ship changes its velocity by \(2v\), toward the Earth, relative to its outward motion. For non-relativistic speeds, this turnaround would result in the ship heading toward Earth at speed \(v\). For relativistic and superluminal speeds, however, the relativistic velocity addition formula must be used, resulting in more complicated behavior. Specifically, in this scenario, it is straightforward to show following the above logic that \(\Delta t_{tot}=2\Delta t_{out}(1-v^{2}/c^{2})\), so that closed-loop backward time travel events occur for all spaceship speeds of \(v>c\). As before, tracking spacecraft and image locations show that pair events also may occur.
This brings up the question: why does it matter against what the spaceship’s relative return velocity is measured – shouldn’t the physics be the same? Coordinate invariance – called general covariance, and inertial frame invariance – called Lorentz invariance – should make the physics the same no matter which coordinates are used for tracking and no matter which inertial frames are used for comparison. Specifically, in this case, closed-loop backward time travel should not depend on whether the spaceship’s return velocity is specified relative to the Earth, or a planet, or the spaceship’s previous velocity, or anything else. The reality of what happens should be same regardless.
The key symmetry-breaking point is that the standard special relativistic velocity transformation, Eq. (\ref{weq}), is not confined to be Lorentz invariant when both subluminal and superluminal speeds are input. Mathematically, the reason is that the denominator of the velocity addition formula goes through a singularity at \(uv=c^{2}\), a singularity that cannot be reversed by a simple coordinate or inertial frame transformation. Physically, turning around relative to a different object may change the scenario – a different physical process may be described.