Deaccelerating to the Speed of Light

Question

The trajectory of an observer with a uniform proper acceleration $a$ (Rindler) in an inertial frame $(t,z)$ can be described by the hyperbola \begin{equation} \left(z+\frac{\gamma_{0}}{a}\right)^{2} - \left(t+\frac{\gamma_{0}\beta_{0}}{a}\right)^{2} = \frac{1}{a^2}\,, \end{equation} where $\beta_{0}$ is the initial velocity and $\gamma_{0} = 1/\sqrt{1-\beta_{0}^{2}}$ the initial Lorentz factor. Plotting this gives the following figure.

There is, of course, another hyperbola when you rotate this hyperbola by $90$ degrees and I'm interested in studying this situation. Purely mathematically, this hyperbola gets described by \begin{equation} \left(z+\frac{\gamma_{0}}{a}\right)^{2} - \left(t+\frac{\gamma_{0}\beta_{0}}{a}\right)^{2} = -\frac{1}{a^2}\,, \end{equation} where there is now a minus sign in the right hand side and is plotted below.

However, this makes not much sense since in this (superluminal) regime, the initial velocity $\beta_{0}$ is larger than one such that the initial Lorentz factor $\gamma_{0}$ becomes imaginary.

My question is if there is a way to work around this. How can we describe this regime such that it physically also makes sense?

Answer 1

This regime does not make physical sense as a worldline. This would be a worldline of a particle moving faster than $c$.

In general, a simple rotation by 90 deg on a spacetime diagram will not transition from one realistic situation to another. The rotation itself is not physically motivated

Answer 2

Kruskal–Szekeres coordinates appear to cover the regime you describe. The time and space coordinates essentially swap roles in the rotated region. Outside the event horizon everything inevitably progresses through time upwards on the chart. Below the event horizon everything inevitably progresses through space towards the the singularity. Moving away from the singularity below the event horizon is just as impossible as trying to move backwards in time above the event horizon.

In the Rindler coordinate system, the space above and below the Rindler regime is just flat space and is not considered to be a valid part of the chart. Although the Rindler coordinate system has some things in common with a real gravitational field it does not exactly duplicate one because the equivalence principle is only valid locally and fails over larger distances where the curvature of space in a real gravitational field becomes obvious.

The hyperbolic curves in the Rindler chart are the worldlines that could be followed by real particles or observers and clocks. The horizontal hyperbolic curves you have added are not physically realistic physical and cannot represent the worldlines of any real particles or observers.