Infinity in Finite Proper Time

[George Jones]

Working on pervect's "messy unsolved" problem has led me to an interesting result. Let $\left( x , t \right)$ be a global inertial coordinate system for Minkowski spacetime.

Consider the worldline given by

$$t \left( \tau \right) = \frac{\tau^3}{3} - \frac{1}{4 \tau}$$

$$x \left( \tau \right) = -\frac{\tau^3}{3} - \frac{1}{4 \tau}.$$

Then

$$\frac{dt}{d \tau} = \tau^{2} + \frac{1}{4 \tau^2}$$

$$\frac{dx}{d \tau} =- \tau^{2} + \frac{1}{4 \tau^2}.$$

Note that $dt/d\tau > 0$, and that

$$\begin{align}<br /> \left( \frac{dt}{d \tau} \right)^2 - \left( \frac{dx}{d \tau} \right)^2 &= \left( \tau^{2} + \frac{1}{4 \tau^2} \right)^2 - \left( - \tau^{2} + \frac{1}{4 \tau^2} \right)^2\\<br /> & = 1.<br /> \end{align}$$

Therefore, $\tau$ is the proper time for a futute-directed timelike worldline.

Note also that when $\tau = -1$, both $t$ and $x$ are finite, but as $\tau \rightarrow 0_-$, both $t$ and $x$ wander off to positive infinity.

The situation is unphysical because the 4-acceleration is unbounded, although there are no hyperlight speeds.

Regards,
George

PS I think I have found an expression for the 4-acceleration of a specific example of pervect's problem, but I have to check to see if my solution really does satisfy the necessary criteria.