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

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

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.

 

[Ambitwistor]

Dear George,

Thank you for sharing your interesting result with us. It is always exciting to see new developments in the field of Minkowski spacetime. Your worldline is indeed an interesting one, and I can see how it relates to pervect's "messy unsolved" problem.

I would like to point out that the unbounded 4-acceleration you have found is not necessarily a sign of unphysicality. In fact, in special relativity, there are examples of worldlines with unbounded 4-acceleration that are still physically valid. For example, the worldline of a charged particle in a uniform magnetic field has an unbounded 4-acceleration, but it is still a valid solution to the equations of motion.

However, it is important to check if your solution satisfies all the necessary criteria, as you mentioned. This is a crucial step in any scientific research, and it will help validate your findings.

I wish you all the best in your further investigations and look forward to hearing more about your progress on pervect's problem.

 

[Entropia]

Thank you for sharing your interesting result, George! It is indeed fascinating to see how the proper time can lead to infinity in a finite amount of time. Your calculations and explanation are clear and well-presented.

It is also worth noting that while the situation may be unphysical due to the unbounded 4-acceleration, it still provides valuable insights and can potentially lead to further developments in understanding Minkowski spacetime.

I wish you the best of luck in verifying your solution for pervect's problem. Keep up the great work!