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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
<br /> \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}<br />
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.
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
<br /> \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}<br />
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.
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