Working on pervect's "messy unsolved" problem has led me to an interesting result. Let [itex]\left( x , t \right)[/itex] be a global inertial coordinate system for Minkowski spacetime.(adsbygoogle = window.adsbygoogle || []).push({});

Consider the worldline given by

[tex]t \left( \tau \right) = \frac{\tau^3}{3} - \frac{1}{4 \tau}[/tex]

[tex]x \left( \tau \right) = -\frac{\tau^3}{3} - \frac{1}{4 \tau}.[/tex]

Then

[tex]\frac{dt}{d \tau} = \tau^{2} + \frac{1}{4 \tau^2}[/tex]

[tex]\frac{dx}{d \tau} =- \tau^{2} + \frac{1}{4 \tau^2}.[/tex]

Note that [itex]dt/d\tau > 0[/itex], and that

[tex]

\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}

[/tex]

Therefore, [itex]\tau[/itex] is the proper time for a futute-directed timelike worldline.

Note also that when [itex]\tau = -1[/itex], both [itex]t[/itex] and [itex]x[/itex] are finite, but as [itex]\tau \rightarrow 0_-[/itex], both [itex]t[/itex] and [itex]x[/itex] 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.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Infinity in Finite Proper Time

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Infinity Finite Proper | Date |
---|---|

I GR observer at infinity | Nov 1, 2017 |

I Is it really possible that relativistic mass tends to reach infinity? | Feb 25, 2017 |

I When observable Universe was the size of a baseball was its gravitational influence bigger? | Aug 15, 2016 |

Example of curvature scalar diverging at infinity? | Nov 12, 2015 |

How do you transition from finite to infinite? | Nov 5, 2015 |

**Physics Forums - The Fusion of Science and Community**