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PC1672 Advanced dynamics

1.3 Proper time

If we cannot find a frame in which an an event occurs at $t=0$ then it does not make sense to talk about its invariant distance from the origin. Instead, if we can find a frame in which an event occurs at the origin, we can use this to define the proper time at which it occurs. The proper time $\tau$ is a (Lorentz invariant) 4-scalar. In a general frame, it can be found using

\begin{displaymath}\tau^2=t^2-{x_1^2+x_2^2+x_3^2\over c^2}\end{displaymath}

Events with the same proper time (with respect to the origin O at $t=0$) lie on the curves

\begin{displaymath}t=\pm\sqrt{\tau^2+x^2/c^2}\end{displaymath}

These are hyperbolas which asymptote to $x=\pm ct$.

Events for which $s^2>0$ are said to have a space-like separation from the origin of space-time while events with $s^2<0$ have a time-like separation. In between there are events with a light-like separation, $s^2=0$.

Textbook references

Home: PC 1672 home page | Up: 1 Relativity | Weekly plan | Help: Guide to using this document |
Next: 1.4 Space-time 4-vectors | Previous: 1.2 Lorentz transformations |

Mike Birse
17th May 2000