# What are the uses of proper length as a parameter?

Proper time is used to parameterize the world line of a moving particle in a way which is Lorentz invariant, which is elegant and powerful. Since space and time are usually treated on the same footing, I'd expect proper length to be a powerful parameter of some sort, yet I've never come across it being used in this way.

What are the uses of Proper length as a parameter?

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Proper distance and proper time are both really the calculation of the metric disitance between two events. In general relativity the infinitesimal metric is:

$ds^2 = g_{\mu\nu}dx^\mu dx^\nu$

In special relativity (with units where $c=1$ ), space-time is flat so the Cartesian Minkowski metric between two points is:

$\Delta s^2 = \Delta t^2 -\Delta x^2 -\Delta y^2 -\Delta z^2$

In general, when $c=1$, the proper time, $\Delta \tau$, and the proper distance, $\Delta L$, are related to the metric by:

$\Delta L^2 = - \Delta \tau^2 = - \Delta s^2$

So for two points that have a time-like separation, such as two points on the worldline of a massive particle the proper time will be a real positive number and is just this metric rewritten as follows:

$\Delta \tau = \sqrt{ \Delta t^2 -(\Delta x^2 +\Delta y^2 +\Delta z^2)/c^2 }$

Where I have put the speed of light in explicitly to make it clear that the proper time has the dimensions of time. Note that all observers will agree on the value of the proper time between two events on the worldline of a massive particle.

Now for two points that are at a space like separation this proper time would become imaginary so it is customary to rewrite it as a real positive proper distance as follows:

$\Delta L = \sqrt{\Delta x^2 +\Delta y^2 +\Delta z^2 - \Delta t^2 c^2 }$

Where, again, I have put the speed of light in explicitly to make it clear that the proper distance has the dimensions of spatial distance not time. if tachyons existed, this would be the proper distance that all observers would agree is the proper distance between two events on the world line of a tachyon. Even without tachyons, the proper distance between two space-like separated events is the spatial distance between those events in a reference frame where they are measured to be simultaneous (for space-like separated events there is always a frame where they will be simultaneous - this the relativity of simultaneity).

Note that for massless particles, such as photons, $\Delta \tau = \Delta L = 0$.

So there really isn't any fundamental distinction between proper time and proper distance - they are both different ways of representing the same metric which happens to give you a real number for events on either time-like or space-like event separations respectively.

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A quick precis: proper time $d\tau$ and proper length $dL$ are basically the same thing and are related by $dL^2 = -d\tau^2$ (in geometrical units). –  John Rennie Feb 25 '13 at 11:47
Thanks @JohnRennie, I added that summary to the answer. –  FrankH Feb 26 '13 at 0:43

Proper length as a parallel for proper time is the Minkowski length of a space-like curve. These are not particularly useful, partly because in general they will loop back and forth in time according to many observers, which is just weird.

It is useful, though, for quantitatively describing the spacelike separation between two events. Being spacelike-separated, the two events can be seen as simultaneous in some rest frame; the distance between them in this frame is their proper separation length. It is the largest separation observable between the two locations, due to length contraction. For co-moving objects, the proper length between them is the only meaningful way of saying how far apart they are.

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