# Confusion on Proper time and Proper Length

so I've been looking at special relativity and I have interpretations of the notions of proper length and proper time. Suppose there are two frames S and S'. S' is moving with respect to S. Therefore;

Proper Time is measured by S' Proper Length is measured by S

This is, at least, my way of understanding these two concepts. Is it correct? Thanks

Proper length and proper time are just different ways of measuring the same thing.

Suppose you have some path through spacetime (not necessarily a straight line path) that connects two points $A$ and $B$, then the proper length is given by:

$$s = \int_A^B ds$$

and the proper time is:

$$\tau = \int_A^B d\tau$$

where:

$$ds^2 = -c^2d\tau^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

The reason we use both concepts is that obviously we want $ds^2$ and $d\tau^2$ to be positive otherwise when we take the square root we will get an imaginary number. For a spacelike path $ds^2$ is positive and $d\tau^2$ is negative, while for a timelike path $d\tau^2$ is positive and $ds^2$ is negative.

So we tend to use the proper time when dealing with timelike intervals and the proper length when dealing with spacelike intervals. However they are just different ways of representing the same quantity.

This concept extends to general relativity in a simple way. We get:

$$ds^2 = -c^2d\tau^2 = g_{\alpha\beta}x^\alpha x^\beta$$

where $g_{\alpha\beta}$ is the metric tensor. In special relativity the metric tensor has the simple form:

$$\mathbf g = \text{diag}(-1, 1, 1, 1)$$

Proper time and Proper length is not a property of the reference frame you are using. It differs for different situations.

However, here are some points that may help:

A stationary observer will see a moving clock run slow, stationary clocks measure the shortest time interval between two events. Some people use the memory aid "moving clocks run slow" to a stationary observer and "moving objects are short" to a stationary observer.