# Special Relativity​ VI: Proper Time and Length

Hello, and welcome back to MPC! Last week, we saw an example of special relativity in our everyday lives by analyzing muons. Today, we will be tying up a few loose ends by discussing the ideas of proper time and proper length.

A few weeks ago, we derived the following formula for time dilation:

To derive this equation, we used the following scenario:

Figure 1: Our original time dilation scenario

Originally, we defined t0 as the time it takes for the particle of light to reach the top of the rocket ship as measured by person 0 and t1 as the time it takes for the particle of light to reach the top of the rocket ship as measured by person 1. Believe it or not, t0 actually has a much more significant meaning than the one we have given it: t0 is the proper time.

An interesting way to think about proper time is as the time measurement of an event taken by a person who is stationary to that “event.” How can one be stationary to an event? Well, let’s think about our original time dilation post. In that post, we were discussing the amount of time it would take for the particle of light reach the top of the rocket ship; the movement of the particle to the top of the rocket ship is the event whose time measurement we are trying to take.

In our scenario, person 1 is watching the particle of light move and up and down, but the particle of light is also traveling to the right at a speed of v. In other words, the event that person 1 is measuring is moving relative to him. **Note:  The event that person 1 is trying to measure the time for is the vertical motion of the particle over a distance of L. For this reason, the fact that the particle is moving up and down is not what why the event being measured is moving relative to person 1 — this vertical motion is the event itself! On the other hand, the horizontal motion is not “a part” of the event, and is therefore considered relative motion).**

Figure 2: The movement of the particle of light as seen by person 1

Does person 0 see anything different? Recall that, according to person 0, the particle of light just appears to be moving up and down — the particle of light does not seem to be moving horizontally at all.  In other words, person 0 only sees the event occurring (the particle moving up and down), which we will interpret as meaning that he is stationary to the event.

Figure 3: The movement of the particle of light as seen by person 1

Because person 0 is stationary to the event, we call his time the proper time. Proper time is typically represented by t0. We call person 1’s time measurement improper time and simply represent it as t.

Looking at equation 2, the proper time is always what the relativistic factor is multiplied by. To translate this equation for the case of our specific scenario, the amount of time person 1 believes it will take for the particle of light to reach the top of the rocket ship (improper time) is equal to the product of the relativistic factor and the amount of time person 0 believes it will take for the particle of light to reach the top of the rocket ship (proper time).

Here’s an interesting question: who was measuring the proper time when we were discussing muons?

Figure 4: Our muon scenario

Recall that we were measuring the “lifespan” of the muons. It may be hard to tell just by looking at figure 4 alone (it is hard to visualize the abstract concept/event of “lifespan”). To make things easier, let’s return to our analogy of muons as miniature clocks:

Figure 5: Our muon clock analogy

With this analogy, we can think of the hand on the clock to rotating around the clock once fully as our event. This, of course, is much simpler to visualize than “lifespan.” Now, we just have to think about each “observer.” The first observer is the person standing on the Earth. This person sees the clock hand rotate, but also sees the clock hand move down towards the Earth (as the entire clock is moving down towards the Earth). This additional motion allows us to conclude that this person is moving relative to the event in question and does not measure proper time. The muon itself, on the other hand, is moving with the clock (as the muon is, for our purposes, the clock) and does not see the clock moving downwards (for the same reason that person 0 in our previous example did not see the horizontal motion of the particle of light). Therefore, there is no relative motion between the muon and the event, the muon is stationary to the event, and the muon will measure proper time (looking at our time dilation equation from last week will confirm this).

As previously mentioned, proper time is always what the relativistic factor is multiplied by. Recall from a previous post that the relativistic factor is always greater than 1. Therefore, it is also correct to say that the proper time is the shortest time measurement for an event (the improper time is the proper time multiplied by a number greater than or equal to 1, which means that the improper time will always be greater than the proper time).

With an understanding of proper time, it should not be too difficult to understand its counterpart: proper length. We will use the rocket ship scenario once again:

Figure 6: Our rocket ship scenario

From this scenario, we previously derived the following equation:

However, there is a small formatting error with this equation. Just as t0 was our proper timeL0 is our proper lengthProper length, like proper time, has to do with the idea of relative motion. Specifically, proper length is the length between two objects as measured by someone who is “stationary” to the length.

How can someone be stationary to a length? Let’s bring back our giant ruler:

Figure 7: Our rocket ship scenario with a ruler

According to person 1, the ruler is stationary (see figure 7), which we will interpret as meaning that he is stationary to the length between the rocket ship’s starting point and the star. This means that person 1 will measure the proper length.

On the contrary, person 0 will see the star approaching him and the ruler moving backward. We can think of this as space itself moving past person 0, which we will call relative motion between person 0 and the length. Because there is relative motion between person 0 and the length, person 0 will measure improper length.

Figure 8: Our rocket ship scenario from person 0‘s perspective

This is where we find our small error with equation 3. Technically, because L0 typically represents the proper length, L1 (which we defined as the length measured by person 1) should be L0 and L0 should be L (the improper length):

As an exercise, can you show that proper length is always the longest distance between two objects? Also, who would measure the proper length in our muon scenario? Post your answers in the comment section below!

That is it for this week’s post! Now that we have a better understanding of improper time and improper length, we are ready to jump into next week’s exciting post on spacetime. See you then!

For more information on proper time and proper length, be sure to check out these resources!: