Application of the Lorentz Transformations

To use the Lorentz Transformations, we must decide how to measure a moving meterstick or a moving rocket ship. It sounds simple enough. But if you will think on this for a moment, you will realize that it is not something you ordinarily do. Consider a freight train moving along a track. How would you measure its length?

In actual practice, you would probably stop it and then measure its length. But we cannot do that; it keeps on moving. We could send two motorcyclists out--one at the front of the engine and the other at the end of the caboose--to make marks at the front and rear of the train at the same time. Then, at our leisure, we can measure the distance between these marks.

Consider a meterstick at rest in the B frame, moving at velocity v with respect to the A frame. The two ends of the meter stick are marked by "events". This may be exploding two flashbulbs or making two chalk marks on a blackboard. They must occur at the same time -- as seen by an observer at rest in the A frame.

Now it is very useful to use the Lorentz transformations for differences in coordinates rather than just coordinates. Since the equations are linear, the differences in coordinates transform just like the coordinates themselves. That is, we can rewrite these as

and

For our moving metersticks, we have tA = 0 (the events marking the ends must occur simultaneously in the A frame). Then, the difference in position between the two events, xB, as measured by B is given by

This is the length as measured by B, the "moving observer"; we will call this Lo. Lo =xB is the "proper length", measured by an observer at rest with respect to the meter stick; we will call this Lo. This "proper length" Lo is the length we commonly measure under ordinary conditions. xA is the length measured by A as the meter stick moves by; we will call this L. Now we can write

or

This tells us that the length L measured for the moving meter stick is shorter than the length Lo measured when it is a test. This difference in length is called the Lorentz-Fitzgerald contraction.

How can this be?

How large is this effect? For ordinary speeds, is as close to unity as you care to calculate. But look at the case of v = 0.9 c. Then

L = 0.44 Lo

For such extremely relativistic speeds, this effect is quite noticeable! Or, to find a contraction of 10% , the speed must be

v = 0.44 c

To observe a contraction of 1% , we must have a speed of

v = 0.14 c

You may also have heard that "moving clocks run slower." This is known as the Einstein time dilation. But what does it mean and how does it come about?

How would you compare your own watch with the watch of a friend and see if one ran faster than the other? You would normally set them side by side and watch them. But you can not do that if one of them is moving. The best you can do is to compare a single, moving clock with two synchronized, stationary clocks as sketched here. Place clock A1 near A's origin and clock B near B's origin. Both read tA = tB = 0 as they pass. What are the readings on clocks A2 and B as they pass each other?

Clock B is at B's origin, so xB = 0. We could call its time reading tB or tB; call it to for the time of the clock which is at rest. This is also known as the proper time, the time indicated by a clock at rest in its own reference frame.

Our equations then gives the reading on clock A2,

t o is the amount of time between two events as measured by a single clock present at both events; that is, t o is measured by a clock at rest with respect to the single location where both events occur. We call this the proper time. Clock B is sitting still in the B frame. tA is the amount of time between two events as measured by two different clocks, synchronized and at rest with respect to each other. tA is larger than to . If it takes 10 seconds for clock B to pass between clocks A1 and A2, as measured by these two clocks, it might only require 7 seconds as measured by B. Thus, A will conclude that B's clock is running slowly. But this effect is symmetric. Let B watch a single clock at rest in A but moving with respect to B. The "stationary" observer still finds the "moving" clock running slower!

The Lorentz Transformations

Velocity Transformations
Return to Ch 27, Special Relativity

(c) Doug Davis, 2002; all rights reserved