By George J. Spix

This paper will derive the Lorentz transformations based upon the postulates of Einstein’s Special Theory of Relativity (1905). By "Special" Einstein meant that he did not include the effects of gravity and accelera tions. He considered only those coordinate systems that are moving in straight line motion in relation to each other.

Einstein based his Special Theory on the 1881 experimental evidence of Michelson showing that the speed of light is constant at many different locations and directions in the earth’s orbit. Also Maxwell had shown in 1864 that the velocity of electromag netic radiation was constant and independent of its source or coordinate system. Einstein used his intuition to extend this information into his Special Theory of Relativity.

In 1899 Lorentz published his transformations to explain Michelson’s findings, but it was later found that these transformations fit Einstein’s theory exactly. We will here derive the Lorentz transformation from Einstein’s Special Theory, but historica lly these discoveries happened in the reverse order.

Lorentz transformations are mathematical statements of the conclusions of the Special Theory. The transformations are algebraic expressions that enable us to transfer measurements of time, length, velocity, mass and light frequencies between a moving coordinate system and a reference coordinate system.

An acquaintance with basic algebra and beginning calculus is helpful (but not required) in following the derivations and examples.

Einstein built his special theory on two postulates.

A coordinate system is the set of x, y and z axes at right angles to each other. The following examples show the use of a reference coordinate system in describing velocity:

When we say a car is traveling 60 mph, we mean the car is moving with respect to a point on the highway. The highway is part of the reference coordinate system and the car is the moving coordinate system.

Sometimes we also use coordinate systems to show that the reference and moving coordinate system are interchangeable when describing velocity. Assume that a boat going downstream and the current of the stream is 3 mph. Also, assume that the boat’s captain knows the boat’s engines will move the boat in still water at 8 mph. The captain judges that, with respect to the water, the boat is moving at 8 mph, and the land is moving past the boat at 11 mph. An observer on shore judges that the boat is moving at 8 mph plus 3 mph =11 mph with respect to the observer’s reference frame. The shore observer can observe the boat move downstream 11 miles in one hour. The stream, boat and shore each define an inertial coordinate system. Each one may be considered at rest (reference frame) and the other systems will be in motion (moving frames).

Inertial coordinate frames are defined as those systems within which Newton’s laws apply. The "Special" in the Special theory includes only inertial frames. Inertial frames differentiate those systems from coordinate frames that are being accelerated or in non-straight line motion. The conservation of momentum law also applies to inertial frames.

The following is an example illustrating Newton’s laws applied to inertial coordinate systems.

Picture a passenger in a railroad car tossing a ball in the air and catching it. The passenger, as observer in his reference frame, sees the ball go straight up and down. The passenger knows the force that was imparted to the ball and, using Newton’s Laws, calculates the maximum height the ball attains and the time the ball is in the air.

Assume there is an observer at the side of the track. The observer has been given the force on the ball and the velocity of the train. Using Newton’s Laws, the trackside observer calculates the same time the ball spends in the air and the parabolic path of the ball.

The conclusions of both observers concerning the ball’s motion are correct. Newton’s Laws apply to both inertial coordinate systems (frames) and there is no preferred frame of reference when considering inertial frames.

Assume that we are riding inside a moving coordinate frame, and a rocket is launched at us from a stationary frame. Assume that we are able to speed up our frame and stay ahead of the rocket. This same action can not be accomplished when a light is flashed at us. The light will catch up with us, pass us, move through our frame, and proceed ahead of us with the exact constant speed of light. And, when we measure the speed of light in all the frames, we discover that it is always the same speed. If we are standing still with respect to the source, the light would proceed past us at the same speed. This fact, that the speed of light is absolutely constant in all frames and in all directions, is an essential postulate in the following derivation of the Lorentz transforms.

In this section we will derive the Lorentz transformations using Einstein’s two postulates and the rules of algebra. There are several steps in the derivation. The progression of the individual steps is not intuitive so that the path of the transform development is not obvious until the conclusion.