The Lorentz Transformation

Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. The Lorentz Transformation

The Lorentz Transformation

We can not at this point forebear introducing a little mathematics to further emphasize the theory and the very logical nature of this contraction hypothesis.

Let us suppose that we were on a world that was absolutely motionless with respect to the ether and were looking at a ray of light. The magnetic and electric fields which form the ray can be described by means of four mathematical expressions which have come to bear the name of “Maxwell’s field equations.” Now suppose that we ask ourselves the question: How must these equations be changed so that they will apply to a ray of light which is being observed by people on a world that is moving with a velocity v through the ether?

The answer is immediate. From the Michelson-Morley experiment we know that we can not tell how fast or how slowly we are moving with respect to the ether. This means that no matter what world we may be upon, the form of the Maxwell field equations will always be the same, even though the second set of axes (or frame of reference) may be moving with high velocity with respect to the first.

Starting from this hypothesis (called in technical language the covariance of the equations with respect to a transformation of coordinates), Lorentz found that the transformation which leaves the field equations unchanged in form was the following:

where k is as on page 92.[246]

And what, now, can be deduced from these very simple looking equations? In the first place we see that the space of x′, y′, z′, t′ is not our ordinary concept of space at all, but a space in which time is all tangled up with length. To put it more concretely, we may deduce from them the interesting fact that whenever an aviator moves with respect to our earth, his shape changes, and if he were to compare his watch with one on the earth, he would find that his time had changed also. A sphere would flatten into an ellipse, a meter stick would shorten up, a watch would slow down and all because, as H. Minkowski has shown us from these very equations, we are really living in a physical world quite different from the world of Euclid’s geometry in which we are accustomed to think we live.

A variety of objections has very naturally been made to this rather radical hypothesis in an attempt to discredit the entire theory, but it is easily seen that any result obtained through the field equations must necessarily be in conformity with the theory of contraction, since this theory is only the physical interpretation of that transformation which leaves the field equations unaltered. Indeed, it is even possible to postulate the Lorentz transformation together with the assumption that each element of charge is a center of uniformly diverging tubes of strain and derive the Maxwell field equations from this, which shows from another point of view the truly fundamental nature of the transformation.

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