Lorentz Transforms
The Metric Equation

The metric equation in special relativity, as it is often written, has two times and one displacement in it, e.g. (c dT)^2 = (c dt)^2 - (dx)^2 where c is the speed of light. This equation is more widely useful in general relativity, and probably in the study of arbitrary hyperspaces, than are the familiar Lorentz equations for transforming between two frames (which of course can be derived therefrom).

This suggests that a strategy for introducing special relativity from the metric equation might have some advantages. One might even see the same benefits that prompt us to discuss single-frame kinematics in classical introductory physics, before we hit kids with the complexities of multiple (i.e. primed and unprimed) frames.

Work here suggests these advantages are real, although the language usually used to introduce relativity may need some updating. Take the assertion, for example, from a popular modern intro-text that the ``principle of relativity means that any kind of experiment (measuring the speed of light for example), performed in a laboratory at rest, must give the same result when performed in a laboratory moving a constant velocity past the first one.'' While students are wondering how this can possibly be true say for the speed of an object NOT moving at the speed of light (like the origin of the second laboratory, which has a finite velocity as measured by an observer in the first laboratory and a zero velocity when measured in the second), the words also prepare them NOT to expect that the rate of momentum change in special relativity (something we are classically used to thinking of as frame-independent) becomes frame-dependent at high speeds (just as the rate of energy change is even classically).

Hence, when one first considers an approach based on the metric equation (we refer to it as a 1-map 2-clock strategy), a bit of cognitive dissonance as well as a bit of jargon might be expected. The good news is that once over the hump, the mathematics of map-based relativity focussed on physical clocks which monitor the two times found in the metric equation, may win you over (as it has me) with one pleasant surprise after another.

You can find more on the surprises we've had, and some suggestions for classroom implementation, here. Here's a more recent page on how one can get Lorentz transforms from the metric equation.

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