# Lorentz factor

The Lorentz factor, commonly assigned the Greek letter γ ("gamma"), is the relationship $\sqrt{ 1 - v^2/c^2 }$, or its inverse, $\frac{1}{\sqrt{1-v^2/c^2}}$ . It is often written in more modern notation as ${(1-v^2/c^2)}^{1/2}$ , or ${(1-v^2/c^2)}^{0.5}$, where the exponent of "one half" represents taking the square root. Putting a minus sign in front of the exponent gives the inverse "one over" relationship.

The equation is best known for its appearance in H.A. Lorentz' 1904 approach to relativising absolute aether theory ("Lorentzian electrodynamics", Lorentz ether theory, LET), which was essentially rederived by Einstein the following year in 1905 (retaining the "relativistic aspect" but without invoking an "aether"), as special relativity.

The Lorentz factor is a profoundly significant equation for mathematicians, and its key role in special relativity makes many believe that the theory has a compelling claim to be an "absolute truth". However, it is also possible, as an exercise, to derive a theory from deliberately bad starting principles (a "flat-moon theory"), and obtain the Lorentz factor and broadly SR-analogous effects from it, so the mathematical beauty of the equation does not necessarily mean that a theory is a good one.

## Variations

Although textbooks often standardise on the "one over" version of the equation, this isn't always appropriate – for instance, if we are discussing the Lorentz redshift of a passing body under special relativity, the Lorentz reddening will be expressed by the "one over" equation if we are talking about the wavelength of the signal, but will need to use the other version is we are talking about the frequency. Since the "one over" version represents an increase with velocity, it's probably an unhelpful version to apply to Lorentz contraction.

#### Inconsistencies

It's not unusual for SR sources to discuss the special theory in terms of a "flat-spacetime" Doppler propagation shift combined with a Lorentz redshift, while the accompanying equations actually describe a stationary-emitter Doppler effect coupled with a Lorentz blueshift. * Although the final result is the same in both cases, this can create confusion over the individual shift components involved.

## "Lorentzlike" factors

A generalisation of gamma, used for constructing a wider range of potential theories of relativity, can be referred to as a "Lorentzlike factor". This is essentially the same equation but with the exponent replaced by a variable whose value identifies one of a range of different potential theories of relativity, or the distance in "theory-space" between different potential relativistic solutions.

## Notes

Einstein used the "inverted" Doppler and Lorentz equations in a key text on SR, as it simplified the math. While he explained that he was going to be deliberately describing the situation from the emitter's point of view, in the English-language translation this caveat appears at the end of the double-page spread before the equations appear, so a casual reader can miss it, and the inverted equations often seem to be quoted verbatim in SR texts without the caveat.