Derivation of Lorentz Transformations

Question

How can I derive the Lorentz transformations? I don't want to use hyperbolic functions and the fact that the light waves travel by forming spherical wavefronts. Is there a way to derive the Lorentz transformations applying the conditions I have mentioned. I was unable to understand the method given in Landau and lifshitz deeply. That's why I want a method other than the one using hyperbolic functions

Answer 1

Here's a derivation that uses very basic properties of space and time (isotropy, homogeneity, the fact that two Lorentz boosts should compose into another valid Lorentz boost, etc.). The constant maximum speed through space (i.e., the speed of light) is a derived property, not an assumption.

One more derivation of the Lorentz transformation - Jean-Marc Levy-Leblond

Here's a similar one that uses linear algebra after deriving the fact that the transform is linear, with similar results.

Nothing but relativity - Palash B. Pal

These kinds of group-theory-based derivations go back to Vladimir Ignatowski in 1910.

Answer 2

The fastest way to derive the Lorentz factor that I can think of is to say: "We want to construct a theory that generates the same results regardless of whether lightspeed is fixed with respect to the observer's or the emitter's frame."

• A – If it's fixed in the observer's frame, the Doppler and visible length-change effects are E'/E = Len'/Len = c/(c+v) for recession and E'/E = 1 for transverse motion.

• B – If it's fixed in the emitter's frame, the Doppler and visible length-change effects are E'/E = Len'/Len = (c-v)/c for recession and E'/E = 1 - v²/c² for transverse motion (Lodge, 1909).

So to create a hybrid that gives the same answer in both cases, we can multiply both conflicting predictions together and square root to get their geometric mean. That gives us the predictions of special relativity: $SR = \sqrt{A×B}$

Alternatively, since the new intermediate prediction must always differ from both the previous predictions by exactly the same proportion, we can derive this proportion by dividing B by A (to find the original discrepancy that we wanted to get rid of), and then square rooting (to find the difference between the new prediction and both the old predictions). This can then be applied as a "correction factor" to either of the original predictions to get the new result.

This necessary correction factor, to be multiplied against A or divided out of B, comes out as $\sqrt{B/A} = \sqrt{1-v^2/c^2}$ In other words, the Lorentz factor.

The derivation is probably nonstandard, but it's a heck of a lot shorter than Einstein's, and it doesn't use any of the mathematical overhead that people often try to attach to SR. Hyperbolic functions?!? Pfft.