Definition of the Lorentz transformations

Question

Until very recently I believed that the Lorentz transformations were defined as "the transformations that carry one inertial reference frame into another". In Wikipedia's page we find something along these lines, since the starting sentence of the article is

In physics, the Lorentz transformation (or transformations) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other.

Now, Griffiths introduces the topic in his Electrodynamics book in the following way:

Suppose that we know the coordinates $(t,x,y,z)$ of a particular event $E$ in one inertial system $\mathcal{S}$, and we would like to calculate the coordinates $(\overline{t},\overline{x},\overline{y},\overline{z})$ of that same event in some other inertial system $\overline{\mathcal{S}}$. What we need is a "dictionary" for translating from the language of $\mathcal{S}$ to the language of $\overline{\mathcal{S}}$.

He then uses quite simple arguments to convert measurements made in $\mathcal{S}$ to measurements made in $\overline{\mathcal{S}}$. In the end it all comes down to the application of length contraction and time dilation in a correct way. In that sense, following Griffiths: if we accept the postulates of relativity as a consequence we have time dilation and length contraction and then if we pick two inertial frames in relative motion we can derive a transformation which is a Lorentz transformations.

This is the idea I always had about the Lorentz transformations, they are simply the only allowed transformation between inertial frames when we accept the postulates of relativity and their consequences.

On the other hand, it appears that this is not the definition. It appears the true definition is that the Lorentz transformations are the transformations which leave the speed of light unaltered or else the ones which preserve Maxwell's equations.

I must confess I still didn't get it. So my question is: what is the real definition of the Lorentz transformations? How they are defined and what is the motivation for the definition?

Answer 1

There are lots of ways of approaching special relativity. My own preferred approach is the invariance of the line element.

Suppose you move a small distance in spacetime $(dt, dx, dy, dz)$ then the length of the line element $ds$ is defined by:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{1} $$

This equation is known as the metric equation and is derived from a geometric object called the metric tensor that describes the geometry of spacetime. The quantity $ds$ is an invariant, that is all observers no matter how they are moving will agree on it's value.

From the invariance of $ds$ we can derive time dilation and Lorentz contraction, and the Lorentz transformations are precisely the linear transformations that preserve $ds$.

The reason I've rambled on about this is to try and explain how all the features of SR that you mention are linked through Lorentz invariance. In my view the most elegant approach is to start with the metric and derive everything from that. However because everything is linked you can start with dilation/contraction or the Lorentz transformations and derive everything else. Griffiths has just chosen the route that he considers most intuitively obvious.

Response to comment:

The simplest justification for equation (1) is that it works. I don't know a lot about the history of SR but I believe that Minkowski suggested the equation, building on the work of Poincaré, because it worked or more precisely because it offered an elegant way to formulate Einstein's theory.

I believe the best way to justify equation (1) is that it is a solution to Einstein's equations of general relativity. If we take GR and look for a solution when no matter is present (a vacuum solution) then special relativity emerges as the solution with the lowest ADM energy. It isn't the only vacuum solution because somewhat surprisingly black holes are also vacuum solutions, but it is the lowest energy vacuum solution.

Of course this only defers the question, because the next question is why GR requires the line element to be invariant. The answer to this is that GR is one of a large class of mathematical objects called metric theories that work with an invariant line element. GR is so naturally formulated as a metric theory that I'm sure Einstein cursed himself for not realising it years earlier!

Answer 2

In a sense, the two definitions you mention are the same. One of the postulates of special relativity is that the speed of light is the same in all reference frames, so the definition of "the coordinate transformation according to the postulates of special relativity" is the same definition as "the coordinate transformation under which the speed of light is constant". Historically though, they arose from Maxwell's equations, and Maxwell's equations imply the Lorentz transformations.