My favorite way of deriving the Lorentz transformation is to start from symmetry principles (an approach originated in Ignatowsky 1911; cf. Pal 2003), and one of my steps is to prove a lemma stating that the Lorentz transformation has to preserve volume in spacetime, i.e., in fancy language, it has to have a Jacobian determinant of 1. My visual proof of the lemma (for 1+1 dimensions) is given at the link above, in a figure and its caption.
Intuitively, the idea behind the proof is that it would be goofy if a boost could, say, double the area, since then what would a boost in the opposite direction do? If it's going to undo the first boost, it has to halve the area, but then we'd be violating parity symmetry. The actual proof is actually a little more complicated than this, though. If a rigorous proof could be this simple, I'd actually be satisfied and say the result was so simple and obvious that we should call it a day. But in fact if you want to get it right it becomes a little more involved, as shown at the link where I give the actual proof. One way to see that the intuitive argument I've given here above doesn't quite suffice is that it seems to require that area in the $x$-$y$ plane be preserved under a boost in the $x$ direction, and that's not true.
This is different from the approach based on Einstein's 1905 axiomatization. In that approach, you derive the Lorentz transformation and then prove the unit Jacobian as an afterthought.
Although my proof of the unit Jacobian from symmetry principles works, I've always felt that there must be some deeper reason or better physical interpretation of this fact. Is there?
Takeuchi 2010 says on p. 92:
This conservation of spacetime area maintains the symmetry between […] frames, since each is moving at the exact same speed when observed from the other frame, and ensures that the correspondence between the points on the two diagrams is one-to-one."
This seems clearly wrong to me, since you can have one-to-one functions that don't preserve area. (Takeuchi is writing for an audience of liberal arts students.)
Mermin has a very geometrical pedagogy that he's honed over the years for a similar audience, and he interprets space-time intervals as areas of "light rectangles" (Mermin 1998). Since he uses the 1905 Einstein axiomatization, conservation of area, which is equivalent to conservation of spacetime intervals, comes as an afterthought.
One thing that bugs me is that the area-preserving property holds for Galilean relativity (and my proof of it holds in the Galilean case without modification). So any interpretation, such as Mermin's light rectangles, that appeals specifically to something about SR seems unsatisfying. The notion of area here is really just the affine one, not the metrical one. (In Galilean relativity we don't even have a metric.)
Another way of getting at this is that if you start with a square in the $x$-$t$ plane and apply a Lorentz transformation, it becomes a parallelogram, and the factors by which the two diagonals change are the forward and backward Doppler shifts. Conservation of area then follows from the fact that these Doppler shifts must be inverses of one another.
Laurent 2012 is an unusual coordinate-free presentation of SR. He interprets $\epsilon_{abcd}U^aB^bC^cD^d$ as a 3-volume measured by observer whose normalized velocity vector is $U$. Restricting to 1+1 dimensions, $\epsilon_{ab}U^aB^b$ is the length of vector $B$ according to this observer. He gives the example of associating affine volume with the number of radioactive decays in that volume. The implications of this are vague to me.
What is the best way of interpreting the preservation of spacetime volume by Lorentz transformation?
Please do not reply with answers that start from the known form of the Lorentz transformation and calculate the Jacobian determinant to be 1. I know how to do that, and it's not what I'm interested in.
W.v. Ignatowsky, Phys. Zeits. 11 (1911) 972
Bertel Laurent, Introduction to spacetime: a first course on relativity
Mermin, "Space-time intervals as light rectangles," Amer. J. Phys. 66 (1998), no. 12, 1077; the ideas can be found at links from http://people.ccmr.cornell.edu/~mermin/homepage/ndm.html , esp. http://www.ccmr.cornell.edu/~mermin/homepage/minkowski.pdf
Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1
Takeuchi, An Illustrated Guide to Relativity
Best Answer
First, you have to distinguish mathematical proofs from physics "proofs". You can't really rigorously prove statements about the real world at the same level of rigor that one has in maths.
The unimodularity (Jacobian = 1 or -1, to agree with the usual definitions of the adjective) of the Lorentz transformations is trivial to prove in maths, whatever definition of the Lorentz transformations we adopt.
Something else is the "proof" that the transformation responsible for changing the frame in physics is unimodular. Such a "proof" has to accept some physical assumptions that are ultimately justified empirically.
The usual "proof" is simple and you sketched it at the beginning.
In the empty space, the transformation $B$ (boost) from an observer at rest to an observer moving by the speed $v$ in the $z$ direction is given by $$ (t,x,y,z)\mapsto (t',x',y',z')$$ Because of time- and space-translational symmetry, this map has to be linear (with a possible shift, i.e. an inhomogeneous term, generalizing Lorentz to Poincaré transformations). Because it is linear, its Jacobian is constant, so it is just one number-valued function of $v$.
By using the rotational symmetry, one can write $$ B(-v) = R_\pi B(v) R_\pi $$ In words, the boost by the opposite speed may be obtained by rotating the system by $\pi$ around e.g. $x$-axis, boosting it by $+v$, and rotating back. The Jacobian is a determinant so for the product transformation above, it's just the product of the determinants and the determinants for the rotations are one (one could have also used parity so that the signs from these two determinants would cancel as well but I chose to avoid negative-determinant transformations).
It follows that $$\det B(-v) = \det B(v)$$ i.e. the determinant is an even function of $v$. At the same moment, the boost by $-v$ is nothing else than the transformation switching back to the original frame, i.e. $$B(-v) = B(v)^{-1}$$ which also means (in combination with the previous identity, to eliminate $\det B(-v)$) $$\det B(v) = \det B(v)^{-1}$$ and the determinant is therefore $\pm 1$. Because it's $1$ for $v=0$ (identity) and it is a continuous function, we must have $\det B(v)=1$ for all $v$.