关于洛仑兹变换，网上讨论很多。我找了两个版本的推导过程。不过是英文的。

本人在阅读中文版的高等教材觉得很吃力。怪自己的语文不好，也怪翻译得有问题。

论讨争议最大的是第一个版本。我找的这篇原文写得很清楚。想必可以解决一些疑惑。

 版本1

FOR the relative orientation of the co-ordinate systems indicated in Fig. 2, the x-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the x-axis. Any such event is represented with respect to the co-ordinate system K by the abscissa x and the time t, and with respect to the system k' by the abscissa x' and the time t'. when x and t are given.	1
A light-signal, which is proceeding along the positive axis of x, is transmitted according to the equation x = ct or Since the same light-signal has to be transmitted relative to k' with the velocity c, the propagation relative to the system k' will be represented by the analogous formula Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation （很明显，如果公式3 成立，公式1，2就会成立） is fulfilled in general, where indicates a constant; for, according to (3), the disappearance of (x – ct) involves the disappearance of (x' – ct').	2
If we apply quite similar considerations to light rays which are being transmitted along the negative x-axis, we obtain the condition	3
By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants a and b in place of the constants and where and we obtain the equations	4
We should thus have the solution of our problem, if the constants a and b were known. These result from the following discussion.	5
For the origin of k' we have permanently x' = 0, and hence according to the first of the equations (5)	6
If we call v the velocity with which the origin of k' is moving relative to K, we then have	7
The same value v can be obtained from equation (5), if we calculate the velocity of another point of k' relative to K, or the velocity (directed towards the negative x-axis) of a point of K with respect to K'. In short, we can designate v as the relative velocity of the two systems.	8
Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is at rest with reference to k' must be exactly the same as the length, as judged from K', of a unit measuring-rod which is at rest relative to K. In order to see how the points of the x'-axis appear as viewed from K, we only require to take a “snapshot” of k' from K; this means that we have to insert a particular value of t (time of K), e.g. t = 0. For this value of t we then obtain from the first of the equations (5) x' = ax.	9
Two points of the x'-axis which are separated by the distance x'=1 when measured in the k' system are thus separated in our instantaneous photograph by the distance	10
But if the snapshot be taken from K'(t' = 0), and if we eliminate t from the equations (5), taking into account the expression (6), we obtain	11
From this we conclude that two points on the x-axis and separated by the distance 1 (relative to K) will be represented on our snapshot by the distance	12
But from what has been said, the two snapshots must be identical; hence x in (7) must be equal to x' in (7a), so that we obtain	13
The equations (6) and (7b) determine the constants a and b. By inserting the values of these constants in (5), we obtain the first and the fourth of the equations given in Section XI>.	14
Thus we have obtained the Lorentz transformation for events on the x-axis. It satisfies the condition	15
The extension of this result, to include events which take place outside the x-axis, is obtained by retaining equations (8) and supplementing them by the relations In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light of arbitrary direction, both for the system K and for the system K'. This may be shown in the following manner.	16
We suppose a light-signal sent out from the origin of K at the time t = 0. It will be propagated according to the equation or, if we square this equation, according to the equation	17
It is required by the law of propagation of light, in conjunction with the postulate of relativity, that the transmission of the signal in question should take place—as judged from K'—in accordance with the corresponding formula r' = ct' or, In order that equation (10a) may be a consequence of equation (10), we must have	18
Since equation (8a) must hold for points on the x-axis, we thus have = 1; for (11) is a consequence of (8a) and (9), and hence also of (8) and (9). We have thus derived the Lorentz transformation.	19
The Lorentz transformation represented by (8) and (9) still requires to be generalised. Obviously it is immaterial whether the axes of K' be chosen so that they are spatially parallel to those of K. It is also not essential that the velocity of translation of K' with respect to K should be in the direction of the x-axis. A simple consideration shows that we are able to construct the Lorentz transformation in this general sense from two kinds of transformations, viz. from Lorentz transformations in the special sense and from purely spatial transformations, which corresponds to the replacement of the rectangular co-ordinate system by a new system with its axes pointing in other directions.	20
Mathematically, we can characterise the generalised Lorentz transformation thus: It expresses x', y', z', t', in terms of linear homogeneous functions of x, y, z, t, of such a kind that the relation is satisfied identically. That is to say: If we substitute their expressions in x, y, z, t, in place of x', y', z', t', on the left-hand side, then the left-hand side of (11a) agrees with the right-hand side.

版本2

4. Lorentz Transformations

We are now prepared to deal with the problem of how to relate an arbitrary and coordinate in an unmoving frame to an arbitrary and coordinate in a moving frame. For this we imagine an experiment in which we get rid of the mirror altogether, and simply have the flash bulb at and the photodetector at . The flash bulb flashes at an arbitrary time at , and the light is received at time at . By choosing the initial flash time carefully, it is possible to arrange an arbitrary time at which the light is received. Since the light travels at speed , we have

That is what happens in the primed frame. What happens in the unprimed frame? The light starts at some time and travels to the final point at time . Since the light travels at , and the photodetector is moving away from the flash at velocity , and the length of the separation is in this reference frame, the time it takes is given by

At the end of the experiment, the flash bulb has moved a distance from the origin, and the photodetector is a distance beyond that, so

Our goal is to obtain equations relating the coordinates and to and . We already have three equations, but we have five unknowns: , , , , and . We need two more equations. Fortunately, since the initial flash was at , we can use the time dilation relation

and the Lorentz contraction relation

to find the desired relationships. From here on out it is all algebra.

First, combine equations (E) and (C) to find a relation for :

That was relatively easy.

Much harder is finding the expression for . We start by solving equation (A) for :

We then use (D) to substitute for and (E) to solve for :

We now solve (B) for , yielding

If we now substitute equation (C) for , we have an expression for exclusively in terms of and , namely

We have the transformation properties of and in the two coordinate systems. What we have not been careful about is the transformations of the other coordinates and . Since the motion is only in the direction, it is not surprising that these coordinates are not affected by Lorentz transformations. So the full equations for the Lorentz transformations are given by

These equations will be discussed at some length in the next lecture