Chapter VIII: Appendix: I

SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION

(SUPPLEMENTARY TO SECTION XI)

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′_. We require to find _x′_ and _t′_ when _x_ and _t_ are given.

A light-signal, which is proceeding along the positive axis of _x_, is transmitted according to the equation

_x_ = _ct_

or

_x_ – _ct_ = 0 . . . . . (1).

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

_x′_ – _ct′_ = 0 . . . . . (2)

Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation

(_x′_ – _ct′_) = λ(_x_ – _ct_) . . . (3).

is fulfilled in general, where λ indicates a constant; for, according to (3), the disappearance of (_x_ – _ct_) involves the disappearance of (_x′_ – _ct′_).

If we apply quite similar considerations to light rays which are being transmitted along the negative _x_-axis, we obtain the condition

(_x′_ + _ct′_) = (_x + ct_) . . . (4).

By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants _a_ and _b_ in place of the constants λ and μ where

image038

and

image039

we obtain the equations

image040

We should thus have the solution of our problem, if the constants _a_ and _b_ were known. These result from the following discussion.

For the origin of _K′_ we have permanently _x′_ = 0, and hence according to the first of the equations (5)

image041

If we call _v_ the velocity with which the origin of _K′_ is moving relative to _K_, we then have

image042

The same value _v_ can be obtained from equations (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.

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_

Two points of the _x′_-axis which are separated by the distance Δ_x′_ = 1 when measured in the _K′_ system are thus separated in our instantaneous photograph by the distance

image043

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

image044

From this we conclude that two points on the _x_-axis separated by the distance 1 (relative to _K_) will be represented on our snapshot by the distance

image045

But from what has been said, the two snapshots must be identical; hence Δ_x_ in (7) must be equal to Δ_x′_ in (7_a_), so that we obtain

image046

The equations (6) and (7_b_) 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.

image047

Thus we have obtained the Lorentz transformation for events on the _x_-axis. It satisfies the condition

_x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (8a).

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

image048

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.

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

image049

or, if we square this equation, according to the equation

_x_2 + _y_2 + _z_2 – _c_2_t_2 = 0 . . . . . (10).

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,

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_).

In order that equation (10_a_) may be a consequence of equation (10), we must have

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2) (11).

Since equation (8_a_) must hold for points on the _x_-axis, we thus have σ = 1. It is easily seen that the Lorentz transformation really satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_) and (9), and hence also of (8) and (9). We have thus derived the Lorentz transformation.

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.

Mathematically, we can characterise the generalised Lorentz transformation thus:

It expresses _x′, y′, x′, t′_, in terms of linear homogeneous functions of _x, y, x, t_, of such a kind that the relation

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2 (11_a_).

is satisficd identically. That is to say: If we substitute their expressions in _x, y, x, t_, in place of _x′, y′, x′, t′_, on the left-hand side, then the left-hand side of (11_a_) agrees with the right-hand side.