Appendix I. Simple Derivation of the Lorentz Transformation [Supplementary to Section XI]

Relativity: The Special and General TheoryAlbert Einstein

For the relative orientation of the co-ordinate systems indicated in Fig. 2, the x-axes of both systems pernumently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the xx-axis. Any such event is represented with respect to the co-ordinate system KK by the abscissa xx and the time tt, and with respect to the system K1K^1 by the abscissa xx' and the time tt'. We require to find xx' and tt' when xx and tt are given.

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

x=ctx = ct


xct=0(1)\tag{1} x - ct = 0

Since the same light-signal has to be transmitted relative to K1K^1 with the velocity cc, the propagation relative to the system K1K^1 will be represented by the analogous formula

xct=0(2)\tag{2} x' - ct' = 0

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

(xct)=λ(xct)(3)\tag{3} (x' - ct') = \lambda (x - ct)

is fulfilled in general, where λ\lambda indicates a constant; for, according to (3), the disappearance of (xct)(x - ct) involves the disappearance of (xct)(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)=μ(x+ct)(4)\tag{4} (x' + ct') = \mu (x + ct)

By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants aa and bb in place of the constants λ\lambda and μ\mu, where

a=λ+μ2a = \frac{\lambda+\mu}{2}


a=λμ2a = \frac{\lambda-\mu}{2}

we obtain the equations

x=axbctct=actbx}(5)\tag{5} \left. \begin{array}{rcl} x' &=& ax-bct \\ ct' &=& act-bx \end{array} \right\}

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

For the origin of K1K^1 we have permanently x=0x' = 0, and hence according to the first of the equations (5)

x=bcatx = \frac{bc}{a}t

If we call vv the velocity with which the origin of K1K^1 is moving relative to KK, we then have

v=bca(6)\tag{6} v=\frac{bc}{a}

The same value vv can be obtained from equations (a5), if we calculate the velocity of another point of K1K^1 relative to KK, or the velocity (directed towards the negative xx-axis) of a point of KK with respect to KK'. In short, we can designate vv as the relative velocity of the two systems.

Furthermore, the principle of relativity teaches us that, as judged from KK, the length of a unit measuring-rod which is at rest with reference to K1K^1 must be exactly the same as the length, as judged from KK', of a unit measuring-rod which is at rest relative to KK. In order to see how the points of the xx-axis appear as viewed from KK, we only require to take a “snapshot” of K1K^1 from KK; this means that we have to insert a particular value of tt (time of KK), e.g. t=0t = 0. For this value of tt we then obtain from the first of the equations (5)

x=axx' = ax

Two points of the xx'-axis which are separated by the distance Δx=I\Delta x' = I when measured in the K1K^1 system are thus separated in our instantaneous photograph by the distance

Δx=Ia(7)\tag{7} \Delta x = \frac{I}{a}

But if the snapshot be taken from K(t=0)K'(t' = 0), and if we eliminate tt from the equations (a5), taking into account the expression (6), we obtain

x=a(Iv2c2)xx' = a \left( I - \frac{v^2}{c^2} \right) x

From this we conclude that two points on the xx-axis separated by the distance II (relative to KK) will be represented on our snapshot by the distance

Δx=a(Iv2c2)(7a)\tag{7a} \Delta x' = a \left( I - \frac{v^2}{c^2} \right)

But from what has been said, the two snapshots must be identical; hence Δx\Delta x in (7) must be equal to Δx\Delta x' in (7a), so that we obtain

a=IIv2c2(7b)\tag{7b} a = \frac{I}{I-\frac{v^2}{c^2}}

The equations (6) and (7b) determine the constants aa and bb. By inserting the values of these constants in (5), we obtain the first and the fourth of the equations given in Section XI.

x=xvtIv2c2 t=tvc2xIv2c2}(8)\tag{8} \left. \begin{array}{rcl} x' &=& \frac{x-vt}{\sqrt{I-\frac{v^2}{c^2}}} \\ ~ \\ t' &=& \frac{t-\frac{v}{c^2}x}{\sqrt{I-\frac{v^2}{c^2}}} \end{array} \right\}

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

x2c2t2=x2c2t2(8a)\tag{8a} x'^2 - c^2t'^2 = x^2 - c^2t^2

The extension of this result, to include events which take place outside the xx-axis, is obtained by retaining equations (8) and supplementing them by the relations

y=yz=z}(9)\tag{9} \left. \begin{array}{rcl} y' &=& y \\ z' &=& z \end{array} \right\}

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 KK and for the system KK'. This may be shown in the following manner.

We suppose a light-signal sent out from the origin of KK at the time t=0t = 0. It will be propagated according to the equation

r=x2+y2+z2=ctr = \sqrt{x^2+y^2+z^2} = ct

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

x2+y2+z2=c2t2=0(10)\tag{10} x^2 + y^2 + z^2 = c^2t^2 = 0

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 K1K^1—in accordance with the corresponding formula

r=ctr' = ct'


x2+y2+z2c2t2=0(10a)\tag{10a} x'^2 + y'^2 + z'^2 - c^2t'^2 = 0

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

x2+y2+z2c2t2=σ(x2+y2+z2c2t2)(11)\tag{11} x'^2 + y'^2 + z'^2 - c^2t'^2 = \sigma (x^2 + y^2 + z^2 - c^2t^2)

Since equation (8a) must hold for points on the xx-axis, we thus have σ=I\sigma = I. It is easily seen that the Lorentz transformation really satisfies equation (11) for σ=I\sigma = I; for (11) is a consequence of (8a) 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 K1K^1 be chosen so that they are spatially parallel to those of KK. It is also not essential that the velocity of translation of K1K^1 with respect to KK should be in the direction of the xx-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,tx', y', x', t', in terms of linear homogeneous functions of x,y,x,tx, y, x, t, of such a kind that the relation

x2+y2+z2c2t2=x2+y2+z2c2t2(11a)\tag{11a} x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2

is satisficd identically. That is to say: If we substitute their expressions in x,y,x,tx, y, x, t, in place of x,y,x,tx', y', x', t', on the left-hand side, then the left-hand side of (11a) agrees with the right-hand side.

Appendix II. Minkowski’s Four-Dimensional Space (“World”) [Supplementary to Section XVII]

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