Relativity - The Special and General Theory Part 10

eq. 32: file eq32.gif

If we call v the velocity with which the origin of K1 is moving relative to K, we then have

eq. 33: file eq33.gif

The same value v can be obtained from equations (5), if we calculate the velocity of another point of K1 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 K1 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 K1 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 Dx' = I when measured in the K1 system are thus separated in our instantaneous photograph by the distance

eq. 34: file eq34.gif

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

eq. 35: file eq35.gif

From this we conclude that two points on the x-axis separated by the distance I (relative to K) will be represented on our snapshot by the distance

eq. 36: file eq36.gif

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

eq. 37: file eq37.gif

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 11.

eq. 38: file eq38.gif

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

x'2 - c^2t'2 = x2 - c^2t2 . . . (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

eq. 39: file eq39.gif

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

eq. 40: file eq40.gif

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

x2 + y2 + z2 = c^2t2 = 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 K1 -- in accordance with the corresponding formula

r' = ct'

or,

x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a).

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

x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11).

Since equation (8a) must hold for points on the x-axis, we thus have s = I. It is easily seen that the Lorentz transformation really satisfies equation (11) for s = 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 K1 be chosen so that they are spatially parallel to those of K. It is also not essential that the velocity of translation of K1 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 h.o.m.ogeneous functions of x, y, x, t, of such a kind that the relation

x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2 (11a).

is satisficd identically. That is to say: If we subst.i.tute 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 (11a) agrees with the right-hand side.

APPENDIX II

MINKOWSKI'S FOUR-DIMENSIONAL s.p.a.cE ("WORLD") (SUPPLEMENTARY TO SECTION 17)

We can characterise the Lorentz transformation still more simply if we introduce the imaginary eq. 25 in place of t, as time-variable. If, in accordance with this, we insert

x[1] = x x[2] = y x[3] = z x[4] = eq. 25

and similarly for the accented system K1, then the condition which is identically satisfied by the transformation can be expressed thus :

x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 (12).

That is, by the afore-mentioned choice of " coordinates," (11a) [see the end of Appendix II] is transformed into this equation.

We see from (12) that the imaginary time co-ordinate x[4], enters into the condition of transformation in exactly the same way as the s.p.a.ce co-ordinates x[1], x[2], x[3]. It is due to this fact that, according to the theory of relativity, the " time "x[4], enters into natural laws in the same form as the s.p.a.ce co ordinates x[1], x[2], x[3].

A four-dimensional continuum described by the "co-ordinates" x[1], x[2], x[3], x[4], was called "world" by Minkowski, who also termed a point-event a " world-point." From a "happening" in three-dimensional s.p.a.ce, physics becomes, as it were, an " existence " in the four-dimensional " world."

This four-dimensional " world " bears a close similarity to the three-dimensional " s.p.a.ce " of (Euclidean) a.n.a.lytical geometry. If we introduce into the latter a new Cartesian co-ordinate system (x'[1], x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are linear h.o.m.ogeneous functions of x[1], x[2], x[3] which identically satisfy the equation

x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2