Relativity - The Special and General Theory by Albert Einstein (early reader books txt) 📕

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 coordinate 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^2t‘2 = x2 + y2 + z2 - c^2t2 (11a).

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 (11a) agrees with the right-hand side.

APPENDIX II

MINKOWSKI’S FOURDIMENSIONAL SPACE (“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 coordinate x[4], enters into the condition of transformation in exactly the same way as the space coordinates 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 space co ordinates x[1], x[2], x[3].

A fourdimensional continuum described by the “coordinates” 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 space, physics becomes, as it were, an ” existence ” in the fourdimensional ” world.”

This fourdimensional ” world ” bears a close similarity to the three-dimensional ” space ” of (Euclidean) analytical geometry. If we introduce into the latter a new Cartesian coordinate system (x’[1], x’[2], x’[3]) with the same origin, then x’[1], x’[2], x’[3], are linear homogeneous 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

The analogy with (12) is a complete one. We can regard Minkowski’s ” world ” in a formal manner as a fourdimensional Euclidean space (with an imaginary time coordinate) ; the Lorentz transformation corresponds to a ” rotation ” of the coordinate system in the fourdimensional ” world.”

APPENDIX III

THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY

From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction. Theories are evolved and are expressed in short compass as statements of a large number of individual observations in the form of empirical laws, from which the general laws can be ascertained by comparison. Regarded in this way, the development of a science bears some resemblance to the compilation of a classified catalogue. It is, as it were, a purely empirical enterprise.

But this point of view by no means embraces the whole of the actual process ; for it slurs over the important part played by intuition and deductive thought in the development of an exact science. As soon as a science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the ” truth ” of the theory lies.

Corresponding to the same complex of empirical data, there may be several theories, which differ from one another to a considerable extent. But as regards the deductions from the theories which are capable of being tested, the agreement between the theories may be so complete that it becomes difficult to find any deductions in which the two theories differ from each other. As an example, a case of general interest is available in the province of biology, in the Darwinian theory of the development of species by selection in the struggle for existence, and in the theory of development which is based on the hypothesis of the hereditary transmission of acquired characters.

We have another instance of far-reaching agreement between the deductions from two theories in Newtonian mechanics on the one hand, and the general theory of relativity on the other. This agreement goes so far, that up to the preseat we have been able to find only a few deductions from the general theory of relativity which are capable of investigation, and to which the physics of pre-relativity days does not also lead, and this despite the profound difference in the fundamental assumptions of the two theories. In what follows, we shall again consider these important deductions, and we shall also discuss the empirical evidence appertaining to them which has hitherto been obtained.

(a) Motion of the Perihelion of Mercury

According to Newtonian mechanics and Newton’s law of gravitation, a planet which is revolving round the sun would describe an ellipse round the latter, or, more correctly, round the common centre of gravity of the sun and the planet. In such a system, the sun, or the common centre of gravity, lies in one of the foci of the orbital ellipse in such a manner that, in the course of a planet-year, the distance sun-planet grows from a minimum to a maximum, and then decreases again to a minimum. If instead of Newton’s law we insert a somewhat different law of attraction into the calculation, we find that, according to this new law, the motion would still take place in such a manner that the distance sun-planet exhibits periodic variations; but in this case the angle described by the line joining sun and planet during such a period (from perihelion—closest proximity to the sun—to perihelion) would differ from 360^0. The line of the orbit would not then be a closed one but in the course of time it would fill up an annular part of the orbital plane, viz. between the circle of least and the circle of greatest distance of the planet from the sun.

According also to the general theory of relativity, which differs of course from the theory of Newton, a small variation from the Newton-Kepler motion of a planet in its orbit should take place, and in such away, that the angle described by the radius sun-planet between one perhelion and the next should exceed that corresponding to one complete revolution by an amount given by

eq. 41: file eq41.gif

(N.B. — One complete revolution corresponds to the angle 2p in the absolute angular measure customary in physics, and the above expression giver the amount by which the radius sun-planet exceeds this angle during the interval between one perihelion and the next.) In this expression a represents the major semi-axis of the ellipse, e its eccentricity, c the velocity of light, and T the period of revolution of the planet. Our result may also be stated as follows : According to the general theory of relativity, the major axis of the ellipse rotates round the sun in the same sense as the orbital motion of the planet. Theory requires that this rotation should amount to 43 seconds of arc per century for the planet Mercury, but for the other Planets of our solar system its magnitude should be so small that it would necessarily escape detection. *

In point of fact, astronomers have found that the theory of Newton does not suffice to calculate the observed motion of Mercury with an exactness corresponding to that of the delicacy of observation attainable at the present time. After taking account of all the disturbing influences exerted on Mercury by the remaining planets, it was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained perihelial movement of the orbit of Mercury remained over, the amount of which does not differ sensibly from the above mentioned +43 seconds of arc per century. The uncertainty of the empirical result amounts to a few seconds only.

(b) Deflection