The algebraical formulæ embodied in the above construction are as follows: From the point of view of =O=, let an event occur at a distance _x_ along the railway, and at a time _t_ after the beginning of the journey (when =O′= was at =O=). From the point of view of =O′=, let the same event occur at a distance _x′_ along the railway, and at a time _t′_ after the beginning of the journey. Let _c_ be the velocity of light, and _v_ the velocity of =O′= relative to =O=. Put _c_ β = ———————————— √(_c_² - _v_²) Then _x′_ = β(_x_ - _vt_) ( _vx_ ) _t′_ = β(_t_ - —————) ( _c²_ ) This is the Lorentz transformation, from which everything in this chapter can be deduced. CHAPTER VII: INTERVALS IN SPACE-TIME The special theory of relativity, which we have been considering hitherto, solved completely a certain definite problem: to account for the experimental fact that, when two bodies are in uniform relative motion, all the laws of physics, both those of ordinary dynamics and those connected with electricity and magnetism, are exactly the same for the two bodies. “Uniform” motion, here, means motion in a straight line with constant velocity. But although one problem was solved by the special theory, another was immediately suggested: what if the motion of the two bodies is not uniform? Suppose, for instance, that one is the earth while the other is a falling stone. The stone has an accelerated motion: it is continually falling faster and faster. Nothing in the special theory enables us to say that the laws of physical phenomena will be the same for an observer on the stone as for one on the earth. This is particularly awkward, as the earth itself is, in an extended sense, a falling body: It has at every moment an acceleration[4] towards the sun, which makes it go round the sun instead of moving in a straight line. As our knowledge of physics is derived from experiments on the earth, we cannot rest satisfied with a theory in which the observer is supposed to have no acceleration. The general theory of relativity removes this restriction, and allows the observer to be moving in any way, straight or crooked, uniformly or with an acceleration. In the course of removing the restriction, Einstein was led to his new law of gravitation, which we shall consider presently. The work was extraordinarily difficult, and occupied him for ten years. The special theory dates from 1905, the general theory from 1915. [4] This does not mean that its velocity is increasing, but that it is changing its direction. The only sort of motion which is called “unaccelerated” is motion with uniform velocity _in a straight line_.