Page:The Meaning of Relativity - Albert Einstein (1922).djvu/46

34 a form which is logically most satisfactory when expressed as laws in the four-dimensional space-time continuum. Upon this depends the great advance in method which the theory of relativity owes to Minkowski. Considered from this standpoint, we must regard $$x_1, x_2, x_3, t$$ as the four co-ordinates of an event in the four-dimensional continuum. We have far less success in picturing to ourselves relations in this four-dimensional continuum than in the three-dimensional Euclidean continuum; but it must be emphasized that even in the Euclidean three-dimensional geometry its concepts and relations are only of an abstract nature in our minds, and are not at all identical with the images we form visually and through our sense of touch. The non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space co-ordinates with the time co-ordinate. On the contrary, we must remember that the time co-ordinate is defined physically wholly differently from the space co-ordinates. The relations (22) and (22a) which when equated define the Lorentz transformation show, further, a difference in the rôle of the time co-ordinate from that of the space co-ordinates; for the term $$\Delta t^2$$ has the opposite sign to the space terms, $$\Delta x_1^2, \Delta x_2^2, \Delta x_3^2$$.

Before we analyse further the conditions which define the Lorentz transformation, we shall introduce the light-time, $$l = ct$$, in place of the time, $$t$$, in order that the constant $$c$$ shall not enter explicitly into the formulas to be developed later. Then the Lorentz transformation is defined in such a way that, first, it makes the equation