Page:The principle of relativity (1920).djvu/88



Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

(x_{1}) = (a_{11} a_{12} a_{13} a_{14})(x_{1}´) (x_{2})  (a_{21} a_{22} a_{23} a_{24})(x_{2}´) (x_{3})  (a_{31} a_{32} a_{33} a_{34})(x_{3}´) (x_{4})  (a_{41} a_{42} a_{43} a_{44})(x_{4}´)

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while a_{41}, a_{42}, a_{43}, are purely imaginary, but a_{44} is real and > 0, and x_{1}^2 + x_{2}^2 + x_{3}^2 + x_{4}^2 transforms into x_{1}´^2 + x_{2}´^2 + x_{3}´^2 + x_{4}´^2. The operation shall be called a general Lorentz transformation.

If we put x_{1}´ = x´, x_{2}´ = y´, x_{3}´ = z´, x_{4}´ = it´, then immediately there occurs a homogeneous linear transformation of (x, y, z, t) to (x´, y´, z´, t´) with essentially real co-efficients, whereby the aggregrate -x^2 - y^2 - z^2 + t^2 transforms into -x´^2 - y´^2 - z´^2 + t´^2, and to every such system of values x, y, z, t with a positive t, for which this aggregate > 0, there always corresponds a positive t';

[Footnote: This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski's notation, x_{1} = a_{11}x_{1}´ + a_{12}x_{2}´+ a_{13}x_{3}´+ a_{14}x_{4}´.]