Page 181 of 212

[25] For the radius R of the universe we obtain the equation

image037

The use of the C.G.S. system in this equation gives 2/k = 1.08 x

1027; ρ is the average density of the matter and k is a constant

connected with the Newtonian constant of gravitation.

APPENDICES

APPENDIX I

SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION

(SUPPLEMENTARY TO SECTION XI)

For the relative orientation of the co-ordinate systems indicated

in Fig. 2, the x-axes of both systems permanently coincide. In

the present case we can divide the problem into parts by

considering first only events which are localised on the x-axis.

Any such event is represented with respect to the co-ordinate

system K by the abscissa x and the time t, and with respect to

the system K′ by the abscissa x′ and the time t′. We require to

find x′ and t′ when x and t are given.

A light-signal, which is proceeding along the positive axis of x,

is transmitted according to the equation

x = ct

or

x - ct = 0 . . . . . (1).