Mechanics. — "Further remarh on the solutions of the fiehl- 

 equations of Einstrin's theory of gravitation ' . By Prof. W. 

 DE Sitter. 



(Communicated in the meeting of April 26, 1917). 



I. Einstein has recently^) enounced the postulate, that a solution 

 of the field-equations 



G/j,, — \l g,j,j = — X 7'^v + i H gij.v T . . . . . (1) 

 in order to be admissible for the actual physical world, must have 

 no discontinuities "at finite distances". In particular the determinant 

 g must for all points at finite distances be different from zero. 

 This postulate is not fulfilled by my solution B, as Einstein very 

 correctly points out, and as is also shown very cieuriy in my 

 communications. This postulate, however, in the form in which it 

 is enounced by Einstein, is a ^Vu/owp/^em/, or metaphysical, postulate. 

 To make it a physical one, the words: "all points at finite 

 distances' must be replaced by "all physically accessible points". 

 And if the postulate is thus formulated, my solution B does fulfil it. 

 For the discontinuity arises for 



7' = rj = I jr R. 

 This is at a finite distance in space, but it is physically in- 

 accessible, as I have already pointed out '). The time needed by a 

 ray of light, and a fortiori by a moving material point, to travel 

 from any point ?•, ip, i> to a point v^, tfj,, {^^ {\\,^ and .9^, being 

 arbitrary) is infinite. The singularity at r=zr^ can thus never affect 

 any physical experiment, or as I expressed it I.e., the paradoxical 

 phenomena, or rather al)sence of phenomena, resulting from this 

 singularity, can only happen before the beginning, or after the 

 end of eternity. 



2. A similar remark has been made by Prof. Felix Klein, in 

 a letter to the present writer dated 1918 April 19. He writes: 



^) Kriiisches zu einer vo?i Herrn dk Sitter gegebenen Lösung der Gravilalions- 

 gleichungen, Sitzungsber. Berlin, 7 March 1918, page 270. 



') On Einstein's theory or gravitation and its asii'onomical consequences, third 

 paper, Monthly Notices of the R. A. S. Vol. LXXVIII, page 17—18. 



On the curvature of space, these Pioceedings, Vol XX, p. 229. 



90 



Proceedings Royal Acad. Amsterdam. Vol. XX. 



