1 7 6 D O C . 1 5 8 O N R I E M A N N C U R V A T U R E T E N S O R (2) . After eliminating Ȝ, these equations assume the form[6] . If one assumes that the gravitational field and the electromagnetic field are the only real objects in physics, and also assumes that the Maxwellian tensor (3) is put into equations (3),[7] then the scalar T vanishes identically, making the field equations take on the form (2a) . It is clear that in the case where just tensor (3) is on the right-hand side, equa- tions (2a) are preferable to equations (1). For the latter have the equation R = 0 as the consequence,[8] whose general validity is improbable. Furthermore, the equa- tions (2a) permit the existence of electrons with continuously distributed charge, in contrast to the equations (1).1) The fact that the equations (2a) have been given little attention is due to two cir- cumstances. First, all of our efforts were directed toward reaching a theory along the path taken by Weyl and Eddington, if not along a similar one, a theory that would merge the gravitational field and the electromagnetic field into a single for- mal whole. However, multiple failures have now persuaded me that one cannot come closer to the truth by following this path.[10] Second, the combination seems unnatural from a mathematical point of view I hope to dismiss this second objection with the following observations. As Mr Rainich2) has shown in an interesting note, in a continuum of four dimen- sions, the Riemann curvature tensor can be decomposed into two parts with different symmetry properties. For each surface element ( ) at a point P there is another ( ) perpendicular to it. We now decompose the curvature tensor into two terms (4) . [12] 1) Alas, later analyses have revealed to me that this way does not lead to a satisfactory theory of electrons.[9] 2) G. Y. Rainich, Nature No. 2892, 115 (1925), p. 498.[11] Rim 1 2 -- - gimR–2gimλ 1 -- - kTim –= Rim 1 4 --gimR - k Tim–4gimT¹ 1 -- - © § · –= Tim 1 4 --gimϕαβϕαβ - ϕiαϕmα = Rim 1 4 --gimR - kTim –= Rim 1 4 -- - gimR Rik lm , f ik f ik [p. 101] Rik lm , Rik lm , Sik lm , Aik, lm +=
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