D O C U M E N T 3 8 7 C O M M E N T O N T R E F F T Z 3 3 1 In (2), x signifies the radial naturally measured distance from one of the two mass points (up to an additive constant, ),[5] the naturally measured circum- ference divided by of a sphere having a constant value x that separates and cen- trally surrounds each of the two masses. The surfaces of both spherically shaped masses would be expressed by two equations and , between which there is empty space. Mr. Trefftz gives as a general solution to the problem: (3) where initially C can be set equal to 1 without limiting the generality. According to (2), one can hence set . For negative A and vanishing B this yields the well-known Schwarzschild solution for the field of a material point. Thus the constant A will have to be chosen as neg- ative here as well, in accordance with the fact that only positive gravitating masses exist. The constant B corresponds to the λ-term of equation (1a). A positive λ cor- responds to a negative B and vice versa. If equation system (3) really represents the field of two spherical masses, then this world obviously has to behave metrically as follows. From the first sphere , the circumference divided by of concentric spheres x = const, which are expressed by , must initially grow, then, upon approaching the second sphere, diminish, provided we are dealing with a closed world with the topology of a spherical world.[6] Therefore, somewhere in the empty space between the two material spheres[7] must be valid. But there, according to (3), would vanish. According to (1), is the running speed of a standard clock that is positioned at rest at that location. fx(x) x X1 = x X2 = X1 x X2) ( [p. 449] x = wd 1 A w --- - Bw2 + + ---------------------------------- f2 = w2 f4 = C 2 1 A w --- - Bw 2 + + , ds2 1 A w --- - Bw2 + + dt2 dw2 1 A w --- - Bw2 + + ----------------------------- - w2(dϑ2 sin2ϑdφ2) + = x X1 = w( = f2) dw dx ------ - 1 A w --- - Bw2 = 0 + + = f4 f4
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