February 17, 192 1] 



NATURE 



801 



itself (i.e. independently of the material con- 

 tent of the universe), a vector "equal" to itself. 

 According to Einstein, however, the guiding field 

 (Fuhrungsfeld) is a physical reality which is de- 

 pendent on the state of matter, and mani- 

 fests itself only infinitesimally (as a tendency of 

 persistence which carries over the vectors from 

 one point to " indefinitely neighbouring " ones). 

 The immense success of Einstein's theory is based 

 on the fact that the effects of gravitation also 

 belong to the gTiiding field, as we should expect 

 a priori from our experience of the equality of 

 gravitational and inertial mass. The planets 

 follow exactly the orbit destined to them by the 

 guiding field ; there is no special " gravitational 

 force " necessary, as in Newton's theory, to cause 

 them to deviate from their Galilean orbit. In 

 general, the parallel displacement is "non-inte- 

 grable " ; i.e. if we transfer a vector at P along 

 two different paths to a point P' at a finite distance 

 from P, then the vectors, which were coincident 

 at P, arrive at P' in two different end-positions 

 after travelling these two paths. 



The " afTine connection " is not an original 

 characteristic of the universe, but arises from a 

 more deeply lying condition of things — the 

 "metrical field." There exists an infinitesimal 

 "light-cone" [Lichtkegel) at every position P in 

 the world, which separates past and future in the 

 immediate vicinity of the point P. In other words, 

 this light-cone separates those world-points which 

 can receive action from P from those from which 

 an "action" can arrive at P. This "cone of 

 light " renders it possible to compare two line- 

 elements at P with each other by measurement ; 

 all vectors of equal measure represent one and 

 the same distance at P. In addition to the deter- 

 mination of measure at a point P (the "relation 

 of action " of P with its surroundings), we have 

 now the "metrical relation," which determines the 

 congruent transference of an arbitrary distance at 

 P to all points indefinitely near to P. 



Just as the point of view of Einstein leads back 

 to that of Galileo and Newton when we assume 

 the transference of vectors by parallel displace- 

 ment to be integrable, so we fall back on Einstein 

 when the transference of distances by congruent 

 transference is integrable. But this particular 

 assumption does not appear to me to be in the 

 least justified (apart from the progress of the 

 historical development). It appears to me rather 

 as a gross inconsistency. For the "distances " the 

 old point of view of a determination of magni- 

 tudes in terms of each other is maintained, this 

 being independent of matter and taking place 

 directly at a distance. This is just as much in 

 conflict with the principle of the relativity of 

 magnitude as the point of view of Newton 

 and f'lalileo is with the principle of the relativity 

 of motion. If, in the case in point, we pro- 

 ceed in earnest with the idea of the continuity 

 f)f action, then "magnitudes of condition" occur 

 in the mathematical description of the world- 

 metrics in just sufficient number and in such a 

 romhination a.s is necessary for the description of 

 NO. 2677, VOL. 106] 



the electromagnetic and of the gravitational field. 

 We saw above that, besides inertia (the reten- 

 tion of the vector-direction), gravitation was also 

 included in the guiding field, as a slight variation 

 of this, as a whole, constant inertia. So in the 

 present case, in addition to the force which 

 conserves space- and time-lengths, electromag- 

 netism is also included in the metrical relation. 

 Unfortunately, this cannot be made clear so 

 readily as in the case of gravitation. Eor the 

 phenomena of gravitation are easily obtained from 

 the Galilean principle, according to which the 

 world-direction of a mass-point in motion follows 

 at every instant the parallel displacement. Now 

 it is by no means the case that the ponderomotive 

 force of the electromagnetic field should be in- 

 cluded in our Galilean law of motion, as well as 

 gravitation, for a charged mass-point does not 

 follow the guiding field. On the contrary, the 

 correct equations of motion are obtained only by 

 the establishment of a definite and concrete law 

 of Nature, which is possible within the frame- 

 work of the theory, and not from the general 

 principles of the theory. 



The form of the law of Nature on which the 

 condition of the metrical field is dependent is 

 limited by our conception of the nature of gravita- 

 tion and electricity in still greater measure than 

 it is by Einstein's general principle of relativity. 

 When the metrical connection alone is virtually 

 varied, the most simple of the assumptions pos- 

 sible leads exactly to the theory of Maxwell. 

 Thus, whereas Einstein's theory of gravitation 

 gave certain inappreciable deviations from the 

 Newtonian theory, such as coukl be tested by 

 experiment, our interpretation of electricity — one 

 is almost tempted to say unfortunately — results 

 in the complete confirmation of Maxwell's laws. 

 If we supplement Maxwell's "magnitude of 

 action " (li'irkung.'sgrosse) by the simplest addi- 

 tional term which also allows of the virtual varia- 

 tion of the "relation of action," we then arrive 

 at Einstein's laws of the gravitational field, 

 from which, however, there are two small 

 deviations : 



(i) That cosmological term appears which 

 Einstein appended later to his equations, 

 and which results in the spatial closure 

 (Gcschlosscnheit) of the universe. .\ hypothesis 

 conceived ad hoc by Einstein to explain the gene- 

 rally prevailing equilibrium of masses results here 

 of necessity. Whereas Einstein has to assume 

 a pre-stabilised harmony between the "cosmo 

 logical constant " which is characteristic for his 

 modified law of gravitation, and the total mass 

 fortuitously present in the universe, in our case, 

 where no such constant occurs, the world-mass 

 determines the curvature of the universe in 

 virtue of the laws of equilibrium. Only in 

 this way, it appears to me, is Einstein's 

 cosmology at all possible from a physical point 

 of view. 



(2) In the case where an electromagnetic field 

 is present, Einstein's cosmological term must be 

 supplemented bv an additional term of similar 



