D O C . 1 5 8 O N R I E M A N N C U RVAT U R E T E N S O R 1 7 5 158. “On the Formal Relation between the Riemann Curvature Tensor and the Field Equations of the Gravitational Field” [Einstein 1927a] Received 9 January 1926 Published 1927 In: Mathematische Annalen 97 (1927): 99–103.[1] The field equations of gravitation are usually written in the form (1) , where Tim is the energy tensor of matter and of the electromagnetic field. The phys- ical justification for the second term on the left-hand side lies in the fact that it makes the divergence of the left-hand side vanish identically.[2] This seems neces- sary for the interpretation of the conservation equation for matter. Furthermore, it is well known that the variation of makes the integral over the curvature scalar yield not the tensor , but the tensor . Finally, Herglotz showed that the tensor has a natural mathematical interpretation: If is an arbitrary direc- tion, then is the curvature scalar of the three-dimensional section of the four-dimensional continuum that is perpendicular to .[3] However, strong reasons can be presented that is in fact the tensor that has decisive significance for a deeper understanding of the law of gravity.[4] For if one wants to do full justice to the fundamental idea of rel- ativity, one is compelled to assume the spacelike sections of the world as finite. This is also necessary if one intends to ascribe to the matter in the world a finite mean density.[5] One can do justice to these conditions by introducing the so-called cosmological term, so that instead of equation (1) we obtain [p. 99] Rim 1 2 -- - gimR kTim –= gμν Rim Rim 1 2 -- - gimR ξi) ( Rikξiξk ξi) ( Rim 1 4 --gimR - [p. 100]
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