290 DOC. 158 ON R I E M A N N C U R V A T U R E T E N S O R Riemannscher Tensor und Gravitationsgleichungen. 103 dessen Symmetrieeigenschaften dieselben sind wie die des Riemannschen Krümmungstensors. Man bilde ferner den „elektromagnetisch ergänzten Krümmungstensor“ (16 ) R* ik ik,lm = R i k,lm + k E i k ,lm und verlange, daß der im Sinne von Rainich gebildete antisymmetrische Be- standteil A i *k,lm von Ri*k,lm verschwinde: Genau wie oben beweist man, daß diese Bedingung mit dem Gleichungs- system (18) G*im = R*m - 1 \ 4 gimR* = 0 äquivalent ist, welches mit (2a), (3) übereinstimmt. Dadurch ist gezeigt, daß die durch das kosmologische Problem und durch den Bau des elektromagnetischen Energietensors nahegelegten Glei- chungen (2a ) des Gravitationsfeldes eine einfache mathematische Inter- pretation zulassen. (Eingegangen am 9. 1 1926.) P u blish ed in a special issue o f M a th e m a tis c h e A n n a le n , v o l. 97, d ed ica ted to the centenary o f B ern - hard R iem a n n ’ s birth o n 17 S eptem ber 1826. [1]E instein w as requested to con tribute to the special issue o f M a th e m a tis c h e A n n a le n b y O tto B lu - m enthal in his letter o f 15 D e ce m b e r 1924 (V ol. 14, D o c . 3 95 ). In 1926, E instein w a s a co e d ito r o f the jou rn a l, togeth er w ith D a v id H ilbert, O tto B lu m en th al, and C onstantin Carathéodory. [2]E instein here refers to the con tracted B ia n ch i identity. H e h ad b e e n unaw are o f these identities w h en h e first in trod u ced the final fie ld equation s o f gen eral relativity in 1915, but u sed them in d eriv- ing the con serv a tion o f e n e rg y -m o m e n tu m o f m atter in 1919. S ee note 8 o f E in s te in 1 9 1 9 a (V ol. 7, D o c . 17) fo r historical d iscu ssion s, see also P a is 1 9 8 2 , pp. 2 7 4 -2 7 6 and R o w e 2 0 0 2 . [3] H e r g lo tz 191 6 see also P a u li 1 9 2 1 , pp. 5 9 6 -5 9 7 and J a n s s e n 1992. In the sentence d irectly fo l- lo w in g the c o lo n it sh ou ld say that d im en sion al slice p erp en d icu lar to ξ i see H e rg lo tz 1 9 1 6 , p. 202. [4]E instein had first in trod u ced alternative fie ld equation s w h ere this tra ce-free gravitation tensor form s the left-h an d side o f the gravitational fie ld equation s in E in s te in 1 9 1 9 a (V ol. 7 , D o c . 1 7) and co n sid ered them again in E in s te in 1 9 2 2 r (V ol. 13, D o c . 387). [5]E instein refers to the co s m o lo g ica l m o d e l in trod u ced in E in s te in 1 9 1 7 b (V ol. 6 , D o c . 4 3). [6]T h is fo rm o f w riting the tra ce -fre e fie ld equation s is m ore gen eral than w h at E in stein h ad intro- d u ce d in E in s te in 1 9 1 9 a (V ol. 7 , D o c . 17). T here, to o , h e h ad substituted the E in stein ten sor o n the lefthand side o f the E in stein equation s b y the trace-free tensor R jk-t,.1/4gikR However h e m ade he is the m ed ia n curvature o f the three- (Rik -1/ 2 gikRξξ ) i k