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By Andrzej Trautman (auth.), Peter G. Bergmann, Venzo De Sabbata (eds.)

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E: 1. 27) yield a 'a 'y + w <5. Y 1. WaS 1. 59) 0a OaS with the constants E: and W . 61) 30 F. W. 62) Thus, in an M4 , the local P-transformation degenerates into the global P-transformation, as it is supposed to do. We have found in this lecture that spacetime ought to be described by a Riemann-Cartan geometry, the geometrical gauge variables being the potentials (eia,r~aS). 6. 1 COUPLING OF MATTER TO SPACETIME AND THE TWO GENERAL GAUGE FIELD EQUATIONS Matter Lagrangian in a U4 In the last lecture we concentrated on working out the geometry of spacetime.

From a gauge-theoretical point of view this is a pure coincidence. Probably this fact is a distinguishing feature of the ECSKtheory as compared to other theories in the PG-framework. Consequently a second and perhaps more satisfactory procedure for suppressing rotons consists in picking a field lagrangian proportional to the U4-curvature scalar. 5 Propagating Gravitons and Rotons After so much suppression it is time to liberate the rotons. How could we achieve this goal? By just giving them enough kinetic energy -[(ae)2 + (ar)2] in order to enable them to get away.

X ) e. 49) the so-called metric postulate of spacetime physics. -. 49) with respect to r ij , we get -. ·k r .. 46) into account, the corresponding relation for the anholonomic connection: r aBy .. 51) We have introduced here the Christoffel symbol components of the torsion tensor, ··k F.. lJ k··a = e ·aF.. lJ - •• k = 2r [ lJ .. 52) and the object of anholonomity rl •• y i J' •• y e rl as'. = e ·a· B ij . 10), we get • ·a F .. lJ • -ex. • S • -cx = rl.. 55) a formula which will play a key role in discussing macroscopic gravity: the object of anholonomity mediates between torsion and the anholonomic connection.

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