Nonmetrical dynamics. I

Einstein's nonvacuum gravitational field equations are examined. The following nonmetrical existence theorem is proved. It is shown that given any analytic symmetric contravariant energy--momentum tensor density as a function of the space--time coordinates, a solution to the gravitational field equations always exists. Furthermore, this solution is such that the law of conservation of energy--momentum is satisfied. The new proof of the present paper makes no use of the coordinate transformation method used in a previous paper. The Cauchy--Kowalewsky existence theorem is used in the proof, and the literature on the Cauchy--Kowalewsky existence theorem is briefly reviewed. (Riquier's existence theorem is avoided except for a brief discussion in the Appendix.) The initial value problem of Einstein's equations is examined as part of the proof. The physical and mathematical meaning of this new proof is discussed. It is noted that the nonmetrical existence theorem disappears when Einstein's equations are linearized whereas the familiar Lichnerowicz-type existence theorems survive linearization. This suggests that the nonmetrical existence theorem may add a new dimension to our understanding of the physical meaning of nonlinearity in Einstein's equations