Motion and Relativity by Leopold Infeld and Jerzy Plebanski (Auth.)

By Leopold Infeld and Jerzy Plebanski (Auth.)

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Because of the Bianchi identities © ^ = 0 we have, as was shown before: N \4-l Λ Λ oo -oo Β -d N -d/,p co A A s s ^ m ( 0 ) f α8Αό{Α)(χ-ξ)Ω°ί8Α). 2) are a consequence of the field equations forming the condition for their integrability. Thus, we cannot solve the field equations with arbitrary motion. 1). 2). Therefore, the usual procedure of solving the field equations by assuming arbitrary motion, a useful procedure in all linear field theories, is of no avail here. How can we avoid this difficulty?

B. T. The field and motion are intrinsically connected with each other. The I. β THE EQUATIONS OF MOTION AND THE FIELD EQUATIONS 46 equations of motion are a condition of integrability for the field equations. We shall now give the mathematical details justifying these remarks and those of Section 1. Once more we write down Einstein's field equations for a system of point particles: N oo A Aa * A -OO ,4-1 We form the covariant derivative # of the left-hand side. 2) where 2°* is the energy-momentum tensor forming the right-hand side of Einstein's field equations.

3) I. 6) «1 where To find OS, let us remark that although \ a \ is not a tensor, its variation dla\ is. Indeed | a y >JL e io^ is the change of the vector Ap in a parallel displacement from of to of+dx?. Therefore [<3J " l l ^ d a ^ is the difference between two vectors at the same point x? + daf. Both of them are obtained by a parallel displacement; one of them in the old field g^, the other in the varied field gafl+ dg^. The difference between two vectors at the same point is a vector. Since Αψ and

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