On the existence of integrals of Einsteins gravitational by Thomas T.Y.

By Thomas T.Y.

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Let the new coordinates u1 ; u2 ; u3 be de®ned by specifying the Cartesian coordinates (x1 ; x2 ; x3 ) as functions of (u1 ; u2 ; u3 †: x1 ˆ f …u1 ; u2 ; u3 †; x2 ˆ g…u1 ; u2 ; u3 †; x3 ˆ h…u1 ; u2 ; u3 †; …1:54† where f, g, h are assumed to be continuous, di€erentiable. A point P (Fig. 16) in space can then be de®ned not only by the rectangular coordinates (x1 ; x2 ; x3 ) but also by curvilinear coordinates (u1 ; u2 ; u3 ). If u2 and u3 are constant as u1 varies, P (or its position vector r) describes a curve which we call the u1 coordinate curve.

Alternatively dV can be written as     @…x1 ; x2 ; x3 †  @r @r @r   du du du ;  Á  du du du ˆ dV ˆ  @u1 @u2 @u3  1 2 3  @…u1 ; u2 ; u3 †  1 2 3 where   @x1   @u1   @…x1 ; x2 ; x3 †  @x2 ˆ Jˆ @…u1 ; u2 ; u3 †  @u1   @x3   @u1 @x1 @u2 @x2 @u2 @x3 @u2 …1:57†  @x1   @u3   @x2   @u3   @x3   @u3  is called the Jacobian of the transformation. 54) is one to one in the neighborhood of a point. We are now ready to express the gradient, divergence, and curl in terms of u1 ; u2 , and u3 .

Tensors of second rank From two contravariant vectors A and B we may form the N 2 quantities A B . This is known as the outer product of tensors. These N 2 quantities form the components of a contravariant tensor of the second rank: any aggregate of N 2 quantities T  which, under a coordinate change, transform like the product of 48 BASIC OPERATIONS WITH TE NS ORS two contravariant vectors T  ˆ @x @x H T ; @x H @x H …1:97† is a contravariant tensor of rank two. We may also form a covariant tensor of rank two from two covariant vectors, which transforms according to the formula T ˆ @x H @x H H T : @x @x …1:98† Similarly, we can form a mixed tensor T   of order two that transforms as follows: T  ˆ @x @x H H T : @x H @x …1:99† We may continue this process and multiply more than two vectors together, taking care that their indexes are all di€erent.

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