The Theory of General Relativity And Gravitation Based On a Course of Lectures
The Theory of General Relativity And Gravitation Based On a Course of Lectures
Ludwik Silberstein
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In fact, by (24) and (25), dx! d But the Xi, # 2, etc. , being mutually independent, the bracketed expression (to be summed over all i) vanishes for all K=F\ and equajs 1 for K = X. Whence, A' l B t ' = A K B K = A l B t, (26) which was to be proved. Any invariant, S= S f, is also called a scalar or a tensor of rank zero, since, in a manifold of n dimensions, it has n com- ponents, i. E. But one component. Similarly, a vector or tensor of rank one, has n l = n, in our case four, components. The... question whether a scalar is a contravariant or a covariant tensor is idle. For it transforms into itself. Vice versa, it can easily be proved that if B K be four (generally, n) magnitudes such that A* B K is invariant for any contravariant A", then B K is a covariant vector. And SECOND RAN^ TENSORS 43 the same thing is true if 'covariant' and ' contra variant ' be exchanged with one another. The product of a vector by a scalar is, obviously, again a vector of the same kind, and any number of vectors of the same kind multiplied by scalars and added together give again a vector of the same kind.
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