This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ...without proof, that VT (VZrgA") = div(A"), (48) called the divergence of the contravariant vector A', is a scalar or invariant. 24. The Riemann-Chris toff el tensor is of such capital importance for Einstein's gravitation theory, and for the geometry of any riemannian manifold, as to deserve to be treated at some ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ...without proof, that VT (VZrgA") = div(A"), (48) called the divergence of the contravariant vector A', is a scalar or invariant. 24. The Riemann-Chris toff el tensor is of such capital importance for Einstein's gravitation theory, and for the geometry of any riemannian manifold, as to deserve to be treated at some length. It is a metrical tensor of rank four, built up of glK and their first and second derivatives, known to the general geometers since the time of Riemann. It expresses the so-called curvature properties of a manifold or re-space whose metrical relations are fixed by the tensor g-, and to Einstein it served as the material for building up his gravitational field-equations. In order to arrive at this all-important tensor let us start from an arbitrary covariant vector Al and let us write down its second covariant derivative, that is to say the covariant derivative of the tensor AlK which is the covariant derivative of Al, i.e., by (42), where A, K is as in (41). Similarly, transposing K and X, let us write the second covariant derivative Either being a third-rank tensor, so will be their difference?d--k-/ x )A + " AA. This is, by (41), Ei: ': -: +': " In the second term the indices a and /? over which the sum is to be taken can be interchanged. Thus A--A is the inner product of an arbitrary covariant vector Aa into the sum of the two bracketed expressions. This sum, therefore, (49) is a mixed tensor of rank four. This is the Riemann-Christoffel tensor which, for reasons to appear presently, may as well be called the curvature tensor. Strictly speaking, Riemann's own system of four-index symbols (i/x. Xk), discovered in 1861 in connection with a problem in heat conduction (Mathematische Werke, 2nd ed., p. 391), is the pu...
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