Page:The Meaning of Relativity - Albert Einstein (1922).djvu/92

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THE MEANING OF RELATIVITY

is also a vector. Since this is the case for an arbitrary choice of the , it follows that

(71)

is a tensor, which we designate as the co-variant derivative of the tensor of the first rank (vector). Contracting this tensor, we obtain the divergence of the contra-variant tensor . In this we must observe that according to (70),

(72)

If we put, further,

(73)

a quantity designated by Weyl as the contra-variant tensor density [1] of the first rank, it follows that,

(74)

is a scalar density.

We get the law of parallel displacement for the co-variant vector by stipulating that the parallel displacement shall be effected in such a way that the scalar

remains unchanged, and that therefore

  1. This expression is justified, in that has a tensor character. Every tensor, when multiplied by , changes into a tensor density. We employ capital Gothic letters for tensor densities.