Page:The principle of relativity (1920).djvu/202

 Rotation of the (covariant) four-vector.

The second member in (26) is symmetrical in the indices μ, and ν. Hence A_{μν} - A_{νμ} is an antisymmetrical tensor built up in a very simple manner. We obtain

(36) B_{μν} = [part]A_{μ}/[part]x_{ν} - [part]A_{ν}/[part]xμ  Antisymmetrical Extension of a Six-vector.

If we apply the operation (27) on an antisymmetrical tensor of the second rank A_{μ{ν^2}}ν}, and ] and form all the equations arising from the cyclic interchange of the indices μ, ν, σ, and add all them, we obtain a tensor of the third rank

(37) B_{μνσ} = A_{μνσ} + A_{νσμ} + A_{σμν} = [part]A_{μν}/[part]x_{σ}        + [part]A_{νσ}/[part]x_{μ} + [part]A_{σμ}/[part]x_{ν}  from which it is easy to see that the tensor is antisymmetrical. Divergence of the Six-vector.

If (27) is multiplied by g^{μα} g^{νβ} (mixed multiplication), then a tensor is obtained. The first member of the right hand side of (27) can be written in the form

[part]/[part]x_{σ} (g^{μα} g^{νβ} A_{μν}) - g^{μα} ([part]g^{νβ}/[part]x_{σ}) A_{μν} - g^{νβ} ([part]g^{μα}/[part]x_{σ}) A_{μν}.