Page:The Meaning of Relativity - Albert Einstein (1922).djvu/95

Rh is Euclidean (more generally, if by a suitable choice of co-ordinates the $$g_{\mu\nu}$$ are constants) then the vector obtained at $$G$$ as a result of this displacement does not depend upon the choice of the curve joining $$P$$ and $$G$$. But otherwise, the result depends upon the path of the displacement. In this case, therefore, a vector suffers a change, $$\Delta A^\mu$$ (in its direction, not its magnitude), when it is carried from a point $$P$$ of a closed curve, along the



curve, and back to $$P$$. We shall now calculate this vector change:

As in Stokes' theorem for the line integral of a vector around a closed curve, this problem may be reduced to the integration around a closed curve with infinitely small linear dimensions; we shall limit ourselves to this case.