Page:The Meaning of Relativity - Albert Einstein (1922).djvu/89

That there are, nevertheless, in the general case, invariant differential operations for tensors, is recognized most satisfactorily in the following way, introduced by Levi-Civita and Weyl. Let $A^{\mu }$ be a contra-variant vector whose components are given with respect to the co-ordinate system of the $x_{\nu }$. Let $P_{1}$ and $P_{2}$ be two infinitesimally near points of the continuum. For the infinitesimal region surrounding the point $P_{1}$, there is, according to our way of considering the matter, a co-ordinate system of the $X_{\nu }$ (with imaginary $X_{\nu }$-co-ordinates) for which the continuum is Euclidean. Let $A_{(1)}^{\mu }$ be the co-ordinates of the vector at the point $P_{1}$. Imagine a vector drawn at the point $P_{2}$, using the local system of the $X_{\nu }$, with the same co-ordinates (parallel vector through $P_{2}$, then this parallel vector is uniquely determined by the vector at $P_{1}$ and the displacement. We designate this operation, whose uniqueness will appear in the sequel, the parallel displacement of the vector $A^{\mu }$ from $P_{1}$ to the infinitesimally near point $P_{2}$ If we form the vector difference of the vector $(A^{\mu })$ at the point $P_{2}$ and the vector obtained by parallel displacement from $P_{1}$ to $P_{2}$, we get a vector which may be regarded as the differential of the vector $(A^{\mu })$ for the given displacement $(dx_{\nu })$.

This vector displacement can naturally also be considered with respect to the co-ordinate system of the $x_{\nu }$. If $A^{\nu }$ are the co-ordinates of the vector at $P_{1}$, $A^{\nu }+\delta A^{\nu }$ the co-ordinates of the vector displaced to $P_{2}$ along the interval $(dx_{\nu })$, then the $\delta A^{\nu }$ do not vanish in this case. We know of these quantities, which do not have a vector