Page:The Meaning of Relativity - Albert Einstein (1922).djvu/55

A physical entity which is specified by four quantities, $A_{\nu }$, in an arbitrary inertial system of the $x_{1},x_{2},x_{3},x_{4}$, is called a 4-vector, with the components $A_{\nu }$, if the $A_{\nu }$ correspond in their relations of reality and the properties of transformation to the $\Delta x_{\nu }$; it may be of the nature of a space or of a time. The sixteen quantities, $A_{\mu \nu }$ then form the components of a tensor of the second rank, if they transform according to the scheme

$A_{\mu \nu }'=b_{\mu \alpha }b_{\nu \beta }A_{\alpha \beta }.$

It follows from this that the $A_{\mu \nu }$ behave, with respect to their properties of transformation and their properties of reality, as the products of components, $U_{\mu }V_{\nu }$, of two 4-vectors, $(U)$ and $(V)$. All the components are real except those which contain the index 4 once, those being purely imaginary. Tensors of the third and higher ranks may be defined in an analogous way. The operations of addition, subtraction, multiplication, contraction and differentiation for these tensors are wholly analogous to the corresponding operations for tensors in three-dimensional space.

Before we apply the tensor theory to the four-dimensional space-time continuum, we shall examine more particularly the skew-symmetrical tensors. The tensor of the second rank has, in general, $16=4\cdot 4$ components. In the case of skew-symmetry the components with two equal indices vanish, and the components with unequal indices are equal and opposite in pairs. There exist, therefore, only six independent components, as is the case in the electromagnetic field. In fact, it will be shown