Tensor Analysis

TENSOR ANALYSIS. The concept of tensors and the knowledge of some of their properties can be traced back to Gauss, Riemann and Christoffel, but their algebra and analysis have been shaped into a systematic method only recently, 1900, by Ricci and Levi-Civita, who coined for this Rowerful branch of mathematics the name of absolute differential calculus. Its chief aim is to construct and discuss relations or laws generally covariant; such, that is, as remain valid in passing from one to any other system of co-ordinates. It has especially become the object of a very widespread interest since the advent of generalized relativity (1916), whose principal requirement is precisely such unrestricted covariance of physical laws.

Definitions, Algebra of Tensors.

Consider a continuous n-dimensional manifold or "space" S, (see MANIFOLDS), whose ele ment or point P (x,) is determined by assigning the values of is real independent variables or co-ordinates x,. Let Q(x,±dx,) be another point of S „. Then the ordered point-pair P, Q or the set of differ entials dx, is called a vector. (To begin with, the idea of "size" or "length" is foreign to this concept, S„ being thus far a non-met rical manifold.) Let the x, be transformed into any other system of n co-ordinates the former being continuous functions of the latter with continuous derivatives ax,/av, and non-vanishing, to be summed over a =1 to n, and vice versa, dx, = (0x,/ Ox'a)dx'a. (The convention will be adopted that every term in which an index occurs twice, is to be summed over all its values.) Any set of n magnitudes A', functions of the x, which are transformed by this rule, i.e., into is called a contravariant tensor of rank one, the A' being its a' components. For contravariant tensors upper indices are used, an exception being made for dx,, the prototype of all such tensors. Next, any n magnitudes A, which are transformed as the dif ferentiators a/ax,, i.e., into form a covariant tensor of rank r, lower indices being used for such tensors. These two kinds of tensors, of rank 1, are also termed vectors; e.g., three-vectors, four-vectors (such as the rela tivistic four-velocity or four-potential, in space-time, S4), etc.,

according as n = 3, 4, etc. Similarly, any magnitudes ./1, transformable into form a covariant, and any A' transformable into form a contravariant tensor of rank two. Again, magnitudes A,' transformable into are said to form a mixed tensor of rank 2, covariant in t. and contravariant in K. The extension to any rank is obvious. Any magnitudes with lower and upper indices, which are transformed according to the rule form a mixed tensor of rank r = This is the most general concept of a tensor.

A tensor of rank zero, called also a scalar, is a single function of the x, invariant with respect to any transformations of co ordinates, f' = f. A,. is symmetrical if A.= A., and antisym metrical or a skew tensor if A,K= — A., implying A.= o. Simi larly for ASK. Analogous definitions hold for mixed tensors, and for higher ranks. Symmetry and antisymmetry are invariant properties.

The transformed tensor components being linear homogeneous functions of the original ones, the sums of corresponding com ponents of tensors of same rank and kind form again a tensor.

Thus, A,, B, being covariant vectors, so is C,. Simi larly, kid-Bt.= etc. The addends, functions of xt, must be taken at the same point of If a tensor vanishes in one, it will vanish also in any other co-ordinate system. Consequently, any tensor equation, if valid in one, holds also in any other co-ordi nate system. This is the chief reason of the importance of tensors in pure geometry and relativistic physics.

The outer product of two tensors of ranks r and s, i.e., the array of n'+' products of their components is again a tensor, of rank r+s, with rid- covariant and contravariant indices. Thus A,B.= C., A.Bx= C,Kx, A.Bx = C. The contraction, an opera tion of almost magical efficiency, applicable to any mixed tensor, consists in equating one of its lower to one of its upper in dices and summing over it from 1 to n. The result is again a tensor, with covariant and i contravariant indices; e.g., the contraction of A,* gives A",K= A,, a vector; con tracted once becomes = and this yields .8',= B, a scalar.