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UNIFIED FIELD THEORY 337

Ghern made two statements which apply equally well to the present section: 'It

is a strange feeling to speak on a topic of which I do not know half the title', and

'I soon saw the extreme difficulty of his [Einstein's] problem and the difference

between mathematics and physics.' Otherwise the overlap between this section and

Chern's paper is minor. Chern deals mainly with modern global problems of dif-

ferential geometry, such as the theory of fiber bundles, subjects which Einstein

himself never wrote about or mentioned to me. My own aim is to give an account

of unified field theory in Einstein's day, when the concerns were uniquely with

local differential geometry and when the now somewhat old-fashioned (and glob-

ally inadequate) general Ricci calculus was the main tool. Hence the main pur-

pose of this section: to give the main ideas of this calculus in one easy lesson.* A

simple way of doing this is first to consider a number of standard equations and

results of Riemannian geometry, found in any good textbook on general relativity,

and then to generalize from there.

In Riemannian geometry, we have a line element

is a tensor of the second rank. Covariant derivatives of higher covariant tensors

are deduced in the standard way. In particular, Q^, defined by

"The interested reader is urged to read Schroedinger's wonderful little book on this subject [S3].

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invariant under all continuous point transformations x' —*• x^1 ' = x'\x}) and a

connection Fj, related to the g^ by

ror later purposes i aistmguisn two groups 01 properties.

The First Group

\. A covariant vector field Af and a contravariant vector field B^1 ' transform as

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from which one deduces the transformation of higher-rank tensors by the standard

rules.

2. Contraction of a tensor of rank n (> 2) yields a tensor of rank n — 2.


3. The covariant derivative of A,., defined bv

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