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where 1"^ is defined by the rhs ot Eq. 17.21 (it follows Irom Eq. 17.25 that A^
is a true tensor). He hoped to be able to identify A\ with the electromagnetic
potential, but even for weak fields he was unable to find equations in which grav-

*See a letter from Cartan to Einstein [C6] (in which Cartan also notes that he had alluded to this
geometry in a discussion with Einstein in 1922) reproduced in the published Cartan-Einstein cor
respondence [Dl]. In 1929, Einstein wrote a review of this theory [E51] to which, at his suggestion,
Cartan added a historical note [C7].

UNIFIED FIELD THEORY 345

parallelism). Transcribed in the formalism of the previous section, this geometry

looks as follows. Consider a contravariant Vierbein field, a set of four orthonormal

vectors h"a, a = 1,2, 3, 4; a numbers the vectors, v their components. Imagine that

it is possible for this Vierbein as a whole to stay parallel to itself upon arbitrary

displacement, that is, h'av = 0 for each a, or, in longhand,

for each a. If this is possible, then one can evidently define the notion of a straight

line (not to be confused with a geodesic) and of parallel lines. Let hm be the nor-

malized minor of the determinant of the h"a. Then (summation over a is under-

stood)

(17.52)

(17.53)

The notation is proper since h,a is a covariant vector field. From Eqs. 17.52 and

17.53 we can solve for the connection:

(17.54)

H7.57^1 )

(17.56)

(17.55)

from which one easily deduces that

Thus distant parallelism is possible only for a special kind of nonsymmetric con-

nection in which the sixty-four FJ,, are expressible in terms of sixteen fields and

in which the curvature tensor vanishes. When Einstein discovered this, he did not

know that Cartan was already aware of this geometry.*

All these properties are independent of any metric. However, one can define an

invariant line element ds^2 = g^dx"dx" with

The resulting geometry, a Riemannian geometry with torsion, was the one Ein-
stein independently invented. A week later he proposed to use this formalism for
unification [E51a]. Of course, he had to do something out of the ordinary since he
had no Ricci tensor. However, he had found a new tensor A^, to play with, defined
by