It is possible to map the -space locally into a q-space via an infinitesimal transformation
with coefficient functions which are not integrable in the sense of Eq. (22), i.e.,
Such a mapping will be called nonholonomic. There exists no single-valued function for which . Nevertheless, we shall write (25) in analogy to (22) as
since this equation involves only the differential . This violation of mathematical conventions will not cause any problems.
From Eq. (25) we see that the image space of a nonholonomic mapping carries torsion. The connection has a nonzero antisymmetric part, called the torsion tensor :
In contrast to , the antisymmetric part is a proper tensor under holonomic coordinate transformations. The contracted tensor
transforms like a vector, whereas the contracted connection does not. Even though is not a tensor, we shall freely lower and raise its indices using contractions with the metric or the inverse metric, respectively: , , . The same thing will be done with .
In the presence of torsion, the connection is no longer equal to the Christoffel symbol. In fact, by rewriting trivially as
and using , we find the decomposition
where the combination of torsion tensors
is called the contortion tensor. It is antisymmetric in the last two indices so that
In Einstein's theory of gravitation, torsion is assumed to be absent, i.e., the integrability condition for is not violated as in (26). The main effect of matter in Einstein's theory of gravitation manifests itself in the violation of the integrability condition for the derivative of the coordinate transformation , namely,
A transformation for which itself is integrable, while the first derivatives are not, carries a flat-space region into a purely curved one. The quantity which records the nonintegrability is the Cartan curvature tensor
Working out the derivatives using (19) we see that can be written as a covariant curl of the connection,
In the last term we have used a matrix notation for the connection. The tensor components are viewed as matrix elements , so that we can use the matrix commutator
Einstein's original theory of gravity assumes the absence of torsion. The space properties are completely specified by the Riemann curvature tensor formed from the Riemann connection (the Christoffel symbol)
The relation between the two curvature tensors is
In the last term, the 's are viewed as matrices . The symbols denote the covariant derivatives formed with the Christoffel symbol. Covariant derivatives act like ordinary derivatives if they are applied to a scalar field. When applied to a vector field, they act as follows:
The effect upon a tensor field is the generalization of this; every index receives a corresponding additive contribution.
In the presence of torsion, there exists another covariant derivative formed with the affine connection rather than the Christoffel symbol which acts upon a vector field as
The two derivatives (38) and (39) are equally covariant under holonomic coordinate transformations. Thus, in conventional differential geometry it is not clear which of them should play a more fundamental role in physics. They do differ, however, in their transformation behavior under nonholonomic transformations, and there (39) is definitely the preferred object for reasons of simplicity.
From either of the two curvature tensors, and , one can form the once-contracted tensors of rank 2, the Ricci tensor
and the curvature scalar
The celebrated Einstein equation for the gravitational field postulates that the tensor
the so-called Einstein tensor, is proportional to symmetric energy-mo-mentum tensor of all matter fields. This postulate was made only for spaces with no torsion, in which case and are both symmetric. As mentioned in the Introduction, it is not yet clear how Einstein's field equations should be generalized in the presence of torsion since the experimental consequences are as yet too small to be observed. In this paper, we are not concerned with the generation of curvature and torsion but only with their consequences upon the motion of point particles.