It is possible to map the x-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.
It does not lead
to a
single-valued function 
.
Nevertheless, we shall write (25) in analogy to
(22)
as
since this equation involves only the differential 
.
Our departure from 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 general 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 the presence of torsion, the shortest and straightest lines are no longer equal. Since the two types of lines play geometrically an equally favored role, the question arises as to which of them describes the correct classical particle orbits. The answer will be given at the end of this section.
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
This will be of use later.
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
the symmetric energy-momentum tensor of all matter fields.
This postulate was made only for spaces with no torsion, in which case
and 
are
both symmetric. As mentioned before, 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.
The generation of defects illustrated in Fig. 1 provides us with two simple examples for nonholonomic mappings which show us in which way these mappings are capable of generating a space with curvature and torsion from a euclidean space. The reader not familiar with this subject is advised to consult the standard literature on this subject quoted in Ref. [8, 9].
Consider first the upper example in which a dislocation is generated, characterized by a missing or additional layer of atoms (see Fig. 10.1). In two dimensions, it may be described differentially by the transformation
with the multi-valued function
The triads reduce to dyads, with the components
and the torsion tensor has the components
If we differentiate (44) formally,
we find
. This, however, is incorrect at the origin.
Using Stokes' theorem we see that
for any closed circuit around the origin,
implying that there is a 
-function singularity at the
origin with
By a linear superposition of such mappings we can generate an arbitrary torsion in the q-space. The mapping introduces no curvature. When encircling a dislocation along a closed path C, its counter image C' in the ideal crystal does not form a closed path. The closure failure is called the Burgers vector
It specifies the direction and thickness of the layer of additional atoms. With the help of Stokes' theorem, it is seen to measure the torsion contained in any surface S spanned by C:
where
is the projection of an oriented
infinitesimal area element onto the plane 
.
The above example has the Burgers vector
A corresponding closure failure appears when mapping a closed contour C in the ideal crystal into a crystal containing a dislocation. This defines a Burgers vector:
By Stokes' theorem, this becomes a surface integral
the last step following from (20).
The second example is the nonholonomic mapping
in the lower part of Fig. 1 generating
a disclination
which corresponds to
an entire section of angle 
missing in an ideal
atomic array.
For an infinitesimal angel , this
may be described, in two dimensions, by the differential mapping
with the multi-valued function (44).
The symbol 
denotes the
antisymmetric Levi-Cività
tensor.
The transformed metric
is single-valued and has commuting derivatives.
The torsion tensor vanishes since
is proportional to 
.
The local
rotation
field
,
on the other hand,
is equal to the multi-valued function
,
thus having the
noncommuting derivatives:
To lowest order in 
, this determines
the curvature tensor,
which in two dimensions posses only one independent component, for instance
.
Using the fact that 
has commuting derivatives,
can be written as
