Additionally, we include a lorentzian metric tensor, g μν , which allows to measure
lengths, volumes and so on. Hence it is possible to talk about the module of a vector
that is not necessarily non-negative, due to the lorentzian signature. Those vectors
that are not trivial but have zero norm determine the lightlike paths and, then, light
cones that define the casual structure of the spacetime.
Another fundamental notion that can be defined, even in the absence of metric, is
parallelism. The motivation for this additional concept is the following. Consider the
Euclidean space ℝ
D and a couple of vectors in different points, p and q (Fig. 5.1). If
we want to compare them, we simply take, for example, the one in p and move it to
q keeping the vector parallel to itself and without changing the module. And, finally,
we subtract both vectors to see the difference.
However, if the manifold is general, the initial vectors live in different spaces (the
tangent spaces at p and q, respectively, T p M and T q M) and there is no natural way
to relate them (Fig. 5.2). In the Euclidean space, both tangent spaces can be identified
making the comparison trivial. In the general case, we need to introduce an additional structure that carries the information about parallelism, the affine structure,
whose fundamental object is the (affine) connection Γ
σ
μν . Once we have a connection,
given a curve between two points, we have a rule to relate vectors in them: the
parallel transport associated to the connection.
The connection permits the definition of a covariant derivative (“covariant”
means that once applied to a tensor, the result is also a tensor), and other intrinsic
geometrical properties of the spacetime, such as the curvature and the torsion,
respectively:
Fig. 5.1 Comparison of
vectors in Euclidean space
(natural notion of
parallelism)
50
B. Janssen et al.
lengths, volumes and so on. Hence it is possible to talk about the module of a vector
that is not necessarily non-negative, due to the lorentzian signature. Those vectors
that are not trivial but have zero norm determine the lightlike paths and, then, light
cones that define the casual structure of the spacetime.
Another fundamental notion that can be defined, even in the absence of metric, is
parallelism. The motivation for this additional concept is the following. Consider the
Euclidean space ℝ
D and a couple of vectors in different points, p and q (Fig. 5.1). If
we want to compare them, we simply take, for example, the one in p and move it to
q keeping the vector parallel to itself and without changing the module. And, finally,
we subtract both vectors to see the difference.
However, if the manifold is general, the initial vectors live in different spaces (the
tangent spaces at p and q, respectively, T p M and T q M) and there is no natural way
to relate them (Fig. 5.2). In the Euclidean space, both tangent spaces can be identified
making the comparison trivial. In the general case, we need to introduce an additional structure that carries the information about parallelism, the affine structure,
whose fundamental object is the (affine) connection Γ
σ
μν . Once we have a connection,
given a curve between two points, we have a rule to relate vectors in them: the
parallel transport associated to the connection.
The connection permits the definition of a covariant derivative (“covariant”
means that once applied to a tensor, the result is also a tensor), and other intrinsic
geometrical properties of the spacetime, such as the curvature and the torsion,
respectively:
Fig. 5.1 Comparison of
vectors in Euclidean space
(natural notion of
parallelism)
50
B. Janssen et al.