Chapter 5
Affine Connections and Geodesics
Abstract In a general Riemann space the concepts of straight lines and parallel
vectors must be generalized from those familiar in Euclidian geometry. The fundamental objects needed for the generalization are affine connections. With affine
connections we are naturally led to a deeper view of spacetime and the behavior
of objects in it.
5.1 Affine Connections, Component View
Most of our considerations in Chap. 4 involved vectors and tensors associated with a
single point. Now we study how to compare vectors and tensors at different points in
a Riemann space, and how to move them (Misner 1973; Adler 1975; Schutz 2009).
This is necessary in order to study tensor fields, that is tensors defined as functions
of position in regions of space; these fields may be denoted for example as
φ(x
μ
) scalar, V
∝
(x
μ
) vector, T
∝β
(x
μ
) 2nd rank tensor.
(5.1)
This is not a trivial process since vector spaces at different points in a Riemann space
are a priori independent and any connection between them requires analysis. The
key concept is that of affine connections, for which we will motivate a definition;
then on the basis of the definition we may obtain their transformation law.
Much of the work in this chapter is based on the classic component view, but we
will relate it to the abstract view in the last section.
Consider first a constant vector field in Euclidean 3-space with Cartesian coordinates; the definition of such a constant vector field is obviously that the components
are constant, as shown in Fig. 5.1a. But it is also clear that in spherical coordinates
constant components do not correspond to what we think of as a constant vector
field, as in Fig. 5.1b. Clearly, we should not define a constant vector field as one
with constant components. The terms “constant” and “parallel” remain to be defined
precisely. The proper definition of a constant vector field will introduce the concept
of affine connections as an elegant generalization of the notion of parallel vectors.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_5
59
Affine Connections and Geodesics
Abstract In a general Riemann space the concepts of straight lines and parallel
vectors must be generalized from those familiar in Euclidian geometry. The fundamental objects needed for the generalization are affine connections. With affine
connections we are naturally led to a deeper view of spacetime and the behavior
of objects in it.
5.1 Affine Connections, Component View
Most of our considerations in Chap. 4 involved vectors and tensors associated with a
single point. Now we study how to compare vectors and tensors at different points in
a Riemann space, and how to move them (Misner 1973; Adler 1975; Schutz 2009).
This is necessary in order to study tensor fields, that is tensors defined as functions
of position in regions of space; these fields may be denoted for example as
φ(x
μ
) scalar, V
∝
(x
μ
) vector, T
∝β
(x
μ
) 2nd rank tensor.
(5.1)
This is not a trivial process since vector spaces at different points in a Riemann space
are a priori independent and any connection between them requires analysis. The
key concept is that of affine connections, for which we will motivate a definition;
then on the basis of the definition we may obtain their transformation law.
Much of the work in this chapter is based on the classic component view, but we
will relate it to the abstract view in the last section.
Consider first a constant vector field in Euclidean 3-space with Cartesian coordinates; the definition of such a constant vector field is obviously that the components
are constant, as shown in Fig. 5.1a. But it is also clear that in spherical coordinates
constant components do not correspond to what we think of as a constant vector
field, as in Fig. 5.1b. Clearly, we should not define a constant vector field as one
with constant components. The terms “constant” and “parallel” remain to be defined
precisely. The proper definition of a constant vector field will introduce the concept
of affine connections as an elegant generalization of the notion of parallel vectors.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_5
59