64
5 Affine Connections and Geodesics
Now we choose the array A
j
ml to be the negative of the affine connection array in
the unbarred system, and thereby cause the connections at P to be zero in the barred
system; we may do this if and only if the connections are symmetric. The coordinate
system where the affine connections vanish is termed the geodesic system (Adler
1975).
Besides being elegant mathematics the Weyl Theorem has important implications
for physics. We will see in Sect. 7.3 that in the geometric description of gravity the
affine connections play a role analogous to Newtonian forces, and the Weyl Theorem
thus tells us that gravitational effects may be transformed away at a point by a choice
of coordinates! This is a very profound fact in general relativity theory and a cornerstone of the geometric view of gravity. As we will discuss in Sect. 7.2 equivalence
principle (EP) experiments indicate that it is true in nature to an accuracy better
than a part in 10
13 (Wiki STEP). Because of this agreement with nature and because
of mathematical elegance we will generally assume that the affine connections are
symmetric.
If the affine connections are not taken to be symmetric a more general theory of
gravity can be developed, the most well-known of which is the Einstein-Cartan theory.
The effects of the non-symmetry of the connections are termed torsion. There is at
present no experimental evidence for torsion to motivate such theories, but some
theorists believe torsion may be necessary in a future theory of quantum gravity
(Trautman 2006).
5.3 Parallel Displacement
The law of vector transplantation (5.5) introduced in the preceding section provides a
way to compare vectors at different nearby points in space. By repeated iterations we
could also compare vectors at widely separated points. Our considerations have so
far been quite general and we made no assumptions about how the connections might
be specified. We now specialize to obtain the specific connections used in relativity
theory; this provides a strikingly elegant generalization of the idea of moving a vector
parallel to itself in Euclidean geometry, and is called parallel displacement. Parallel
displacement is basic to the idea and definition of space curvature that we will develop
in Chap. 8. It also allows us to define geodesic curves, which are the generalization
of straight lines to general Riemann spaces.
Suppose that we transplant two vectors to a nearby point using the law of vector
transplantation. There is no a priori reason that the inner product of the two will remain
unchanged; however we may consider this to be a naturally compelling demand to be
imposed so as to make the transplantation analogous to the parallel displacement of
vectors in Euclidean geometry. In the special case of Euclidean space the demand for
parallelism implies that the lengths of various vectors and the angles between them
remain unchanged as they are transplanted. We thus impose this demand and refer
to this special case of vector transplantation as generalized parallel displacement,
or simply parallel displacement for brevity. Remarkably, the connections are then
5 Affine Connections and Geodesics
Now we choose the array A
j
ml to be the negative of the affine connection array in
the unbarred system, and thereby cause the connections at P to be zero in the barred
system; we may do this if and only if the connections are symmetric. The coordinate
system where the affine connections vanish is termed the geodesic system (Adler
1975).
Besides being elegant mathematics the Weyl Theorem has important implications
for physics. We will see in Sect. 7.3 that in the geometric description of gravity the
affine connections play a role analogous to Newtonian forces, and the Weyl Theorem
thus tells us that gravitational effects may be transformed away at a point by a choice
of coordinates! This is a very profound fact in general relativity theory and a cornerstone of the geometric view of gravity. As we will discuss in Sect. 7.2 equivalence
principle (EP) experiments indicate that it is true in nature to an accuracy better
than a part in 10
13 (Wiki STEP). Because of this agreement with nature and because
of mathematical elegance we will generally assume that the affine connections are
symmetric.
If the affine connections are not taken to be symmetric a more general theory of
gravity can be developed, the most well-known of which is the Einstein-Cartan theory.
The effects of the non-symmetry of the connections are termed torsion. There is at
present no experimental evidence for torsion to motivate such theories, but some
theorists believe torsion may be necessary in a future theory of quantum gravity
(Trautman 2006).
5.3 Parallel Displacement
The law of vector transplantation (5.5) introduced in the preceding section provides a
way to compare vectors at different nearby points in space. By repeated iterations we
could also compare vectors at widely separated points. Our considerations have so
far been quite general and we made no assumptions about how the connections might
be specified. We now specialize to obtain the specific connections used in relativity
theory; this provides a strikingly elegant generalization of the idea of moving a vector
parallel to itself in Euclidean geometry, and is called parallel displacement. Parallel
displacement is basic to the idea and definition of space curvature that we will develop
in Chap. 8. It also allows us to define geodesic curves, which are the generalization
of straight lines to general Riemann spaces.
Suppose that we transplant two vectors to a nearby point using the law of vector
transplantation. There is no a priori reason that the inner product of the two will remain
unchanged; however we may consider this to be a naturally compelling demand to be
imposed so as to make the transplantation analogous to the parallel displacement of
vectors in Euclidean geometry. In the special case of Euclidean space the demand for
parallelism implies that the lengths of various vectors and the angles between them
remain unchanged as they are transplanted. We thus impose this demand and refer
to this special case of vector transplantation as generalized parallel displacement,
or simply parallel displacement for brevity. Remarkably, the connections are then