5.6 Affine Connections, Abstract View
75
The above treatment of the geodesic curve has several interesting features that
should be noted. First, the connections and the metric may be treated independently;
in the general case it is not necessary to have a relation between the two. Any
affine space can thus admit geodesic curves. Secondly, the connections according to
(5.49) need not be symmetric in the lower indices, unlike the Christoffel connections,
so torsion is admissible; that is, the connections may have antisymmetric parts.
However, according to (5.55) the anti-symmetric part of the connection cancels out
of the geodesic equation. We will see later in Chap. 7 that the geodesic equation
determines the motion of bodies in relativity theory, so torsion would have no effect
on such motion. The physical relevance of torsion in the context of relativity theory
is indeed not obvious (Trautman 2006).
Appendix 1: A Special Coordinate System
Recall that in Chap. 4 we stated the Signature Theorem, that at any point P there exists
a special coordinate system in which the metric is diagonal and has diagonal elements
equal to 1 or −1 or 0. The special system may be reached by a linear transformation.
This form of the metric is called the Cayley-Sylvester canonical form. We proved
the theorem for the case of two dimensions in Appendix 4.1 (Perlis 1952).
In this chapter we obtained another special coordinate system, the geodesic
system, in which the affine connections vanish at any given point P. If the connections are zero, then from the definition of the Christoffel connections (5.19) this
clearly means that the first derivatives of the metric must also be zero. We can in fact
combine these transformations and for any given point P find a coordinate system
in which the metric has the Cayley-Sylvester canonical form and also has vanishing
first derivatives and thus vanishing connections. To do this we merely apply the two
transformations together with the point P taken to be the origin,
x
j
= L
j
k x
k
+
1
2
A
i
jl (L
j
n x
n
)
L
l
m x
m
.
(5.56)
The L array makes the transformation to the system in which the metric has the
Cayley-Sylvester canonical form, and the A array specifies the transformation to the
geodesic system. The coordinate system thus obtained is very special: the axes are
orthogonal, the metric is Lorentz, and the connections vanish, so physics is locally
much like that of special relativity, but of course only in a vanishingly small region
near P.
75
The above treatment of the geodesic curve has several interesting features that
should be noted. First, the connections and the metric may be treated independently;
in the general case it is not necessary to have a relation between the two. Any
affine space can thus admit geodesic curves. Secondly, the connections according to
(5.49) need not be symmetric in the lower indices, unlike the Christoffel connections,
so torsion is admissible; that is, the connections may have antisymmetric parts.
However, according to (5.55) the anti-symmetric part of the connection cancels out
of the geodesic equation. We will see later in Chap. 7 that the geodesic equation
determines the motion of bodies in relativity theory, so torsion would have no effect
on such motion. The physical relevance of torsion in the context of relativity theory
is indeed not obvious (Trautman 2006).
Appendix 1: A Special Coordinate System
Recall that in Chap. 4 we stated the Signature Theorem, that at any point P there exists
a special coordinate system in which the metric is diagonal and has diagonal elements
equal to 1 or −1 or 0. The special system may be reached by a linear transformation.
This form of the metric is called the Cayley-Sylvester canonical form. We proved
the theorem for the case of two dimensions in Appendix 4.1 (Perlis 1952).
In this chapter we obtained another special coordinate system, the geodesic
system, in which the affine connections vanish at any given point P. If the connections are zero, then from the definition of the Christoffel connections (5.19) this
clearly means that the first derivatives of the metric must also be zero. We can in fact
combine these transformations and for any given point P find a coordinate system
in which the metric has the Cayley-Sylvester canonical form and also has vanishing
first derivatives and thus vanishing connections. To do this we merely apply the two
transformations together with the point P taken to be the origin,
x
j
= L
j
k x
k
+
1
2
A
i
jl (L
j
n x
n
)
L
l
m x
m
.
(5.56)
The L array makes the transformation to the system in which the metric has the
Cayley-Sylvester canonical form, and the A array specifies the transformation to the
geodesic system. The coordinate system thus obtained is very special: the axes are
orthogonal, the metric is Lorentz, and the connections vanish, so physics is locally
much like that of special relativity, but of course only in a vanishingly small region
near P.