Chapter 6
Tensor Analysis
Abstract The ideas of classical vector analysis in Euclidian space generalize naturally to Riemann space. Affine connections are the key to this generalization. Moreover much of classical vector analysis becomes more clear and simple; the divergence
and Laplacian are prime examples.
6.1 Covariant Derivatives, Component View
We know that the derivative of a scalar function is a covariant vector from Chap. 4,
so it has well-defined tensor transformation properties. The derivative of a vector
field is not so simple however. In the preceding chapter we learned how to displace
a vector parallel to itself in an elegant and general way, and we now use the parallel
displacement concept to form a new kind of derivative. Consider the vector field
W
i
x
j
. In going from a point in space x
j to a nearby point x
j
+ dx
j the field
changes according to
W
i
x
l
+ dx
l
= W
i
x
l
+ W
i
,k
x
l
dx
k
.
(6.1)
If it were parallel displaced to the new point it would change according to
W
∗i
x
l
+ dx
l
= W
i
x
l
−
i
k j
x
l
dx
k
.
(6.2)
If the vector field were constant, in the sense of being parallel to itself, then these
two would be equal. Thus we may think of the relevant change in the field as the
difference between the actual value of the field at x
j
+ dx
j and the value it would
have if parallel displaced there from x
j . This is shown in Fig. 6.1.
Accordingly we define the covariant derivative of W
i in terms of this difference
via
W
i
x
l
+ dx
l
− W
∗i
x
l
+ dx
l
=
W
i ,k
x
l
+
i
k j W
j
x
l
dx
k
= W
i ;k dx
k
,
(6.3a)
© 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_6
81
Tensor Analysis
Abstract The ideas of classical vector analysis in Euclidian space generalize naturally to Riemann space. Affine connections are the key to this generalization. Moreover much of classical vector analysis becomes more clear and simple; the divergence
and Laplacian are prime examples.
6.1 Covariant Derivatives, Component View
We know that the derivative of a scalar function is a covariant vector from Chap. 4,
so it has well-defined tensor transformation properties. The derivative of a vector
field is not so simple however. In the preceding chapter we learned how to displace
a vector parallel to itself in an elegant and general way, and we now use the parallel
displacement concept to form a new kind of derivative. Consider the vector field
W
i
x
j
. In going from a point in space x
j to a nearby point x
j
+ dx
j the field
changes according to
W
i
x
l
+ dx
l
= W
i
x
l
+ W
i
,k
x
l
dx
k
.
(6.1)
If it were parallel displaced to the new point it would change according to
W
∗i
x
l
+ dx
l
= W
i
x
l
−
i
k j
x
l
dx
k
.
(6.2)
If the vector field were constant, in the sense of being parallel to itself, then these
two would be equal. Thus we may think of the relevant change in the field as the
difference between the actual value of the field at x
j
+ dx
j and the value it would
have if parallel displaced there from x
j . This is shown in Fig. 6.1.
Accordingly we define the covariant derivative of W
i in terms of this difference
via
W
i
x
l
+ dx
l
− W
∗i
x
l
+ dx
l
=
W
i ,k
x
l
+
i
k j W
j
x
l
dx
k
= W
i ;k dx
k
,
(6.3a)
© 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_6
81