34
4 Riemann Spaces and Tensors
Fig. 4.1 Some 2-dimensional spaces: a sphere, a torus, and an odd shaped 2-surface with
coordinates lines shown
x
j
= x
j
x
n
, x
k
= x
k
x
j
.
(4.2)
The square array of derivatives we denote as
∂ x
j
∂ x n ,
∂ x
j
∂ x .
(4.3)
These are the transformation or Jacobian matrices familiar from elementary calculus.
Loosely speaking a space coordinatized in several different ways by n real numbers
as we use here is called an n-dimensional manifold. A manifold is defined as a space
which locally resembles a Euclidean space and in which we can perform the usual
analytic operations as in Euclidean space. Thus we can for example set up systems of
differential equations in a manifold. See Appendix 1 for a more detailed discussion
of the manifold idea.
The spaces of interest in physics usually have a well-defined distance between any
two points. We therefore assume that between two nearby points in the space, separated by small coordinate distances dx
μ , there is a distance with physical meaning,
or line element, given by a quadratic form
ds
2
=
μν
g μν dx
μ dx
ν
= g μν dx
μ dx
ν
.
(4.4)
Here we use as usual the Einstein summation convention, wherein a repeated index
or dummy index is to be summed over. The line element is a direct generalization
of the Pythagorean theorem of Euclidean geometry on a differential scale. The array
g μν is called the metric or metric tensor; the Lorentz metric of special relativity is
one important example. A space with such a distance measure or metric we will call
a Riemann metric space. The phrase Riemannian manifold is more specific and often
used, as elucidated in Appendix 1.
Since the expression (4.4) for the line element completely determines the metric
tensor we will often refer to the line element as the metric.
The summation convention is very powerful in the sense that it simplifies the
look of an equation; it is important to remember that repeated or dummy indices
are summed over so they may be denoted by any convenient symbol. After a little
practice the “index juggling” we will encounter in tensor equations becomes easy.
4 Riemann Spaces and Tensors
Fig. 4.1 Some 2-dimensional spaces: a sphere, a torus, and an odd shaped 2-surface with
coordinates lines shown
x
j
= x
j
x
n
, x
k
= x
k
x
j
.
(4.2)
The square array of derivatives we denote as
∂ x
j
∂ x n ,
∂ x
j
∂ x .
(4.3)
These are the transformation or Jacobian matrices familiar from elementary calculus.
Loosely speaking a space coordinatized in several different ways by n real numbers
as we use here is called an n-dimensional manifold. A manifold is defined as a space
which locally resembles a Euclidean space and in which we can perform the usual
analytic operations as in Euclidean space. Thus we can for example set up systems of
differential equations in a manifold. See Appendix 1 for a more detailed discussion
of the manifold idea.
The spaces of interest in physics usually have a well-defined distance between any
two points. We therefore assume that between two nearby points in the space, separated by small coordinate distances dx
μ , there is a distance with physical meaning,
or line element, given by a quadratic form
ds
2
=
μν
g μν dx
μ dx
ν
= g μν dx
μ dx
ν
.
(4.4)
Here we use as usual the Einstein summation convention, wherein a repeated index
or dummy index is to be summed over. The line element is a direct generalization
of the Pythagorean theorem of Euclidean geometry on a differential scale. The array
g μν is called the metric or metric tensor; the Lorentz metric of special relativity is
one important example. A space with such a distance measure or metric we will call
a Riemann metric space. The phrase Riemannian manifold is more specific and often
used, as elucidated in Appendix 1.
Since the expression (4.4) for the line element completely determines the metric
tensor we will often refer to the line element as the metric.
The summation convention is very powerful in the sense that it simplifies the
look of an equation; it is important to remember that repeated or dummy indices
are summed over so they may be denoted by any convenient symbol. After a little
practice the “index juggling” we will encounter in tensor equations becomes easy.