Chapter 4
Riemann Spaces and Tensors
Abstract In this chapter we begin to study the mathematics needed for general relativity. Quite general spaces such as we will use to describe spacetime and gravity were
first developed by nineteenth century mathematicians such as Gauss and Riemann.
The most important mathematical objects in such spaces are vectors and tensors. We
will treat these using both the classic index notation and a more modern abstract
notation.
4.1 Riemann Spaces
We now make the transition from the Minkowski spacetime of special relativity to
more general spaces and coordinate systems and the mathematical objects in them,
vectors and tensors and forms. There are standard reference texts dating over many
years (Pauli 1958; Bergmann 1942; Rindler 1969; Weinberg 1972; Misner 1973;
Adler 1975; Kenyon 1990). Here we will rely heavily on examples to illustrate the
basic ideas.
Think of a physical space such as the surface of a blackboard or sphere or torus
as in Fig. 4.1, or the Euclidean 3-space of classical physics. In particular include
the spacetime of special relativity that we studied in Part I. We imagine a marker
system or labeling system or coordinate system to specify the points in the space
with a set of real numbers. In general there will be many ways to set up such a
marker system, and we assume that there will be transformations between them. An
excellent example to remember is the Euclidean 3-space of classical geometry and
physics, labeled by Cartesian or spherical coordinates. We denote the transformation
between two coordinate systems, denote them as unprimed and primed, by a set of
functions
x
j
= f
j
x
n
.
(4.1)
The functions f
j are assumed to be continuous monotonic one-to-one and differentiable as often as needed. The transformation therefore has an inverse. For brevity
we usually denote the transformation and its inverse in shorthand notation,
© 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_4
33
Riemann Spaces and Tensors
Abstract In this chapter we begin to study the mathematics needed for general relativity. Quite general spaces such as we will use to describe spacetime and gravity were
first developed by nineteenth century mathematicians such as Gauss and Riemann.
The most important mathematical objects in such spaces are vectors and tensors. We
will treat these using both the classic index notation and a more modern abstract
notation.
4.1 Riemann Spaces
We now make the transition from the Minkowski spacetime of special relativity to
more general spaces and coordinate systems and the mathematical objects in them,
vectors and tensors and forms. There are standard reference texts dating over many
years (Pauli 1958; Bergmann 1942; Rindler 1969; Weinberg 1972; Misner 1973;
Adler 1975; Kenyon 1990). Here we will rely heavily on examples to illustrate the
basic ideas.
Think of a physical space such as the surface of a blackboard or sphere or torus
as in Fig. 4.1, or the Euclidean 3-space of classical physics. In particular include
the spacetime of special relativity that we studied in Part I. We imagine a marker
system or labeling system or coordinate system to specify the points in the space
with a set of real numbers. In general there will be many ways to set up such a
marker system, and we assume that there will be transformations between them. An
excellent example to remember is the Euclidean 3-space of classical geometry and
physics, labeled by Cartesian or spherical coordinates. We denote the transformation
between two coordinate systems, denote them as unprimed and primed, by a set of
functions
x
j
= f
j
x
n
.
(4.1)
The functions f
j are assumed to be continuous monotonic one-to-one and differentiable as often as needed. The transformation therefore has an inverse. For brevity
we usually denote the transformation and its inverse in shorthand notation,
© 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_4
33