Chapter 8
Curved Space and Gravity
Abstract In this chapter we return to mathematics and study curvature in a Riemann
space. Einstein’s general relativistic field equations of gravity follow in an intuitive
way from a study of the Riemann tensor and the geometric view of gravity.
8.1 Curved Space and the Riemann Tensor
When we studied vectors and tensors in Part II we often referred to curved 2-surfaces,
depending on geometrical intuition for the meaning of curvature. Now however we
must deal with the more sophisticated idea of a general curved space, because in
general relativity gravity is described by a curved 4-dimensional spacetime; this is
the natural outcome of the discussion of the last section on the geometric view of
gravity (Misner 1973; Adler 1975; Schutz 2009). Indeed, we have already had an
example of how to handle the analysis and definition of curvature when we parallel
displaced a vector around a triangle on a plane and on a spherical surface in Chap. 5.
Consider first the familiar 2 and 3-dimensional spaces of Euclidean geometry. We
call such a space a Euclidean space; a Euclidean space is defined by the property
that there is a coordinate system in which the metric is equal to the identity matrix
everywhere; thus a Euclidean space has signature (1, 1, … 1). For Euclidean 3-space,
for example, the metric in the special system is
g i j =
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎠ , Euclidean 3-space.
(8.1)
We may of course describe the space with other coordinate systems, such as spherical.
Next recall the space of special relativity, Minkowski space, which is usually
coordinatized with ct and Cartesian coordinates. This is similar to a Euclidean space,
but is distinguished by the minus signs in the metric. A pseudo-Euclidean space is
defined by the property that there is a coordinate system in which the metric is
equal everywhere to a diagonal matrix with +1 or −1 on the diagonal. For example,
Minkowski space has in the special system the Lorentz metric
© 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_8
109
Curved Space and Gravity
Abstract In this chapter we return to mathematics and study curvature in a Riemann
space. Einstein’s general relativistic field equations of gravity follow in an intuitive
way from a study of the Riemann tensor and the geometric view of gravity.
8.1 Curved Space and the Riemann Tensor
When we studied vectors and tensors in Part II we often referred to curved 2-surfaces,
depending on geometrical intuition for the meaning of curvature. Now however we
must deal with the more sophisticated idea of a general curved space, because in
general relativity gravity is described by a curved 4-dimensional spacetime; this is
the natural outcome of the discussion of the last section on the geometric view of
gravity (Misner 1973; Adler 1975; Schutz 2009). Indeed, we have already had an
example of how to handle the analysis and definition of curvature when we parallel
displaced a vector around a triangle on a plane and on a spherical surface in Chap. 5.
Consider first the familiar 2 and 3-dimensional spaces of Euclidean geometry. We
call such a space a Euclidean space; a Euclidean space is defined by the property
that there is a coordinate system in which the metric is equal to the identity matrix
everywhere; thus a Euclidean space has signature (1, 1, … 1). For Euclidean 3-space,
for example, the metric in the special system is
g i j =
⎛
⎝
1 0 0
0 1 0
0 0 1
⎞
⎠ , Euclidean 3-space.
(8.1)
We may of course describe the space with other coordinate systems, such as spherical.
Next recall the space of special relativity, Minkowski space, which is usually
coordinatized with ct and Cartesian coordinates. This is similar to a Euclidean space,
but is distinguished by the minus signs in the metric. A pseudo-Euclidean space is
defined by the property that there is a coordinate system in which the metric is
equal everywhere to a diagonal matrix with +1 or −1 on the diagonal. For example,
Minkowski space has in the special system the Lorentz metric
© 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_8
109