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.

Appendices

Appendix 1: Tangent Spaces

Consider a 2-surface $S$ imbedded in 3-dimensional Euclidean space. Intuition tells us that if S is reasonably smooth at some point $P$ then there will be a flat plane $T$ which coincides with it. The two spaces will be quite similar in a small region near $P$, as shown in Fig. 8.2. We call $T$ the tangent plane at $P$.

Fig. 8.2

The 2-surface $S$ and the tangent plane $T$ at point P, showing the special orthogonal coordinate systems

We emphasize that the two spaces $S$ and $T$ are different spaces which closely coincide (or osculate) only at the point $P$.

We can view this relation in the context of Appendix 2 in Chap. 4 and Appendix 1 in Chap. 5; there we showed that there exists a coordinate system in $S$ for which the metric has the Cayley-Sylvester canonical form and vanishing first derivatives, so the connections are zero. The axes in this coordinate system are orthogonal and it clearly coincides closely with a global Cartesian coordinate system in the tangent plane $T$, as shown in Fig. 8.2. Notice the additional interesting fact that in the special coordinate system in $S$ the covariant derivatives are the same as ordinary derivatives since the connections vanish at P.

This relation between a curved 2-dimensional Riemann space and a flat tangent plane can be generalized to higher dimensions and any signature. In general relativity theory a curved Riemann space, analogous to $S,$ corresponds to a gravitational field. The space analogous to the tangent plane $T$ is a flat Lorentz space and the coordinates may be taken to be the Minkowski coordinates; special relativity holds in this tangent Lorentz space, which coincides locally with the curved Riemann space. There is a gravitational field in the curved space, while there is none in the tangent Lorentz space.

Appendix 2: The Riemann Tensor as a 6 by 6 Matrix

There is an elegant way to view the Riemann tensor as a matrix in which the number of independent components becomes quite clear. It is also useful in classifying spacetimes (Petrov 1969). Think of the first pair of indices $\alpha \beta$ as a single index $A$. Since the Riemann tensor is antisymmetric in this pair only 6 values of the pair occur (Adler 1975)

$$\begin{aligned} & {\text{tensor indices}}\quad \alpha \beta = \begin{array}{*{20}c} {23} & {31} & {12} & {01} & {02} & {03} \\ \end{array} \\ & {\text{matrix indices}}\quad A = \begin{array}{*{20}c} {\;\;1} & {\;\;2} & {\;\;3} & {\;\;4} & {\;\;5} & {\;\;6} \\ \end{array} \\ \end{aligned} $$

(8.53)

Similarly for the pair $\gamma ,\delta$ we may associate a 6 valued matrix index$B$. This allows us to think of the Riemann tensor as a 6 by 6 matrix $R_{AB}$. But the symmetry (8.18c) means that the matrix is symmetric in $A$ and $B$ so it has at most 21 independent components. The final symmetry in (8.19) is one more relation on the components and reduces the number of independent components to 20, much less than the total of 256.

Exercises

• 8.1 How many independent components does the Riemann tensor have in two dimensions, three dimensions, and four dimensions?

• 8.2 Consider a metric in two dimensions with coordinates $x,y$ that has the special form $\text{d}s^{2} = \text{d}x^{2} + G^{2} \left( x \right)\text{d}y^{2} .$ Show that one of the components of the Riemann tensor is

  $$ {R^{1}}_{212} = G\left( {\frac{{\text{d}^{2} G}}{{\text{d}x^{2} }}} \right). $$

  Obtain all the nonzero components from this one. The metric which we will use for cosmology will be analogous to this. See also Exercise 8.7.

• 8.3 What is the Riemann tensor for the 2-dimensional surface of a sphere? What is it for the surface of a cylinder? (Is Exercise 8.2 any help?)

• 8.4 What is the Riemann scalar for the surface of a sphere? Is this a surprise?

• 8.5 Prove that the Ricci tensor is symmetric.

• 8.6 Prove Theorem 5, that the Einstein tensor is zero if and only if the Ricci tensor is zero.

• 8.7 Consider a 4-dimensional spacetime with a particularly simple metric form,

  $$ \text{d}s^{2} = \left( {\text{d}x^{0} } \right)^{2} - g_{ik} \text{d}x^{i} \text{d}x^{k} ,\quad i = 1,2,3, $$

  where the $g_{ik}$ are independent of the time marker $x^{0}$. That is the 4-space contains a 3-space in a simple way. How are the 4-space connections related to the 3-connections?

• 8.8 For the metric of Exercise 8.7, what is the relation between the Riemann tensor in 4-space and that in 3-space? What is the relation between the Ricci tensor in 4-space and that in 3-space? What is the relation between the Riemann scalar in 4-space and that in 3-space?

• 8.9 Show that the gravity-free pseudo-Euclidean space of special relativity is a solution of the Einstein equations in vacuum. (This is as easy as it sounds.)

• 8.10 The sign of the constant C in the field equation (8.36) is negative according to (8.48). This is a drawback of our choice of the overall metric sign in Part I. Think through the development of the field equations in Chap. 8 and convince yourself that all is in order with either sign of the metric.