122
8 Curved Space and Gravity
Fig. 8.2 The 2-surface S and the tangent plane T at point P, showing the special orthogonal
coordinate systems
After these brief comments on the equivalence principle and its various forms and
interpretations it is well to note a cautionary remark by Nordtvedt, that “Principles
are for when you do not yet have a theory.”
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. 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 αβ as a single index A. Since
8 Curved Space and Gravity
Fig. 8.2 The 2-surface S and the tangent plane T at point P, showing the special orthogonal
coordinate systems
After these brief comments on the equivalence principle and its various forms and
interpretations it is well to note a cautionary remark by Nordtvedt, that “Principles
are for when you do not yet have a theory.”
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. 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 αβ as a single index A. Since