If we subtract these, we get the commutator of the covariant derivative operators α andβ , written in the same notation as we would employ in quantum mechanics:

[Let us pause to recall that Vμ,α is the partial derivative of the component Vμ, so by the laws of partial differentiation the partial derivatives must commute. On the other hand,

reason (from differential calculus) why it must be symmetric on α and β. We have proved, by showing that Eq. (6.75) is nonzero, that the double covariant derivative generally is not symmetric.] Now, in this frame (where μαβ = 0 at P), we can compare Eq. (6.75) with Eq. (6.63) and see that at P

Now, this is a valid tensor equation, so it is true in any coordinate system: the Riemann tensor gives the commutator of covariant derivatives. We can drop the restriction to locally inertial coordinates: they were simply a convenient way of arriving at a general tensor expression for the commutator. What this means is that in curved spaces, we must be

careful to know the order in which covariant derivatives are taken: they do not commute.

This can be extended to tensors of higher rank. For example, a 11 tensor has

That is, each index gets a Riemann tensor on it, and each one comes in with a + sign. (They must all have the same sign because raising and lowering indices with g is unaffected byα , since g = 0.)

Eq. (6.77) is closely related to our original derivation of the Riemann tensor from parallel-transport around loops, because the parallel-transport problem can be thought of

as computing, first the change of V in one direction, and then in another, followed by subtracting changes in the reverse order: this is what commuting covariant derivatives also does.

Geodesic deviation

We have often mentioned that in a curved space, parallel lines when extended do not remain parallel. This can now be formulated mathematically in terms of the Riemann tensor. Con-

where the second equality follows from Eq. (6.63). The final expression is frame invariant, and A was an arbitrary point, so we have, in any basis,

Geodesics in flat space maintain their separation; those in curved spaces don’t. This is called the equation of geodesic deviation and shows mathematically that the tidal forces of a gravitational field (which cause trajectories of neighboring particles to diverge) can be represented by curvature of a spacetime in which particles follow geodesics.

6.6 B i a n c h i i d e n t i t i e s : R i cc i a n d E i n s t e i n t e n s o r s

Let us return to Eq. (6.63) for the Riemann tensor’s components. If we differentiate it with respect to xλ (just the partial derivative) and evaluate the result in locally inertial coordinates, we find

From this equation, the symmetry gαβ = gβα and the fact that partial derivatives commute, we can show that

But this is a tensor equation, valid in any system. It is called the Bianchi identities, and will be very important for our work.

The Ricci tensor

Before pursuing the consequences of the Bianchi identities, we shall need to define the Ricci tensor Rαβ :

It is the contraction of Rμανβ on the first and third indices. Other contractions would in principle also be possible: on the first and second, the first and fourth, etc. But because Rαβμν is antisymmetric on α and β and on μ and ν, all these contractions either vanish identically or reduce to ±Rαβ . Therefore the Ricci tensor is essentially the only contraction of the Riemann tensor. Note that Eq. (6.69) implies it is a symmetric tensor (Exer. 25, § 6.9).

Similarly, the Ricci scalar is defined as

The Einstein tensor

Let us apply the Ricci contraction to the Bianchi identities, Eq. (6.90):

Therefore, gαμ and gβν can be taken in and out of covariant derivatives at will: indexraising and -lowering commutes with covariant differentiation. The second fact is that

accounting for the second term in Eq. (6.93). Eq. (6.93) is called the contracted Bianchi identities. A more useful equation is obtained by contracting again on the indices β and ν:

gβν "Rβν;λ − Rβλ;ν + Rμβνλ;μ# = 0

or