157	6.5 The curvature tensor

Since the Christoffel symbols α μβ are known functions of the coordinates {xα }, this is a nonlinear (quasi-linear), second-order differential equation for xα (λ). It has a unique solution when initial conditions at λ = λ0 are given: x0α = xα (λ0) and U0α = (dxα /dλ)λ0 . So, by giving an initial position (x0α ) and an initial direction (U0α ), we get a unique geodesic.

Recall that if we change parameter, we change, mathematically speaking, the curve (though not the points it passes through). Now, if λ is a parameter of a geodesic (so that Eq. (6.51) is satisfied), and if we define a new parameter

φ = aλ + b,

(6.52)

where a and b are constants (not depending on position on the curve), then φ is also a parameter in which Eq. (6.51) is satisfied:

d2xα			dxμ dxβ
	+ α	μβ				= 0.
dφ2			dφ		dφ

Generally speaking, only linear transformations of λ like Eq. (6.52) will give new parameters in which the geodesic equation is satisfied. A parameter like λ and φ above is called an affine parameter. A curve having the same path as a geodesic but parametrized by a nonaffine parameter is, strictly speaking, not a geodesic curve.

A geodesic is also a curve of extremal length between any two points: its length is unchanged to first order in small changes in the curve. The student is urged to prove this by using Eq. (6.7), finding the Euler–Lagrange equations for it to be an extremal for fixed λ0 and λ1, and showing that these reduce to Eq. (6.51) when Eq. (6.32) is used. This is a very instructive exercise. We can also show that proper distance along the geodesic is itself an affine parameter (see Exers. 13–15, § 6.9).

6.5 T h e c u r va t u re t e n s o r

At last we are in a position to give a mathematical description of the intrinsic curvature of a manifold. We go back to the curious example of the parallel-transport of a vector around a closed loop, and take it as our definition of curvature. Let us imagine in our manifold a very small closed loop (Fig. 6.5) whose four sides are the coordinate lines

	B
	A	C
	D	x1 = a + δa
	x2 = b	x1 = a
	x2 = b + δb
Figure 6.5	Small section of a coordinate grid.

158			Curved manifolds
	x1 = a, x1 = a + δa, x2 = b, and		x2 = b + δb. A vector V defined at A is							parallel-
								=
	transported to B. The parallel-transport law						e1 V		0 has the component form
				∂Vα
					= − α μ1Vμ.					(6.53)
				∂x1
	Integrating this from A to B gives						'A	∂x1 dx1
		Vα (B) = Vα (A) +								(6.54)
							B ∂Vα

'

= Vα (A) − α μ1Vμdx1,

x2=b

where the notation ‘x2 = b’ under the integral denotes the path AB. Similar transport from B to C to D gives

Vα (C) = Vα (B) −	'x1=a+δa	α μ2Vμ dx2,	(6.55)
Vα (D) = Vα (C) +	'x2=b+δb	α μ1Vμ dx1.	(6.56)

The integral in the last equation has a different sign because the direction of transport from C to D is in the negative x1 direction. Similarly, the completion of the loop gives

'

Vα (Afinal) = Vα (D) + α μ2Vμ dx2. (6.57)

x1=a

The net change in Vα (A) is a vector δVα , found by adding Eqs. (6.54)–(6.57):

δVα = Vα (Afinal) − Vα (Ainitial)

= 'x1=a	α μ2Vμdx2 − 'x1=a+δa		α μ2Vμdx2
+ 'x2=b+δb		α μ1Vμdx1 − 'x2=b	α μ1Vμdx1.	(6.58)

Notice that these would cancel in pairs if α μν and Vμ were constants on the loop, as they would be in flat space. But in curved space they are not, so if we combine the integrals over similar integration variables and work to first order in the separation in the paths, we get to lowest order,

δVα −	'b	+	δb		δa ∂x1 ( α μ2Vμ) dx2
	b						∂
+	'a	+		δb	∂x2 ( α μ1Vμ) dx1					(6.59)
	a		δa		∂
≈ δa δb		$− ∂x1			( α μ2Vμ) + ∂x2 ( α				μ1Vμ)! .	(6.60)
				∂				∂

This involves derivatives of Christoffel symbols and of Vα . The derivatives Vα can be eliminated using Eq. (6.53) and its equivalent with 1 replaced by 2. Then Eq. (6.60) becomes

δVα = δa δb " α μ1,2 − α μ2,1 + α ν2 ν μ1 − α ν1 ν μ2

# Vμ.

(6.61)

159	6.5 The curvature tensor

(To obtain this, we need to relabel dummy indices in the terms quadratic in s.) Notice that this turns out to be just a number times Vμ, summed on μ. Now, the indices 1 and 2 appear because the path was chosen to go along those coordinates. It is antisymmetric in 1 and 2 because the change δVα would have to have the opposite sign if we went around the loop in the opposite direction (that is, interchanging the roles of 1 and 2). If we used general coordinate lines xσ and xλ, we would find

δVα = change in Vα due to transport, first δa eσ , then δb eλ, then −δa aσ , and finally −δb eλ

Now, δV	α	= δa δb " α μσ ,λ − α μλ,σ + α νλ ν μσ − α νσ ν μλ# Vμ.	(6.62)
		depends on δa δb, the coordinate ‘area’ of the loop. So it is clear that if the

length of the loop in one direction is doubled, δVα is doubled. This means that δVα depends linearly on δa eσ and δb eλ. Moreover, it certainly also depends linearly in Eq. (6.62) on

			˜
Vα itself and on ωα , which is the basis one-form that gives δVα from the vector δV. Hence
we have the following result: if we define
			Rα βμν := α βν,μ − α βμ,ν + α σ μ σ βν − α σ ν σ βμ,				(6.63)
ω		δa eμ, δb eν , gives δV , the
then Rα		βμν	must be components of the 31			tensor which, when supplied with arguments
˜	α , V,			α	component of the change in V after parallel-transport

around a loop given by δa eμ and δb eν . This tensor is called the Riemann curvature tensor R.1

It is useful to look at the components of R in a locally inertial frame at a point P. We have α μν = 0 at P, but we can find its derivative from Eq. (6.32):

α μν,σ = 21 gαβ gβμ,νσ + gβν,μσ − gμν,βσ .	(6.64)
Since second derivatives of gαβ don’t vanish, we get at P
Rα βμν = 21 gασ (gσβ,νμ + gσ ν,βμ − gβν,σ μ
− gσβ,μν − gσ μ,βν + gβμ,σ ν ).	(6.65)
Using the symmetry of gαβ and the fact that
gαβ,μν = gαβ,νμ,	(6.66)
because partial derivatives always commute, we find at P
Rα βμν = 21 gσ σ gσ ν,βμ − gσ μ,βν + gβμ,σ ν − gβν,σ μ .	(6.67)

If we lower the index α, we get (in the locally flat coordinate system at its origin P)

1As with other definitions we have earlier introduced, there is no universal agreement about the overall sign of the Riemann tensor, or even on the placement of its indices. Always check the conventions of whatever book you read.

160		Curved manifolds
		Rαβμν := gαλRλβμν = 21 gαν,βμ − gαμ,βν + gβμ,αν − gβν,αμ .	(6.68)

In this form it is easy to verify the following identities:

Rαβμν = −Rβαμν = −Rαβνμ = Rμναβ ,	(6.69)
Rαβμν + Rανβμ + Rαμνβ = 0.	(6.70)

Thus, Rαβμν is antisymmetric on the first pair and on the second pair of indices, and symmetric on exchange of the two pairs. Since Eqs. (6.69) and (6.70) are valid tensor equations true in one coordinate system, they are true in all bases. (Note that an equation like Eq. (6.67) is not a valid tensor equation, since it involves partial derivatives, not covariant ones. Therefore it is true only in the coordinate system in which it was derived.)

It can be shown (Exer. 18, § 6.9) that the various identities, Eqs. (6.69) and (6.70), reduce the number of independent components of Rαβμν (and hence of Rα βμν ) to 20, in four dimensions. This is, not coincidentally, the same number of independent gαβ,μν that we found at the end of § 6.2 could not be made to vanish by a coordinate transformation. Thus Rα βμν characterizes the curvature in a tensorial way.

A flat manifold is one which has a global definition of parallelism: a vector can be moved around parallel to itself on an arbitrary curve and will return to its starting point unchanged.

This clearly means that
Rα βμν = 0 flat manifold.	(6.71)

(Try showing that this is true in polar coordinates for the Euclidean plane.)

An important use of the curvature tensor comes when we examine the consequences of

taking two covariant derivatives of a vector field V. We found in § 6.3 that first derivatives were like flat-space ones, since we could find coordinates in which the metric was flat to first order. But second derivatives are a different story:

α β Vμ = α (Vμ;β )
= (Vμ;β ),α + μσ α Vσ ;β − σ βα Vμ;σ .	(6.72)

In locally inertial coordinates whose origin is at P, all the s are zero, but their partial derivatives are not. Therefore we have at P

α β Vμ = Vμ,βα + μνβ,α Vν .

(6.73)

Bear in mind that this expression is valid only in this specially chosen coordinates system, and that is true also for Eqs. (6.74) through (6.76) below. These coordinates make the computation easier: consider now Eq. (6.73) with α and β exchanged:

β α Vμ = Vμ,αβ + μνα,β Vν .

(6.74)