Curved manifolds

(5)A number of manipulations of components of tensor fields are called ‘permissible tensor operations’ because they produce components of new tensors:

(i)Multiplication by a scalar field produces components of a new tensor of the same type.

(ii)Addition of components of two tensors of the same type gives components of a new tensor of the same type. (In particular, only tensors of the same type can be added.)

(iii)Multiplication of components of two tensors of arbitrary type gives components of a new tensor of the sum of the types, the outer product of the two tensors.

(iv)Covariant differentiation (to be discussed later) of the components of a tensor of

(v)Contraction on a pair of indices of the components of a tensor of type MN pro-

N−1 . (Contraction is only defined between

M 1

an upper and lower index.)

(6)If an equation is formed using components of tensors combined only by the permissible tensor operations, and if the equation is true in one basis, then it is true in any other. This is a very useful result. It comes from the fact that the equation (from (5) above) is simply an equality between components of two tensors of the same type, which (from (4)) is then true in any system.

6.2 R i e m a n n i a n m a n i f o l d s

So far we have not introduced a metric on to the manifold. Indeed, on certain manifolds a metric would be unnecessary or inconvenient for whichever problem is being considered. But in our case the metric is absolutely fundamental, since it will carry the information about the rates at which clocks run and the distances between points, just as it does in

called Riemannian; indefinite metrics, like SR and GR, are called pseudo-Riemannian. This is a distinction that we won’t bother to make.) It is important to understand that in picking out a metric we ‘add’ structure to the manifold; we shall see that the metric completely defines the curvature of the manifold. Thus, by our choosing one metric g the manifold gets a certain curvature (perhaps that of a sphere), while a different g would give it a different curvature (perhaps an ellipsoid of revolution). The differentiable manifold itself is ‘primitive’: an amorphous collection of points, arranged locally like the points of Euclidean space, but not having any distance relation or shape specified. Giving the metric g gives it a specific shape, as we shall see. From now on we shall study Riemannian manifolds, on which a metric g is assumed to be defined at every point.

(For completeness we should remark that it is in fact possible to define the notion of curvature on a manifold without introducing a metric (so-called ‘affine’ manifolds). Some

In a general gravitational field this will be impossible, because otherwise it would imply the existence of coordinates for which Eq. (6.2) is true everywhere: a global Lorentz frame. However, having found a basis at a particular point P for which Eq. (6.2) is true, it is possible to find coordinates such that, in the neighborhood of P, Eq. (6.2) is ‘nearly’ true. This is embodied in the following theorem, whose (rather long) proof is at the end of this section. Choose any point P of the manifold. A coordinate system {xα } can be found whose origin is at P and in which:

That is, the metric near P is approximately that of SR, differences being of second order in the coordinates. From now on we shall refer to such coordinate systems as ‘local Lorentz frames’ or ‘local inertial frames’. Eq. (6.3) can be rephrased in a somewhat more precise way as:

but generally

∂2

∂xγ ∂xμ gαβ (P) =0

for at least some values of α, β, γ , and μ if the manifold is not exactly flat.

The existence of local Lorentz frames is merely the statement that any curved space has a flat space ‘tangent’ to it at any point. Recall that straight lines in flat spacetime are the world lines of free particles; the absence of first-derivative terms (Eq. (6.5)) in the metric of a curved spacetime will mean that free particles are moving on lines that are locally straight in this coordinate system. This makes such coordinates very useful for us, since the equations of physics will be nearly as simple in them as in flat spacetime, and if constructed by the rules of § 6.1 will be valid in any coordinate system. The proof of this theorem is at the end of this section, and is worth studying.

Le ngths and volumes

The metric of course gives a way to define lengths of curves. Let dx be a small vector displacement on some curve. Then dx has squared length ds2 = gαβ dxα dxβ . (Recall that we call this the line element of the metric.) If we take the absolute value of this and take its square root, we get a measure of length: dl ≡ |gαβ dxα dxβ |1/2. Then integrating it gives

as the length of the arbitrary curve.

The computation of volumes is very important for integration in spacetime. Here, we mean by ‘volume’ the four-dimensional volume element we used for integrations in Gauss’ law in § 4.4. Let us go to a local Lorentz frame, where we know that a small fourdimensional region has four-volume dx0 dx1 dx2 dx3, where {xα } are the coordinates which at this point give the nearly Lorentz metric, Eq. (6.3). In any other coordinate system {xα } it is a well-known result of the calculus of several variables that:

This would be a rather tedious way to calculate the Jacobian, but there is an easier way using the metric. In matrix terminology, the transformation of the metric components is

where (g) is the matrix of gαβ , (η) of ηαβ , etc., and where ‘T’ denotes transpose. It follows that the determinants satisfy

This is a very useful result. It is also conceptually an important result because it is the first example of a kind of argument we will frequently employ, an argument that uses locally flat coordinates to generalize our flat-space concepts to analogous ones in curved space. In this case we began with dx0 dx1 dx2 dx3 = d4x in a locally flat coordinate system. We argue that this volume element at P must be the volume physically measured by rods and clocks, since the space is the same as Minkowski space in this small region. We then find that the value of this expression in arbitrary coordinates {xμ } is Eq. (6.18), (−g)1/2 d4x , which is thus the expression for the true volume in a curved space at any point in any coordinates. We call this the proper volume element.

It should not be surprising that the metric comes into it, of course, since the metric measures lengths. We only need remember that in any coordinates the square root of the negative of the determinant of (gαβ ) is the thing to multiply by d4x to get the true, or proper, volume element.

Perhaps it would be helpful to quote an example from three dimensions. Here proper volume is (g)1/2, since the metric is positive-definite (Eq. (6.14) would have a + sign). In spherical coordinates the line element is dl2 = dr2 + r2dθ 2 + r2 sin2 θ dφ2, so the metric is