It now turns out that parallel transfer of an arbitrary vector along an arbitrary closed curve is locally integrable (i.e. X'p is necessarily the same as Xp for each only if R1^ = 0 at all points of JV\ in this case we say that the connection is fiat.

By contracting the curvature tensor, one can define the Ricci tensor as the tensor of type (0,2) with components

2.6 The metric

A metric tensor g at a point p is a symmetric tensor of type (0, 2)

at p, so a Cr metric on Jt is a Cr symmetric tensor field g. The metric g at p assigns a 'magnitude' (|<7(X,X)|)i to each vector ~XeTp and defines the ' cos angle' ,v

(|?(X,X).^Y,Y)|)i between any vectors X, YeTp such that g(X, X). g(Y, Y) # 0; vectors X, Y will be said to be orthogonal if g(X, Y) = 0.

The components of g with respect to a basis {E0} are i.e. the components are simply the scalar products of the basis vectors E0. If a coordinate basis {8/8xa} is used, then g = gabdtf'®dxt>. (2.23)

Tangent space magnitudes defined by the metric are related to magnitudes on the manifold by the definition: the path length between points p = y(a) and q = y(b) along a C°, piecewise C1 curve y(t) with tangent vector 8j8t such that g(dldt, 8/8t) has the same sign at all points along y(t), is the quantity

We may symbolically express the relations (2.23), (2.24) in the form d«2 = gi}dxidx'

used in classical textbooks to represent the length of the' infinitesimal' arc determined by the coordinate displacement xi-*xi + dxi.

The metric is said to be non-degenerate at p if there is no non-zero vector XeTp such that g(X, Y) = 0 for all vectors YeTp. In terms of components, the metric is non-degenerate if the matrix (gat,) of components of g is non-singular. We shall from now on always assume the metric tensor is non-degenerate. Then we can define a unique symmetric tensor of type (2, 0) with components g°*> with respect to the basis {Ea} dual to the basis {E°}, by the relations

^"fftc = K, i.e. the matrix (gai>) of components is the inverse of the matrix (gab). It follows that the matrix (g"b) is also non-singular, so the tensors g"b, gab can be used to give an isomorphism between any covariant tensor argument and any contravariant argument, or to 'raise and lower indices'. Thus, if Xa are the components of a contravariant vector, then Xa are the components of a uniquely associated covariant vector, where Xa = gabXb, Xa — gabXb; similarly, to a tensor Tab of type (0,2) we can associate unique tensors Tab = gacTcb, Tab = gbcTac, Tab = gacgbdTcd. We shall in general regard such associated covariant and contravariant tensors as representations of the same geometric object (so in particular, gab, Sab and g^ may be thought of as representations (with respect to dual bases) of the same geometric object g), although in some cases where we have more than one metric we shall have to distinguish carefully which metric is used to raise or lower indices.

The signature of g at p is the number of positive eigenvalues of the matrix (gab) at p, minus the number of negative ones. If g is non-degenerate and continuous, the signature will be constant on by suitable choice of the basis {Ea}, the metric components can at any point p be brought to the form ffab = diag(+l, +1,..., +1, -1,..., -1), £(n-M) terms — s) terms where s is the signature of g and n is the dimension of In this case the basis vectors {E0} form an orthonormal set at p, i.e. each is a unit vector orthogonal to every other basis vector.

A metric whose signature is n is called a positive definite metric, for such a metric, g(X, X) = 0 => X = 0, and the canonical form is ffab ~ diag (+1 +1).

n terms

A positive definite metric is a 'metric' on the space, in the topological sense of the word.

A metric whose signature is (n — 2) is called a Lorentz metric; the canonical form is ffab = diag(+l +1,-1).

With a Lorentz metric on JK, the non-zero vectors at p can be divided into three classes: a vector XeTp being said to be timelike, null, or spacelike according to whether g(X, X) is negative, zero, or positive, respectively. The null vectors form a double cone in Tp which separates the timelike from the spacelike vectors (see figure 8). If X, Y are any two non-spacelike (i.e. timelike or null) vectors in the same half of the light cone at p, then g(X, Y) < 0, and equality can only hold if X and Y are parallel null vectors (i.e. if X = aY, g(X, X) = 0).

Any paracompact Cr manifold admits a Cr_1 positive definite metric (that is, one defined on the whole of JK). To see this, let {/a} be a partition of unity for a locally finite atlas 0a}. Then one can define g by

a where ( , ) is the natural scalar product in Euclidean space Rn; thus one uses the atlas to determine the metric by mapping the

Timelike vectors lie inside the null cones

Timelike vectors lie inside the null cones

Null cone

Null vectors lie on the null cones

Spacelike vectors lie outside the null cones En-,

Hyperplane spanned by Ei E„_,

Null cone

Figure 8. The null cones defined by a Lorentz metric.

Null cone

Null vectors lie on the null cones

Spacelike vectors lie outside the null cones En-,

Hyperplane spanned by Ei E„_,

Null cone

Figure 8. The null cones defined by a Lorentz metric.

Euclidean metric into . This is clearly not invariant under change of atlas, so there are many such positive definite metrics on .

In contrast to this, a Cr paracompact manifold admits a Cr~J Lorentz metric if and only if it admits a non-vanishing Cr_1 line element field; by a line element field is meant an assignment of a pair of equal and opposite vectors (X, — X) at each pointy of JK, i.e. a line element field is like a vector field but with undetermined sign. To see this, let £ be a Cr"1 positive definite metric defined on the manifold. Then one can define a Lorentz metric g by at each point p, where X is one of the pair (X, — X) at p. (Note that as X appears an even number of times, it does not matter whether X or — X is chosen.) Then y(X,X) = — $(X, X), and if Y, Z are orthogonal to X with respect to they are also orthogonal to X with respect to g and g(Y, Z) = §(Y, Z). Thus an orthonormal basis for £ is also an orthonormal basis for g. As £ is not unique, there are in fact many

Lorentz metrics on Jt if there is one. Conversely, if g is a given Lorentz metric, consider the equation ga\,Xb = AcjabXb where £ is any positive definite metric. This will have one negative and (n—1) positive eigenvalues. Thus the eigenvector field X corresponding to the negative eigenvalue will locally be a vector field determined up to a sign and a normalizing factor; one can normalize it by gab XaXb = — 1, so defining a line element field on

In fact, any non-compact manifold admits a line element field, while a compact manifold does so if and only if its Euler invariant is zero (e.g. the torus T2 does, but the sphere S2 does not, admit a line element field). It will later turn out that a manifold can be a reasonable model of space-time only if it is non-compact, so there will exist many Lorentz metrics on ^.

So far, the metric tensor and connection have been introduced as separate structures on . However given a metric g on , there is a unique torsion-free connection on defined by the condition: the covariant derivative of g is zero, i.e.

With this connection, parallel transfer of vectors preserves scalar products defined by g, so in particular magnitudes of vectors are invariant. For example if 8j8t is the tangent vector to a geodesic, then g(8l8t, 8/8t) is constant along the geodesic. From (2.25) it follows that

+<7(Y, Vx Z) = <7(VxY, Z)+<7(Y, Vx Z) holds for arbitrary C1 vector fields X, Y, Z. Adding the similar expression for Y(g(Z, X)) and subtracting that for Z(g(K, Y)) shows g{ Z, VXY) = K-^(X,Y))+ Y(g(Z,X)) + X(g(Y,Z))

+ g{ Z, [X, Y]) + <7(Y, [Z, X]) —<7(X, [Y, Z])}. Choosing X, Y, Z as basis vectors, one obtains the connection components ^ = ^(Ea, VBt Ec) = 9adr«bc in terms of the derivatives of the metric components gab = gr(Ea, Eb), and the Lie derivatives of the basis vectors. In particular, on using a coordinate basis these Lie derivatives vanish, so one obtains the usual Christoffel relations ra6e = mj^ + 8gjdx» - 8gJ8x"} (2.26)

for the coordinate components of the connection.

From now on we will assume that the connection on^is the unique Cr-1 torsion-free connection determined by the Cr metric g. Using this connection, one can define normal coordinates (§2.5) in a neighbourhood of a point q using an orthonormal basis of vectors at q. In these coordinates the components gab of g at q will be + and the components r^ of the connection will vanish at q. By 'normal coordinates', we shall in future mean normal coordinates defined using an orthonormal basis.

The Riemann tensor of the connection defined by the metric is a Cr~2 tensor with the symmetry

in addition to the symmetries (2.21); as a consequence of (2.21) and (2.27a), the Riemann tensor is also symmetric in the pairs of indices {ah}, {cd}, i.e. R^R^. (2.276)

This implies that the Ricci tensor is symmetric:

The curvature scalar R is the contraction of the Ricci tensor:

With these symmetries, there are ^w2(n2—1) algebraically independent components of Rabcd, where n is the dimension ofM; \n{n +1) of them can be represented by the components of the Ricci tensor. If n= 1, Rabcd = 0; if n = 2 there is one independent component of Rabcd> which is essentially the function R. If n = 3, the Ricci tensor completely determines the curvature tensor; if n > 3, the remaining components of the curvature tensor can be represented by the Weyl tensor Cabcd, defined by

2 2 Cabcd = Rabcd + ^Tjj {ydd+ 9blc^Sia} + _ 1) (n _ 2) ^dcdiib-

As the last two terms on the right-hand side have the curvature tensor symmetries (2.21), (2.27), it follows that Cabcd also has these symmetries. One can easily verify that in addition,

Cabad = 0, i.e. one can think of the Weyl tensor as that part of the curvature tensor such that all contractions vanish.

An alternative characterization of the Weyl tensor is given by the fact that it is a conformal invariant. The metrics g and § are said to be conformai if g=fi2g (2 2g)

for some non-zero suitably differentiable function ft. Then for any vectors X, Y, V, W at a point p, g(X, Y) _ 3(X,Y) <7(V,W) 0(V,W)'

so angles and ratios of magnitudes are preserved under conformal transformations; in particular, the null cone structure in Tp is preserved by conformal transformations, since ff(X,X) > 0, = 0, < 0 =>0(X,X) > 0, = 0, < 0, respectively. As the metric components are related by

= &gab, §°b = n-V6, the coordinate components of the connections defined by the metrics (2.28) are related by

Calculating the Riemann tensor of one finds where fi»6: = 2(0-i).c(0-i).d^6;

the covariant derivatives in this equation are those determined by the metric g. Then (assuming n > 2)

the last equation expressing the fact that the Weyl tensor is con-formally invariant. These relations imply

£ = n-2i2-2(n-l)n-3ii:cd^d-(n-l)(n-4)ii-4n;ci2.,j^. (2.30)

Having split the Riemann tensor into a part represented by the Ricci tensor and a part represented by the Weyl tensor, one can use the Bianchi identities (2.22) to obtain differential relations between the Ricci tensor and the Weyl tensor: contracting (2.22) one obtains

and contracting again one obtains

From the definition of the Weyl tensor, one can (if n > 3) rewrite (2.31) in the form

If n ^ 4, (2.31) contain all the information in the Bianchi identities (2.22), so if n = 4, (2.32) are equivalent to these identities.

A diffeomorphism <f>: will be said to be an isometry if it carries the metric into itself, that is, if the mapped metric g is equal to g at every point. Then the map preserves scalar products, as jrtX.YJl, = <7(0* X, Y)| = <7(0* X^Y)^,.

If the local one-parameter group of diffeomorphisms generated by a vector field K is a group of isometries (i.e. for each t, the transformation is an isometry) we call the vector field K a Killing vector field. The Lie derivative of the metric with respect to K is iKg = lim^(g-i6i*g) = 0, ¿-►o t since g = g for] each t. But from (2.17), L^gab = 2Kia.b), so a Killing vector field K satisfies Killing's equation

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