ñòð. 3 |

other words, can we find a set of coordinates x a such that 2 gab / x c x d P = 0?

This condition consists of N 2 N + 1 2 /4 independent equations, but we have only

N 2 N + 1 N + 2 /6 free values in 2 gab / x c x d P , so these equations cannot

in general be satisfied. This means that there are N 2 N 2 âˆ’ 1 /12 â€˜degrees of

freedomâ€™ among the second derivatives 2 gab / x c x d P , i.e. in general at least

this number of second derivatives will not vanish.

Although we have shown, in principle, that it is always possible to define local

Cartesian coordinates at any given point P, we have not shown explicitly how to

find such coordinates. We will return to this point in Chapter 3.

2.12 Tangent spaces to manifolds

To aid our intuition of local Cartesian coordinates, it is useful to consider the

simple example of a two-dimensional Riemannian manifold, which we can often

45

2.13 Pseudo-Riemannian manifolds

P

TP

Figure 2.5 The tangent plane TP to the curved surface at the point P.

consider as a generally curved surface embedded in three-dimensional Euclidean

space. A simple example is the surface of a sphere, shown in Figure 2.2. As we

have shown, at any arbitrary point P we can find coordinates x and y (say) such

that in the neighbourhood of P we have

ds2 = dx2 + dy2

It thus follows that a Euclidean two-dimensional space (a plane) will match the

manifold locally at P. This Euclidean space is called the tangent space TP to the

manifold at P. In other words, in terms of our embedding picture a plane can

always be drawn at any arbitrary point on a two-dimensional Riemannian surface

in such a way that it is locally tangential to the surface (see Figure 2.5). Although

the tangent plane to a surface at P gives a useful way of visualising the tangent

space of a manifold at a point, this view can be misleading. As we stressed

earlier, a manifold should be regarded as an entity in itself: there is no need for

a higher-dimensional space in which it and its tangent spaces are embedded.

We may extend the idea of tangent spaces to higher dimensions. At an arbitrary

point P in an N -dimensional Riemannian manifold we can find a coordinate

system such that in the neighbourhood of P the line element is Euclidean. Thus,

an N -dimensional Euclidean space matches the manifold locally at P. Just as each

point P of an embedded two-dimensional surface has its tangent plane, making

contact with the surface at P, so each point P of a manifold has a tangent space

TP attached to it.

2.13 Pseudo-Riemannian manifolds

Thus far we have confined our attention almost exclusively to strictly Rieman-

nian manifolds, in which ds2 > 0 always. In a pseudo-Riemannian manifold,

however, ds2 can be either positive, negative or zero and it is therefore much

46 Manifolds and coordinates

harder to visualise even two-dimensional manifolds of this type. Nevertheless,

the mathematical tools we have developed so far are straightforwardly applied to

pseudo-Riemannian manifolds with little modification.

The simplest way to understand pseudo-Riemannian manifolds is to consider

the transformation to local â€˜Cartesianâ€™ coordinates at some arbitrary point P. You

will notice from Section 2.11 that our argument showing that the condition (2.20)

holds for the derivatives of the metric functions in a Riemannian manifold can

be extended immediately to the pseudo-Riemannian case. Let us assume that

the coordinate system xa already satisfies this condition. However, the condition

(2.19) on the values of the metric functions themselves requires further investi-

gation. Let us now attempt to obtain a new coordinate system x a in which (2.19)

is also satisfied. We note in passing that, in order for (2.20) to remain valid, the

new coordinates x a must be related to the old ones xa by a linear transformation,

x a = X a b xb , where the X a b are constants.

In general, at a point P the metric functions in the new coordinate system are

given in terms of the original metric functions by

xc xd

gab P = (2.21)

gcd P

xa xb

P P

Let us define symmetric matrices G and G having elements gab P and gab P

respectively. Similarly, we can define a matrix X having elements xa / x b P .

Then, in matrix notation, (2.21) can be written as

G = XT GX

Since G is symmetric, it can be diagonalised by this similarity transformation,

provided that we choose the columns of X to be the normalised eigenvectors

of G. Then G = diag 1 2 N , where a is the ath eigenvalue of G (the

eigenvalues must all be real).

In a strictly Riemannian manifold, ds2 = gab dxa dxb is always positive at any

point P. Thus the matrix G â‰¡ gab at any point must be positive definite, i.e. all

its eigenvalues must be positive. At an arbitrary point in a pseudo-Riemannian

manifold, however, ds2 can be positive, negative or zero, depending on the

direction in which one moves from P. Correspondingly, some of the eigenvalues

of G are negative.

If we now scale our coordinates according to x a â†’ x a / a (note that here

there is no sum on a), we obtain at the point P

G = diag Â±1 Â±1 Â±1

where the + and âˆ’ signs depend on whether the corresponding eigenvalue is

positive or negative. Thus, at any arbitrary point P in a pseudo-Riemannian

47

2.14 Integration over general submanifolds

manifold, it is always possible to find a coordinate system x a such that in the

neighbourhood of P we have

gab x = ab + x âˆ’ xP 2

where ab = diag Â±1 Â±1 Â±1 . The number of positive entries minus the

number of negative entries in ab is called the signature of the manifold and is

the same at all points.

It follows that, at any arbitrary point P in a pseudo-Riemannian manifold, an

N -dimensional space with line element

ds2 = Â± dx1 2 Â± dx2 2 Â± Â· Â· Â· Â± dxN 2

will match the manifold locally at P. Such a space is called pseudo-Euclidean and

is the tangent space TP to the pseudo-Riemannian manifold at P. An example of a

pseudo-Euclidean space is the four-dimensional Minkowski spacetime of special

relativity, which has the line element

ds2 = d ct 2 âˆ’ dx2 âˆ’ dy2 âˆ’ dz2

when expressed in coordinates corresponding to a Cartesian inertial frame.

Minkowski spacetime thus has a signature of âˆ’2.

2.14 Integration over general submanifolds

In Section 2.10, we restricted our calculation of â€˜volumesâ€™ to coordinate systems

xa that were orthogonal and to submanifolds that were obtained simply by allowing

some of the coordinates to be constants. In fact neither of these simplifications is

necessary, and we are now in a position to consider the general case.

Let us begin by calculating the full N -dimensional volume element dN V in an

N -dimensional (pseudo-)Riemannian manifold. From Section 2.10, we know that

if we are working in an orthogonal coordinate system then this volume element

is given by

dN V = g11 g22 Â· Â· Â· gNN dx1 dx2 Â· Â· Â· dxN

For such a coordinate system the matrix G is given by

G â‰¡ gab = diag g11 g22 gNN

so that its determinant is simply the product of the diagonal elements,

det G = g11 g22 Â· Â· Â· gNN

48 Manifolds and coordinates

It is usual to denote det G simply by the symbol g. Thus, we may rewrite the

volume element as

dN V = g dx1 dx2 Â· Â· Â· dxN (2.22)

What we will now show is that this expression remains valid for an arbitrary

coordinate system.

The key to proving the general result (2.22) for the volume element at some

arbitrary point P in the manifold is to transform to local Cartesian coordinates

x a at P. We know that a small N -dimensional region at P will have volume

dN V = dx 1 dx 2 Â· Â· Â· dx N . In any other general coordinate system xa it is a well-

known result that

dx 1 dx 2 Â· Â· Â· dx N = J dx1 dx2 Â· Â· Â· dxN (2.23)

where the Jacobian factor J is given by

xa

J = det

xb

If, as in Section 2.13, we use X to denote the transformation matrix xa / x b then

J = det Xâˆ’1 = det X âˆ’1 . Defining matrices G and G as those having elements

gab P and gab P respectively, we have (see Section 2.13)

G = XT GX (2.24)

Taking determinants of both sides of (2.24) and denoting det G by g and det G

by g we obtain

1

g = det X 2 g = 2 g

J

Since the x a are locally Cartesian coordinates, G = diag Â±1 Â±1 Â±1 , where

the number of positive and negative signs depends on the signature of the manifold.

Thus we have g = Â±1, so that g = Â±J 2 . Hence, we obtain the required result:

dN V = dx 1 dx 2 Â· Â· Â· dx N = g dx1 dx2 Â· Â· Â· dxN

We now turn to the question how to integrate over submanifolds that are not

defined simply by setting some of the coordinates xa to be constant. Consider

some M-dimensional subspace of an N -dimensional manifold. In general, the

subspace can be defined by the N parametric equations

xa = xa u1 u2 uM

where the ui i = 1 2 M may be considered simply as a set of coordinates

that parameterise the subspace. If we consider two neighbouring points in the

49

2.15 Topology of manifolds

subspace whose parameters differ by dui then the coordinate separation between

these points is simply

xa i

dx = i du

a

u

Thus the distance ds between the points is given by

xa xb i j

ds = gab dx dx = gab

2 a b

du du

ui uj

which we may write as

ds2 = hij dui duj

where the hij are the induced metric functions on the subspace and are given by

xa xb

hij = gab (2.25)

ui uj

Thus we can now work simply in terms of this subspace and regard it as a

manifold in itself. Thus the volume element for integrals over this subspace is

given in terms of the parameters ui by

dM V = h du1 du2 Â· Â· Â· duM

where h = det hij .

It is also worth noting here that the relation (2.25) is the key to determin-

ing whether one can embed a given manifold in another manifold of higher

dimension. Suppose we begin with an M-dimensional manifold possessing the

metric hij u when labelled with the coordinates ui i = 1 2 M . In order to

embed this manifold in an N -dimensional manifold (where N > M) with metric

gab x in the coordinates xa a = 1 2 N , then one must be able to satisfy the

relation (2.25).

2.15 Topology of manifolds

In this chapter we have discussed only the local geometry of manifolds, which

is defined at any point by the line element (2.4) giving the distance between

points with infinitesimal coordinate separations. In addition to this local geometry

a manifold also has a global geometry or topology. The topology of a manifold

is defined by identifying certain sets of points, that is, regarding them as being

coincident. For example, in Figure 2.1, we identified the line AA with the line

BB . This property can be detected by a â€˜bugâ€™ on the surface, since by continuing

in a straight line in a certain direction, it can get back to where it started. Thus a

50 Manifolds and coordinates

topology (in this case the fact that the space is periodic in one of the coordinates)

is an intrinsic property of a manifold.

We shall see that general relativity is a â€˜localâ€™ theory, in which the local

geometry (or curvature) of the four-dimensional spacetime manifold at any point

is determined by the energy density of matter and/or radiation at that point. The

field equations of general relativity do not constrain the global topology of the

spacetime manifold.

Exercises

2.1 In three-dimensional Euclidean space R3 , write down expressions for the change of

coordinates from Cartesian coordinates xa = x y z to spherical polar coordinates

xa = r . Obtain expressions for the transformation and inverse transformation

matrices in terms of the primed coordinates. By calculating the Jacobians J and J for

the transformation and its inverse, find where the transformation is non-invertible.

2.2 Write down the line element for three-dimensional Euclidean space in spherical

polar coordinates xa and cylindrical polar coordinates x a . Hence identify the metric

functions in each coordinate system and show that they obey

xa xb

gcd x = gab x

xc xd

2.3 In three-dimensional Euclidean space a coordinate system x a is related to the Carte-

sian coordinates xa by

x1 = x 1 + x 2 x2 = x 1 âˆ’ x 2 x3 = 2x 1 x 2 + x 3

Describe the coordinate surfaces in the primed system. Obtain the metric functions

gab in the primed system and hence show that these coordinates are not orthogonal.

Calculate the volume element dV in the primed coordinate system.

2.4 Consider the surface of a 2-surface embedded in three-dimensional Euclidean space.

In a stereographic projection, one assigns coordinates to each point on the

surface of the sphere. The -coordinate is the standard azimuthal polar angle. The

-coordinate of each point is obtained by drawing a straight line in three dimensions

from the south pole of the sphere through the point in question and extending the line

until it intersects the tangent plane to the north pole of the sphere; the -coordinate

is then the distance in the tangent plane from the north pole to the intersection point.

Show that the line element for the surface of the sphere in these coordinates is

d2 2

ds2 = + 2

d

1 + 2 /a2 1+

2 2 /a2

At what point(s) on the sphere are these coordinates degenerate? If instead one works

in terms of the Cartesian coordinates x and y in the tangent plane at the north pole,

what is the corresponding form of the line element? At what point(s) on the sphere

are these new coordinates degenerate?

51

Exercises

2.5 Consider the surface of the Earth, which we assume for simplicity to be a 2-sphere

of radius a. In terms of standard polar coordinates , the longitude of a point,

in radians, rather than the usual degrees, is simply (measured eastwards from the

Greenwich meridian), whereas its latitude = /2 âˆ’ radians. Show that the line

element on the Earthâ€™s surface in these coordinates is

ds2 = a2 d + cos2 d

2 2

To make a map of the Earthâ€™s surface, we introduce the functions x = x and

y=y and use them as Cartesian coordinates on a flat rectangular piece of

paper. Each choice of the two functions corresponds to a different map projection.

The Mercator projection is defined by

W H

x= y= +

ln tan

2 2 4 2

where W and H are the width and height of the map respectively. Find the line

element for this projection.

2.6 For the general map projection discussed in Exercise 1.5, show that the angle between

two directions at some point on the Earthâ€™s surface will equal the angle between the

corresponding directions on the map, provided that the functions x and y are chosen

such that

x y dx2 + dy2 = a2 d + cos2

2 2

d

for some function x y . Show that the Mercator projection satisfies this condition.

Write down the general requirement on x and y for an equal-area projection, in

which the area of any region of the map is proportional to the corresponding area on

the Earthâ€™s surface. Find such a projection. Is it possible to obtain a projection that

simultaneously is equal-area and preserves angles?

2.7 A conformal transformation is not a change of coordinates but an actual change in

the geometry of a manifold such that the metric tensor transforms as

gab x =

Ëœ 2

x gab x

where x is some non-vanishing scalar function of position. In a pseudo-

Riemannian manifold, show that if xa is a null curve with respect to gab (i.e.

ds2 = 0 along the curve), then it is also a null curve with respect to gab . Is this true

Ëœ

for timelike curves?

2.8 A curve on the surface of a 2-sphere of radius a is defined parametrically by = u,

= 2u âˆ’ , where 0 â‰¤ u â‰¤ . Sketch the curve and show that its total length is

L=a 1 + 4 sin2 u du

0

Show that, in general, the length of a curve is independent of the parameter used to

describe it.

52 Manifolds and coordinates

2.9 Show that the line element of a 3-sphere of radius a embedded in four-dimensional

Euclidean space can be written in the form

ds2 = a2 d + sin2 + sin2 d

2 2 2

d

Hence, in this three-dimensional non-Euclidean space, calculate the area of the

2-sphere defined by = 0 . Also find the total volume of the three-dimensional

space.

2.10 Consider the three-dimensional space with line element

dr 2

ds = + r2 d + sin2 d

2 2 2

1 âˆ’ 2 /r

and calculate the following quantities:

area of a sphere of coordinate radius r = R;

(a) the

3-volume of a sphere of coordinate radius r = R;

(b) the

radial distance between the sphere r = 2 and the sphere r = 3 ;

(c) the

(d) the 3-volume contained between the two spheres in part (c).

â†’ 0.

Verify that your answers reduce to the usual Euclidean results in the limit

2.11 Prove the following results used in Section 2.11:

(a) xa / x b P has N 2 independent values;

(b) 2 xa / x b x c P has 2 N 2 N + 1 independent values;

1

(c) 3 xa / x b x c x d P has 1 N 2 N + 1 N + 2 independent values;

6

(d) gab P has 2 N N + 1 independent values;

1

(e) gab / x c P has 2 N 2 N + 1 independent values;

1

(f) 2 gab / x c x d P has 4 N 2 N + 1 2 independent values.

1

Hence show that, in a general Riemannian manifold, at least N 2 N 2 âˆ’ 1 /12 of the

second derivatives 2 gab / x c x d P will not vanish in any coordinate system.

2.12 Consider the two-dimensional space with line element

dr 2

ds = + r2 d

2 2

1 âˆ’ 2 /r

Using the result (2.25), show that this geometry can be embedded in three-

dimensional Euclidean space, and find the equations for the corresponding two-

dimensional surface.

2.13 By identifying a suitable coordinate transformation, show that the line element

ds2 = c2 âˆ’ a2 t2 dt2 âˆ’ 2at dt dx âˆ’ dx2 âˆ’ dy2 âˆ’ dz2

where a is a constant, can be reduced to the Minkowski line element.

3

Vector calculus on manifolds

The notion of a vector is extremely useful in describing physical processes and is

employed in nearly all branches of mathematical physics. The reader should be

familiar with vector calculus in two- and three-dimensional Euclidean spaces and

with the description of vectors in terms of their components in simple coordinate

systems such as Cartesian or spherical polar coordinates.

The concept of vectors is also very useful in both special and general relativity,

and we now consider how to generalise our familiar Euclidean ideas in order to

define vectors in a general (pseudo-)Riemannian manifold and in arbitrary coor-

dinate systems. For illustration, however, we will often consider two-dimensional

Riemannian manifolds that can be envisaged as surfaces embedded in three-

dimensional Euclidean space. An example is the surface of a sphere, which we

might take to be the surface of the Earth (remembering to consider ourselves as

truly two-dimensional â€˜bugsâ€™!).

3.1 Scalar fields on manifolds

Before considering vector fields on manifolds, let us briefly discuss scalar fields.

A real (or complex) scalar field defined on (some region of) a manifold assigns

a real (or complex) value to each point P in (that region of) ; an example is

the air temperature on the surface of the Earth. If one labels the points in

using some coordinate system xa then one can express the value at each point

as a function of the coordinates xa . The value of the scalar field at any point

P does not depend on the chosen coordinate system. Thus, under an arbitrary

coordinate transformation xa â†’ x a , the scalar field is described by a different

x a of the new coordinates, such that

function

xa = xa

Indeed, this is the defining characteristic for a scalar field.

53

54 Vector calculus on manifolds

3.2 Vector fields on manifolds

A vector field defined on (some region of) a manifold assigns a single vector

to each point P in (that region of) . The vector at P is often drawn as an

extended directed line segment with its base at P, but this convention requires

careful interpretation on general manifolds. Once again it is convenient to illustrate

our discussion by considering a two-dimensional manifold such as the spherical

surface of the Earth. Let us consider, for example, the vector field defined by the

wind velocity (at ground level). Wind velocity is measured at a given observation

point and refers solely to that point, despite the visual convenience of showing it

on a chart as an arrow apparently extending for a long distance. It is an example

of a local vector. Other examples include momentum, current density and velocity

in general. Such vectors are defined at a given point P. More accurately, they can

be measured by an observer (bug) in a laboratory covering a small region of the

manifold in the neighbourhood of P.

At an arbitrary point P in the manifold, any local vector v lies in the tangent

space TP to the manifold at P. Indeed, TP consists of the set of all (local) vectors

at the point P. This may be visualised simply for two-dimensional manifolds by

embedding them as surfaces in three-dimensional Euclidean space (see Figure 3.1),

but the idea is easily extended to higher dimensions and can be defined indepen-

dently of any embedding. As we discussed in Chapter 2, the tangent space at any

point of a (pseudo-)Riemannian manifold is a (pseudo-)Euclidean space of the

same dimensionality. Moreover, at an arbitrary point P, local vectors obey all the

usual rules of vector algebra in (pseudo-)Euclidean geometry.

It is important to realise, however, that local vectors defined at different points

P and Q in the manifold lie in different tangent spaces. Thus there is no way of

adding local vectors at different points. Other notions that must be abandoned are

those of position vectors and displacement vectors, which clearly are not locally

P

TP

Figure 3.1 Local vectors defined at the point P lie in the tangent space TP to

the manifold at that point.

55

3.3 Tangent vector to a curve

Q

P

Figure 3.2 The displacement vector between two general points P and Q does

not lie in the manifold , unless the manifold is itself Euclidean.

defined. Using an embedding picture of a two-dimensional manifold, this is may be

visualised as shown in Figure 3.2. The â€˜displacement vectorâ€™ connecting the points

P and Q does not lie in the manifold and thus has no intrinsic geometrical meaning.

Heuristically, however, we can define the displacement vector s between two

nearby points P and Q, since this is a local quantity. In the limit Q â†’ P, the

vector s lies in the tangent space at P.

Clearly, if the original manifold is itself (pseudo-)Euclidean then the tangent

space at any point coincides with the manifold. Thus vectors defined at different

points in the manifold do lie in the same space, and the notions of position and

displacement vectors are valid. This reflects our common experience of vector

algebra.

3.3 Tangent vector to a curve

The most obvious example of a vector field defined on (a subregion of) a manifold

is the tangent vector to a curve , which is defined at each point along . The

notion of a tangent vector to a curve is also central to our subsequent development

of basis vectors, described below.

Consider a curve in an N -dimensional manifold. This curve may be described

by the N parametric equations xa u , where u is some general parameter that

varies along the curve. At any point P along , the tangent vector t to the curve,

with respect to the parameter value u, is defined as

s

t = lim (3.1)

u

uâ†’0

where s is the infinitesimal separation vector between the point P and some

nearby point Q on the curve corresponding to the parameter value u + u. Clearly

t will lie in the tangent space TP at the point P; this is illustrated in Figure 3.3.

56 Vector calculus on manifolds

t

TP P

Figure 3.3 The tangent vector t to the curve at a point P.

Although the heuristic approach we have adopted here is perfectly adequate for

our purposes, in a general manifold the formal mathematical device for construct-

ing the tangent vector to some curve at P is to identify t with the directional

derivative operator along the curve at that point. This is discussed further in

Appendix 3A and, in fact, enables one to give a precise mathematical meaning to

the general notion of a vector in a non-Euclidean manifold.

3.4 Basis vectors

As we have seen, a vector field on a manifold is defined simply by giving, in

a smooth manner, a prescription for a local vector v x at each point in the

manifold. At each point P the vector lies in the tangent space TP at that point. This

vector is a geometrical entity, defined independently of any coordinate system

with which we choose to label points in the manifold. Nevertheless, at each point

P we can define a set of basis vectors ea for the tangent space TP , the number of

such vectors being equal to the dimension of TP and hence of (how this may

be achieved will be discussed shortly). Any vector at P can then be expressed

as a linear combination of these basis vectors, provided that they are linearly

independent, which we will assume is always the case. Thus, we can express the

local vector field v x at each point in terms of basis vectors ea x defined at

each point:

v x = va x ea x

The numbers va x are known as the contravariant components of the vector field

v x in the basis ea x .

For any set of basis vectors ea x , we can define a second set of vectors called

the dual basis vectors. Instead of denoting the dual basis vectors by some other

kernel letter, it is the convention to denote a member of this second basis set by

57

3.4 Basis vectors

ea x . Although the positioning of the index may seem odd (not least because of

the possible confusion with powers), it enables effective use of the summation

convention that we shall adopt in due course. At any point P, the dual basis

vectors are defined by the relation

ea x Â· eb x = a

(3.2)

b

so that ea and ea form reciprocal systems of vectors.

The dual basis vectors at P also lie in the tangent space TP and form an

alternative basis for it.1 Thus, we can also express the local vector field v x at

each point as a linear combination of the dual basis vectors ea x defined at that

point:

v x = va x ea x

The numbers va x are known as the covariant components of the vector v x in

the basis ea x .

Using the relation (3.2) we can find simple expressions for the contravariant

and covariant components of a vector v. For example,2

v Â· ea = vb eb Â· ea = vb = va

a

b

where we have used the fact that b can be used to replace one index with another.

a

a = v Â· ea . Similarly, we may show that v = v Â· e . We now

Thus we may write v a a

consider how a set of basis vectors (and their duals) may be constructed at each

point P in the manifold.

Coordinate basis vectors

An obvious basis in which to describe local vectors is the coordinate basis. In any

particular coordinate system xa , we can define at every point P of the manifold

a set of N coordinate basis vectors

s

ea = lim (3.3)

xa

x â†’0

a

where s is the infinitesimal vector displacement between P and a nearby point Q

whose coordinate separation from P is xa along the xa coordinate curve. Thus

ea is the tangent vector to the xa coordinate curve at the point P. This set of

vectors provides a basis for the tangent space TP at the point P (see Figure 3.4).

âˆ—

1

More precisely, these vectors define the dual tangent space TP at P, but this subtlety need not concern us

here.

2

From now on we will no longer make explicit the dependence of the basis vectors and components on the

position x in the manifold, except where including this argument makes the explanation clearer.

58 Vector calculus on manifolds

x2

e2

e1

TP P

x1

Figure 3.4 The coordinate basis vectors ea at a point P in a manifold are the

tangent vectors to the coordinate curves in the manifold and form a basis for the

tangent space at P.

From the definition (3.3), we see that if two nearby points P and Q have

coordinates xa and xa + dxa respectively, where now we allow dxa to be non-zero

for all a, then their infinitesimal vector separation is given by

ds = ea x dxa (3.4)

We can use this expression to relate the inner product of the coordinate basis

vectors at some arbitrary point P to the value of the metric functions gab x at

that point. From (3.4), we have

ds2 = ds Â· ds = dxa ea Â· dxb eb = ea Â· eb dxa dxb

Comparing this with the standard expression ds2 = gab x dxa dxb , (2.4), for the

line element, we find that

ea x Â· eb x = gab x (3.5)

Thus, quite generally, in a coordinate basis the scalar product of two vectors is

given by

v Â· w = va ea Â· wb eb = gab va wb

If the basis ea x is dual to a coordinate basis ea x then the a-coordinate

distance between two nearby points separated by the displacement vector ds is

given by

dxa = ea Â· ds

Moreover, in this case we may use the dual basis vectors to define the quantities

g ab x = ea x Â· eb x (3.6)

59

3.5 Raising and lowering vector indices

which, as we will show, form the contravariant components of the metric tensor

and are in general different from the quantites gab x ; we will return to these

later.

Orthonormal basis vectors at a point

At any given point P in a manifold, it is often useful to define a set of orthonormal

Ë†

basis vectors ea in TP , which are chosen to be of unit length and orthogonal to

one another. This is expressed mathematically by the requirement that at P

Ë†Ë†

ea Â· eb = (3.7)

ab

where ab = diag Â±1 Â±1 Â±1 is the Cartesian line element of the tangent

space TP and depends on the signature of the (in general) pseudo-Riemannian

manifold (see Section 2.13). These orthonormal basis vectors need not be related

to any particular coordinate system that we are using to label the manifold,

although they can always be defined by, for example, giving their components

in a coordinate basis. Moreover, it is clear from (3.7) that the orthonormal basis

Ë†

vectors ea at P are in fact the coordinate basis vectors of a coordinate system for

which gab P = ab (or gab P = ab for a strictly Riemannian manifold).

3.5 Raising and lowering vector indices

Unless otherwise stated, we will assume that we are working with a coordinate

basis, as discussed above, and its dual. The contravariant and covariant compo-

nents in these bases are equally good ways of specifying a vector. The link

between them is found by considering the different ways in which one can write

the scalar product v Â· w of two vectors. First, we can write

v Â· w = va ea Â· wb eb = ea Â· eb va wb = gab va wb

where we have used the contravariant components of the two vectors. Similarly,

using the covariant components, we can write the scalar product as

v Â· w = va ea Â· wb eb = ea Â· eb va wb = g ab va wb

Finally, we could express the scalar product in terms of the contravariant compo-

nents of one vector and the covariant components of the other,

v Â· w = va ea Â· wb eb = va wb ea Â· eb = va wb = va wa

b

a

similarly, we could write

v Â· w = va ea Â· wb eb = va wb ea Â· eb = va wb = va wa

a

b

60 Vector calculus on manifolds

By comparing these four alternative expressions for the scalar product of two

vectors, we can deduce one of the most useful properties of the quantities gab and

g ab . Since gab va wb = va wa holds for any arbitrary vector v, it follows that

gab wb = wa

which illustrates the fact that the quantities gab can be used to lower an index.

In other words, we can obtain the covariant components of a vector from its

contravariant components. By a similar argument, we have

g ab wb = wa

so that the quantities g ab can be used to perform the reverse process of raising

an index. It is straightforward to show that the coordinate and dual basis vectors

themselves are related in an analogous way by

ea = gab eb ea = g ab eb

and

We will now prove the useful result that the matrix g ab containing the

contravariant components of the metric tensor is the inverse of the matrix gab that

contains its covariant components. Using the index-lowering and index-raising

action of gab and g ab on the components of an arbitrary vector v, we find that

= va = g ab vb = g ab gbc vc

ac

cv

but since v is arbitrary we must have

g ab gbc = a

(3.8)

c

Ëœ

Denoting the matrix gab by G and the matrix g ab by G, this equation

Ëœ Ëœ

can be written in matrix form as GG = I. Hence G and G are inverse matrices.

3.6 Basis vectors and coordinate transformations

Let us consider a coordinate transformation xa â†’ x a on a manifold. There is

a simple relationship between the coordinate basis vectors ea associated with

the coordinate system xa and the coordinate basis vectors ea associated with the

new system of coordinates x a . It can be found by considering the infinitesi-

mal displacement vector ds between two nearby points P and Q. Clearly, this

displacement cannot depend on the coordinate system being used, so we must have

ds = dxa ea = dx a ea

61

3.7 Coordinate-independent properties of vectors

Noting that dxa = xa / x b dx b , we find that at any point P the two sets of

coordinate basis vectors are related by

xb

ea = (3.9)

e

xa b

where the partial derivative is evaluated at the point P. Repeating this calculation

using the dual basis vectors, we find that

xa b

e = be

a

(3.10)

x

Using (3.9) and (3.10), we can now calculate how the components of any

general vector v must transform under the coordinate transformation. Since a

vector is a geometrical entity that is independent of the coordinate system, we

have (for example)

v = va ea = v a ea

So, the new contravariant components are given by

xa b xa b

v = e Â·v = b e Â·v â‡’ v = bv

a a a

x x

Similarly, the new covariant components are given by

xb xb

v a = ea Â· v = e Â·v â‡’ va = v

xb b xa b

3.7 Coordinate-independent properties of vectors

As we have seen, in a coordinate basis and its dual the scalar product v Â· w of two

vectors at each point P of the manifold can be written in four ways:

gab va wb = g ab va wb = va wa = va wa

Using the transformation properties of the metric coefficients gab and those of the

vector components, it is straightforward to show that these expressions yield the

same result in any other coordinate system.

In a strictly Riemannian manifold the scalar product is positive definite, which

means that gab va vb â‰¥ 0 for all vectors va , with gab va vb = 0 only if va = 0. In a

pseudo-Riemannian space, however, this condition is relaxed and leads to some

rather odd properties, such as the possibility of non-zero vectors having zero

62 Vector calculus on manifolds

length. We must therefore make definitions that allow us to deal with such prop-

erties in a way that extends and generalises familiar concepts in Euclidean space.

The length of a vector v is defined in terms of its components by

= g ab va vb = va v a

1/2 1/2 1/2

gab va vb

A unit vector has length unity. As remarked above, in a pseudo-Riemannian

manifold we can have va va 1/2 = 0 for va = 0, in which case the vector v is

described as null.

The angle between two non-null vectors v and w is defined by

va wa

cos =

vb vb 1/2 wc wc 1/2

In a pseudo-Riemannian manifold, this formula can lead to cos > 1, resulting

in a non-real value for .

Two vectors are orthogonal if their scalar product is zero. This definition makes

sense even if one or both of the vectors is or are null. In fact, a null vector is a

non-zero vector that is orthogonal to itself.

3.8 Derivatives of basis vectors and the affine connection

As we have said, local vectors at different points P and Q in a manifold lie

in different tangent spaces, so there is no way of adding or subtracting them.

In order to define the derivative of a vector field, however, one must compare

vectors at different points, albeit in the limit where the distance between the points

tends to zero. We will adopt here an intuitive approach that is sufficient for our

purposes in developing vector calculus on curved manifolds and provides a simple

geometrical picture. Specifically, on this occasion, we will assume the manifold

to be embedded in a higher-dimensional (pseudo-)Euclidean space, which thus

allows vectors at different points to be compared.3

In some arbitrary coordinate system xa on the manifold, let us consider the basis

vectors ea at two nearby points P and Q with coordinates xa and xa + xa respec-

tively (see Figure 3.5). In general, the basis vectors at Q will differ infinitesimally

from those at P, so that

ea Q = ea P + ea

3

It is worth noting that one can embed any four-dimensional torsionless (pseudo-)Riemannian manifold in

some (pseudo-)Euclidean space of sufficiently higher dimension; see, for example, J. Nash, The imbedding

problem for Riemannian manifolds, Annals of Mathematics 63, 20â€“63, 1956 and C. Clarke, On the global

isometric embedding of pseudo-Riemannian manifolds, Proceedings of the Royal Society A314, 417â€“28,

1970. Indeed, recent theoretical work on braneworld models suggests that our spacetime may indeed be

embedded in some higher-dimensional manifold! Alternatively, one can define the derivative of a vector field

on a general manifold without using an embedding picture, but in a rather more formal manner; see, for

example, R.M. Wald, General Relativity, University of Chicago Press, 1984.

63

3.8 Derivatives of basis vectors and the affine connection

ea(P)

ea(Q)

Q

P

Figure 3.5 The basis vectors ea P and ea Q lie in the tangent spaces to the

manifold at the points P and Q respectively.

The standard partial derivative of the basis vector is given by ea / xc in the

limit xc â†’ 0. In general, however, the resulting vector will not lie in the tangent

space to the manifold at P. We thus define the derivative in the manifold of the

coordinate basis vector by projecting into the tangent space at P,

ea ea

â‰¡ lim (3.11)

xc xc

x â†’0

c

TP

Now we can expand this derivative vector in terms of the basis vectors ea P at

the point P, and write

ea

= b

(3.12)

ac eb

xc

where the N 3 coefficients ac are known collectively as the affine connection or,

b

in older textbooks, the Christoffel symbol (of the second kind) at the point P.

From (3.11), it is also clear that the derivative operator obeys Leibnitzâ€™ theorem.

By taking the scalar product of (3.12) with the dual basis vector ed and using the

reciprocity relation (3.2), we can also write the affine connection as4

= eb Â· c ea

b

(3.13)

ac

Furthermore, by differentiating the reciprocity relation ea Â· eb = b with respect

a

to the coordinate xc , we find that

ea Â· eb = Â· eb + ea Â· =0

a

ce c eb

c

4

From now on, we shall often use the shorthand c to denote / xc . We also note here that, in some textbooks,

an even more terse notation is used, in which partial differentation is denoted by a comma. For example, the

partial derivative c va of the contravariant components of a vector would be written va c .

64 Vector calculus on manifolds

Then, on using (3.13), we find that the derivatives of the dual basis vectors with

respect to the coordinates are given by

=âˆ’

a a b

(3.14)

ce bc e

The expressions (3.12â€“3.14) will be used extensively in our subsequent discus-

sions.

3.9 Transformation properties of the affine connection

From the expression (3.13) for the affine connection,

eb

bc = e Â·

a a

(3.15)

xc

we see that, in some new coordinate system x a , it is given by

eb

bc = e Â·

a a

xc

Substituting the expressions (3.9) and (3.10) for the new basis and dual basis

vectors, we find

xa d xf

= deÂ· c

a

e

bc

xb f

x x

2 xf

xa d xf ef

= deÂ· + c b ef

xb xc

x xx

x a 2 xf

x a xf xg d ef

=d e Â· g+ d ed Â· ef

b xc c xb

xx x xx

x a 2 xd

x a xf xg

=d fg +

d

(3.16)

x xb xc xd x c x b

where in the last line we have used the reciprocity relation (3.2) between the

basis and dual basis vectors. We will see later that, because of the presence of

the last term on the right-hand side of (3.16), the a bc do not transform as the

components of a tensor.

By swapping derivatives with respect to x and x in the last term on the

right-hand side of (3.16), we arrive at an alternative (but equivalent) expression:

xd xf 2 x a

x a xf xg

bc = fg âˆ’

a d

(3.17)

xd x b x c x b x c xd xf

65

3.10 Relationship of the connection and the metric

3.10 Relationship of the connection and the metric

The observant reader will have noticed that there was some arbitrariness in how

we introduced the affine connection in (3.12). We could just as easily have written

(3.12) with b ac replaced by b ca , i.e. with the two subscripts interchanged. In

a general Riemannian manifold, these two sets of quantities are not necessarily

equal to one another. In fact, one can show that the quantities

T b ac = ac âˆ’

b b

(3.18)

ca

are the components of a third-rank tensor (see Chapter 4) called the torsion tensor.

For our considerations of standard general relativity, however, we can assume

that our manifolds are torsionless, so that T b ac = 0 in any coordinate system.5

Hence, from here onwards, we will assume (unless otherwise stated) that the

affine connection is symmetric in its last two indices, i.e.

=

b b

(3.19)

ac ca

In a manifold that is torsionless, so that (3.19) is satisfied, there is a simple

relationship between the affine connection b ac and the metric functions gab ,

which we now derive. From (3.5) we have gab = ea Â· eb . Differentiating this

expression with respect to xc , we obtain

= Â· eb + ea Â·

c ea c eb

c gab

= ac ed Â· eb + ea Â·

d d

bc ed

= ac gdb +

d d

(3.20)

bc gad

By cyclically permuting the indices a b c, we obtain two equivalent expressions,

= cb gda +

d d

b gca ab gcd

= ba gdc +

d d

a gbc ca gbd

Using these three expressions, we now form the combination

c gab + b gca âˆ’ a gbc

= ac gdb + bc gad + cb gda + ab gcd âˆ’ ba gdc âˆ’ =2

d d d d d d d

ca gbd cb gad

where, in obtaining the last line, we have used the assumed symmetry properties

(3.19) of the affine connection and the symmetry of metric functions. Multiplying

5

It is straightforward to show that any (pseudo-)Riemannian manifold that can be embedded in some (pseudo-)

Euclidean space of higher dimension must be torsionless.

66 Vector calculus on manifolds

through by g ea , recalling from (3.8) that g ea gad = e and relabelling indices, we

a

finally obtain

= 2 g ad b gdc + c gbd âˆ’ d gbc

1

a (3.21)

bc

In fact, the quantity defined by the right-hand side in (3.21) is properly called the

a

metric connection and is often denoted by the symbol bc . In a manifold with

torsion, it will differ from the affine connection defined by (3.11). As we have

shown, however, in a torsionless manifold the affine and metric connections are

equivalent, and so a bc is usually referred to simply as the connection. Unless

otherwise stated, we will follow this convention from now on.

Equation (3.21) is very important, because it tells us how to compute the

connection at any point in a manifold. In other words, if one knows the metric gab

in some coordinate system xa then one can form the derivatives of gab appearing

in (3.21) and hence calculate all the numbers a bc at any point.

We finish this section by establishing a few useful formulae involving the

connection a bc and the related quantities

â‰¡ gad d

abc bc

= g ad

a

It is straightforward to show that dbc . From (3.21), we find that

bc

= b gac + c gba âˆ’ a gbc

1

(3.22)

abc 2

The quantity abc is traditionally known as a Christoffel symbol of the first kind.

Adding bac to abc gives

= abc + bac (3.23)

c gab

which allows us to express partial derivatives of the metric components in terms

of the connection coefficients. If we denote the value of the determinant det gab

by g then the cofactor of the element gab in this determinant is ggab (note that

g is not a scalar: changing coordinates changes the value of g at any point). It

follows that c g = gg ab c gab , so from (3.23) we have

= gg ab abc + bac =g bc + = 2g

b a a

(3.24)

cg ac ac

The implied summation over a is an example of a contraction over a pair of

indices (see Chapter 4); a ac means simply 1 1c + 2 2c + Â· Â· Â· + N Nc . Thus the

contraction of the connection coefficients (3.21) is given by

= 2 g âˆ’1 b g =

1 1

a

2 b ln (3.25)

g

ab

67

3.11 Local geodesic and Cartesian coordinates

the modulus signs being needed if the manifold is pseudo-Riemannian. Alterna-

tively, we can write

1

= g=

a

b ln (3.26)

g

ab b

g

3.11 Local geodesic and Cartesian coordinates

In Chapter 2, we showed that at any point P in a pseudo-Riemannian manifold it

is possible in principle to find local Cartesian coordinates x a such that

gab P = (3.27)

ab

gab

=0 (3.28)

xc P

where ab = diag Â±1 Â±1 Â±1 . The number of positive entries in ab

minus the number of negative entries is the signature of the manifold. Supposing

that we start with some general system of coordinates xa , we now show how to

obtain local Cartesian coordinates in practice.

Let us begin by demanding that our new coordinate system x a satisfies the

condition (3.28) but not necessarily the condition (3.27). From our expression

(3.20) for the derivative of the metric in terms of the connection, we see that

the condition (3.28) will be satisfied if the connection coefficients in the new

coordinate system vanish at P, i.e.

P =0

a

(3.29)

bc

Conversely, from (3.21) we see that the condition (3.28) implies (3.29). The

condition (3.29) makes much simpler the mathematics of parallel transport, covari-

ant differentiation and intrinsic differentiation (see later). Coordinates for which

(3.29) holds are generally referred to as geodesic coordinates about P, but this is

not always appropriate since they need not be based on geodesics (which we will

also discuss later).

Suppose that we start with some arbitrary coordinate system xa , the â€˜originalâ€™

a

system, in which the point P has coordinates xP . Let us now define a new system

of coordinates x a by

x a = xa âˆ’ xP + 2 P xb âˆ’ xP xc âˆ’ xP

1

a b c

a (3.30)

bc

68 Vector calculus on manifolds

where the a bc P are the connection coefficients at P in the original coordinate

system. Clearly, the origin of the new coordinate system is at P. Differentiation

of (3.30) with respect to xd yields

xa

= d+ P xc âˆ’ xP

a a c

dc

xd

so that, at the point P, x a / xd = its inverse is given by xa / x d =

a a

d; d.

Differentiating again we obtain

2x a

= =

a c a

P P

dc de

e

xe xd

If we now substitute these results into the expression (3.17) for the transformation

properties of the connection, we find that

afgd

P= Pâˆ’ P= Pâˆ’ P =0

a df a a a

bc fg df bc bc

dbc bc

So in the new (primed) coordinate system the connection coefficients at P are

zero, and from (3.29) we have a system of geodesic coordinates at P.

The metric functions gab P in the geodesic coordinates x a will not necessarily

satisfy the condition (3.27). Nevertheless, we can obtain such a system of local

Cartesian coordinates by making a second linear coordinate transformation

= Xab x b

a

x

where the coefficients X a b are constants. Thus we can bring the metric gab P

in these coordinates into the form (3.27) without affecting its derivatives, so that

(3.28) will still be satisfied. The required values of the coefficients X a b were

discussed in Section 2.13.

3.12 Covariant derivative of a vector

Suppose that a vector field v x is defined over some region of a manifold. We

will consider the derivative of this vector field with respect to the coordinates

labelling the points in the manifold. Let us begin by writing the vector in terms

of its contravariant components v = va ea . We thus obtain

= ea + va

a

(3.31)

bv b ea

bv

where the second term arises because, in an arbitrary coordinate system, the coor-

dinate basis vectors vary with the position in the manifold. If we defined locally

Cartesian coordinates at some point P in the manifold then in the neighbourhood

of this point the coordinate basis vectors are constant and so the second term would

vanish at P (but not elsewhere, unless the manifold is (pseudo-)Euclidean, so

that the whole of can be covered by a Cartesian coordinate system).

69

3.12 Covariant derivative of a vector

Using (3.13), we may write (3.31) as

= ea + va

a c

bv ab ec

bv

Since a and c are dummy indices in the last term on the right-hand side, we may

interchange them to obtain

= ea + vc = + vc

a a a a

bv cb ea ea

bv bv cb

The reason for interchanging the dummy indices is that we may then factor out

ea . Thus, at any point P, we now have an expression for the derivative of a vector

field with respect to the coordinates in terms of the basis vectors of the coordinate

system at P. The quantity in brackets is called the covariant derivative of the

vector components, and the standard notation for it is6

â‰¡ +

a a a c

(3.32)

bv bv cb v

Thus the derivative of the vector field v can be written in the compact notation

= a

bv ea

bv

We note that, in local geodesic coordinates about some point P, the second term

in the covariant derivative (3.32) vanishes at P and thus reduces to the ordinary

partial derivative.

So far we have considered only the covariant derivative of the contravariant

components va of a vector. The corresponding result for the covariant components

va may be found in a similar way, by considering the derivative of v = va ea and

using (3.14) to obtain

= b va âˆ’

c

(3.33)

b va ab vc

Comparing the expressions (3.32) and (3.33) for the covariant derivatives of

the contravariant and covariant components of a vector respectively, we see that

there are some similarities and some differences. It may help to remember that

the index with respect to which the covariant derivative is taken (b in this case) is

also the last subscript on the connection; the remaining indices can then only be

arranged in one way without raising or lowering them. Finally, the sign difference

must be remembered: for a contravariant index (superscript) the sign is positive,

whereas for a covariant index (subscript) the connection carries a minus sign.

We conclude this section by considering the covariant derivative of a scalar.

The covariant derivative differs from the simple partial derivative only because

the coordinate basis vectors change with position in the manifold. However, a

6 a

In some textbooks, the covariant derivative is denoted by a semicolon, so that the covariant derivative bv

would be written as va b .

70 Vector calculus on manifolds

scalar does not depend on the basis vectors at all, so its covariant derivative

must be the same as its partial derivative, i.e.

= (3.34)

b b

3.13 Vector operators in component form

The equations of electromagnetism, fluid mechanics and many other areas of

classical physics make use of vector calculus in three-dimensional Euclidean

and the Laplacian 2 of scalar fields, together

space, employing the gradient

with the divergence Â· v and the curl Ã— v of a vector field. Explicit forms for

these are given in many texts for useful coordinate systems such as Cartesian,

cylindrical polar, spherical polar (typically the 11 coordinate systems in which

Laplaceâ€™s equation separates). The covariant derivative provides a unified picture

of all these derivatives and a direct route to the explicit forms in an arbitrary

coordinate system. Moreover, it allows for the generalisation of these operators

to more general manifolds.

Gradient

The gradient of a scalar field is given simply by

= ea = ea (3.35)

a a

since the covariant derivative of a scalar is the same as its partial derivative.

Divergence

Replacing the partial derivatives that occur in local Cartesian coordinates by

covariant derivatives, which are valid in arbitrary coordinate systems, the diver-

gence of a vector field is given by the scalar quantity

Â·v = = +

a a a b

av av ab v

Using the result (3.26) we can rewrite the divergence as

1

Â·v â‰¡ av =

a

g va (3.36)

a

g

where g is the determinant of the matrix gab .

71

3.14 Intrinsic derivative of a vector along a curve

Laplacian

in Â· v then we obtain the Laplacian 2 . From (3.35),

If we replace v by

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