we require the contravariant components va . These may be obtained by raising

the index with the metric to give

va = g ab vb = g ab b

Subtituting this into (3.36), we obtain

1

≡ =

2 a

g g ab

a a b

g

It is worth noting that the symbol used for the Laplacian operator often depends

on the dimensionality of the manifold being used. In particular, the triangular

(three-sided) symbol 2 that is commonly used in the three-dimensional (and

N -dimensional cases) is replaced by the box-shaped (four-sided) symbol 2 in

four-dimensional spacetimes, in which case it is called the d™Alembertian operator.

Curl

The special form of the curl of a vector field, which is itself a vector, exists

only in three dimensions. In its more general form, which is valid in higher

dimensions, the curl is defined as a rank-2 antisymmetric tensor (see Chapter 4)

with components

= a vb ’

curl v b va

ab

In fact this difference of covariant derivatives can be simplified, since

a vb ’ = a vb ’ ba vc ’ b va + = a vb ’ b va

c c

b va ab vc

where the connections have cancelled because of their symmetry properties.

3.14 Intrinsic derivative of a vector along a curve

Normally, we think of vector fields as functions of the coordinates xa defined

over some region of the manifold. However, we can also encounter vector fields

that are defined only on some subspace of the manifold, and an extreme example

occurs when the vector field v u is defined only along some curve xa u in the

manifold; an example might be the spin 4-vector s of a single particle along

its worldline in spacetime. We now consider how to calculate the derivative of

such a vector with respect to the parameter u along the curve.

72 Vector calculus on manifolds

Let us begin by writing the vector field at any point along the curve as

v u = va u ea u

where the ea u are the coordinate basis vectors at the point on the curve corre-

sponding to the parameter value u. Thus, the derivative of v along the curve is

given by

dv dva dva c

a ea dx

a dea

= e +v = e +v

du a du a xc du

du du

where we have used the chain rule to rewrite the last term on the right-hand side;

this is a valid procedure since the basis vectors ea are also defined away from

the curve . Using (3.13) to write the partial derivatives of the basis vectors in

terms of the connection, we obtain

dv dva c

a dx

= e+ b

eb

ac v

du a

du du

Interchanging the dummy indices a and b in the last term, we may factor out the

basis vector, and we find that

dva c Dva

dv b dx

= + ea ≡

a

(3.37)

e

bc v

Du a

du du du

The term in parentheses is called the intrinsic (or absolute) derivative of the

components va along the curve and is often denoted by Dva /Du as indicated.

Similarly, the intrinsic derivative of the covariant components va of a vector is

given by

dxc

Dva dva

= ’ b

ac vb

Du du du

A convenient way to remember the form of the intrinsic derivative is to pretend

that the vector v is in fact defined throughout (some region of) the manifold, i.e.

not only along the curve . In some cases of interest, this may in fact be true

anyway; for example, v might denote the 4-velocity of some distributed fluid. We

can now differentiate the components va (say) with respect to the coordinates xa .

Thus we can write

dva va dxc

=c

du x du

Substituting this into (3.37), we can then factor out dxc /du and recognise the

other factor as the covariant derivative c va . Thus we can write

Dva dxc

= a

(3.38)

cv

Du du

73

3.15 Parallel transport

and similarly for the intrinsic derivatives of the covariant components. It must be

remembered, however, that if v is only defined along the curve then formally

(3.38) is not defined and acts merely as an aide-memoire.

3.15 Parallel transport

Let us again consider some curve in the manifold, given parameterically in

some general coordinate system by xa u . Moreover, let O be some initial point

on the curve with parameter u0 at which a vector v is defined. We can now think

of ˜transporting™ v along in such a way that

dv

=0 (3.39)

du

is satisfied at each point along the curve. The result is a ˜parallel™ field of vectors

at each point along , generated by the parallel transport of v.

In a (pseudo-)Euclidean manifold, the parallel transport of a vector has the

simple geometrical interpretation that the vector v is transported without any

change to its length or direction. This is illustrated in Figure 3.6 for a curve

in a two-dimensional Euclidean space (i.e. a plane). If the coordinates xa are

Cartesian, it is clear that the components va of the vector field satisfy

dva

=0 (3.40)

du

v

O

Figure 3.6 A parallel field of vectors v u generated by parallel transport along

a curve parameterised by u.

74 Vector calculus on manifolds

In an arbitrary coordinate system in the plane, however, (3.40) is no longer valid,

and from (3.37) we see that it must be generalised to

Dva dva c

a b dx

≡ + =0 (3.41)

bc v

Du du du

From the basic requirement (3.39), it is clear that (3.41) is equally valid for the

parallel transport of a vector along a curve in any (pseudo-)Riemannian manifold

in some arbitrary coordinate system xa , although the geometrical interpretation is

more subtle in this case. If one is willing to adopt a picture in which the (pseudo-)

Riemannian manifold is embedded in a (pseudo-)Euclidean space of sufficiently

higher dimension, then one can recover a simple geometrical interpretation of

parallel transport. Consider some curve in the (pseudo-)Riemannian manifold

given in terms of some coordinate system in the manifold by xa u . Let P and

Q be two neighbouring points on the curve with affine parameter values u and

u + u respectively. Starting with the vector v at P, which lies in the tangent

space TP , shift the vector to the neighbouring point Q while keeping it parallel

to itself. In a Euclidean embedding space, this simply means transporting the

vector without changing its length or direction. At the point Q the vector will

not, in general, lie in the tangent space TQ , on account of the curvature of the

embedded manifold. Nevertheless, by considering only that part of the vector

that is tangential to the embedded manifold at Q, we obtain a definite vector

lying in TQ . It is straightforward to show that this vector coincides with the

parallel-transported vector at Q according to (3.41).

If we rewrite (3.41) as

dva c

b dx

=’ a

(3.42)

bc v

du du

then we can see that, if we specify the components va at some arbitrary point

along the curve, equation (3.43) fixes the components of va along the entire

length of the curve. If you are worried about whether the transportation is really

parallel, simply consider an infinitesimal displacement of the vector from some

point P. For a small displacement we can choose locally Cartesian coordinates at

P, in which the s vanish, and so setting the covariant derivative equal to zero

describes an infinitesimal displacement which keeps the vector parallel (dva = 0).

We note here that, in at least one respect, parallel transport along curves in a

general (pseudo-)Riemannian manifold is significantly different from that along

curves in a (pseudo-)Euclidean space, in that it is path dependent: the vector

obtained by transporting a given vector from a point P to a remote point Q

depends on the route taken from P to Q. This path dependence is also apparent

in transporting a vector around a closed loop, where on returning to the starting

75

3.16 Null curves, non-null curves and affine parameters

point the direction of the transported vector is (in general) different from the

vector™s initial direction. This path dependence can be demonstrated on a curved

two-dimensional surface, and in general can be expressed mathematically in terms

of the curvature tensor of the manifold. We will return to this topic in Chapter 7.

3.16 Null curves, non-null curves and affine parameters

So far, we have treated all curves in a manifold on a equal footing. In pseudo-

Riemannian manifolds, however, it is important to distinguish between null curves

and non-null curves. In the former, the interval ds between any two nearby points

on the curve is zero, whereas in the latter case ds is non-zero. The distinction

between these two types of curve may also be defined in terms of their tangent

vectors, and this leads to the identification of a class of privileged parameters,

called affine parameters, in terms of which the curves may be defined.

Consider some curve xa u in a general manifold. As discussed earlier, the

tangent vector t to the curve at some point P, with respect to the parameter value

u, is defined by (3.1). In a given coordinate system, we can write s = ea xa ,

where the ea are coordinate basis vectors at P. We then obtain

dxa

t= (3.43)

e

du a

From this expression, we see that the length of the tangent vector t to the curve

xa u at the point P is given by

1/2 1/2

dxa dxb gab dxa dxb ds

t = gab t t = gab = =

a b 1/2

du du du du

where ds is the distance measured along the curve at P that corresponds to the

parameter interval du along the curve.

A non-null curve is one for which the tangent vector at every point is not null,

i.e. t = 0. For such a curve, the length of the tangent vector at each point depends

on the parameter u and, in general, can vary along the curve. However, we see

that if the curve is parameterised in terms of a parameter u that is related to the

distance s measured along the curve by u = as + b, where a and b are constants,

with a = 0, then the length of the tangent vector will be constant along the curve.

In this case u is called an affine parameter along the curve. Moreover, if we take

u = s then the tangent vector (with components dxa /ds) is always of unit length.

A null curve is one for which the tangent vector is null, t = 0, at every point

along the curve; equivalently, the distance ds between any two points on a null

curve is zero. Since s does not vary along the curve, we clearly cannot use it as

76 Vector calculus on manifolds

a parameter. We are, however, free to use any other non-zero scalar parameter u

that does vary along the curve. Moreover, even for null curves it is still possible

to define a privileged family of affine parameters. The definition of an affine

parameter for a null curve is best introduced through the study of geodesics.

3.17 Geodesics

A geodesic in Euclidean space is a straight line, which has two equivalent defining

properties. First, its tangent vector always points in the same direction (along

the line) and, second, it is the curve of shortest length between two points. We

can use generalisations of either property to define geodesics in more general

manifolds. The fixed direction of their tangent vectors can be used to define both

non-null and null geodesics in a pseudo-Riemannian manifold, whereas clearly

the extremal length can only be used to define non-null geodesics. In a manifold

that is torsionless (so that (3.19) is satisfied) these two defining properties are

equivalent, for non-null geodesics, and lead to the same curves.7

Let us begin by characterising a geodesic as a curve xa u described in terms

of some general parameter u by the fixed direction of its tangent vector t u . The

equations satisfied by the functions xa u are thus determined by the requirement

that, along the curve,

dt

= (3.44)

ut

du

where u is some function of u. From (3.41), we see that the components ta of

the tangent vector in the coordinate basis must satisfy

Dta dta c

b dx

= + =

a

u ta

bc t

Du du du

Since the components of the tangent vector are ta = dxa /du we find that the

equations satisfied by a geodesic are

d 2 xa dxb dxc dxa

+ =

a

(3.45)

u

bc

du2 du du du

Equation (3.45) is valid for both null and non-null geodesics parameterised

in terms of some general parameter u. If the curve is parameterised in such a

way that u vanishes, however, then u is a privileged parameter called an

affine parameter. From (3.44), we see that this corresponds to a parameterisation

7

In a manifold with torsion, the two properties lead to different curves: a curve whose length is stationary with

respect to small variations in the path is called a metric geodesic, whereas a curve whose tangent vector is

constant along the path is an affine geodesic.

77

3.18 Stationary property of non-null geodesics

in which the tangent vector is the same at all points along the curve (i.e. it is

parallel-transported), so that

Dta

dt

=0 ’ =0 (3.46)

du Du

The equations satisfied by an affinely parameterised geodesic are thus

d 2 xa dxb dxc

+ =0

a

(3.47)

bc

du2 du du

Since one is always free to choose an affine parameter, we shall henceforth

restrict ourselves to this simplified form. In particular, for non-null geodesics

a convenient affine parameter is the distance s measured along the curve. The

geodesic equation (3.47) is one of the most important results for our study of

particle motion in general relativity.

Finally, we note how affine parameters are related to one another. If we change

the parameterisation from an affine parameter u to some other parameter u then

the functions xa u describing in terms of the new parameter will differ from

the original functions xa u . If, for some arbitrary new parameter u , we rewrite

(3.47) in terms of derivatives with respect to u then the geodesic equation does

not, in general, retain the form (3.47) but instead becomes

d 2 xa d2 u/du 2

dxb dxc dxa

+ =

a

(3.48)

bc

du 2 du du du/du du

It is clear from (3.48) that if u is an affine parameter then so too is any linearly

related parameter u = au + b, where a and b are constants (i.e. they do not

depend on position along the curve) and a = 0.

3.18 Stationary property of non-null geodesics

Let us now consider non-null geodesics as curves of extremal length between two

fixed points A and B in the manifold. Suppose that we describe the curve xa u

in terms of some general (not necessarily affine) parameter u. The length along

the curve is

B B

L= ds = ™™ 1/2

gab xa xb du

A A

where the overdot is a shorthand for d/du. Now consider the variation in path

xa u ’ xa u + xa u , where A and B are fixed. The requirement for xa u to be

a geodesic is that L = 0 with respect to the variation in the path. This is a calculus-

of-variations problem, (3.66), in which the integrand F = s = gab xa xb 1/2 .

™ ™™

78 Vector calculus on manifolds

If we substitute this form for F directly into the Euler“Lagrange equa-

tions (3.67), i.e.

d F F

’ c =0

™

xc

du x

then we obtain

™

gac xa 1

d

’ xa xb = 0

™™ (3.49)

c gab

™ 2™

du s s

Noting that gac =

™ ™

xb , the u-derivative is given by

b gac

™ ¨

gac xa 1

d s

= gac xa xb + gac xa ’ gac xa

™™ ¨ ™

b

™ ™ ™

du s s s

Substituting this expression back into (3.49) and rearranging yields

¨

s

gac xa +

¨ b gac x x ’ 2

™a ™b 1 c gab x x =

™a ™b ™

gac xa (3.50)

™

s

b gac x x = 2 b gac +

™a ™b 1

By interchanging dummy indices, we can write

™a ™b dc

a gbc x x . Substituting this into (3.50), multiplying the whole equation by g

and remembering that g dc gac = d , we find that

c

¨

sd

xd + 2 g dc

¨ b gac + a gbc ’ c gab xa xb =

™™ ™

1

x

™

s

Finally, using the expression (3.21) for the connection in terms of the metric and

relabelling indices, we obtain

¨

sa

xa +

¨ bc x x =

™b ™c ™

a

(3.51)

x

™

s

Comparing this equation with (3.48) we see that the two are equivalent. We also

see that, for a non-null geodesic, an affine parameter u is related to the distance

s measured along the curve by u = as + b, where a and b are constants a = 0 .

3.19 Lagrangian procedure for geodesics

In order to obtain the parametric equations xa = xa u of an affinely parameterised

geodesic, we must solve the system of differential equations (3.47). Bearing in

mind that the equations (3.21), which define the a bc , are already complicated,

it would seem a formidable procedure to set up the geodesic equations, let alone

solve them. Nevertheless, in the previous section we found that the equations

79

3.19 Lagrangian procedure for geodesics

for a non-null geodesic arise very naturally from a variational approach. Looking

back at the derivation of (3.51), however, we note this requires that s = 0. Thus

™

the proof is not valid for null geodesics. Fortunately, it is possible to set up a

variational procedure which generates the equations of an affinely parameterised

geodesic and which remains valid for null geodesics. This very neat procedure

also produces the connection coefficients a bc as a spin-off.

In standard classical mechanics, one can describe a system in terms of a set of

generalised coordinates xa that are functions of time t. These coordinates define

a space with a line element

ds2 = gab dxa dxb

which, in classical mechanics, is called the configuration space of the system. One

can form the Lagrangian for the system from the kinetic and potential energies,

L = T ’ V = 2 gab xa xb ’ V x

™™

1

where xa ≡ dxa /dt. By demanding that the action

™

tf

S= Ldt

ti

is stationary with respect to small variations in the functions xa t , the equations

of motion of the system are then found as the Euler“Lagrange equations

d L L

’ a =0

™

xa

dt x

This should all be familiar to the reader (but is discussed in more detail in

Chapter 19). Less familiar, perhaps, is how the equations of motion look if we

write them out in full:

xa +

¨ ™ x = ’g ab b V

™

a bc

bc x

These are just the equations of an affinely parameterised geodesic with a force

term on the right-hand side. In this case, the a bc are the metric connections of

the configuration space. If the forces vanish then Lagrange™s equations say that

˜free™ particles move along geodesics in the configuration space.

Thus, by analogy, in an arbitrary pseudo-Riemannian manifold we may obtain

the equations for an affinely parameterised (null or non-null) geodesic xa u by

considering the ˜Lagrangian™

L = gab xa xb

™™

80 Vector calculus on manifolds

where xa ≡ dxa /du and we have omitted the irrelevant factor 2 . As can be shown

™ 1

directly, substituting this Lagrangian into the Euler“Lagrange equations

d L L

’ a =0 (3.52)

™

xa

du x

yields, as required,

xa +

¨ ™ x =0

™

a bc

(3.53)

bc x

Performing this calculation, one finds that nowhere does it require s = 0 and so

™

is valid for both null and non-null geodesics. Thus the Euler“Lagrange equations

provide a useful way of generating the geodesic equations, and the connection

coefficients may be extracted from the latter.

We note that, in seeking solutions of the geodesic equations (3.53), it often

helps to make use of the first integral of the equations. For null geodesics, the

first integral is simply

gab xa xb = 0

™™ (3.54)

whereas, for non-null geodesics, if we choose the parameter u = s then

gab xa xb = 1

™™ (3.55)

These results can prove extremely useful in solving the geodesics equations.

Demonstrating the equivalence of the geodesic and Euler“Lagrange equations

allows us to make a useful observation. If the gab do not depend on some particular

coordinate xd (say) then (3.52) shows that

L

= gdb xb = constant

™

™ d

x

However, xb = tb , where t is the tangent vector to the geodesic, and so we find that

™

td = constant

Thus, we have the important result that if the metric coefficients gab do not depend

on the coordinate xd then the dth covariant component td of the tangent vector is

a conserved quantity along an affinely parameterised geodesic. We will use this

result often in our discussion of particle motion in general relativity.

81

Appendix 3A: Vectors as directional derivatives

3.20 Alternative form of the geodesic equations

The most common form of the geodesic equations is that given in (3.53). It is

sometimes useful, however, to recast the geodesic equations in different forms.

Thus, we note here an alternative way of writing them that will be of particular

practical use when we come to study particle motion in general relativity.

From (3.46), for a geodesic we have dt/du = 0. In some coordinate system

we may write this equation in terms of the intrinsic derivative of the covariant

components of the tangent vector as

dxc

Dta dta

≡ ’ ac tb =0

b

Du du du

Remembering that tc = xc = dxc /du, we thus have

™

™

ta = b c

ac tb t

b

which, on rewriting the connection coefficients using (3.21), becomes

ac

™

ta = 2 g bd a gdc + c gad ’ d gac tb tc = a gdc + c gad ’ d gac

1 1

td tc

2

Using the symmetry of the metric tensor, we see that the last two terms in the

summation on d and c cancel. Thus, we obtain a useful alternative form of the

geodesic equations,

™

ta = 1

tc td (3.56)

a gcd

2

From this equation, we may immediately verify our earlier finding that if the

metric gcd does not depend on the coordinate xa then ta = constant.

Appendix 3A: Vectors as directional derivatives

In an arbitrary manifold, the formal mathematical definition of a tangent vector

to a curve at some point P is in terms of the directional derivative along the

curve at that point. In particular, let us consider some curve defined in terms of

an arbitrary coordinate system by xa u . In addition, suppose that some arbitrary

scalar function f xa is defined on the manifold. At any point P on the curve, the

directional derivative of f is defined simply as

f dxa

df

=a

du x du

at that point. However, ta ≡ dxa /du gives the components of a tangent vector to

the curve at P and, since f is arbitrary, we may write

d

= ta a

du x

82 Vector calculus on manifolds

Thus, the components ta define a unique directional derivative, which we may

identify as the tangent vector t. Moreover, it follows that the differential operators

/ xa are the coordinate basis vectors ea at P, i.e. they are the tangent vectors to

the coordinate curves at this point.

In fact, any set of vector components va defines a unique directional derivative

va (3.57)

xa

and, conversely, this directional derivative defines a unique set of components va .

We may thus identify (3.57) as the vector v. Thus the definition of a vector as a

directional derivative replaces the more familiar notion of a directed line segment,

which cannot be generalised to non-Euclidean manifolds. It is straightforward

to verify that all the usual rules of vector algebra and the behaviour of the

components va under coordinate transformations follow immediately from (3.57).

Appendix 3B: Polar coordinates in a plane

As a simple example of the material presented in this chapter, let us consider

the special case of a two-dimensional Euclidean plane. The most common way

of labelling points in a plane is by using Cartesian coordinates x y , but it is

sometimes convenient to use plane polar coordinates . The two coordinate

systems are related by the equations

= tan’1 y/x

= x2 + y 2 1/2

and their inverses

x = cos y = sin

The transformation matrices relating these two sets of coordinates are

⎛ ⎞⎛ ⎞

⎜ ⎟ ⎜ cos ⎟

sin

⎜x y⎟=⎜ ⎟

⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝1 ⎠

1

’ sin cos

x y

and ⎛ ⎞ ⎛ ⎞

x x

⎜ ⎟ ⎜cos ’ sin ⎟

⎜ ⎟⎜ ⎟

⎟=

⎜

y⎠ ⎝ ⎠

⎝y

sin cos

which are easily shown to be inverses of one another. For convenience, in the

following we will sometimes refer to the polar coordinates as the coordinate

system xa a = 1 2 .

83

Appendix 3B: Polar coordinates in a plane

ey

ex eφ

ey

eρ eρ

ex

eφ

eφ

ey

eρ

ex

Figure 3.7 Labelling points in a plane with Cartesian coordinates and plane

polar coordinates. Examples of basis vectors for the two systems are also shown.

Basis vectors Let us now consider the coordinate basis vectors in each system.

The coordinate curves for each system are shown as dotted lines in Figure 3.7

and the basis vectors are tangents to these curves. For the Cartesian coordinates,

ex and ey have the special property that they are the same at every point P in the

plane. They are of unit length and point along the x- and y-directions respectively,

and we can write

ds = dx ex + dy ey

In plane polar coordinates this becomes

ds = cos d ’ sin d ex + sin d + cos d (3.58)

ey

and so, using the definition (3.3) of the coordinate basis vectors, we obtain

e = cos ex + sin ey (3.59)

e = ’ sin ex + cos ey (3.60)

Alternatively, we could have arrived at the same result using the transformation

equations (3.9) for basis vectors. The basis vectors e and e are shown in

Figure 3.7.

Metric components Substituting the expressions (3.59) and (3.60) into the result

gab = ea · eb , we find that in polar coordinates

1 0

gab = 2

0

84 Vector calculus on manifolds

Thus, we have

ds2 = ds · ds = gab dxa dxb = d +

2 2 2

d

which matches the result obtained using (3.58) directly. The matrix g ab is the

inverse of the matrix gab and thus is given by

1 0

g ab = 2

0 1/

Dual basis The dual basis vectors given by ea = g ab eb are

e = g e +g e =e

1

e =g e +g e= e

2

where no summation is implied over or . These dual basis vectors are easily

shown to obey the reciprocity relation ea · eb = a .

b

Derivatives of basis vectors Since ex and ey are constant vector fields, the

derivatives of the polar coordinate basis vectors are easily found as

e

= cos ex + sin ey = 0

e 1

= cos ex + sin ey = ’ sin ex + cos ey = e

These have a simple geometrical picture. At each of two nearby points P and

Q the vector e must point away from the origin, and so in slightly different

directions. The derivative of e with respect to is just the difference between

between e at P and Q divided by (the angle between them). The difference

in this case is clearly a vector parallel to e , which makes the above results

reasonable. Similarly,

e 1

= ’ sin ex + cos ey = ’ sin ex + cos ey = e

e

= ’ sin ex + cos ey = ’ cos ex ’ sin ey = ’ e

The student is encouraged to explain these formulae geometrically.

85

Appendix 3B: Polar coordinates in a plane

Connection coefficients Using the general formula c ea = b ac eb , we can now

read off the connection coefficients in plane polar coordinates:

e

=0 ’ =0 =0

e 1 1

= ’ =0 =

e

e 1 1

= ’ =0 =

e

e

=’ ’ =’ =0

e

where no summation is assumed over repeated indices. Thus, although we

computed the derivatives of e and e by using the constancy of ex and ey , the

Cartesian basis vectors do not appear in the above equations. The connection™s

importance is that it enables one to express these derivatives without using

any other coordinates than polar. We can alternatively calculate the connection

coefficients from the metric using the general result (3.21). For example,

= 2 ga ga + ga ’

1

ag

= 0 and g = 1/ 2 ,

where summation is implied only over the index a. Since g

we have

1 1 1 1

= + ’ = = =

2

g g g g

22 22 22

This is the same expression for as that derived above. Indeed, this method of

computing the connection is generally far more straightforward than calculating

the derivatives of basis vectors.

Covariant derivative Given the connection coefficients, we can calculate the

covariant derivative of a vector field in polar coordinates. As an example of its

use, let us find an expression for the divergence · v of a vector field. This is

given by

·v = = +

a a a b

av av ba v

Now, the contracted connection coefficients are given by

1

= + =

a

a

= + =0

a

a

86 Vector calculus on manifolds

so we have

1 1

v v v

·v = + v+ = v+

This formula may not be immediately familiar. The reason for this is that most

ˆ

often a vector v is expressed in terms of the normalised basis vectors e = e

ˆ

and e = e / . In this normalised basis the vector components are v = v and

ˆ

v = v , and the divergence takes its more usual form

ˆ

ˆ

1 1v

·v = v+

ˆ

Geodesics Finally, let us consider a geodesic in a plane. We already know that

the answer is a straight line, and this is trivially proven in Cartesian coordinates.

For illustration, however, let us perform the calculation the hard way, i.e. in plane

polar coordinates. There are two geodesic equations,

d 2 xa b c

a dx dx

+ bc =0

ds2 ds ds

for a = , where we are using the arclength s as our parameter along the

= ’ and =

geodesic. The only non-zero connection coefficients are

= 1/ . Thus, writing out the geodesic equations for a = and a = ,

we have

2

d2 d

’ =0 (3.61)

ds2 ds

d2 2d d

+ =0 (3.62)

ds2 ds ds

Also, since in a Euclidean plane we can only have non-null geodesics, a first

integral of these equations is provided by

2 2

dxa dxb d d

=1 ’ + =1

2

(3.63)

gab

ds ds ds ds

Of course, this could have been obtained simply by dividing through ds2 =

d 2 + 2 d 2 by ds2 .

Equation (3.62) can be written as

1d 2d

=0

2 ds ds

from which we obtain

2d

= k = constant (3.64)

ds

87

Appendix 3C: Calculus of variations

Inserting this into (3.63), we find that

1/2

k2

d

= 1’ 2 (3.65)

ds

The shape of the geodesic is what really interests us, i.e. as a function of or

vice versa. Dividing (3.64) by (3.65), we obtain

’1/2

k2

d k

= 2 1’ 2

d

which can be integrated easily to give

k

’1

= 0 + cos

where is the integration constant. The shape of the geodesic is given by

0

’ =k

cos 0

which, on expanding the cosine and using x = cos and y = sin , gives

0 + y sin =k

x cos 0

This is the general equation of a straight line. Thus we recover the familiar result

in an unfamiliar coordinate system.

Appendix 3C: Calculus of variations

The calculus of variations provides a means of finding a function (or set of

functions) that makes an integral dependent on the function(s) stationary, i.e.

makes the value of the integral a local maximum or minimum. Let us consider

the path integral

B

I= ™

F xa xa u du (3.66)

A

where A, B and the form of the integrand F are fixed, but the ˜curve™ or path

xa u has to be chosen so as to make stationary the value of I. From (3.66),

we see that we are considering quite a general case, in which the integrand

F is a function of the 2N independent functions xa and xa ≡ dxa /du and the

™

parameter u.

Now consider making an arbitrary variation xa u ’ xa u + xa u in the

path, keeping the endpoints A and B fixed. The corresponding first-order variation

in the value of the integral is

B B F F

I= F du = xa + a xa du

™

™

xa x

A A

88 Vector calculus on manifolds

Integrating the last term by parts and requiring the variation I to be zero, we

obtain

B B

F F d F

I= + ’ xa du = 0

xa

™ ™

a xa du xa

x A

A

Since A and B are fixed, the first term vanishes. Then, since xa is arbitrary, our

required extremal curve xa u must satisfy the N equations

d F F

’ a =0 (3.67)

™

xa

du x

These are the Euler“Lagrange equations for the problem.

Exercises

3.1 Show that, in general, ea = gab eb and ea = g ab eb . Show also that, under a coordinate

transformation,

xb xa b

ea = ea =

and

e e

xa b xb

3.2 Calculate the coordinate basis vectors ea of the coordinates system x a in Exercise 2.3

in terms of the coordinate basis vectors ea of the Cartesian system. Hence verify

that the metric functions gab agree with those found earlier. Calculate the dual basis

vectors e a in the primed system and hence the quantities g ab . Find the contravariant

and covariant components of e1 in the primed basis. Hence verify that e1 is of unit

length.

3.3 For any metric gab show that g ab gab = N , where N is the dimension of the manifold.

3.4 Show that the affine connection can be written as b ac = eb · c ea . Show further that,

in a torsionless manifold, c ea = a ec .

3.5 Show that, under a coordinate transformation, the affine connection transforms as

xd xf 2 x a

x a xf xg

=d fg ’

a d

bc

x xb xc x b x c xd xf

3.6 For a diagonal metric gab , show that the connection coefficients are given by (with

a = b = c and no summation over repeated indices)

1

=0 =’

a b

b gaa

bc aa

2gbb

= = =

a a a

ln ln

gaa gaa

ba ab b aa a

3.7 Let g be the determinant of the matrix gab . By considering the cofactor of the

element gab in this determinant, or otherwise, show that c g = gg ab c gab .

89

Exercises

3.8 In a manifold with non-zero torsion, show that the affine connection defined by

(3.11) may be written as

1

a

= ’ T a cb + Tc a b ’ Tbc a

a

bc

2

bc

a

where bc is the metric connection defined by the right-hand side of (3.21) and T a bc

is the torsion tensor defined in (3.18). Defining an index symmetrisation operation

such that a bc ≡ 2 a bc + a cb , show further that

1

a

= + T bc

a a

bc

bc

In a manifold with non-zero torsion, show that the condition a bc = 0 implies that

3.9

a gbc = 0 but not vice versa. Show further that, under a coordinate transformation

of the form

1

x a = xa ’ xP + a bc P xb ’ xP xc ’ xP

a b c

2

the affine connection at the point P in the new coordinate system is given by

1

P= T

a a

P

bc bc

2

and hence the transformation does not yield a set of geodesic coordinates. Is it still

possible to define local Cartesian coordinates in a manifold with non-zero torsion?

3.10 Show that, for the covariant components va of a vector, the covariant derivative and

the intrinsic derivative along a curve are given respectively by

dxc

Dva dv

= b va ’ = a’

c b

and

b va ab vc ac vb

Du du du

3.11 Show that for a vector field with contravariant components vb to have a vanishing

covariant derivative a vb everywhere in a manifold, it must satisfy the relation

’ + ’ va = 0

d d e d e d

b ac c ab ac eb ab ec

Hint: Use the fact that partial derivatives commute.

3.12 If a vector field va vanishes on a hypersurface S that bounds a region V of an

N -dimensional manifold, show that

√

’g dN x = 0

a

av

V

3.13 On the surface of a unit sphere, ds2 = d 2 + sin2 d 2 . Calculate the connection

coefficients in the coordinate system. A vector v of unit length is defined

at the point 0 0 as parallel to the circle = 0. Calculate the components of v

after it has been parallel-transported around the circle = 0 . Hence show that,

in general, after parallel transport the direction of v is different but its length is

unchanged.

90 Vector calculus on manifolds

3.14 If the two vectors with contravariant components va and wa are each parallel-

transported along a curve, show that va wa remains constant along the curve. Hence

show that if a geodesic is timelike (or null or spacelike) at some point, it is timelike

(or null or spacelike) at all points.

3.15 An affinely parameterised geodesic xa u satisfies

d 2 xa dxb dxc

+ =0

a

bc

du2 du du

Show that the form of this equation remains unchanged by an arbitrary coordinate

transformation xa ’ x a . Find the form of the geodesic equation for a geodesic

described in terms of some general (non-affine) parameter . Hence show that all

affine parameters are related by a linear transformation with constant coefficients.

3.16 If x is an affinely parameterised geodesic, show that

Du

=0

D

where u = dx /d . Hence show that the geodesic equations can be written as

du 1

= g uu

2

d

3.17 By substituting the ˜Lagrangian™ L = gab xa xb into the Euler“Lagrange equations,

™™

show directly that

xa +

¨ ™ x =0

™

a bc

bc x

where the dots denote differentiation with respect to an affine parameter.

3.18 By transforming from a local inertial coordinate system in which

ds2 = c2 d =

2

d d

to a general coordinate system x , show that freely falling particles obey the geodesic

equations of motion

d2 x dx dx

+ =0

d2 dd

where

2

x

=

xx

3.19 By considering the ˜Lagrangian™ L = gab xa xb , derive the equations for an affinely

™™

parameterised geodesic on the surface of a sphere in the coordinates . Hence

show that, of all the circles of constant latitude on a sphere, only the equator is a

geodesic. Use your geodesic equations to pick out the connection coefficients in

this coordinate system.

3.20 In the 2-space with line element

dr 2 + r 2 d 2

r 2 dr 2

ds = ’2

2

r 2 ’ a2 r ’ a2 2

91

Exercises

where r > a, show that the differential equation for the geodesics may be written as

2

dr

+ a2 r 2 = Kr 4

2

a

d

where K is a constant such that K = 1 if the geodesic is null. By setting r d /dr =

tan , show that the space is mapped onto a Euclidean plane in which r are

taken as polar coordinates and the geodesics are mapped to straight lines.

4

Tensor calculus on manifolds

The coordinates with which one labels points in a manifold are entirely arbitrary.

For example, we could choose to parameterise the surface of a sphere in terms

of the coordinates , taking any point as the north pole, or we could use

any number of alternative coordinate systems. It is also clear, however, that our

description of any physical processes occurring on the surface of the sphere should

not depend on our chosen coordinate system. For example, at any point P on the

surface one can say that, for example, the air temperature has a particular value

or that the wind has a certain speed in a particular direction. These respectively

scalar and vector physical quantities do not depend on which coordinates are used

to label points in the surface. Thus in, order to describe these physical fields

on the surface, we must formulate our equations in a way that is valid in all

coordinate systems. We have already dealt with such a description for scalar and

vector quantities on manifolds, but now we turn to the generalisation of these

ideas to quantities that cannot be described as a scalar or a vector. This requires

the introduction of the concept of tensors.

4.1 Tensor fields on manifolds

Let us begin by considering vector fields in a slightly different manner. Suppose

we have some arbitrary vector field, defining a vector t at each point of a manifold.

How can we obtain from t a scalar field? Clearly, the only way to do this is to

take the scalar product of t with a vector v from another vector field. Thus, at

each point P in the manifold, we can think of vector t in TP as a linear function

t · that takes another vector in TP as its argument and produces a real number.

We can denote the number produced by the action of t on a particular vector v by

t v ≡ t·v (4.1)

92

93

4.2 Components of tensors

It is now clear how we can generalise the notion of a vector: in the tangent

space TP , we can define a tensor t as a linear map from some number of vectors

to the real numbers. The rank of the tensor is the number of vectors it has for its

arguments. For example, we can write a third-rank tensor as t · · · . Once again,

we denote the number that the tensor t produces from the vectors u, v and w by

tuvw

The tensor is defined by the precise set of operations applied to the vectors u, v

and w to produce a scalar. Notice, however, that the definition of a tensor does

not mention the components of the vectors; a tensor must give the same real

number independently of the reference system in which the vector components

are calculated. If at each point P in some region of the manifold we have a tensor

defined then the result is a tensor field in this region.

In fact we have already encountered examples of tensors. Clearly, from our

above discussion, any vector is a rank-1 tensor. Higher-rank tensors thus constitute

a generalisation of the concept of a vector. For example, a particularly important

second-rank tensor is the metric tensor g, which we have already met. This defines

a linear map of two vectors into the number that is their inner product, i.e.

g u v ≡ u·v

We will investigate the properties of this special tensor shortly. Finally, we note

also that a scalar function of position x is a real-valued function of no vectors

at all, and is therefore classified as a zero-rank tensor.

The fact that a tensor is a linear map of the vectors into the reals is particularly

useful. For simplicity, let us consider a rank-1 tensor. Linearity means that, for

general vectors u and v and general scalars and ,

u+ v = t u + t v

t

Similar expansions may be performed for tensors of higher rank. For a second-rank

tensor, for example, we can write

u+ v w+ z = t u w+ z + t v w+ z

t

= tuw+ tuz+ tvw+ tvz

4.2 Components of tensors

When a tensor is evaluated with combinations of basis and dual basis vectors

it yields its components in that particular basis. For example, the covariant and

94 Tensor calculus on manifolds

contravariant components of the rank-1 tensor (vector) in (4.1) in the basis ea are

given by

t ea = ta t ea = ta

and

Consider now a second-rank tensor t · · . Its covariant and contravariant compo-

nents are given by

t ea eb = tab t ea eb = tab

and

For tensors of rank 2 and higher, however, we can also define sets of mixed

components. For a rank-2 tensor there are two possible sets of mixed components,

t ea eb = ta b t ea eb = ta b

and

For a general rank-2 tensor these two sets of components need not be equal.

The contravariant, covariant and mixed components of higher-rank tensors can be

obtained in an analogous manner.

The components of a tensor in a particular basis set specify the action of the

tensor on any other vectors in terms of their components. For example, using the

linearity property, we find that

t u v = t ua ea vb eb = tab ua vb

To obtain this result, we expressed u and v in terms of their contravariant compo-

nents. We could have written either vector in terms of its contravariant or covariant

components, however. Hence we find that there are numerous equivalent expres-

sions for t u v in component notation:

t u v = tab ua vb = tab ua vb = ta b ua vb = ta b ua vb

This illustrates the general rule that the subscript and superscript positions of a

dummy index can be swapped without affecting the result.

4.3 Symmetries of tensors

A second-rank tensor t is called symmetric or antisymmetric if, for all pairs of

vectors u and v,

t u v = ±t v u

95

4.3 Symmetries of tensors

with the plus sign for a symmetric tensor and the minus sign for an antisym-

metric tensor. Setting u = ea and v = eb , we see that the covariant components

of a symmetric or antisymmetric tensor satisfy tab = ±tba . By using different

combinations of basis and dual basis vectors we also see that, for such a tensor,

tab = ±tba and ta b = ±tb a .

An arbitrary rank-2 tensor can always be split uniquely into the sum of its

symmetric and antisymmetric parts. For illustration let us work with the covariant

components tab of the tensor in some basis. We can always write

tab = tab + tba + 2 tab ’ tba

1 1

2

which is clearly the sum of a symmetric and an antisymmetric part. A notation

frequently used to denote the components of the symmetric and antisymmetric

parts is

t ab ≡ tab + tba t ab ≡ tab ’ tba

1 1

and

2 2

In an analogous manner, a general rank-N tensor t u v w is symmetric

or antisymmetric with respect to some permutation of its vector arguments if

its value after permuting the arguments is equal to respectively plus or minus

its original value. From an arbitrary rank-N tensor, however, we can always

obtain a tensor that is symmetric with respect to all permutations of its vector

arguments and one that is antisymmetric with respect to all permutations. In terms

of the tensor™s covariant components, these symmetric and antisymmetric parts are

given by

1

= sum over all permutations of the indices a b

t ab c

c

N!

1

= alternating sum over all permutations of the indices a b

t ab c

c

N!

For example, the covariant components of the totally antisymmetric part of a

third-rank tensor are given by

t abc = tabc ’ tacb + tcab ’ tcba + tbca ’ tbac

1

6

We may extend the notation still further in order to define tensors that are

symmetric or antisymmetric to permutations of particular subsets of their indices.

96 Tensor calculus on manifolds

To illustrate this, let us consider the covariant components tabcd of a fourth-rank

tensor. Typical expressions might include:

t ab cd = tabcd + tbacd

1

2

ta b c d = tabcd ’ tadcb

1

2

t a b cd = tabcd + tabdc + tdbac + tdbca + tcbda + tcbad

1

6

= tab cd ’ tba cd

1

t ab cd 2

= 2 2 tabcd + tabdc ’ 2 tbacd + tbadc

11 1

= 4 tabcd + tabdc ’ tbacd ’ tbadc

1