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v = va ea = a ea , so the covariant components are va = a . In (3.36), however,
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





ey

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

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