1’ 2 1’ 1+ 1’

1/2 1/2 1/2 1/2

te

AB

This ratio must be the reciprocal of the ratio of the photon™s frequency as measured

by and respectively, and thus we obtain the familiar Doppler-effect formula

1/2

1’

= (1.12)

1+

1.12 Addition of velocities in special relativity

If a particle™s worldline is described by giving x, y and z as functions of t in

some inertial frame S then the components of its velocity in S at any point are

dx dy dz

ux = uy = uz =

dt dt dt

The components of its velocity in some other inertial frame S are usually obtained

by taking differentials of the Lorentz transformation. For inertial frames S and S

related by a boost v in standard configuration, we have from (1.3)

dt = dt ’ v dx/c2 dx = dx ’ v dt dy = dy dz = dz

v v

where we have made explicit the dependence of on v. We immediately obtain

ux ’ v

dx

ux = =

1 ’ ux v/c2

dt

uy

dy

uy = = (1.13)

1 ’ ux v/c2

dt v

dz uz

uz = =

1 ’ ux v/c2

dt v

These replace the ˜common sense™ addition-of-velocities formulae of Newtonian

mechanics. The inverse transformations are obtained by replacing v by ’v.

The special-relativistic addition of velocities along the same direction is

elegantly expressed using the rapidity parameter (Section 1.4). For example,

consider three inertial frames S, S and S . Suppose that S is related to S by a

boost of speed v in the x-direction and that S is related to S by a boost of speed

u in the x -direction. Using (1.5), we quickly find that

ct = ct cosh v+ ’ x sinh v+

u u

19

1.13 Acceleration in special relativity

x = ’ct sinh v+ + x cosh v+

u u

y =y

z =z

where tanh v = v/c and tanh u = u /c. This shows that S is connected to S

by a boost in the x-direction with speed u, where u/c = tanh v + u . Thus we

simply add the rapidities (in a similar way to adding the angles of two spatial

rotations about the same axis). This gives

tanh v + tanh u +v

u

u = c tanh v+ =c =

u

1 + tanh v tanh 1 + u v/c2

u

which is the special-relativistic formula for the addition of velocities in the same

direction.

1.13 Acceleration in special relativity

The components of the acceleration of a particle in S are defined as

duy

dux duz

ax = ay = az =

dt dt dt

and the corresponding quantities in S are obtained from the differential forms of

the expressions (1.13). For example,

dux

dux =

1 ’ ux v/c2

2 2

v

Also, from the Lorentz transformation (1.3) we find that

dt = dt ’ v dx/c2 = 1 ’ ux v/c2 dt

v v

So, for example, we have

1

dux

ax = = (1.14)

ax

1 ’ ux v/c2

3 3

dt v

Similarly, we obtain

duy uy v

1

ay = = a+ a

2y 1 ’ ux v/c2 3 x

1 ’ ux v/c2

2 c2 2

dt v v

1

du uz v

az = z = az + a

1 ’ ux v/c2 3 x

1 ’ ux v/c2

2 2 c2 2

dt v v

20 The spacetime of special relativity

We see from these transformation formulae that acceleration is not invariant

in special relativity, unlike in Newtonian mechanics, as discussed in Section 1.2.

However, it is clear that acceleration is an absolute quantity, that is, all observers

agree upon whether a body is accelerating. If the acceleration is zero in one

inertial frame, it is necessarily zero in any other frame.

Let us investigate the worldline of an accelerated particle. To make our illus-

tration concrete, we consider a spaceship moving at a variable speed u t relative

to some inertial frame S and suppose that an observer B in the spaceship makes

a continuous record of his accelerometer reading f as a function of his own

proper time .

We begin by introducing an instantaneous rest frame (IRF) S , which, at each

instant, is an inertial frame moving at the same speed v as the spaceship, i.e. v = u.

Thus, at any instant, the velocity of the spaceship in the IRF S is zero, i.e. u = 0.

Moreover, from the above discussion of proper time, it should be clear that at any

instant an interval of proper time is equal to an interval of coordinate time in the

= t . An accelerometer measures the rate of change of velocity, so

IRF, i.e.

that, during a small interval of proper time , B will record that his velocity has

changed by an amount f . Therefore, at any instant, in the IRF S we have

du du

= =f

dt d

From (1.14), we thus obtain

3/2

u2

du

= 1’ 2 f

dt c

However, since d = 1 ’ u2 /c2 1/2 dt, we find that

u2

du

= 1’ 2 f

d c

which integrates easily to give

= c tanh

u

= 0f d and we have taken u = 0 to be zero. Thus we have

where c

an expression for the velocity of the spaceship in S as a function of B™s proper time.

To parameterise the worldline of the spaceship in S, we note that

’1/2

u2

dt

= 1’ 2 = cosh

d c

’1/2

u2

dx

= u 1’ 2 = c sinh (1.15)

d c

Integration of these equations with respect to gives the functions t and x .

21

1.14 Event horizons in special relativity

1.14 Event horizons in special relativity

The presence of acceleration can produce surprising effects. Consider for simplic-

ity the case of uniform acceleration. By this we mean we do not mean that

du/dt = constant, since this is inappropiate in special relativity because it would

imply that u ’ as t ’ , which is not permitted. Instead, uniform acceler-

ation in special relativity means that the accelerometer reading f is constant.

A spaceship whose engine is set at a constant emission rate would be uniformly

accelerated in this sense.

Thus, if f = constant, we have = f /c. The equations (1.15) are then easily

integrated to give

c f

t = t0 + sinh

f c

c2 f

x = x0 + ’1

cosh

f c

where t0 and x0 are constants of integration. Setting t0 = x0 = 0 gives the path

shown in Figure 1.13. The worldline takes the form of a hyperbola.

Imagine that an observer B has the resources to maintain an acceleration f

indefinitely. Then there will be events that B will never be able to observe.

The events in question lie on the future side of the asymptote to B™s hyperbola;

this asymplote (which is a null line) is the event horizon of B. Objects whose

worldlines cross this horizon will disappear from B™s view and will seem to take

ct

A

B

B never sees A

after this event

on

iz

or

H

x

Figure 1.13 The worldline of a uniformly accelerated particle B starting from

rest from the origin of S. If an observer A remains at x = 0, then the worldline

of A is simply the t-axis. No message sent by A after t = c/f will ever reach B.

22 The spacetime of special relativity

for ever to do so. Nevertheless, the objects themselves cross the horizon in a finite

proper time and still have an infinite lifetime ahead of them.

Appendix 1A: Einstein™s route to special relativity

Most books on special relativity begin with some sort of description of the

Michelson“Morley experiment and then introduce the Lorentz transformation. In

fact, Einstein claimed that he was not influenced by this experiment. This is

disputed by various historians of science and biographers of Einstein. One might

think that these scholars are on strong ground, especially given that the experiment

is referred to (albeit obliquely) in Einstein™s papers. However, it may be worth

taking Einstein™s claim at face value.

Remember that Einstein was a theorist “ one of the greatest theorists who has

ever lived “ and he had a theorist™s way of looking at physics. A good theorist

develops an intuition about how Nature works, which helps in the formulation

of physical laws. For example, possible symmetries and conserved quantities are

considered. We can get a strong clue about Einstein™s thinking from the title of

his famous 1905 paper on special relativity. The first paragraph is reproduced

below.

On the Electrodynamics of Moving Bodies

by A. Einstein

It is known that Maxwell™s electrodynamics “ as usually understood at the present time “

when applied to moving bodies, leads to asymmetries which do not appear to be inherent

in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet

and a conductor. The observable phenomenon here depends only on the relative motion

of the conductor and the magnet, whereas the customary view draws a sharp distinction

between the two cases in which either the one or the other of these bodies is in motion.

For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood

of the magnet an electric field with a certain definite energy, producing a current at the

places where parts of the conductor are situated. But if the magnet is stationary and

the conductor in motion, no electric field arises in the neighbourhood of the magnet. In

the conductor, however, we find an electromotive force, to which in itself there is no

corresponding energy, but which gives rise “ assuming equality of relative motion in the

two cases discussed “ to electric currents of the same path and intensity as those produced

by the electric forces in the former case.

You see that Einstein™s paper is not called ˜Transformations between inertial

frames™, or ˜A theory in which the speed of light is assumed to be a universal

constant™. Electrodynamics is at the heart of Einstein™s thinking; Einstein realized

that Maxwell™s equations of electromagnetism required special relativity.

23

Appendix 1A: Einstein™s route to special relativity

Maxwell™s equations are

·D = ·B = 0

B D

—E = ’ —H = j +

t t

where D = 0 E + P and B = 0 H + M , P and M being respectively the polari-

sation and the magnetisation of the medium in which the fields are present. In free

space we can set j = 0 and = 0, and we then get the more obviously symmetrical

equations

·E = 0 ·B = 0

B E

—E = ’ —B = 00

t t

— E, applying the relation

Taking the curl of the equation for

— —E = ·E ’ 2

E

— B, we derive the

and performing a similar operation for B in the equation for

equations for electromagnetic waves:

2E 2B

E= B= 0 0 2

2 2

00

t2 t

These both have the form of a wave equation with a propagation speed c =

√

1/ 0 0 . Now, the constants 0 and 0 are properties of the ˜vacuum™:

the permeability of a vacuum, equals 4 — 10’7 Hm’1

0

the permittivity of a vacuum, equals 8 85 — 10’12 Fm’1

0

This relation between the constants 0 and 0 and the speed of light was one of

the most startling consequences of Maxwell™s theory. But what do we mean by a

˜vacuum™? Does it define an absolute frame of rest? If we deny the existence of an

absolute frame of rest then how do we formulate a theory of electromagnetism?

How do Maxwell™s equations appear in frames moving with respect to each other?

Do we need to change the value of c? If we do, what will happen to the values

of 0 and 0 ?

Einstein solves all of these problems at a stroke by saying that Maxwell™s

equations take the same mathematical form in all inertial frames. The speed of light

c is thus the same in all inertial frames. The theory of special relativity (including

amazing conclusions such as E = mc2 ) follows from a generalisation of this

simple and theoretically compelling assumption. Maxwell™s equations therefore

require special relativity. You see that for a master theorist like Einstein, the

24 The spacetime of special relativity

Michelson“Morley experiment might well have been a side issue. Einstein could

˜see™ special relativity lurking in Maxwell™s equations.

Exercises

1.1 For two inertial frames S and S in standard configuration, show that the coordinates

of any given event in each frame are related by the Lorentz tranformations (1.3).

1.2 Two events A and B have coordinates tA xA yA zA and tB xB yB zB respectively.

Show that both the time difference t = tB ’ tA and the quantity

r 2 = x 2 + y 2 + z2

are separately invariant under any Galilean transformation, whereas the quantity

s 2 = c 2 t 2 ’ x 2 ’ y 2 ’ z2

is invariant under any Lorentz transformation.

1.3 In a given inertial frame two particles are shot out simultaneously from a given

point, with equal speeds v in orthogonal directions. What is the speed of each particle

relative to the other?

1.4 An inertial frame S is related to S by a boost of speed v in the x-direction, and S

is related to S by a boost of speed u in the x -direction. Show that S is related to

S by a boost in the x-direction with speed u, where

u = c tanh v+ u

tanh v = v/c and tanh u = u /c.

1.5 An inertial frame S is related to S by a boost v whose components in S are vx vy vz .

Show that the coordinates ct x y z and ct x y z of an event are related by

⎛ ⎞⎛ ⎞⎛ ⎞

’ ’ ’

ct ct

x y z

⎜ x ⎟ ⎜’ ⎟⎜ ⎟

1+ 2

⎜⎟⎜ z ⎟⎜x⎟

⎜ ⎟=⎜ ⎟⎜ ⎟

x x y x

x

⎝ y ⎠ ⎝’ z ⎠⎝y ⎠

1+ 2

y y x y

y

’ 1+ 2

z z

y z x z y z

where = v/c = 1 ’ 2 ’1/2 and = ’ 1 / 2 . Hint: The transformation

must take the same form if both S and S undergo the same spatial rotation.

1.6 An inertial frame S is related to S by a boost of speed u in the positive x-direction.

Similarly, S is related to S by a boost of speed v in the y -direction. Find the

transformation relating the coordinates ct x y z and ct x y z and hence

describe how S and S are physically related.

1.7 The frames S and S are in standard configuration. A straight rod rotates at a uniform

angular velocity about its centre, which is fixed at the origin of S . If the rod lies

along the x -axis at t = 0, obtain an equation for the shape of the rod in S at t = 0.

25

Exercises

1.8 Two events A and B have coordinates tA xA yA zA and tB xB yB zB respec-

tively in some inertial frame S and are separated by a spacelike interval. Obtain an

expression for the boost v required to transform to a new inertial frame S in which

the events A and B occur simultaneously.

1.9 Derive the Doppler effect (1.12) directly, using the Lorentz transformation

formulae (1.3).

1.10 Two observers are moving along trajectories parallel to the y-axis in some inertial

frame. Observer A emits a photon with frequency A that travels in the positive

x-direction and is received by observer B with frequency B . Show that the Doppler

shift B / A in the photon frequency is the same whether A and B travel in the same

direction or opposite directions.

Astronauts in a spaceship travelling in a straight line past the Earth at speed v = c/2

1.11

wish to tune into Radio 4 on 198 kHz. To what frequency should they tune at the

instant when the ship is closest to Earth?

1.12 Draw a spacetime diagram illustrating the coordinate transformation corresponding

to two inertial frames S and S in standard configuration (i.e. where S moves at a

speed v along the positive x-direction and the two frames coincide at t = t = 0).

Show that the angle between the x- and x - axes is the same as that between the t-

and t - axes and has the value tan’1 v/c .

1.13 Consider an event P separated by a timelike interval from the origin O of your

diagram in Exercise 1.12. Show that the tangent to the invariant hyperbola passing

through P is a line of simultaneity in the inertial frame whose time axis joins P

to the origin. Hence, from your spacetime diagram, derive the formulae for length

contraction and time dilation.

1.14 Alex and Bob are twins working on a space station located at a fixed position in

deep space. Alex undertakes an extended return spaceflight to a distant star, while

Bob stays on the station. Show that, on his return to the station, the proper time

interval experienced by Alex must be less than that experienced by Bob, hence Bob

is now the elder. How does Alex explain this age difference?

1.15 A spaceship travels at a variable speed u t in some inertial frame S. An observer

on the spaceship measures its acceleration to be f , where is the proper time.

If at = 0 the spaceship has a speed u0 in S show that

u ’ u0

= c tanh

1 ’ u u0 /c2

= 0f

where c d . Show that the velocity of the spaceship can never reach c.

1.16 If the spaceship in Exercise 1.15 left base at time t = = 0 and travelled forever

in a straight line with constant acceleration f , show that no signal sent by base

later than time t = c/f can ever reach the spaceship. By sketching an appropriate

spacetime diagram show that light signals sent from the base appear increasingly

redshifted to an observer on the spaceship. If the acceleration of the spaceship is g

(for the comfort of its occupants), how long by the spaceship clock does it take to

reach a star 10 light years from the base?

2

Manifolds and coordinates

Our discussion of special relativity has led us to model the physical world as a

four-dimensional continuum, called spacetime, with a Minkowski geometry. This

is an example of a manifold. As we shall see, the more complicated spacetime

geometries of general relativity are also examples of manifolds. It is therefore

worthwhile discussing manifolds in general. In the following we consider general

properties of manifolds commonly encountered in physics, and we concentrate in

particular on Riemannian manifolds, which will be central to our discussion of

general relativity.

2.1 The concept of a manifold

In general, a manifold is any set that can be continuously parameterised. The

number of independent parameters required to specify any point in the set uniquely

is the dimension of the manifold, and the parameters themselves are the coor-

dinates of the manifold. An abstract example is the set of all rigid rotations of

Cartesian coordinate systems in three-dimensional Euclidean space, which can be

parameterised by the Euler angles. So the set of rotations is a three-dimensional

manifold: each point is a particular rotation, and the coordinates of the point

are the three Euler angles. Similarly, the phase space of a particle in classical

mechanics can be parameterised by three position coordinates q1 q2 q3 and

three momentum coordinates p1 p2 p3 , and thus the set of points in this phase

space forms a six-dimensional manifold. In fact, one can regard ˜manifold™ as just

a fancy word for ˜space™ in the general mathematical sense.

In its most primitive form a general manifold is simply an amorphous collection

of points. Most manifolds used in physics, however, are ˜differential manifolds™,

which are continuous and differentiable in the following way. A manifold is

continuous if, in the neighbourhood of every point P, there are other points whose

coordinates differ infinitesimally from those of P. A manifold is differentiable if

it is possible to define a scalar field at each point of the manifold that can be

differentiated everywhere. Both our examples above are differential manifolds.

26

27

2.3 Curves and surfaces

The association of points with the values of their parameters can be thought of

as a mapping of the points of a manifold into points of the Euclidean space of the

same dimension. This means that ˜locally™ a manifold looks like the corresponding

Euclidean space: it is ˜smooth™ and has a certain number of dimensions.

2.2 Coordinates

An N -dimensional manifold of points is one for which N independent real

coordinates x1 x2 xN are required to specify any point completely.1 These

N coordinates are entirely general and are denoted collectively by xa , where it is

understood that a = 1 2 N.

As a technical point, we should mention that in general it may not be possible

to cover the whole manifold with only one non-degenerate coordinate system,

namely, one which ascribes a unique set of N coordinate values to each point,

so that the correspondence between points and sets of coordinate values (labels)

is one-to-one. Let us consider, for example, the points that constitute a plane.

These points clearly form a two-dimensional manifold (called R2 ). An example

of a degenerate coordinate system on this manifold is the polar coordinates r

in the plane, which have a degeneracy at the origin because is indeterminate

there. For this manifold, we could avoid the degeneracy at the origin by using,

for example, Cartesian coordinates. For a general manifold, however, we might

have no choice in the matter and might have to work with coordinate systems that

cover only a portion of the manifold, called coordinate patches. For example, the

set of points making up the surface of a sphere forms a two-dimensional manifold

(called S 2 ). This manifold is usually ˜parameterised™ by the coordinates and

, but is degenerate at the poles. In this case, however, it can be shown that

there is no coordinate system that covers the whole of S 2 without degeneracy; the

smallest number of patches needed is two. In general, a set of coordinate patches

that covers the whole manifold is called an atlas.

Thus, in general, we do not require the whole of a manifold to be covered

by a single coordinate system. Instead, we may have a collection of coordinate

systems, each covering some part of , and all these are on an equal footing.

We do not regard any one coordinate system as in some way preferred.

2.3 Curves and surfaces

Given a manifold, we shall be concerned with points in it and with subsets of

points that define curves and surfaces. We shall frequently define these curves

1

The reason why the coordinates are written with superscripts rather than subscripts will become clear later.

28 Manifolds and coordinates

and surfaces parametrically. Thus, since a curve has one degree of freedom,

it depends on one parameter and so we define a curve in the manifold by the

parametric equations

xa = xa u a=1 2 N

where u is some parameter and x1 u x2 u xN u denote N functions of u.

Similarly, since a submanifold or surface of M dimensions M < N has M

degrees of freedom, it depends on M parameters and is given by the N parametric

equations

xa = xa u1 u2 a=1 2

uM (2.1)

N

If, in particular, M = N ’ 1 then the submanifold is called a hypersurface. In this

case, the N ’ 1 parameters can be eliminated from these N equations to give one

equation relating the coordinates, i.e.

xN = 0

f x1 x2

From a different but equivalent point of view, a point in a manifold is charac-

terised by N coordinates. If the point is restricted to lie in a particular hypersurface,

i.e. an N ’ 1 -dimensional subspace, then the point™s coordinates must satisfy

one constraint equation, namely

xN = 0

f x1 x2

Similarly, points in an M-dimensional subspace M < N must satisfy N ’ M

constraints

xN = 0

f1 x1 x2

xN = 0

f2 x1 x2

xM = 0

fN ’M x1 x2

which is an alternative to the parametric representation (2.1).

2.4 Coordinate transformations

To locate a point in a manifold we use a system of N coordinates, but the choice of

these coordinates is arbitrary. The important idea is not the ˜labels™ but the points

themselves and the geometrical and topological relationships between them.

29

2.4 Coordinate transformations

We may relabel the points of a manifold by performing a coordinate transfor-

mation xa ’ x a expressed by the N equations

x a = x a x1 x2 a=1 2

xN (2.2)

N

giving each new coordinate as a function of the old coordinates. Hence we

view a coordinate transformation passively as assigning the new primed coor-

dinates x 1 x 2 x N to a point of the manifold whose old coordinates are

x1 x2 xN .

We will assume that the functions involved in (2.2) are single-valued, contin-

uous and differentiable over the valid ranges of their arguments. Thus by differ-

entiating each equation in (2.2) with respect to each of the old coordinates xb we

obtain the N — N partial derivatives x a / xb . These may be assembled into the

N — N transformation matrix 2

⎛ ⎞

1 1 1

x x x

⎜1 ⎟

···

⎜x xN ⎟

x2

⎜ ⎟

⎜2 2⎟

2

⎜x x⎟

x

⎜1 ⎟

···

a

⎜x xN ⎟

x x2

=⎜ ⎟

⎜ ⎟

xb

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎝ xN N⎠

N

x x

···

x1 x2 xN

so that rows are labelled by the index in the numerator of the partial derivative

and columns by the index in the denominator. The elements of the transforma-

tion matrix are functions of the coordinates, and so the numerical values of the

matrix elements are in general different when evaluated at different points in the

manifold. The determinant of the transformation matrix is called the Jacobian of

the transformation and is denoted by

xa

J = det

xb

Clearly, the numerical value of J also varies from point to point in the manifold.

If J = 0 for some range of the coordinates xb then it follows that in this region

we can (in principle) solve the equations (2.2) for the old coordinates xb and

obtain the inverse transformation equations

xa = xa x 1 x 2 a=1 2

xN N

2

In general the notation denotes the matrix containing the elements within the square brackets.

30 Manifolds and coordinates

In a similar manner to the above, we define the inverse transformation matrix

xa / x b and the Jacobian of the inverse transformation J = det xa / x b .

Using the chain rule, it is easy to show that the inverse transformation matrix

is the inverse of the transformation matrix, since

if a = c

N

x a xb xa 1

= = =

a

c

if a = c

xb x c xc 0

b=1

a

where we have defined the Kronecker delta and used the fact that

c

xa xa

= c =0 if a = c

xc x

because the coordinates in either the unprimed or the primed set are independent.

Since the two transformation matrices are inverses of one another, it follows that

J = 1/J .

If we consider neighbouring points P and Q in the manifold, with coordinates

a and xa + dxa respectively, then in the new, primed, coordinate system the

x

infinitesimal coordinate separation between P and Q is given by

xa 1 xa 2 xa N

dx = 1 dx + 2 dx + · · · + N dx

a

x x x

where it is understood that the partial derivatives on the right-hand side are

evaluated at the point P. We can write this more economically as

N

xa b

dx =a

(2.3)

dx

xb

b=1

2.5 Summation convention

Our notation can be made more economical still by adopting Einstein™s summation

convention: whenever an index occurs twice in an expression, once as a subscript

and once as a superscript, this is understood to imply a summation over the index

from 1 to N , the dimension of the manifold.

Thus we can write (2.3) simply as

xa b

dx = b dx

a

x

where, once again, it is understood that all the partial derivatives are evaluated at

P. The index a appearing on each side of this equation is said to be a free index

and may take on separately any value from 1 to N . We consider a superscript that

31

2.6 Geometry of manifolds

appears in the denominator of a partial derivative as a subscript (and vice versa).

Thus the index b on the right-hand side in effect appears once as a subscript

and once as a superscript, and hence there is an implied summation from 1 to

N . An index that is summed over in this way is called a dummy index, because

it can be replaced by any other index not already in use. For example, we may

write

xa b xa c

dx = c dx

xb x

since c was not already in use in the expression.

Note that the proper use of the summation convention requires that, in any

term, an index should not occur more than twice and that any repeated index must

occur once as a subscript and once as a superscript.

2.6 Geometry of manifolds

So far, we have considered manifolds only in a very primitive form. We have

assumed that the manifold is continuous and differentiable, but aside from these

properties it remains an amorphous collection of points. We have not yet defined

its geometry.

Consider two infinitesimally separated points P and Q in the manifold, with

coordinates xa and xa + dxa respectively a = 1 2 N . The local geometry

of the manifold at the point P is determined by defining the invariant ˜distance™

or ˜interval™ ds between P and Q. In general, the distance between the points can

be assigned to be any reasonably well-behaved function of the coordinates and

their differentials, i.e.3

ds2 = f xa dxa

Clearly this function contains information on both the local geometry of the

manifold at P and our chosen coordinate system. It is the assignment at each

point in the manifold of a distance between points with infinitesimally different

values of the coordinates that determines the local geometry of the manifold. To

choose an example at random, a two-dimensional manifold, beloved of differential

geometers for its richness, is the Finsler geometry, in which one may define

coordinates and such that

ds2 = d +d

4 4 1/2

3

It is conventional to give the expression for ds2 rather than ds.

32 Manifolds and coordinates

2.7 Riemannian geometry

For developing general relativity, we are not interested in the most general geome-

tries and can confine our attention to manifolds in which the interval is given by

an expression of the form4 (assuming the summation convention)

ds2 = gab x dxa dxb (2.4)

Thus, such an interval is quadratic in the coordinate differentials. We shall see

below that the gab x are the components of the metric tensor field in our chosen

coordinate system. For the moment, however, we can consider them simply as

a set of functions of the coordinates that determine the local geometry of the

manifold at any point. Manifolds with a geometry expressible in the form (2.4) are

called Riemannian manifolds. Strictly speaking, the manifold is only Riemannian

if ds2 > 0 always. If ds2 can be positive or negative (or zero), as is the case

in special relativity and general relativity, then the manifold should properly be

called pseudo-Riemannian but is usually simply referred to as Riemannian.

The metric functions gab x can be considered as the elements of a position-

dependent N — N matrix. The metric functions can always be chosen so that

gab x = gba x , i.e the matrix is symmetric. Suppose for argument™s sake that the

functions gab were not symmetric in a and b. Then we could always decompose

the metric function into parts that are symmetric and antisymmetric respectively

in a and b, i.e.

gab x = gab x + gba x + 2 gab x ’ gba x

1 1

2

The contribution to ds2 from the antisymmetric part would be 2 gab x ’ 1

gba x dxa dxb , which vanishes identically, as is easily confirmed on swapping

indices in one of the terms, so that any antisymmetric part of gab can safely be

neglected. Thus in an N -dimensional Riemannian manifold there are 2 N N + 1

1

independent metric functions gab x .

It is important to remember that the form of the metric functions can always

be changed by making a change of coordinates. Since the interval between two

points in the manifold is invariant under a coordinate transformation, using (2.4)

and (2.3) we have

ds2 = gab x dxa dxb

xa xb

= gab dx c dx d

x c xd

x

= gcd x dx c dx d (2.5)

4

As we shall see in Chapter 7, this is a consequence of the equivalence principle.

33

2.8 Intrinsic and extrinsic geometry

where the new metric functions gab x in the primed coordinate system are related

to those in the unprimed coordinate system by

xa xb

gcd x = gab x x

xc xd

Clearly, the metric functions gab x describe the same local geometry of the

manifold as do the functions gab x .

Since there are N arbitrary coordinate transformations there are really only

2 N N + 1 ’ N = 2 N N ’ 1 independent degrees of freedom associated with the

1 1

gab x .

2.8 Intrinsic and extrinsic geometry

It is important to realise that the local geometry or curvature characterised by

(2.4) is an intrinsic property of the manifold itself, i.e. it is independent of whether

the manifold is embedded in some higher-dimensional space.

It is, of course, difficult (or impossible) to imagine higher-dimensional curved

manifolds, so it is instructive to consider two-dimensional Riemannian manifolds,

which can often be visualised as a surface embedded in a three-dimensional

Euclidean space. It is important to make a distinction, however, between the

extrinsic properties of the surface, which are dependent on how it is embedded

into a higher-dimensional space, and properties that are intrinsic to the surface

itself.

This distinction is traditionally made clear by considering the viewpoint of

some two-dimensional being (called a ˜bug™) confined exclusively to the two-

dimensional surface. Such a being would believe that it is able to look and measure

in all directions, whereas it is in fact limited to making measurements of distance,

angle etc. only within the surface. For example, it would receive light signals that

had travelled within the two-dimensional surface. Properties of the geometry that

are accessible to the bug are called intrinsic, whereas those that depend on the

viewpoint of a higher-dimensional creature (who is able to see how the surface

is shaped in the three-dimensional space) are called extrinsic.

The bug is able to define a coordinate system and measure distances in the

surface (e.g. by counting how many steps it has to take) from one point to another.

It can thus define a set of metric functions gab x that characterise the intrinsic

geometry of the surface (as expressed in the bug™s chosen coordinate system).

Consider, for example, a two-dimensional plane surface, such as a flat sheet

of paper, in our three-dimensional Euclidean space. The bug can label the entire

sheet using rectangular Cartesian coordinates, so that the distance ds measured

34 Manifolds and coordinates

over the surface between any pair of points whose coordinate separations are dx

and dy is given by

ds2 = dx2 + dy2

If this sheet is then rolled up into a cylinder, the bug would not be able to detect

any differences in the geometrical properties of the surface (see Figure 2.1).

To the bug, the angles of a triangle still add up to 180 , the circumference of a

circle is still 2 r etc. The proof of this fact is simple “ the surface can simply be

unrolled back to a flat surface without buckling, tearing or otherwise distorting

it. A more mathematical approach is to note that if one parameterises the surface

of the cylinder (of radius a) using cylindrical coordinates z , the distance ds

measured over the surface between any two points whose coordinate separations

are dz and d is given by

ds2 = dz2 + a2 d 2

By making the simple change of variables x = z and y = a we recover the

expression ds2 = dx2 + dy2 , which is valid over the whole surface, and so the

intrinsic geometry is identical to that of a flat plane. Thus the surface of a cylinder

is not intrinsically curved; its curvature is extrinsic and a result of the way it is

embedded in three-dimensional space. Even if one were to crumple up the sheet

of paper (without tearing it), so that its extrinsic geometry in three-dimensional

space was very complicated, its intrinsic geometry would still be that of a plane.

The situation is somewhat different for a 2-sphere, i.e. a spherical surface,

embedded in three-dimensional Euclidean space. Once again the surface is mani-

festly curved extrinsically on account of its embedding. Additionally, however,

it cannot be formed from a flat sheet of paper without tearing or deformation.

Its intrinsic geometry “ based on measurements within the surface “ differs from

the intrinsic (Euclidean) geometry of the plane. This problem is well known to

A' B'

Q

Q

P

P

A B

Figure 2.1 Rolling up a flat sheet of paper into a cylinder.

35

2.8 Intrinsic and extrinsic geometry

cartographers. Mathematically, if we parameterise a sphere (of radius a) by the

usual angular coordinates then

ds2 = a2 d + sin2 d

2 2

which cannot be transformed to the Euclidean form ds2 = dx2 + dy2 over the

whole surface by any coordinate transformation. Thus the surface of a sphere is

intrinsically curved.

We note, however, that locally at any point A on the spherical surface we

can define a set of Cartesian coordinates, so that ds2 = dx2 + dy2 is valid in the

neighbourhood of A. For example, the street layout of a town can be accurately

represented by a flat map, whereas the entire globe can only be represented by

performing projections that distort distance and/or angles. As an idea of what can

happen to local Cartesian coordinate systems far from the point A where they are

defined, consider Figure 2.2. If a bug starts at A and travels in the locally defined

x-direction to B, it observes that C still lies in the y-direction. If instead the bug

travels from A to C, it finds that B still lies in the x-direction. The non-Euclidean

geometry of the spherical surface is also apparent from the fact that the angles of

the triangle ABC sum to 270 .

We may take our discussion one step further, dispense with the three-

dimensional space and embedding-related extrinsic geometry and consider the

surfaces in isolation. Intrinsic geometry is all that remains with any meaning.

For example, when we talk of the curvature of spacetime in general relativity,

we must resist any temptation to think of spacetime as embedded in any ˜higher™

space. Any such embedding, whether or not it is physically realised, would

be irrelevant to our discussion. Nevertheless, in developing our intuition for

A

y

x

B C

Figure 2.2 A two-dimensional spherical surface.

36 Manifolds and coordinates

curved manifolds it oftens remains useful to imagine two-dimensional surfaces

embedded in three-dimensional Euclidean space.

2.9 Examples of non-Euclidean geometry

Let us develop our intuition for non-Euclidean geometry by considering in more

detail the surface of a sphere. We begin by imagining the usual Cartesian coor-

dinate system (x, y, z) defining a Euclidean three-dimensional space with line

element

ds2 = dx2 + dy2 + dz2 (2.6)

Now, suppose that we have a sphere of radius a with its centre at the origin of

our coordinate system. We will now ask the following question: what is the line

element on the surface of the sphere?

The equation defining the sphere is

x2 + y2 + z2 = a2

So, differentiating this equation, we obtain

2x dx + 2y dy + 2z dz = 0

and we can write an equation for dz,

x dx + y dy ’ x dx + y dy

dz = ’ = (2.7)

1/2

a2 ’ x2 + y2

z

Thus, equation (2.9) provides a constraint on dz that keeps us on the surface of

the sphere if we are displaced by small amounts dx and dy from an arbitrary

point on the sphere (for example, the point A in Figure 2.2). Substituting for dz

in (2.6) gives us the interval for such constrained displacements:

x dx + y dy 2

ds = dx + dy + 2

2 2 2

(2.8)

a ’ x2 + y 2

which is the line element for the surface of the sphere in terms of our chosen

coordinates (as shown in Figure 2.2), taking A as the origin of x and y. We

see that this line element reduces to the Euclidean form ds2 = dx2 + dy2 in

the neighbourhood of A. Practically, one could construct the coordinate curves

x = constant and y = constant on the surface of the sphere by creating a standard

x y coordinate grid in the tangent plane at A and ˜projecting™ vertically down

onto the spherical surface.

37

2.9 Examples of non-Euclidean geometry

We may obtain an alternative form for the line element by making the substi-

tutions

x = cos y = sin

and after a little algebra we obtain5

a2 d 2

ds = 2 +

2 2 2

(2.9)

d

a’ 2

As above, one can construct the and coordinate curves on the sphere by creat-

ing a standard coordinate system in the tangent plane at A and projecting

vertically down onto the surface. We also note that this line element contains

a ˜hidden symmetry™, namely our freedom to choose an arbitrary point on the

sphere as the origin = 0.

The observant reader will have noticed that the line elements (2.8) and (2.9)

have singularities at x2 + y2 = a, or, equivalently, = a, corresponding to the

equator of the sphere (relative to A). From our embedding picture, it is clear

why the x y and coordinates cover the surface of the sphere uniquely

only up to this point. We note, however, that there is nothing pathological in the

intrinsic geometry of the 2-sphere at the equator. What we have observed is only

a coordinate singularity, which has resulted simply from choosing coordinates

with a restricted domain of validity. Although the embedding picture we have

adopted gives both the x y and coordinate systems a clear geometrical

meaning in our three-dimensional Euclidean space, it is important to realise that

a bug confined to the two-dimensional surface of the sphere could, if it wished,

have defined these coordinate systems to describe the intrinsic geometry without

any reference to an embedding in higher dimensions.

We can make an analogous construction to find the metric for a 3-sphere embed-

ded in four-dimensional Euclidean space. The metric for the four-dimensional

Euclidean space is

ds2 = dx2 + dy2 + dz2 + dw2 (2.10)

and, by analogy with the example above, the equation defining a 3-sphere is

x2 + y2 + z2 + w2 = a2

Differentiating as before gives

2x dx + 2y dy + 2z dz + 2w dw = 0

5

Note that the line elements (2.8) and (2.9) look different from the metric we would write down using

standard spherical polars, ds2 = a2 d 2 + a2 sin2 d 2 . Nonetheless, both are valid line elements for the

two-dimensional surface of a sphere.

38 Manifolds and coordinates

and so substituting for dw in (2.10) gives the line element:

2

x dx + y dy + z dz

ds = dx + dy + dz + 2

2 2 2 2

a ’ x2 + y2 + z2

Transforming to spherical polar coordinates

x = r sin cos

y = r sin sin

z = r cos

we obtain an alternative form for the line element:

a2

ds = 2 dr 2 + r 2 d + r 2 sin2 d

2 2 2

(2.11)

a ’r 2

Notice that, in the limit a ’ , the metric tends to the form

ds2 = dr 2 + r 2 d + r 2 sin2 d

2 2

which is simply the metric of ordinary Euclidean three-dimensional space ds2 =

dx2 + dy2 + dz2 , rewritten in spherical polar coordinates. The line element (2.11)

therefore describes a non-Euclidean three-dimensional space. We note that this

line element also has a singularity, this time at r = a. As one might expect from

our discussion above, this is once again just a coordinate singularity, although our

existence as three-dimensional ˜bugs™ makes the geometric reason for this less

straightforward to visualise!

2.10 Lengths, areas and volumes

For a given set of metric functions gab x , (2.4), it is useful to know how to

compute the lengths of curves and the ˜areas™ and ˜volumes™ of subregions of the

manifold.

The lengths of curves follow immediately from the line element. Suppose that

the points A and B are joined by some path; then the length of this curve is given by

B B

LAB = ds = 1/2

gab dxa dxb

A A

where the integral is evaluated along the curve. As indicated, the absolute

value of ds is taken before the square root is evaluated when considering

39

2.10 Lengths, areas and volumes

pseudo-Riemannian manifolds. If the equation of the curve xa u is given in

terms of some parameter u then

1/2

dxa dxb

uB

LAB = (2.12)

gab du

du du

uA

where uA and uB are the values of the parameter u at the endpoints of the curve.

For the calculation of areas and volumes, let us begin by considering the simple

case where the metric is diagonal, i.e. gab x = 0 for a = b.6 In this case the line

element takes the form

ds2 = g11 dx1 2 + g22 dx2 2 + · · · + gNN dxN 2

(2.13)

Such a system of coordinates is called orthogonal since, at all points in the

manifold, any pair of coordinate curves cross at right angles, as is clear from

(2.13). Thus, in orthogonal coordinate systems the ideas of area and volume can be

built up simply. Consider, for example, an element of area in the x1 x2 -surface

defined by xa = constant for a = 3 4 N . Suppose that the area element is

defined by the coordinate lengths dx1 and dx2 (see Figure 2.3). The proper

√ √

lengths of the two line segments will be g11 dx1 and g22 dx2 respectively.

Thus the element of area is7

dA = g11 g22 dx1 dx2 (2.14)

x1 + dx1 x2 + dx2

x2

x1

Figure 2.3 An element of area, on a manifold , defined by the coordinate

1 2 1 2

intervals dx and dx . The proper lengths dl and dl of these intervals are related

to dx1 and dx2 by the metric functions. If the coordinate lines are orthogonal

then the area of is dl1 dl2 .

6

The general case is discussed in Section 2.14.

7

We have implicitly assumed here that the manifold is strictly Riemannian. If the manifold is pseudo-

Riemannian, some of the elements gab in (2.13) may be negative (see Section 2.13), and then we require the

modulus signs.

40 Manifolds and coordinates

Similarly, for 3-volumes in the x1 x2 x3 -surface defined by xa = constant for

a=4 5 N , we have

d3 V = g11 g22 g33 dx1 dx2 dx3 (2.15)

We may, of course, define 3-volumes for any other three-dimensional subspace.

We can define higher-dimensional ˜volume™ elements in a similar way until we

reach the N -dimensional volume element

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

As examples of working with such metric functions, let us consider the non-

Euclidean spaces discussed in Section 2.9. We begin with the line element (2.9),

a2 d 2

ds = 2 +

2 2 2

(2.16)

d

a’ 2

which describes two-dimensional geometry on the surface of a sphere in terms

of the coordinates , the geometrical meanings of which are illustrated in

Figure 2.4 assuming an embedding in three-dimensional Euclidean space. From

(2.16) we see that this coordinate system is orthogonal, with g = a2 / a2 ’ 2

and g = 2 (no sums on or ).8 Let us consider a circle defined by = R,

O

ρ

φ

P

Figure 2.4 The surface of a sphere parameterised by the coordinates

appearing in the line element (2.16).

8

This form of notation is quite common, once a particular coordinate system has been chosen, and it is usually

clear from the context that no summation is implied.

41

2.10 Lengths, areas and volumes

where R is some constant, and calculate its length, its area and the distance from

its centre to the perimeter.

From (2.12) and (2.16), the distance in the surface from the centre to the

perimeter along a line of constant is

R a R

d = a sin’1

D=

a2 ’ 2 1/2 a

0

while the circumference of the circle is given by

2

C= Rd = 2 R

0

Similarly, from (2.14) we have, for the area of the spherical surface enclosed by C,

1/2

R2

2 R a

A= d d =2 a 1’ 1’ 2

2

a2 ’ 2 1/2 a

0 0

Note that if we rewrite the circumference C and area A in terms of the distance

D then we obtain

D D

C = 2 a sin A = 2 a2 1 ’ cos

and (2.17)

a a

Thus, as D increases, both the circumference and area of the circle increase

until the point when D = a/2, after which both C and A become smaller as D

increases.

In fact there is a slight subtlety here. As noted earlier, if we attempt to param-

eterise points beyond the equator of the sphere using the coordinates ,

the system becomes degenerate, i.e. there is more than one point in the surface

with the same coordinates. The degenerate nature of the coordinate system

means that some care is required, for example, in calculating the total area of the

surface. By symmetry this is given by

2 a a

Atot = 2 d d = 4 a2

a2 ’ 2 1/2

0 0

Although we cannot easily visualise the geometry, we can perform similar

calculations for the line element (2.11),

a2

ds = 2 dr 2 + r 2 d + r 2 sin2 d

2 2 2

(2.18)

a ’r 2

which describes a non-Euclidean three-dimensional space that tends to Euclidean

three-dimensional space as a ’ . Let us consider a 2-sphere of coordinate radius

r = R and calculate the circumference around the equator, the area, the volume

and the distance from its centre to the surface of the sphere.

42 Manifolds and coordinates

From (2.12) and (2.18), the distance from the centre to the surface along a line

= constant, = constant is

R a dr R

= a sin’1

D=

a2 ’ r 2 1/2 a

0

Noting that the equator of the sphere is the curve r = R, = /2, its circumference is

2

C = 0 R d = 2 R while the area of the surface r = R and the volume it

encloses are obtained from (2.14) and (2.15) and read

2

A= R2 sin d d = 4 R2

0 0

ar 2 sin

2 R

V= dr d d

a2 ’ r 2 1/2

0 0 0

§ «

⎨1 2 1/2 ¬

R R R

sin’1

= 4 a3 + 1’

©2 ⎭

a a a

It is not difficult to see that the familiar results of three-dimensional Euclidean

space are recovered when R/a 1. Once again, we can rewrite our results in

terms of D rather than R, and we find that C, A and V all have maximum values

at D = a/2. By analogy with the above two-dimensional example, the total

volume of this space is

ar 2 sin

2 a

Vtot = 2 dr d d = 2 23

a

a2 ’ r 2 1/2

0 0 0

The three-dimensional non-Euclidean space described by the line element (2.18)

thus has a finite volume. We can generate a line element for an infinite non-

Euclidean three-dimensional space by making the substitution a = ib, i.e. choosing

the ˜radius™ of the space to be pure imaginary. The line element (2.18) then becomes

b2

ds = 2 dr 2 + r 2 d + r 2 sin2 d

2 2 2

b +r 2

If we again consider the sphere defined by r = R, we find easily that in this space

C = 2 R and A = 4 R2 as before but the distance from the centre of the sphere

to its surface is now given by D = b sinh’1 R/b . In this case, one finds that C,

A and the volume V of the sphere are all monotonically increasing functions.

2.11 Local Cartesian coordinates

We now introduce a key property of Riemannian manifolds, to which we have

alluded in earlier sections. For the moment we will confine our attention to

43

2.11 Local Cartesian coordinates

manifolds that are strictly Riemannian, so that ds2 > 0 always, but subsequently

we will extend our discussion to pseudo-Riemannian spaces, in which ds2 can be

of either sign (or zero).

For a general Riemannian manifold, it is not possible to perform a coordinate

transformation xa ’ x a that will take the line element ds2 = gab x dxa dxb into

the Euclidean form

ds2 = dx 1 2 + dx 2 2 + · · · + dx N =

2 a

dx b

ab dx

at every point in the manifold. This is clear, since there are N N + 1 /2 inde-

pendent metric functions gab x but only N coordinate transformation functions

x a x . As we shall now demonstrate, however, it is always possible to make a

coordinate transformation such that in the neighbourhood of some specified point

P the line element takes the Euclidean form. In other words, we can always find

coordinates x a such that at the point P the new metric functions gab x satisfy

gab P = (2.19)

ab

gab

=0 (2.20)

xc P

Thus, in the neighbourhood of P, we have

gab x = ab + x ’ xP 2

Such coordinates are called local Cartesian coordinates at P.

From (2.5), the general transformation rule for the metric functions is

xc xd

gab = g

x a x b cd

which we require to satisfy the conditions (2.19) and (2.20) at our chosen point

P. If xa is an arbitrary given coordinate system and x a is the desired system

then there will be some relation xa x connecting the two sets of coordinates.

Although we do not (as yet) know the required transformation, we can define it

in terms of its Taylor expansion about P:

xa

xx= + x b ’ xP

a a b

xP

xb P

2 xa

1

+ x b ’ xP x c ’ xP

b c

b c

2 x x P

3 xa

1

+ x b ’ xP x c ’ xP x d ’ xP + · · ·

b c d

b c d

6 x x x P

44 Manifolds and coordinates

The numbers of free independent variables we have for this purpose are as

follows:

has N 2 independent values

xa / x b P

has 2 N 2 N + 1 independent values

1

2 xa / xb xc P

has 1 N 2 N + 1 N + 2 independent values

3 xa / xb xc xd P 6

where we have made use of the fact that the second set of quantities is symmetric

in b and c and the third set of quantities is totally symmetric in b, c and d. We

may compare this with the number of independent parameters we may want to fix:

has 2 N N + 1 independent values

1

gab P

has 2 N 2 N + 1 independent values

1

gab / x c P

has 4 N 2 N + 1 2

1

2g / xc xd independent values

P

ab

The first question is whether we can satisfy the requirement (2.19). This condition

consists of N N + 1 /2 independent equations, and to satisfy them we have N 2 free

values in xa / x b P . Therefore, they can indeed be satisfied, leaving N N ’ 1 /2

numbers to spare! These spare degrees of freedom correspond exactly to the number

of independent N -dimensional ˜rotations™ that leave ab unchanged.

The next question is whether we can satisfy the requirement (2.20). This

condition consists of N 2 N + 1 /2 independent equations, and we can choose an

equal number of free values 2 xa / x b x c P to satisfy them.