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Chapter 2
Geometry
Our tour of theoretical physics begins with geometry, and there are two reasons
for this. One is that the framework of space and time provides, as it were, the
stage upon which physical events are played out, and it will be helpful to gain a
clear idea of what this stage looks like before introducing the cast. As a matter of
fact, the geometry of space and time itself plays an active role in those physical
processes that involve gravitation (and perhaps, according to some speculative
theories, in other processes as well). Thus, our study of geometry will culminate,
in chapter 4, in the account of gravity offered by Einstein’s general theory of
relativity. The other reason for beginning with geometry is that the mathematical
notions we develop will reappear in later contexts.
To a large extent, the special and general theories of relativity are ‘negative’
theories. By this I mean that they consist more in relaxing incorrect, though
plausible, assumptions that we are inclined to make about the nature of space
and time than in introducing new ones. I propose to explain how this works in
the following way. We shall start by introducing a prototype version of space
and time, called a ‘differentiable manifold’, which possesses a bare minimum of
geometrical properties—for example, the notion of length is not yet meaningful.
(Actually, it may be necessary to abandon even these minimal properties if, for
example, we want a geometry that is fully compatible with quantum theory and
I shall touch briefly on this in chapter 15.) In order to arrive at a structure
that more closely resembles space and time as we know them, we then have to
endow the manifold with additional properties, known as an ‘affine connection’
and a ‘metric’. Two points then emerge: first, the common-sense notions of
Euclidean geometry correspond to very special choices for these affine and metric
properties; second, other possible choices lead to geometrical states of affairs that
have a natural interpretation in terms of gravitational effects. Stretching the point
slightly, it may be said that, merely by avoiding unnecessary assumptions, we
are able to see gravitation as something entirely to be expected, rather than as a
phenomenon in need of explanation.
To me, this insight into the ways of nature is immensely satisfying, and it
6
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The Special andGeneral Theoriesof Relativity
7
is in the hope of communicating this satisfaction to readers that I have chosen to
approach the subject in this way. Unfortunately, the assumptions we are to avoid
are, by and large, simplifying assumptions, so by avoiding them we let ourselves
in for some degree of complication in the mathematical formalism. Therefore, to
help readers preserve a sense of direction, I will, as promised in chapter 1, provide
an introductory section outlining a more traditional approach to relativity and
gravitation, in which we ask how our naıve geometrical ideas must be modified
to embrace certain observed phenomena.
2.0 The Special and General Theories of Relativity
2.0.1 The special theory
The special theory of relativity is concerned in part with the relation between
observations of some set of physical events in two inertial frames of reference
that are in relative motion. By an inertial frame, we mean one in which Newton’s
first law of motion holds:
Every body continues in its state of rest, or of uniform motion in a right line,
unless it is compelled to change that state by forces impressed on it.
(Newton 1686)
It is worth noting that this definition by itself is in danger of being a mere
tautology, since a ‘force’ is in effect defined by Newton’s second law in terms
of the acceleration it produces:
The change of motion is proportional to the motive force impressed; and is
made in the direction of the right line in which that force is impressed.
(Newton 1686)
So, from these definitions alone, we have no way of deciding whether some
observed acceleration of a body relative to a given frame should be attributed, on
the one hand, to the action of a force or, on the other hand, to an acceleration of
the frame of reference. Eddington has made this point by a facetious re-rendering
of the first law:
Every body tends to move in the track in which it actually does move, except
insofar as it is compelled by material impacts to follow some other track than
that in which it would otherwise move.
(Eddington 1929)
The extra assumption we need, of course, is that forces can arise only from the
influence of one body on another. An inertial frame is one relative to which any
body sufficiently well isolated from all other matter for these influences to be
negligible does not accelerate. In practice, needless to say, this isolation cannot
be achieved. The successful application of Newtonian mechanics depends on our
being able systematically to identify, and take proper account of, all those forces
8
Geometry
Figure 2.1. Two systems of Cartesian coordinates in relative motion.
that cannot be eliminated. To proceed, we must take it as established that, in
principle, frames of reference can be constructed, relative to which any isolated
body will, as a matter of fact, always refuse to accelerate. These frames we call
inertial.
Obviously, any two inertial frames must either be relatively at rest or have a
uniform relative velocity. Consider, then, two inertial frames, S and S (standing
for S ystems of coordinates) with Cartesian axes so arranged that the x and x axes
lie in the same line, and suppose that S
moves in the positive x direction with
relative to S .Taking y parallel to y and z parallel to z ,wehavethe
arrangement shown in figure 2.1. We assume that the sets of apparatus used to
measure distances and times in the two systems are identical and, for simplicity,
that both clocks are adjusted to read zero at the moment the two origins coincide.
Suppose that an event at the coordinates
v
(
x
,
y
,
z
,
t
)
relative to S is observed
relative to S . According to the Galilean, or common-sense, view
of space and time, these two sets of coordinates must be related by
(
x ,
y ,
z ,
t )
x =
x
− v
t
y =
y
z =
z
t =
t
.
(2.1)
Since the path of a moving particle is just a sequence of events, we easily find that
its velocity relative to S , in vector notation u
=
d x
/
d t , is related to its velocity
u
=
d x /
d t
relative to S
by u
=
u
− v
, with
v = (v,
0
,
0
)
, and that its
a .
Despite its intuitive plausibility, the common-sense view turns out to be
mistaken in several respects. The special theory of relativity hinges on the fact
that the relation u =
is not true. That is to say, this relation disagrees with
experimental evidence, although discrepancies are detectable only when speeds
are involved whose magnitudes are an appreciable fraction of a fundamental
speed c , whose value is approximately 2
u
− v
10 8 ms − 1 . So far as is known,
light travels through a vacuum at this speed, which is, of course, generally
.
998
×
speed
at
acceleration is the same in both frames, a =
389503815.002.png
The Special andGeneral Theoriesof Relativity
9
µ 0 are called
the permittivity and permeability of free space, respectively) but the theory does
not single out any special frame relative to which this speed should be measured.
For quite some time after the appearance of Maxwell’s theory (published in its
final form in 1864; see also Maxwell (1873)), it was thought that electromagnetic
radiation consisted of vibrations of a medium, the ‘luminiferous ether’, and would
travel at the speed c relative to the rest frame of the ether. However, a number
of experiments cast doubt on this interpretation. The most celebrated, that of
Michelson and Morley (1887), showed that the speed of the Earth relative to the
ether must, at any time of year, be considerably smaller than that of its orbit
round the Sun. Had the ether theory been correct, of course, the speed of the
Earth relative to the ether should have changed by twice its orbital speed over a
period of six months. The experiment seemed to imply, then, that light always
travels at the same speed, c , relative to the apparatus used to observe it.
In his paper of 1905, Einstein makes the fundamental assumption (though
he expresses things a little differently) that light travels with exactly the same
speed, c, relative to any inertial frame . Since this is clearly incompatible with
the Galilean transformation law given in (2.1), he takes the remarkable step of
modifying this law to read
( 0 µ 0 ) − 1 / 2
(in SI units, where
0 and
x =
x
− v
t
y =
y
(
1
− v
2
/
c 2
)
1
2
(2.2)
t
− v
x
/
c 2
z =
z
t =
2 .
(
1
− v
2
/
c 2
)
1
/
These equations are known as the Lorentz transformation , because a set of
equations having essentially this form had been written down by H A Lorentz
(1904) in the course of his attempt to explain the results of Michelson and Morley.
However, Lorentz believed that his equations described a mechanical effect of the
ether upon bodies moving through it, which he attributed to a modification of
intermolecular forces. He does not appear to have interpreted them as Einstein
did, namely as a general law relating coordinate systems in relative motion. The
assumptions that lead to this transformation law are set out in exercise 2.1, where
readers are invited to complete its derivation. Here, let us note that (2.2) does
indeed embody the assumption that light travels with speed c relative to any
inertial frame. For example, if a pulse of light is emitted from the common origin
of S and S
=
t =
c 2 t 2 . Using the transformation (2.2), we
easily find that its equation at time t relative to S is x 2
+
y 2
+
z 2
=
c 2 t 2 .
Many of the elementary consequences of special relativity follow directly
from the Lorentz transformation, and we shall meet some of them in later
chapters. What particularly concerns us at present—and what makes Einstein’s
interpretation of the transformation equations so remarkable—is the change that
+
y 2
+
z 2
=
called the speed of light. Indeed, the speed of light is predicted by Maxwell’s
electromagnetic theory to be
/
0, then the equation of the resulting spherical wavefront
at time t relative to S is x 2
at t
389503815.003.png 389503815.004.png
10
Geometry
these equations require us to make in our view of space and time. On the face of
it, equations (2.1) or (2.2) simply tell us how to relate observations made in two
different frames of reference. At a deeper level, however, they contain information
about the structure of space and time that is independent of any frame of reference.
Consider two events with spacetime coordinates
(
x 1 ,
t 1 )
and
(
x 2 ,
t 2 )
relative to
S . According to the Galilean transformation, the time interval t 2 −
t 1 between
t 1 relative to S . In particular, it
may happen that these two events are simultaneous, so that t 2 −
0, and
this statement would be equally valid from the point of view of either frame
of reference. For two simultaneous events, the spatial distances between them,
|
t 1
=
are also equal. Thus, the time interval between two events
and the spatial distance between two simultaneous events have the same value in
every inertial frame, and hence have real physical meanings that are independent
of any system of coordinates. According to the Lorentz transformation (2.2),
however, both the time interval and the distance have different values relative to
different inertial frames. Since these frames are arbitrarily chosen by us, neither
the time interval nor the distance has any definite, independent meaning. The one
quantity that does have a definite, frame-independent meaning is the proper time
interval
x 2 |
and
|
x 1 −
x 2 |
τ
,definedby
c 2
τ
2
=
c 2
t 2
−
x 2
(2.3)
where
t
=
t 2 −
t 1 and
x
=|
x 2 −
x 1 |
. Byusing(2.2),itiseasytoverifythat
2 .
We see, therefore, that the Galilean transformation can be correct only in
a Galilean spacetime ; that is, a spacetime in which both time intervals and
spatial distances have well-defined meanings. For the Lorentz transformation to
be correct, the structure of space and time must be such that only proper-time
intervals are well defined. There are, as we shall see, many such structures. The
one in which the Lorentz transformation is valid is called Minkowski spacetime
after Hermann Minkowski who first clearly described its geometrical properties
(Minkowski, 1908). These properties are summarized by the definition (2.3) of
proper time intervals. In this definition, the constant c does not refer to the speed
of anything. Although it has the dimensions of velocity, its role is really no more
than that of a conversion factor between units of length and time. Thus, although
the special theory of relativity arose from attempts to understand the propagation
of light, it has nothing to do with electromagnetic radiation as such. Indeed, it
is not in essence about relativity either! Its essential feature is the structure of
space and time expressed by (2.3), and the law for transforming between frames
in relative motion serves only as a clue to what this structure is. With this in
mind, Minkowski (1908) says of the name ‘relativity’ that it ‘...seemsto mevery
feeble’.
The geometrical structure of space and time restricts the laws of motion that
may govern the dynamical behaviour of objects that live there. This is true, at
least, if one accepts the principle of relativity , expressed by Einstein as follows:
t 2
x 2 is also equal to c 2
−
τ
them relative to S is equal to the interval t 2 −
x 1 −
c 2

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