IV. COSMOLOGY AND THE THEORY
OF RELATIVITY
When it became apparent at the end of the nine-
teenth century that pure Newtonian theory (that is,
without the addition of a repulsive term to Newton's
law of
force) could not lead to a static model of the
universe, most scientists
lost interest in the cosmologi-
cal problem
and very little work was done in this field
until the whole subject was
dramatically reopened by
Einstein in 1917, when he published his famous
paper
on relativistic cosmology. New life was suddenly given
to
cosmology by the appearance of this paper, since
it now appeared that the
flaws in Newtonian cosmol-
ogy would be
eliminated with the introduction of the
Einsteinian space-time concept. As
we shall presently
see, this is indeed true, but difficulties still arise
because
a number of different model universes can be obtained
from
general relativity theory, and we are then left
with the problem of
deciding which of these is the
correct model. This is a somewhat
unsatisfactory situa-
tion since one of the
purposes of a theory is to restrict
the theoretical models that can be
deduced from it to
just those that we actually observe in nature; but
in
spite of this drawback, we must turn to the general
theory of
relativity for an understanding of cosmology,
since it is the best theory
of space and time that we
now have and Newtonian theory has certainly
been
disproved. However, before we can discuss relativistic
cosmology
meaningfully, we must understand the basic
concepts of the theory of
relativity itself.
This theory was developed in two stages: the first
(1905) is called the
special or restricted theory of
relativity and the second (1915) is called
the general
theory. The basic feature of the special theory is that
all observers moving with uniform speed in straight
lines relative to the
distant background stars (such
observers are said to be moving in inertial
frames of
reference) are equivalent in the eyes of nature, in the
sense that the laws of nature are the same for all of
them. Put
differently, the special theory states that an
observer in an inertial
frame cannot determine his state
of motion by any kind of experiment (or
observation)
performed entirely in his frame of reference (that is,
without referring to the background stars). Before the
time of Einstein,
this formulation of the special theory
was accepted by physicists only
insofar as it applied
to the laws of Newtonian mechanics. They
believed
that an observer in an inertial frame could not detect
his
uniform motion by means of any mechanical exper-
iment, but they assumed that the principle did not
apply to
optical phenomena and that an inertial ob-
server
could detect his motion through the ether (whose
existence had been postulated to account for the prop-
agation of light) by observing the way light moves
(that
is, by measuring the speed of light) in various directions
in
his frame of reference. Physicists believed this to
be so because the
Newtonian concepts of absolute
space and absolute time lead precisely to
this very
conclusion.
One can deduce from these concepts that the speed
of light is not the same
in all directions, as measured
by a moving observer—the measured
value of the
speed of light should be a maximum for a beam of
light
moving against the motion of the observer and
a minimum for a beam moving
in the same direction
as the observer. This deduction, however, is
contrary
to the experimental evidence. In 1887 Michelson and
Morley
demonstrated experimentally that the speed of
light is the same in all
directions for all inertial ob-
servers. Thus
the constancy of the speed of light for
all such observers must be accepted
as a law of nature.
This means, as emphasized by Einstein, that the
special
theory of relativity must apply to optical phenomena
just as
it does to mechanical phenomena, so that an
observer in an inertial frame
cannot deduce his state
of motion from optical phenomena. Since this is
con-
trary to the deductions from the
Newtonian concepts
of absolute space and absolute time, Einstein
rejected
these absolute Newtonian notions and replaced them
by
relative time and relative space.
To illustrate the essential difference between the two
concepts (absolute
and relative) we may consider two
events separated in space by a certain
distance d and
in time by the interval t as measured by some particular
observer in an inertial
frame. Now, according to the
absolute concepts of Newton, all other inertial ob-
servers recording these two events would find the same
distance d between them and the same interval t. This
is what Einstein denies, for, as we have noted, this
contradicts the observed fact that the speed of light
is the same in all
directions for all observers. This
means that the distance d and the time interval t are
different for observers moving with different speeds,
so that space and
time separately vary as we pass from
one inertial frame to another. The
special theory of
relativity replaces the separate absolute Newtonian
concepts of space and time with a single absolute
space-time concept for
any two events, which is con-
structed as
follows by any observer: Let this observer
measure the distance between
these two events and
square this number to obtain d2. Next, let him measure
the time interval between the two events
and square
this to obtain t2. He now constructs the
numerical
quantity d2-c2t2, where c is the speed of light.
This
quantity, which is called the square of the
space-time
interval between the two events, is the same for all
observers moving in different inertial frames of refer-
ence.
We see from this that the absolute three-dimensional
Newtonian spatial
universe, with its events unfolding
in a unique (absolute) temporal
sequence, is replaced
by a four-dimensional space-time universe in which
the
spatial separation and the time interval between any
two events
vary from observer to observer, but in
which all observers measure the same space-time in-
terval. We may state this somewhat differently by
saying that the universe of the special theory of rela-
tivity is a four-dimensional space-time universe gov-
erned by Euclidean
geometry. The last part of this
statement is important since it is
equivalent to saying
that the square of the space-time interval in a
universe
governed by special relativity is exactly d2-c2t2. In such
a
universe, free bodies (bodies that are not pulled or
pushed by ropes, or
rods, or by some other force) move
in straight lines in space-time.
We must now see how this theory, which is restricted
to observers in
inertial frames of reference, is to be
extended when we introduce
gravitational fields and
observers undergoing any arbitrary kind of
motion
(rotation, linear acceleration, etc.). That the theory as
it
stands (that is, the special theory of relativity) is not
equipped to treat
observers in accelerated frames of
reference or to deal with gravitational
fields can be
seen easily enough if we keep in mind that the special
theory is based on the premiss that all inertial observers
are equal in the
eyes of nature and that there is no
observation, mechanical or optical,
that an inertial
observer can make to indicate how he is moving.
Now it appears at first sight that such a statement
cannot be made about
observers in accelerated frames
of reference since the acceleration causes
objects to
depart from straight line motion. If one is in a train
which is moving at constant speed in a straight line,
objects in the train
behave just as they would if the
train were standing still; thus one can as
easily pour
coffee into a cup when the train is moving with con-
stant speed as when it is at rest. But any
departure
from constant motion (that is, any kind of acceleration)
can
at once be detected, because such things as pouring
liquids from one vessel
into another become extremely
difficult. We should therefore be able to
detect that
we are in an accelerated frame by observing just such
phenomena. It thus appears that inertial frames of
reference and
accelerated frames are not equivalent.
This, then, at first blush, would
seem to eliminate the
possibility of generalizing the theory of relativity.
But
we shall presently see just how Einstein overcame this
difficulty.
That the law of gravity, as stated by Newton, is not
in conformity with the
special theory of relativity, is
evident from the fact that, according to
this theory,
clocks, measuring rods, and masses change when
viewed
from different inertial frames of reference. But,
according to Newton's law
of gravity, the gravitational
force between two bodies is expressed in
terms of the
masses of the bodies and the distance between them
at a
definite instant of time. Hence this force can have
no absolute
meaning—in fact, there is no meaningful
way for an inertial
observer to calculate this force since
he has no way of knowing which values to use for the
masses of
the two bodies and the distance between
them. This breakdown of the
Newtonian law of gravity,
and the impossibility of incorporating
accelerated
frames of reference in the framework of special rela-
tivity, convinced Einstein that a
generalization of the
theory of relativity was not only necessary, but
possi-
ble. For if it were not possible to
generalize the theory,
a whole range of observers and of physical
phenomena
related to gravity would not be expressible in terms
of a
space-time formulation.
To see how Einstein set about generalizing his the-
ory, we may first note that two apparently unrelated
classes of
phenomena—those arising from accelerations
and those arising
from gravitational fields—are ex-
cluded from the special theory. Einstein therefore
proceeded on the
assumption that these two groups
of phenomena must be treated together and
that a
generalization of the theory of relativity must stem
from some
basic relationship between gravitational
fields and accelerated frames of
reference. This basic
relationship is contained in Einstein's famous
principle
of equivalence, a principle which permits one to state
that
all frames of reference (in a small enough region
of space) are equivalent
and that in such a region there
is no way for an observer to tell whether
he is in an
inertial frame of reference, in an accelerated frame,
or
in a gravitational field. Another way of putting this
is that the principle
of equivalence permits one to use
any kind of coordinate system (frame of
reference) to
express the laws of physics. This means, further, that
no law of physics can contain any reference to any
special coordinate
system, for if a law did contain such
a reference, this in itself could be
used by an observer
to determine the nature of his frame of reference.
Thus
all laws must have the same form in all coordinate
systems.
To understand how the principle of equivalence
leads to the general theory,
we must first see just what
the basis of this principle is and what it
states. The
principle itself stems from Galileo's observation that
all
bodies allowed to fall freely (that is, in a vacuum
with nothing impeding
them) fall with the same speed.
This can be stated somewhat differently if
we consider
the mass of a body (the amount of matter the body
contains). This quantity appears in two places in the
laws of Newtonian
physics. On the one hand, it is the
quantity that determines the inertia of
a body (that
is, the resistance a body offers to a force that tries to
move it). For this reason, the quantity is referred to
as the inertial mass of the body. But the concept of
mass
also appears in Newton's formula that expresses
the gravitational force
that one body exerts on another;
this mass is then referred to as the gravitational mass
of the body. The fact that all
bodies fall with the same
speed in a gravitational field means that the inertial
mass and
the gravitational mass of a body must be
equal.
This remarkable fact had been considered as no more
than a numerical
coincidence before Einstein devel-
oped his
general theory of relativity. Einstein started
out on the assumption that
the equality of the inertial
and gravitational masses of a body is not a
coincidence
but, instead, must have a deep significance. To see what
this significance is, consider the way bodies behave in
an accelerated
frame of reference somewhere in empty
space (far away from any masses) and
the way they
behave in a gravitational field (for example, on the
surface of the earth). Owing to their inertial masses,
all the bodies in
the accelerated system behave as
though they were being pulled opposite to
the direc-
tion of the acceleration and they
all respond in exactly
the same way (that is, they all
“fall” with the same
speed). To Einstein, this meant
that there is no way
to differentiate between an accelerated frame of refer-
ence and a frame that is at rest (or
moving with con-
stant speed) in a
gravitational field. This is called the
principle of equivalence. Another
way of stating it is
to say that the apparent force that a body
experiences
when it is in an accelerated frame of reference is
identical with the force this body would experience
in an appropriate
gravitational field; thus inertial and
gravitational forces are
indistinguishable.
Since the principle of equivalence makes it impossi-
ble to assign any special quality or physical significance
to
inertial frames of reference, the special theory
(which is based on the
assumption that inertial frames
are special in the
sense that only in such frames do
the laws of physics have their correct
and simplest
form) must be discarded for a more general theory
which
puts all frames of reference and all coordinate
systems on the same
footing. In such a theory, the laws
of physics must have the same form in
all coordinate
systems. With this in mind, we can now see how
Einstein
constructed his general theory of relativity.
We begin by noting that the
special theory replaces
the concepts of absolute distance d and absolute time
t between events by a single absolute space-time inter-
val whose square is d2-c2t2. Consider now a
freely
moving particle as viewed by an observer in an inertial
frame
of reference in a region of space where no
gravitational fields are
present. If this particle moves
a distance d in a
time t, the quantity d2-c2t2 must be
the same for all
observers in inertial frames. This simply
means that the natural space-time
path of a free parti-
cle for inertial frames
of reference is a straight line
and that the space-time geometry of the
special theory
of relativity is Euclidean. We may take this
formulation
then as the law of motion (and hence a law of nature)
of a
free particle.
Now if we are to carry out our program of extending
the principle of
relativity to cover observers in gravi-
tational fields and in accelerated frames of reference,
we must say
that this same law of motion (straight line
motion) applies to a body
moving freely in a gravita-
tional field
or in an accelerated frame of reference.
But we know that the space-time
path of a free particle
in a gravitational field (or in a rotating system)
appears
to be anything but straight. How, then, are we to
reconcile
this apparent contradiction? We must re-
define
the concept of a straight line! We are ordinarily
accustomed to think of a
straight line in the Euclidean
sense of straightness, because the geometry
of our
world is very nearly Euclidean and we have been
brought up on
Euclidean geometry. In a sense, we
suffer from the same kind of geometrical
bias concern-
ing space-time as does the man
who thinks the earth
is flat because he cannot detect its sphericity in
his
small patch of ground.
To overcome this parochial attitude, we note that
we can replace the
“straightness” concept by the con-
cept of the shortest distance between two points. We
can now state the law of motion of a free particle as
follows:
A free particle moving between two space-time
points always moves in such a
way that its space-time
path between these two points is shorter than any
other
space-time path that can be drawn between the two
points.
This statement of the law of motion makes no refer-
ence to the way the space-time path of the particle
looks, but
refers only to an absolute property of the
path which has the same meaning
for all observers.
If no gravitational fields or accelerated observers
are
present, the shortest space-time path is d2-c2t2 and the
geometry is
Euclidean. But if gravitational fields are
present, the shortest space-time
path of the particle
(that is, its geodesic) is not given by d2-c2t2,
but by
a different combination of d and t because the space-
time geometry is non-Euclidean. The essence of Ein-
stein's general theory is, then, that a gravitational
field
distorts space-time (it introduces a curvature into
space-time)
and the behavior of a free particle (that
is, the departure from Euclidean
straight-line motion)
is not due to a “gravitational
force” acting on the
particle, but rather to the natural
inclination of the
free particle to move along a geodesic. In a sense,
this
is similar to what happens whan a ball is allowed to
roll freely
on a perfectly smooth piece of ground. The
ball appears to us to move in a
“straight line,” but
we know that this cannot be so
because it is following
the contour of the earth, which is spherical.
Actually
the ball is moving along the shortest path on the
smooth
surface, which is the arc of a great circle.
From this discussion we see that in the general
theory of relativity, the space-time path of a freely
moving
particle is not d2-c2t2, but some variation of
this, which depends on the
kind of gravitational fields
that are present, and on the acceleration of
our coordi-
nate system. We can therefore go
from the special
theory to the general theory of relativity by
replacing
the space-time interval (d2-c2t2) by the quantity gd2-
qc2t2, where g and q are
quantities that vary from point
to point. The value of the quantities g and q at any
point for a
given observer will depend on the intensity
of the gravitational field at
that point and on the
acceleration of the frame of reference of the
observer.
Just as the special theory of relativity is based on the
statement that the quantity d2-c2t2 is the same for all
observers in
inertial frames of reference, the general
theory of relativity is based on
the statement that the
quantity gd2-qc2t2 must be the same for all
observers,
regardless of their frames of reference.
Now the use of the latter expression as the absolute
space-time interval
instead of the former means that
we pass from Euclidean to non-Euclidean
geometry
in going from the special to the general theory, and
the
quantities g and q (they are also
referred to as the
Einstein gravitational potentials) determine by how
much the geometry at any point of space-time departs
from Euclidean
geometry—in other words, these
quantities determine the
curvature of space time at
each point. If, then, we know how to find g and q,
we can determine the
nature of the geometry in any
region of space-time and hence the path of a
free
particle in that region. The curvature of space-time
thus becomes
equivalent to the intensity of the gravi-
tational field, so that the gravitational problem is re-
duced to a problem in non-Euclidean geometry.
The
next step, then, in this development was to set down
the law that
determines the quantities g and q,
and
this Einstein did in his famous field equations—a set
of ten partial differential equations that show just how
the quantities g and q (there are actually ten
such
quantities, but in the gravitational field arising from
a body
like the sun, only two of these ten quantities
are different from zero)
depend on the distribution of
matter. These gravitational field equations
are the basis
of all modern cosmological theories which we shall
now
discuss.
