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Dictionary of the History of Ideas | ||

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*IV. COSMOLOGY AND THE THEORY*

OF RELATIVITY

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

*d*2. Next, let him measure

the time interval between the two events and square

this to obtain

*t*2. He now constructs the numerical

quantity

*d*2-

*c*2

*t*2, 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.

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

*d*2-

*c*2

*t*2. 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

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 *d*2-*c*2*t*2. 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 *d*2-*c*2*t*2 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 *d*2-*c*2*t*2 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 *d*2-*c*2*t*2,
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 *d*2-*c*2*t*2, 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 (*d*2-*c*2*t*2) by the quantity *gd*2-*qc*2*t*2, 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 *d*2-*c*2*t*2 is the same for all

observers in
inertial frames of reference, the general

theory of relativity is based on
the statement that the

quantity *gd*2-*qc*2*t*2 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.

Dictionary of the History of Ideas | ||