The Classical Vacuum

[From Scientific American, August 1985, pp 70-78.]

It is not empty. Even when all matter and heat radiation have

been removed from a region of space, the vacuum of classical

physics remains filled with a distinctive pattern of

electromagnetic fields

by Timothy H. Boyer

Aristotle and his followers believed no region of space could

be totally empty: This notion that "nature abhors a vacuum" was

rejected in the scientific revolution of the 17th century;

ironically, though, modern physics has come to hold a similar

view. Today there is no doubt that a region of space can be

emptied of ordinary matter, at least in principle. In the

modern view, however, a region of vacuum is far from being

empty or featureless. It has a complex structure, which cannot

be eliminated by any conceivable means.

This use of words may seem puzzling. If the vacuum is not

empty, how can it be called a vacuum? Physicists today define

the vacuum as whatever is left in a region of space when it has

been emptied of everything that can possibly be removed from it

by experimental means. The vacuum is the experimentally

attainable void. Obviously a first step in creating a region of

vacuum is to eliminate all visible matter, such as solids and

liquids. Gases must also be removed. When all matter has been

excluded, however, space is not empty; it remains filled with

electromagnetic radiation. A part of the radiation is thermal,

and it can be removed by cooling, but another component of the

radiation has a subtler origin. Even if the temperature of a

vacuum could be reduced to absolute zero, a pattern of

fluctuating electromagnetic waves would persist. This residual

radiation, which has been analyzed only in recent years, is an

inherent feature of the vacuum, and it cannot be suppressed.

A full account of the contemporary theory of the vacuum would

have to include the ideas of quantum mechanics, which are

curious indeed. For example, it would be necessary to describe

the spontaneous creation of matter and antimatter from the

vacuum. Nevertheless, some of the remarkable properties of the

vacuum do not depend at all on the peculiar logic of the

quantum theory, and they can best be understood in a classical

description (one that ignores quantum effects). Accordingly I

shall discuss the vacuum entirely in terms of classical ideas.

Even in the comparatively simple world of classical physics the

vacuum is amply strange.

The Discovery of the Vacuum

Aristotle's doctrine that a vacuum is physically impossible was

overthrown in the 17th century. The crucial development was the

invention of the barometer in 1644 by Evangelista Torricelli,

who had been secretary to Galileo. Torricelli poured mercury

into a glass tube closed at one end and then inverted the tube,

with the open end in a vessel filled with mercury. The column

of liquid fell to a height of about 30 inches above the level

of the mercury in the vessel, leaving a space at the top of the

tube. The space was clearly empty of any visible matter;

Torricelli proposed that it was also free of gas and so was a

region of vacuum. A lively controversy ensued between

supporters of the Aristotelian view and those who believed

Torricelli had indeed created a vacuum. A few years later

Blaise Pascal supervised a series of ingenious experiments, all

tending to confirm Torricelli's hypothesis.

In the following decades experiments with the vacuum had a

great vogue. The best-remembered of these demonstrations is one

conducted by Otto von Guericke, the burgomaster of Magdeburg,

who made a globe from two copper hemispheres and evacuated the

space within. Two teams of eight draft horses were unable to

separate the hemispheres. Other experiments of the era were

less spectacular but perhaps more informative. For example,

they led to the discovery that a vacuum transmits light but not

sound.

MAGDEBURG HEMISPHERES made in 1654 by Otto von Guericke

demonstrated the existence of the vacuum, When the hemispheres

were put together and the air was pulled out, two teams of

eight draft horses could not separate them. The apparatus is

now in the Deutsches Museum in Munich.

The understanding of the vacuum changed again in the 19th

century. The nature of the change can be illustrated by a

thought experiment to be done with imaginary ideal apparatus.

Suppose one had a piston and cylinder machined so perfectly

that the piston could move freely and yet nothing could leak

past it. Initially the piston is at the closed end of the

cylinder and there is no vacant space at all. When a steady

force is applied to withdraw the piston against the pressure of

the air outside, the space developed between the piston and

the end of the cylinder is a region of vacuum. If the piston is

immediately released, it moves back into the cylinder,

eliminating the vacuum space. If the piston is withdrawn and

held for some time at room temperature, however, the result is

quite different. External air pressure pushes on the piston,

tending to restore the original configuration. Nevertheless, the

piston does not go all the way back into the cylinder, even if

additional force is applied. Evidently something is inside the

cylinder. What appeared to be an empty space is not empty

after the wait.

The physicists of the 19th century were able to explain this

curious result. During the period when the piston was withdrawn

the walls of the cylinder were emitting heat radiation into the

vacuum region. When the piston was forced back in, the

radiation was compressed. Thermal radiation responds to

compression much as a gas does: both the pressure and the

temperature rise. Thus the compressed radiation exerts a force

opposing the reinsertion of the piston. The piston and cylinder

could be closed again only if one waited long enough for the

higher-temperature radiation to be reabsorbed by the walls of

the cylinder.

The form of thermal radiation is intimately connected with the

structure of the vacuum in classical physics. Nothing in my

discussion so far has indicated that this should be so, and

indeed the physicists of the 19th century were unaware of the

connection.

The Thermal Spectrum

Thermal radiation consists of electromagnetic fields that

fluctuate in the most random way possible. Paradoxically this

maximum randomness gives the radiation great statistical

regularity. Under conditions of thermal equilibrium, in which

the temperature is uniform everywhere, the radiation is both

homogeneous and isotropic: its properties are the same at every

point in space and in every direction. An instrument capable of

measuring any property of the radiation would give the same

reading no matter where it was placed and what direction it was

pointed in.

The one physical quantity that determines the character of

thermal radiation is its temperature. In 1879 the Austrian

physicist Josef Stefan investigated the total energy density

(or energy per unit volume) of thermal radiation and, on the

basis of some preliminary experiments suggested that the energy

density varies as the fourth power of the absolute temperature.

Five years later Stefan's student Ludwig Boltzmann found the

same relation through a theoretical analysis.

The temperature of thermal radiation determines not only its

total energy density but also its spectrum, that is, the curve

defining the amount of radiant energy at each frequency. The

effect of temperature on the thermal spectrum is familiar from

everyday experience; as an object is heated it first glows red

and then white or even blue as the spectrum comes to be

dominated by progressively higher frequencies. The thermal

spectrum is not a monochromatic one, however; a red-hot poker

emits radiation most strongly at frequencies corresponding to

red light, but it also gives off lesser quantities of radiation

at all higher and lower frequencies.

The shape of the thermal spectrum and its relation to

temperature were explored experimentally in the last years of

the 19th century, but the attempt to formulate a consistent

theoretical explanation met with only limited success. The aim

was to find a mathematical expression that would give the

intensity of the radiation as a function of the frequency and the

temperature. In other words, given some specified temperature,

the expression had to predict the intensity of radiation that

would be measured at any chosen frequency.

A sophisticated classical analysis of the thermal spectrum was

given by the German physicist Wilhelm Wien in 1893. Wien based

his analysis on a thought experiment much like the one

described above, but with the added provision that the cylinder

be perfectly insulated so that no heat could be gained or lost.

Wien calculated the change in the spectrum that would be

brought about by an infinitesimal change in the internal volume

of the cylinder. From this calculation he was able to deduce

that the mathematical expression describing the spectrum must

have two factors, which are multiplied to yield the intensity

at a given frequency and temperature. One factor is the cube of

the frequency. The second factor is a function of the absolute

temperature divided by the frequency, but Wien was not able to

determine the correct form of the function. (He made a

proposal, but it was soon shown to be wrong.)

CREATION OF A VACUUM proceeds in stages that parallel the

historical development of ideas about the vacuum. In the 17th

century (a) it was thought a totally empty volume of space

could be created by removing all matter, and in particular all

gases. Late in the 19th century (6) it became apparent that

such a region still contains thermal radiation, but it seemed

the radiation might be eliminated by cooling. Since then both

theory and experiment have shown there is nonthermal radiation

in the vacuum (c), and it would persist even if the temperature

could be lowered to absolute zero. It is called zero-point

radiation.

Classical Electron Theory

The mathematical function needed to describe the thermal

spectrum was suggested by Max Planck in 1900. Planck emphasized

that an understanding of thermal radiation required the

introduction of a new fundamental constant, now called Planck's

constant, with a value of 6.26 x 10**(-27) erg-second. In the

course of his struggle to explain his function for the thermal

spectrum Planck launched the quantum theory. The start of

quantum physics, however, did not mark the end of the story of

classical physics.

Stefan's and Boltzmann's proposal that the total energy density

of the thermal radiation is proportional to the fourth power of

the temperature implies that the energy density falls to zero

at a temperature of absolute zero. The thermal radiation simply

disappears at zero temperature. The possibility of eliminating

all thermal radiation led to a conception of the classical

vacuum that was an extension of the 17th-century view. A

perfect vacuum was still a totally empty region of space, but

to attain this state one had to remove not only all visible

matter and all gas but also all electromagnetic radiation. The

last requirement could be met in principle by cooling the

region to absolute zero.

This conception of the vacuum within classical physics was

embodied in the fundamental physical theory of the time, which

has since come to be known as classical electron theory. It

views electrons as pointlike particles whose only properties

are mass and electric charge. They can be set in motion by

electric and magnetic fields, and their motion in turn gives

rise to such fields. (An electron in steady oscillation, for

example, radiates electromagnetic waves at the frequency of

oscillation.) The interactions between particles and fields are

accounted for by Newton's laws of motion and by James Clerk

Maxwell's equations of electromagnetism. In addition certain

boundary conditions must be specified if the theory is to make

definite predictions. Maxwell's equations describe how an

electromagnetic field changes from place to place and from

moment to moment, but to calculate the actual value of the

field one must know the initial, or boundary, values of the

field, which provide a baseline for all subsequent changes.

It is through the choice of initial conditions that the nature

of the vacuum enters classical electron theory. Since in the

19th-century view the vacuum was empty of all radiation, the

initial conditions set on Maxwell's equations were the absence

of electric and magnetic radiation. Roughly speaking, the

19th-century version of classical electron theory assumed that

at some time in the distant past the universe contained matter

(electrons) but no radiation. All electromagnetic radiation

evolved from the acceleration of electric charges.

The Casimir Effect

Classical electron theory remains a viable field of

investigation today, but it has taken a new form in the 20th

century. The need for a revision is easily seen from an

experiment proposed in 1948 by Hendrik B. G. Casimir of the

Philips Research Laboratories in the Netherlands. Casimir

analyzed the forces that would act on two electrically

conducting, parallel plates mounted a small distance apart in a

vacuum. If the plates carry an electric charge, the laws of

elementary electrostatics predict a force between them, but

Casimir considered the case in which the plates are uncharged.

Even then a force can arise from electromagnetic radiation

surrounding the plates. The origin of this force is not

immediately obvious, but a mechanical analogy serves to make it

clear.

Suppose a smooth cord is threaded snuggly through holes in two

wood blocks, as in the upper illustration on the next page. The

cord is not tied to the blocks, and so at rest it neither

pushes them apart nor pulls them together. Nevertheless, if the

part of the cord between the blocks is made to vibrate

transversely, a force acts on the blocks and they tend to slide

along the cord away from each other. The force arises because

transverse motion of the cord is not possible where it passes

through a block, and so waves in the cord are reflected there.

When a wave is reflected, some of its momentum is transferred

to the reflector

The situation in Casimir's proposed experiment is similar. The

metal plates are analogous to the wood blocks, and the

fluctuating electric and magnetic radiation fields represent

the vibrating cord. The analogue of the hole in the wood block

is the conducting quality of the metal plates; just as waves on

the cord are reflected by the block, so electromagnetic waves

are reflected by a conductor. In this case there is radiation

on both sides of each plate, and thus the forces tend to

cancel. The cancellation is not exact, however; a small

residual force remains. The force is directly proportional to

the area of the plates and also depends on both the separation

between the plates and the spectrum of the fluctuating

electromagnetic radiation.

IDEAL PISTON AND CYLINDER provide the apparatus for a thought

experiment revealing the presence of thermal radiation. The

piston is initially at the closed end of the cylinder, leaving

no free space; then it is withdrawn partway and held in this

position for some time at room temperature. The space enclosed

would seem to be a vacuum, and yet when the piston is released,

it does not return to its initial position; indeed, it cannot

be pushed all tile way back into the cylinder even with

additional force. While the piston was held in the open

position tile walls of the cavity emitted thermal radiation

with a spectrum determined by the temperature. An attempt to

reinsert the piston compresses the radiation, raising its

temperature and tiles altering its spectrum. The hotter

radiation opposes the compression.

So far this analysis is wholly consistent with the 19th-century

view of the vacuum. The force acting on the plates is

attributed to fluctuating thermal radiation. When the

temperatures reduced to absolute zero, both the thermal

radiation and the force between. the plates should disappear.

Experiment contradicts this prediction. In 1958 the Dutch

physicist M. J· Sparnaay carried out a series of experiments

based on Casimir's proposal and found that the force did not

approach zero when the thermal radiation was reduced to low

intensity. Instead there was a residual attractive force that

would persist even at absolute zero.

The residual force is directly proportional to the area of the

plates and inversely proportional to the fourth power of their

separation; the constant of proportionality is 1.3 x 10**(-18)

erg-centimeter. Although such a force is small, it is

measurable if the plates are sufficiently close together. For

plates with an area of one square centimeter separated by 0.5

micrometer the Casimir force is equivalent to the weight of 0.2

milligram.

Whatever the magnitude of the Casimir effect, its very

existence indicates that there is something fundamentally wrong

with the 19th-century idea of the classical vacuum. If one is

to fit classical theory with experiment, then even at zero

temperature the classical vacuum cannot be completely empty; it

must be filled with the classical electromagnetic fields

responsible for the attractive force Sparnaay measured. Those

vacum fields are now referred to as classical electromagnetic

zero-point radiation.

CASIMIR EFFECT demonstrates the existence of electromagnetic

fields in the vacuum. Two metal plates in a vacuum chamber are

mounted parallel to each other and a small distance apart.

Because the plates are conducting, they reflect electromagnetic

waves; for a wave to be reflected there must be a node of the

electric field - a point of zero electric amplitude - at the

surface of the plate. The resulting arrangement of the waves

gives rise to a force of attraction. The origin of the force

can be understood in part through a mechanical analogy. If a

cord threaded through holes in two wood blocks is made to

vibrate, waves is the cord are reflected at tire holes and

generate forces on the blocks. The forces on a single block act

in opposite directions, but a small net force remains. Its

magnitude and direction depend on the separation between the

blocks and the spectrum of waves along the cord.

FORCE OBSERVED IN THE CASIMIR EXPERIMENT has two components. At

high temperature thermal radiation gives rise to a force

directly proportional to the temperature and inversely

proportional to the cube of the distance between the plates.

This force disappears at absolute zero, as the thermal

radiation itself does. The force associated with the zero-point

radiation is independent of temperature and inversely

proportional to the fourth power of the distance between the

plates. The forces shown are for plates with an area of one

square centimeter; the thermal force is an approximation valid

at high temperature.

The Zero-Point Spectrum

What are the characteristics of the zero-point radiation in the

classical vacuum? Much can be deduced from the fact that it

exists in a vacuum: it must conform to accepted basic ideas

about the nature of the vacuum. For example, it seems essential

that the vacuum define no special places or directions, no

landmarks in space or time; it should look the same at all

positions and in all directions. Hence the zero-point

radiation, like thermal radiation, must be homogeneous and

isotropic. Furthermore, the vacuum should not define any

special velocity through space; it. should look the same to any

two observers no matter what their velocity is with respect to

each other, provided the velocity is constant. This last

requirement is expressed by saying the zero-point radiation

must be invariant with respect to Lorentz transformation. (The

Lorentz transformation, named for the Dutch physicist H. A.

Lorentz, is a conversion from one constant-velocity frame of

reference to another, taking into account that the speed of

light is the same in all frames of reference.)

The requirement of Lorentz invariance is a serious constraint.

A railroad passenger may be momentarily unsure whether his

own train or the one on the next track is moving relative to

the earth, but the ambiguity can be resolved simply by

looking at some landmark known to be fixed. Lorentz invariance

implies that there are no such landmarks in the vacuum and

that no experiment could ever reveal an observer's velocity with

respect to the background of zero-point radiation. To meet this

condition the spectrum of the radiation must have quite

specific properties.

Suppose for the moment that the zero-point radiation, as

perceived by some observer, were all in the region of the

electromagnetic spectrum corresponding to green light. No

matter where the observer stood and no matter in what direction

he looked, the vacuum would appear to be filled with uniform

green radiation. Such a spectrum satisfies the requirements of

homogeneity and isotropy for this one observer, but now suppose

there is another observer moving toward the first one at a

constant speed. Because of the Doppler effect, the moving

observer would see the radiation in front of him shifted toward

the blue end of the spectrum and the radiation behind him

shifted toward the red end. The Lorentz transformation also

alters the intensity of the radiation: it would be brighter in

front and dimmer behind. Thus the radiation does not look the

same to both observers; it is isotropic to one but not to the

other.

It turns out that the zero-point spectrum can have only one

possible shape if the radiation is to be Lorentz-invariant. The

intensity of the radiation at any frequency must be

proportional to the cube of that frequency. A spectrum defined

by such a cubic curve is the same for all unaccelerated

observers, no matter what their velocity; moreover, it is the

only spectrum that has this property.

ZERO-POINT SPECTRUM is independent of the observer's velocity

because of compensating changes in frequency and intensity.

When an observer is approaching a source of radiation, all

frequencies are shifted to higher values and all intensities

are increased; moving away from the source has the opposite

effect. Thus a spectrum that has a peak in the green region for

a stationary observer has a larger blue peak for so approaching

observer and a smaller red peak for a receding observer. The

cubic curve that defines the zero-point spectrum balances the

shifts in frequency and intensity. Light that appears green in

the stationary frame of reference becomes blue to an

approaching observer, but its intensity matches that of the

blue light seen by an observer at rest. By the same token,

green light is shifted to red frequencies for a receding

observer, but its intensity is diminished correspondingly.

One immediate objection might be made to the cubic form of the

zero-point spectrum: because the intensity of the radiation

increases steadily at higher frequencies, the spectrum predicts

an infinite energy density for the vacuum. In the 19th century

such a prediction might well have been considered a fatal flaw,

but since the 1940's infinities have turned up in several areas

of physics, and methods have been developed for dealing with

them. In this case the infinite energy is confronted directly

only in the realm of gravitational forces. All other

calculations are based on changes or differences in energy,

which are invariably finite.

If the universe is permeated by classical zero-point radiation,

one might suppose it would make its presence known in phenomena

less subtle than the Casimir effect. For example, one might

think it would alter the outcome of the piston-and-cylinder

experiment by resisting the insertion of the piston even after

all thermal radiation had been eliminated.

Analysis indicates otherwise. Under equilibrium conditions,

when no external force is applied to the piston, there is

radiation both inside and outside the cylinder, and the

radiation pressures acting on the piston are balanced. This

balance holds for both thermal and zero-point radiation. When

the piston is pushed into the cylinder, the radiation is

compressed. Wien's calculation of the change in the spectrum as

a result of a change in volume indicates that the thermal

radiation resists such compression; it increases in temperature

and exerts a greater pressure against the piston. When the

same analysis is made for the zero-point radiation, however,

the result is different: the zero-point spectrum does not change

at all in response to compression. Indeed, a spectrum described

by a cubic curve is the only one that has this remarkable property.

The other experiment in which the cubic zero-point spectrum

should be checked is the Casimir effect itself. A theoretical

calculation based on the spectrum predicts a force between the

plates directly proportional to their area and inversely

proportional to the fourth power of their separation, in

agreement with Sparnaay's results. Again it can be shown that

the spectrum is unique in supporting this prediction; no other

spectral curve yields an inverse-fourth-power dependence on

distance.

The New Classical Electron Theory

The statement that a spectrum described by a cubic curve is

unique refers only to the shape of the curve; actually there

are infinitely many curves with the same shape but different

scales. In all the curves the intensity of the radiation is

proportional to the cube of the frequency, but the magnitude of

the intensity in each spectrum depends on a constant, which

sets the scale of the curve.

The value of the constant cannot be calculated theoretically,

but Sparnaay's measurement of the force in the Casimir effect

allows the value to be determined from experiment. After some

preliminary algebraic manipulation it is found that the

constant is equal to 3.3 x 10**(-27) erg-second, a magnitude

corresponding to one-half of Planck's constant. Thus Planck's

constant, the hallmark of all quantum physics, appears in a

purely classical context.

The introduction of classical zero-point radiation in the

vacuum mandates an important change in classical electron

theory. The revised version of the theory is still based on

Newton's laws of motion for the electrons and Maxwell's

equations for the electromagnetic field, but the boundary

conditions imposed on Maxwell's equations must be altered. No

longer is the vacuum empty of all electromagnetic fields; it is

now filled with randomly fluctuating fields having the

zero-point spectrum. The modified theory is called classical

electron theory with classical electromagnetic zero-point

radiation, a name often shortened to stochastic

electrodynamics.

The altered boundary conditions change the predictions of the

theory. The changes can be understood by considering one of the

favorite models of modern physics: a harmonic oscillator made

up of an electron attached to a perfectly elastic and

frictionless spring. This imaginary mechanical system is to be

set up in the classical vacuum. If the spring is stretched and

then released, the electron oscillates about its equilibrium

position and gives off electromagnetic radiation at the

frequency of oscillation.

HARMONIC OSCILLATOR reveals the effects of zero-point radiation

on matter. The oscillator consists of all electron attached to

an ideal, frictionless spring. When the electron is set in

motion, it oscillates about its point of equilibrium, emitting

electromagnetic radiation at the frequency of oscillation. The

radiation dissipates energy, and so in the absence of

zero-point radiation and at a temperature of absolute zero the

electron eventually comes to rest. Actually zero-point

radiation continually imparts random impulses to the electron,

so that it never comes to a complete stop. Zero-point radiation

gives the oscillator an average energy equal to the frequency

of oscillation multiplied by one-half of Planck's constant.

The harmonic oscillator is a convenient model because the

motion of the electron is readily calculated. Under the older

version of classical electron theory just two forces act on the

electron: the restoring force from the spring and a reaction

force arising from the emission of radiation. Because the

reaction force is directed opposite to the electron's motion,

the theory predicts that the oscillations will be steadily

damped and the electron will eventually come to rest. In the

new version of classical electron theory, however, the

zero-point radiation provides an additional force on the

electron. The charged particle is continually buffeted by the

randomly fluctuating fields of the zero-point radiation, so

that it never comes to rest. It turns out the harmonic

oscillator retains an average energy related to the zero-point

spectrum, namely one-half of Planck's constant multiplied by

the frequency of oscillation.

Up to now the classical vacuum has been described from the

point of view of an observer at rest or moving with constant

velocity. The consequences of zero-point radiation are even

more remarkable for an accelerated observer, that is, one whose

velocity is changing in magnitude or direction.

Effects of Acceleration

Consider an observer in a rocket continuously accelerating with

respect to some frame of reference that can be regarded as

fixed, such as the background of distant stars. What does the

classical vacuum look like to the rocket-borne observer? To

find out, one must perform a mathematical transformation from

the fixed frame of reference to the accelerated one. The

Lorentz transformation mediates between frames that differ in

velocity, but the situation is more complex here because the

velocity of the accelerated observer is continuously changing.

By carrying out Lorentz transformations over some time

interval, however, the vacuum observed from the rocket can be

determined.

One might guess that the spectrum for an accelerated observer

would no longer be isotropic, and in particular that some

difference would be detected between the forward and the

backward directions. The spectrum might also, be predicted to

change as the acceleration continued. In fact the spectrum

remains homogeneous and isotropic, and no change is observed

as long as the rate of acceleration itself does not change.

Nevertheless, the spectrum is not the one seen by an

unaccelerated observer. At any given frequency the intensity of

the radiation is greater in the accelerated frame than it is in the

frame at rest.

The form of the classical electromagnetic spectrum seen by an

accelerated observer is not one immediately familiar to

physicists, but it can be interpreted by analyzing the motion

of a harmonic oscillator carried along in the rocket. The

equation of motion for the accelerated oscillator is much like

the one valid in a fixed frame of reference. There are two

differences: the radiation-reaction force has a new term

proportional to the square of the acceleration, and the

oscillator is exposed to a new spectrum of random radiation

associated with the acceleration. The effect of these changes

is to increase the average energy above the energy associated

with the zero-point motion. In other words, when an oscillator

is accelerated, it jiggles more vigorously than it would if it

were at rest in the vacuum.

One way of understanding the effect of acceleration on the

harmonic oscillator is to ask what additional electromagnetic

spectrum could be added to the zero-point radiation to cause

the extra motion. To answer this question one can turn to the

equivalence principle on which Einstein founded his theory of

gravitation. The principle states that an observer in a small

laboratory supported in a gravitational field makes exactly the

same measurements as an observer in a small accelerating

rocket. The laws of thermodynamics are found to hold in a

gravitational field. From the equivalence principle one

therefore expects the laws of thermodynamics to hold in an

accelerating rocket. There is then only one possible

equilibrium spectrum that can be added to the zero-point

radiation: the additional radiation must have a thermal

spectrum. With any other spectrum the oscillator would not be

in thermal equilibrium with its surroundings, and so it could

serve as the basis of a perpetual-motion machine. By this route

one is led to a remarkable conclusion: a physical system

accelerated through the vacuum has the same equilibrium

properties as an unaccelerated system immersed in thermal

radiation at a temperature above absolute zero.

The mathematical relation connecting acceleration and

temperature was found in about 1976 by William G. Unruh of the

University of British Columbia and P. C· W· Davies of the

University of Newcastle upon Tyne. The effective spectrum seen

by an observer accelerated through the vacuum is the sum of two

parts. One part is the zero-point radiation; the other is the

spectrum of thermal radiation deduced by Planck in 1900. Planck

was able to explain the form of that curve only by introducing

quantum-mechanical ideas, which he did with some reluctance; it

now turns out the curve can be derived from an entirely

classical analysis of radiation in the vacuum.

At least one more intriguing result arises from this line of

inquiry. If one again invokes the equivalence principle

relating an observer in a gravitational field with an

accelerating observer, one concludes that there is a minimum

attainable temperature in a gravitational field. This limit is

an absolute one, quite apart from any practical difficulties of

reaching low temperatures. At the surface of the earth the

limit is 4 x 10**(-20) degree Kelvin, far beyond the

capabilities of real refrigerators but nonetheless greater than

zero.

The discovery of a connection between thermal radiation and the

structure of the classical vacuum reveals an unexpected unity

in the laws of physics, but it also complicates our view of

what was once considered mere empty space. Even with its

pattern of electric and magnetic fields in continual

fluctuation, the vacuum remains the simplest state of nature.

But perhaps this statement reflects more on the subtlety of

nature than it does on the simplicity of the vacuum.

EFFECT OF ACCELERATION through tire vacuum is to change the

spectrum of observed radiation. At a temperature of absolute

zero a harmonic oscillator in a frame of reference at rest or

moving with constant velocity is subject only to zero-point

oscillations. In an accelerated frame the oscillator responds

as if it were at a temperature greater than zero.