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.
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.
Even in the comparatively simple world of
classical physics the vacuum is amply strange.
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.
The bestremembered 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.
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.
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 highertemperature radiation to be reabsorbed by the walls
of the cylinder.
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.
Five years later Stefan’s student
Ludwig Boltzmann found the same relation through a
theoretical analysis.
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 redhot 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.
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.)
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
zeropoint radiation.
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.
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.
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.
All electromagnetic radiation evolved from the
acceleration of electric charges.
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.
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 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.
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.
Those vacuum fields are now referred to as
classical electromagnetic zeropoint radiation.
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.
This force disappears at absolute zero, as the thermal radiation itself does. The force associated with the zeropoint 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.
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 zeropoint 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 zeropoint 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 constantvelocity frame of
reference to another, taking into account that the speed of light is
the same in all frames of reference.)
The Lorentz transformation relates frames of reference that differ in velocity; for radiation to be Lorentzinvariant its spectrum must be unchanged by the transformation. The effect of motion on the spectrum is illustrated by an observer surrounded by peculiar traffic signals, which always indicate the intensity of the zeropoint radiation at three frequencies, namely those of red, green and blue light, Suppose an observer at rest with respect to the array of signals finds they all show green (a), meaning that all the zeropoint radiation is concentrated in the green part of the electromagnetic spectrum. If the observer then begins to move (b), the pattern is altered by the Doppler effect: the signals ahead appear blue and those behind red.
The Lorentz transformation also makes the approaching signals brighter and the receding ones dimmer. It turns out that ’ only one spectral form has the property of Lorentz invariance: the intensity must be proportional to the cube of the frequency.
When the traffic signals are illuminated according
to this rule, an observer at rest (c) and an observer in motion (d)
see the same pattern.
To meet this condition the spectrum of the
radiation must have quite specific properties.
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.
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.
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 zeropoint 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.
This balance holds for both thermal and zeropoint 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 zeropoint radiation, however, the
result is different: the zeropoint 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.
Again it can be shown that the
spectrum is unique in supporting this prediction; no other spectral
curve yields an inversefourthpower dependence on distance.
No longer is the vacuum empty
of all electromagnetic fields; it is now filled with randomly
fluctuating fields having the zeropoint spectrum. The modified
theory is called classical electron theory with classical
electromagnetic zeropoint radiation, a name often shortened to
stochastic electrodynamics.
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.
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 zeropoint radiation and at a
temperature of absolute zero the electron eventually comes to rest.
Actually zeropoint radiation continually imparts random impulses to
the electron, so that it never comes to a complete stop. Zeropoint
radiation gives the oscillator an average energy equal to the
frequency of oscillation multiplied by onehalf of Planck’s
constant.
The charged particle is continually buffeted
by the randomly fluctuating fields of the zeropoint radiation, so
that it never comes to rest. It turns out the harmonic oscillator
retains an average energy related to the zeropoint spectrum, namely
onehalf of Planck’s constant multiplied by the frequency of
oscillation.
The
consequences of zeropoint radiation are even more remarkable for an
accelerated observer, that is, one whose velocity is changing in
magnitude or direction.
What does the classical vacuum look like to the rocketborne 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.
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.
There are two differences: the radiationreaction 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 zeropoint motion. In other words, when an
oscillator is accelerated, it jiggles more vigorously than it would
if it were at rest in the vacuum.
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 zeropoint 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 perpetualmotion 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.
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.
But perhaps this statement reflects more
on the subtlety of nature than it does on the simplicity of the
vacuum.
In an accelerated frame the oscillator responds as if it were at a temperature greater than zero.
