The Classical Vacuum-Zero-Point Energy
The Classical Vacuum-Zero-Point Energy
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
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.
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.
[Picture.]
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.
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 (b) 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.
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 point-like 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.
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.
[Figure.]
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 vacuum fields are now referred to as classical electromagnetic zero-point radiation.
[Figure.]
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.
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.)
[Figure.]
LORENTZ INVARIANCE of the zero-point radiation ensures that the vacuum looks the same to observers moving through it at different velocities, provided each observer’s velocity is constant.
The Lorentz transformation relates frames of reference that differ in velocity; for radiation to be Lorentz-invariant 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 zero-point 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 zero-point 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.
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.
[Figure.]
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 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.
[Figure.]
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.
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.
[Figure.]
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.
by Timothy H. Boyer
Scientific American Magazine, Aug. 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
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.
[Picture.]
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 (b) 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.
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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 point-like 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.
[Figure.]
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 vacuum fields are now referred to as classical electromagnetic zero-point radiation.
[Figure.]
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.)
[Figure.]
LORENTZ INVARIANCE of the zero-point radiation ensures that the vacuum looks the same to observers moving through it at different velocities, provided each observer’s velocity is constant.
The Lorentz transformation relates frames of reference that differ in velocity; for radiation to be Lorentz-invariant 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 zero-point 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 zero-point 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.
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.
[Figure.]
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.
[Figure.]
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.
[Figure.]
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.
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The Classical Vacuum-Zero-Point Energy
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