Unruh effect

# Unruh effect

Discussion

Encyclopedia
The Unruh effect was first described by Stephen Fulling in 1973, Paul Davies
Paul Davies
Paul Charles William Davies, AM is an English physicist, writer and broadcaster, currently a professor at Arizona State University as well as the Director of BEYOND: Center for Fundamental Concepts in Science...

in 1975 and Bill Unruh
Bill Unruh
William George Unruh is a Canadian physicist at the University of British Columbia, Vancouver, who discovered the Unruh effect. Unruh was born in Winnipeg, Manitoba. He obtained his B.Sc. from the University of Manitoba in 1967, followed by an M.A. and Ph.D...

in 1976. It is the prediction that an accelerating observer will observe black-body radiation where an inertial observer
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

would observe none. In other words, the background appears to be warm from an accelerating reference frame; in layman's terms, a thermometer waved around in empty space will record a non-zero temperature. The ground state for an inertial observer is seen as in thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

with a non-zero temperature by the uniformly accelerated observer.

It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are under dispute. There is also some doubt about whether the Unruh effect implies the existence of Unruh radiation.

## The equation

The Unruh temperature, derived by William Unruh in 1976, is the effective temperature experienced by a uniformly accelerating detector in a vacuum field
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...

. It is given by:

where:
is the local acceleration
is the Boltzmann constant
is the reduced Planck's constant
is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

Thus, for example, an acceleration of corresponds to a temperature of 1 K.
The Unruh temperature has the same form as the Hawking temperature  of a black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

, which was derived (by Stephen Hawking
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...

) independently around the same time. It is, therefore, sometimes called the Hawking–Unruh temperature.

## Explanation

Unruh demonstrated theoretically that the notion of vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

depends on the path of the observer through spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

. From the viewpoint of the accelerating observer, the vacuum of the inertial observer will look like a state containing many particles in thermal equilibrium—a warm gas.

Although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the word vacuum is interpreted appropriately, as below.

### Vacuum interpretation

In modern terms, the concept of "vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

" is not the same as "empty space", as all of space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

is filled with the quantized fields that make up a universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

. Vacuum is simply the lowest possible energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

state of these fields, a very different definition from "empty".

The energy states of any quantized field are defined by the Hamiltonian, based on local conditions, including the time coordinate. According to special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, two observers moving relative to each other must use different time coordinates. If those observers are accelerating, there may be no shared coordinate system. Hence, the observers will see different quantum states and thus different vacua.

In some cases, the vacuum of one observer is not even in the space of quantum states of the other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field canonical commutation relations. This is because two mutually accelerating observers may not be able to find a globally defined coordinate transformation relating their coordinate choices.

An accelerating observer will perceive an apparent event horizon forming (see Rindler spacetime). The existence of Unruh radiation could be linked to this apparent event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...

, putting it in the same conceptual framework as Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...

. On the other hand, the theory of the Unruh effect explains that the definition of what constitutes a "particle" depends on the state of motion of the observer.

The (free) field needs to be decomposed into positive and negative frequency
Negative frequency
The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A signed value of frequency indicates both the rate and direction of rotation...

components before defining the creation and annihilation operators. This can only be done in spacetimes with a timelike Killing vector field. This decomposition happens to be different in Cartesian and Rindler coordinates
Rindler coordinates
In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum. The Rindler chart was introduced by Wolfgang Rindler. The Rindler coordinate system or frame describes a uniformly accelerating...

(although the two are related by a Bogoliubov transformation
Bogoliubov transformation
In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an...

). This explains why the "particle numbers", which are defined in terms of the creation and annihilation operators, are different in both coordinates.

The Rindler spacetime has a horizon, and locally any non-extremal black hole horizon is Rindler. So the Rindler spacetime gives the local properties of black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

s and cosmological horizons. The Unruh effect would then be the near-horizon form of the Hawking radiation
Hawking radiation is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein...

.

## Calculations

The theory of the Unruh effect involves the Rindler coordinates
Rindler coordinates
In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum. The Rindler chart was introduced by Wolfgang Rindler. The Rindler coordinate system or frame describes a uniformly accelerating...

and , which have metric

This is just ordinary Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

in relativistic polar coordinates:

The orbit in (1+1) space-time is a regular hyperbola. The parametric equations are of the above form, but with equal to the ratio of square of the speed of light to the proper acceleration. It is therefore a constant. There is only one parameter - the proper time . In the arguments of the hyperbolic function, this should be multiplied by the ratio of the proper acceleration to the speed of light.

A detector moving along a path of constant is uniformly accelerated, and is coupled to field modes which have a definite steady frequency as a function of . These modes are constantly Doppler shifted relative to ordinary Minkowski time as the detector accelerates, and they change in frequency by enormous factors, even after only a short proper time.

Translation in is a symmetry of Minkowski space: It is a boost around the origin. For a detector coupled to modes with a definite frequency in , the boost operator is then the Hamiltonian. In the Euclidean field theory, these boosts analytically continue to rotations, and the rotations close after . So

The path integral for this Hamiltonian is closed with period which guarantees that the H modes are thermally occupied with temperature . This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in at position has a frequency which is determined by the square root of the metric at , the redshift factor. The actual inverse temperature at this point is therefore

Since the acceleration of a trajectory at constant is equal to , the actual inverse temperature observed is:

The temperature observed by a uniformly accelerating particle is (in engineering units):

The Unruh effect could only be seen when the Rindler horizon is visible. If a refrigerated accelerating wall is placed between the particle and the horizon, at fixed Rindler coordinate , the thermal boundary condition for the field theory at is the temperature of the wall. By making the positive side of the wall colder, the extension of the wall's state to is also cold. In particular, there is no thermal radiation from the acceleration of the surface of the Earth, nor for a detector accelerating in a circle, because under these circumstances there is no Rindler horizon in the field of view.

The temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

of the vacuum, seen by an isolated observer accelerated at the Earth's gravitational acceleration of g
Standard gravity
Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...

= 9.81 m/s²
Metre per second squared
The metre per second squared is the unit of acceleration in the International System of Units . As a derived unit it is composed from the SI base units of length, the metre, and the standard unit of time, the second...

, is only 4×10−20 K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

. For an experimental test of the Unruh effect it is planned to use accelerations up to 1026 m/s², which would give a temperature of about 400,000 K.

To put this in perspective, at a vacuum Unruh temperature of 3.978×10−20 K, an electron would have a de Broglie Wavelength of h/√(3mekT) = 540.85 meters, and a proton at that temperature would have a wavelength of 12.62 meters. If electrons and protons were in intimate contact in a very cold vacuum, they would have rather long wavelengths and interaction distances.

At one astronomical unit
Astronomical unit
An astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....

from the sun, the acceleration is GM s/AU² = 0.005932 m/s². This gives an Unruh temperature of 2.41×10−23 kelvin. At that temperature, the electron and proton wavelengths are 21.994 kilometers 513 meters, respectively. Even a uranium atom will have a wavelength of 2.2 meters at such a low temperature.

## Other implications

The Unruh effect would also cause the decay rate of accelerated particles to differ from inertial particles. Stable particles like the electron could have nonzero transition rates to higher mass states when accelerated fast enough.

Although Unruh's prediction that an accelerating detector would see a thermal bath is not controversial, the interpretation of the transitions in the detector in the non-accelerating frame are. It is widely, although not universally, believed that each transition in the detector is accompanied by the emission of a particle, and that this particle will propagate to infinity and be seen as Unruh radiation.

The existence of Unruh radiation is not universally accepted. Some claim that it has already been observed, while others claims that it is not emitted at all. While the skeptics accept that an accelerating object thermalises at the Unruh temperature, they do not believe that this leads to the emission of photons, arguing that the emission and absorption rates of the accelerating particle are balanced.

## Experimental observation of the Unruh effect

Under experimentally achievable conditions for gravitational systems this effect is too small and its observation is very difficult. It was shown by Bell and Leinaas
that if one takes an accelerated observer to be an electron circularly orbiting in a constant external magnetic field, then the experimentally verified Sokolov–Ternov effect coincides with the Unruh effect, see also
.

A recent work by Martín-Martínez, Fuentes and Mann showed that accelerated detectors acquire a geometrical phase due to their movement through spacetime and that this can be used for the direct detection of the Unruh effect in regimes physically accessible with current technology
.

• Pair production
Pair production
Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon . For example an electron and its antiparticle, the positron, may be created...

• Virtual particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...