The Unruh effect is a surprising prediction of quantum field theory: From the point of view of an accelerating observer or detector, empty space contains a gas of particles at a temperature proportional to the acceleration. Direct experimental confirmation is difficult because the linear acceleration needed to reach a temperature 1 K is of order

1020m/s2

, but it is believed that an analog under centripetal acceleration is observed in the spin polarization of electrons in circular accelerators. Furthermore, the effect is necessary for consistency of the respective descriptions of observed phenomena, such as particle decay, in inertial and in accelerated reference frames; in this sense the Unruh effect does not require any verification beyond that of relativistic free field theory itself. The Unruh theory has had a major influence on our understanding of the proper relationship between mathematical formalism and (potentially) observable physics in the presence of gravitational fields, especially those near black holes.

c=ℏ=G==1

Bogolubov transformation

A typical free field theory is governed by a second-order hyperbolic partial differential equation. (Generalizations of this formalism are needed to cover fields describing particles with spin, but the resulting complications are irrelevant to the main points of this article.) When the background geometry and other external conditions are independent of time and of the direction of time, the field equation takes the form

∂2

where

A

is a (second-order, self-adjoint) differential operator acting on the spatial coordinates (

x

), with coefficients that may depend on those coordinates. The most elementary example (in four-dimensional space-time) is the Klein–Gordon equation for a massless scalar field in infinite flat space (“Minkowski space”),

−==++.−=+A

accompanied by the appropriate time dependence:

ϕ(t,x)=(2[(x)+(x],A=,>0.

Here the summation is schematic: in a particular case it may involve integrals or sums over several indices. Also, we have assumed that

ϕ

is real-valued and have inserted some normalization factors for later convenience. In a quantum theory of the field the numerical coefficients are promoted to annihilation and creation operators,

aj

and

a†j

. The standard physical interpretation of such a theory is that (1) the fundamental state is the vacuum

|0⟩

, characterized by

for all

j

; (2) states

describe

n

particles in the single-particle states labeled by

j1jn

; (3) the Hilbert space of all nonsingular physical states consists of limits of linear combinations of all these

n

-particle states. (When the spectrum of

j

is continuous,

should be integrated over a normalizable wave function,

fk(j)

.) This construction is the inverse of second quantization, where a quantum theory of particles is converted to a field theory by combining all possible multiparticle wave functions into one Hilbert space.

In the theory of relativity, a constant gravitational field is equivalent to a uniform acceleration. If in flat space-time one introduces new coordinates by

t=sinh(gτ),z=cosh(gτ),∂2=−ϕ.

The relation between the two coordinate systems is shown, with the two transverse dimensions suppressed, in the figure below. The new coordinates cover only one quadrant of the original space, the region where

|t|c/Ω

; it is more closely related to the creation of particles by rotating black holes than to the Unruh (or the Hawking) effect. The surface

r=c/Ω

is called static limit surface or ergosphere. The rotating detector’s response disappears if the system is enclosed in a conducting cylinder of radius less than

c/Ω

(Levin et al.,1993; Davies et al., 1996).

This analysis (Korsbakken-Leinaas, 2004) extends to arbitrary time-independent accelerated motion (Letaw and Pfautsch, 1981 and related papers), of which linear and circular acceleration are two extreme special cases. (“Time-independent acceleration” means that the worldline is an orbit of a timelike Killing vector field (symmetry generator), so that the geometry of space-time is stationary as perceived by the moving observer.) In the general case there is both an event horizon and an ergosphere, and hence the detector response and effective temperature are a combination of the Fulling-Davies-Unruh and the Letaw-Pfautsch-Bell-Leinaas effects.

The temperature concept is not completely appropriate for describing the effects of nonlinear acceleration. For a system with only two energy levels, such as an electron, a difference in the population of the two states can always be attributed to an effective temperature through the Boltzmann factor

e−(−)/T

. When there are more than two states, their populations and energy gaps may not all be related by the same Boltzmann formula. Thus

T

becomes “energy-dependent” (or “frequency dependent”). Analysis of the Gaussian fluctuations in the field’s Green function as seen in a rotating frame shows that the temperature is indeed frequency-dependent (Unruh, 1998; Korsbakken and Leinaas, 2004). The theoretical analysis of rotating electrons interacting with the electromagnetic field displays many complications (Bell and Leinaas, 1987; Unruh, 1998), to which a naive application of the temperature formula (6) provides only a crude first approximation.

Observability

Because the acceleration necessary to reach a temperature of

1K

through the Unruh effect is of order

1020s2

, one might think that the central experimental problem in this area is to reach and maintain such an acceleration in such a way that the temperature can be observed. There have been numerous proposals to use lasers to achieve this goal (Chen and Tajima, 1999; Brodin et al., 2008). First, however, one must ask exactly what such an experiment would accomplish.

The Unruh effect does not really require any more experimental confirmation than free quantum field theory as a whole does. The Unruh effect is necessary to keep the consistency between inertial and Rindler frame calculations of physical observables. An analogy is the appearance of inertial (centrifugal, Coriolis, etc.) forces in noninertial frames. They do not require any more confirmation than classical mechanics does, because inertial forces are necessary to allow noninertial observers to reproduce the same experimental predictions calculated by inertial ones (e.g., the trajectory of a Foucault pendulum).

This observation has a reverse side (Peña and Sudarsky, 2014): It is hard to see how any experiment carried out in an inertial laboratory could “prove the existence” of the Unruh effect, since it must be possible in principle to analyze the phenomenon entirely in the inertial frame using standard physical theory (unless there is something wrong with the latter, which the theory of the Unruh effect does not claim). The Unruh effect is not really a new phenomenon, but rather an unavoidable consequence of looking at known phenomena from a new point of view.

Nevertheless, a more direct demonstration of the effect would be highly satisfying, and it may be sought in phenomena that are most “naturally” interpreted in the Rindler frame. Such phenomena may or may not have been already experimentally observed or already theoretically predicted in purely inertial terms. We discuss a few examples.

The depolarization of electrons in storage rings was already calculated (Sokolov and Ternov, 1963; Jackson, 1976) and observed (Johnson et al., 1983) before Bell and Leinaas explained it as a rotational analog of the Unruh effect (see Sec. 2).

Another example is the decay of uniformly accelerated protons. According to the particle standard model, inertial protons are stable. Nevertheless, this is not so for noninertial ones, since the accelerating agent may provide the necessary extra energy to allow proton conversion. According to inertial observers, the main decay channel for protons with proper accelerations

g

in the interval

p+ΓW=(g)

Now, one may carry out an independent calculation of

ΓW

with respect to Rindler observers lying at rest with the proton. In this case, the necessary energy for proton conversion comes from the Unruh thermal bath. By describing the proton-neutron system through a semiclassical current, the decay channel (9) associated with inertial observers is replaced by the following three transition channels (Müller, 1997; Vanzella and Matsas, 2001; Suzuki and Yamada, 2003):

e−

where

,

ν

, and

ν¯

in processes (10)-(12) are Rindler particles being absorbed from or emitted to the Unruh thermal bath at temperature

TUnruh=g/2π

. Despite the quite distinct descriptions that accelerated and inertial observers would give to the proton decay phenomenon, we emphasize that inertial and Rindler frame calculations lead to identical results for

ΓW

. (If the Unruh thermal bath were not present, the Rindler-frame calculation would lead to

ΓW

.) In summary, a proton is an excellent realization of the classic two-state Unruh-DeWitt detector!

No signal of the accelerated proton decay is expected to be seen at Earth experiments, e.g, LHC/CERN, because

g

is too small in comparison with the energy scale provided by

mnmp

. But the situation is different for the third example, the electromagnetic process of photon emission (bremsstrahlung) from uniformly accelerated protons:

p+

where the

p+

is assumed to be inertial in the asymptotic past and future. Because photons are massless, this process is not suppressed for small

g

. Indeed, for any nonzero

g

, the corresponding total emission rate

ΓE=(g)

is well known to diverge because of an unbounded emission of soft photons. This is the so called infrared catastrophe (see, e.g., Itzykson and Zuber, 1987 (Secs. 1-3-2, 5-2-4, and 7-2-3)). Thus, let us consider the emission rate

ΓE

per fixed transverse momentum

where

γ

are zero-energy Rindler photons. Although they have zero energy because the proton is static with respect to Rindler observers (no proton recoil is considered in the process), they carry nonzero transverse momentum

k⊥

. Then, the interpretation goes as follows: Each (finite-energy) photon with fixed

k⊥

emitted by the accelerated charge according to inertial observers (see channel (13)) corresponds to either the emission or the absorption of a zero-energy Rindler photon with the same

k⊥

according to Rindler observers (see channels (14)-(15)). Thus, the observed photon emission from accelerated charges serves to demonstrate the reality of the Unruh effect as much as any experiment can.

The fact that, depending on the phenomenon, it may be easier to carry out an analysis in the uniformly accelerated frame than in the inertial one has led the Unruh effect to be used as a “calculational tool” in other areas. This is the case of quantum information science involving accelerated apparatus, where decoherence (Kok and Yurtsever, 2003), sudden death of entanglement (Landulfo and Matsas, 2009) and other features have been analyzed with the help of the Unruh effect. Although conceptually interesting, acceleration should not pose any serious concern to quantum information processing in practice.

Related topics

A number of related effects are frequently mentioned in the same breath as the Unruh effect; sometimes the relations are somewhat overstated. Here they are listed in (roughly) decreasing order of relevance to the Unruh scenario.

Bisognano-Wichmann theorem

Independently of but almost simultaneously with Unruh’s work, a theorem was proved (Bisognano and Wichmann, 1975, 1976; Sewell, 1982) in the framework of axiomatic quantum field theory to the effect that the vacuum state is a thermal (KMS) state with respect to the generator of Lorentz boosts, regarded as a generalized Hamiltonian. Because translation in the Rindler time coordinate

τ

is, in fact, a Lorentz boost, this amounts to a derivation of the Unruh effect under very general conditions (e.g., interacting fields satisfying the axiomatic requirements). The proof makes essential use of a

PCT

transformation relating the Rindler quadrant and its opposite, so it genuinely is an abstract implementation of the basic Unruh construction in Sec. 1.1.

Hawking effect

Essentially that same construction can be applied (Sewell, 1982; Kay and Wald, 1991) to horizons in curved space-times, thereby reproducing the essence of the thermal properties of black holes (Hawking, 1975; Gibbons and Perry, 1978) and de Sitter space (Gibbons and Hawking, 1977). The mathematical relationship between Rindler and Minkowski coordinates in flat space is practically identical to that between Schwarzschild and Kruskal coordinates in a nonrotating black hole, and hence many of the elements of the Unruh theory have counterparts in the black-hole theory, with the important difference that Schwarzschild coordinates become inertial in the limit of large distance and hence the analogs of Unruh-Rindler particles are “real” (at infinity) rather than effects of an observer’s acceleration. At small distance (close to the black hole’s horizon, which is well defined without reference to accelerated worldlines), the thermal effects can, however, be attributed to the acceleration of the curves of constant Schwarzschild radial position, whereas a freely falling observer there sees, approximately, cold empty space (Unruh, 1977b; Fulling, 1977) This is the origin of the thermal emission or ambience, as viewed from afar, of black holes, as already emphasized in Unruh’s original paper (Unruh, 1976). As remarked in Sec. 2, the Unruh-like effects for detectors undergoing motions including rotation have more in common with the ergoregion effect happening near, but outside, the horizon of a rotating black hole (Starobinsky, 1973; Unruh, 1974).

Moore-DeWitt (dynamical Casimir) effect

When a free field is subjected to time-dependent boundary conditions, its vacuum state evolves by a Bogolubov transformation into a superposition of states of various numbers of particles. When the background space-time is flat (except for the boundary), this effect can be localized quite literally as radiation from the moving “mirror” (Moore, 1970; DeWitt, 1975; Fulling and Davies, 1976). When a spherically symmetric system is solved by separation of variables, the origin of coordinates acts mathematically as a mirror in the

r

–

t

plane, and in the case of a black hole this mirror is effectively moving. This allowed the particle creation by black holes to be investigated as an instance of the Moore-DeWitt effect (Davies and Fulling, 1977). The “enhanced Unruh effect” of Scully et al., 2003, also fits into this category.

Parker effect, and other particle creation by external fields

The Moore-DeWitt effect is a boundary-localized instance of the general theory of linear quantum fields in time-dependent external fields. Another important special case of this theory is cosmological particle creation (e.g., Parker, 1969), where the external field is gravitational. Electromagnetic analogs involve, for instance, strong laser fields or collisions of heavy ions (e.g., Greiner et al., 1985). Also in this category are the recently reported experimental demonstrations of analogs of the Moore-DeWitt effect involving changes in the bulk properties of materials (Wilson et al., 2011). It should be noted that the effects in Secs. 4.3 and 4.4 involve time-dependent external conditions and give rise to Bogolubov transformations parametrized by time, while the Unruh effect (and the Hawking effect for a static black hole) involve a single Bogolubov transformation induced by a change of coordinate frame (under circumstances where the coordinates are tightly associated with operational definitions of field or particle observables).

Casimir effect

The original Casimir effect for the electromagnetic field between parallel flat conducting plates corresponds to a spatially homogeneous negative energy density representing a difference between the vacuum states of the field with and without the conductors. It has little to do with the other effects, especially the so-called “dynamical Casimir effect”. However, in more general models, with curved boundaries or nonelectromagnetic fields, there is additional vacuum energy concentrated near the (static) boundaries, and one may say at least picturesquely that in the Moore-DeWitt process some of this energy is being “shaken off” as real particles. Both kinds of (static) Casimir effect involve vacuum “polarization” rather than real particle creation and hence might be regarded, loosely, as akin to the thermal effects in Rindler space and around a black hole in thermal equilibrium. Thus we come full circle.

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