# Unruh effect

Stephen A. Fulling and George E. A. Matsas (2014), Scholarpedia, 9(10):31789. | doi:10.4249/scholarpedia.31789 | revision #143950 [link to/cite this article] |

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 \(10^{20}\ \mathrm{m/s}^2\),
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.

## Contents |

## Theoretical approaches

This article uses natural (Planck) units, \(c=\hbar=G=k_B=1\), except when discussing experimental implications.

### 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 \[ \frac{\partial^2 \phi }{\partial t^2} = -A\phi, \] where \(A\) is a (second-order, self-adjoint) differential operator acting on the spatial coordinates (\(\mathbf x\)), 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"), \[- A_\mathrm{Mink} = \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\,. \] Another example, describing the same kind of field interacting with a constant gravitational field, is \[-A_\mathrm{Rind} = \frac{\partial^2}{\partial \xi^2} + e^{2g\xi}\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \,. \]

The general solution of the field equation is a superposition
of **normal modes**, eigenfunctions of \(A\) accompanied by
the appropriate time dependence:
\[
\phi(t,\mathbf{x}) = \sum_j (2\omega_j)^{-1/2}
\bigl[a_j \phi_j(\mathbf{x})e^{-i\omega_j t}
+ a_j^* \phi_j(\mathbf{x})^*e^{+i\omega_j t}\bigr],
\tag{1}\]
\[ A\phi_j = \omega_j^2 \phi_j\,, \quad \omega_j>0.\]
Here the summation is schematic: in a particular case it
may involve integrals or sums over several indices.
Also, we have assumed that \(\phi\) 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**,
\(a_j\) and \(a_j^\dagger\).
The standard physical interpretation of such a theory is that
(1) the fundamental state is the **vacuum** \(|0\rangle\),
characterized by \(a_j |0\rangle=0\) for all \(j\); (2)
states \(a_{j_1}^\dagger \cdots a_{j_n}^\dagger |0\rangle\)
describe \(n\) **particles** in the single-particle
states labeled by \(j_1\,, \ldots, j_n\,\);
(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, \(a_{j_k}^\dagger\)
should be integrated over a normalizable wave function,
\(f_k(j)\).)
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
\[ t = g^{-1} e^{g\xi} \sinh(g\tau), \quad
z = g^{-1} e^{g\xi} \cosh(g\tau), \]
then the Klein--Gordon equation transforms to
\[ \frac{\partial^2 \phi }{\partial \tau^2} =
-A_\mathrm{Rind}\phi. \]
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 $|t|<z$ (called **Rindler space**).
The principles of general relativity (equivalence
and general covariance)
require that the \((t,x,y,z)\) and \((\tau,x,y,\xi)\)
are equally valid descriptions of physics interior
to that quadrant (Unruh, 1977a).
The surface \(|t| =z\) is a **horizon** beyond which
lie space-time regions that are causally isolated,
in a certain sense, from each accelerated worldline,
\(\xi=\mathrm{const.}\)

Within the quadrant there holds a field expansion of the type (1), \[ \phi(\tau,x,y,\xi) = \sum_{k} (2\Omega_{k})^{-1/2} \bigl[b_k \psi_k(x,y,\xi)e^{-i\Omega_k \tau} + b_k^\dagger \psi_k(x,y,\xi)^*e^{+i\Omega_k \tau}\bigr], \tag{2}\] \[ A_\mathrm{Rind}\psi_k = \Omega_k^2 \psi_k\,, \quad \Omega_k>0.\] Meanwhile, the standard field expansion in the entire space-time continues to be precisely (1) with \(a_j^*\) replaced by \(a_j^\dagger\) and with \(A=A_\mathrm{Mink}\). (In (2) the schematic sum is actually an integral over Fourier variables in the \(x\) and \(y\) directions and over \(\Omega\); the dependence of \(\psi_k\) on \(\xi\) and \(\Omega\) involves a Bessel function of imaginary order, \(i\Omega\), and imaginary argument, proportional to \(e^{g\xi}\). In (1) the eigenfunctions are plane waves and the \(j\) sum is actually a Fourier transform in all three spatial dimensions.) One can then show that \[ b_k = \sum_j [u_{kj} a_j + v_{kj} a_j^\dagger], \tag{3}\] for certain rather singular functions \(u\) and \(v\). For the details of such calculations we refer to Crispino et al., 2008; Birrell and Davies, 1982 (Sec. 4.5); Mukhanov and Winitski, 2007 (Chap. 8); Korsbakken and Leinaas, 2004 (Sec. III); and the original research papers, Fulling, 1973; Davies, 1975; Unruh, 1976. They are usually carried out either by directly inverting the eigenfunction transform (2) and substituting (1), or by examining the analyticity properties of the normal modes.

To explain the last remark, it is necessary to note that the Rindler coordinate system has a natural extension to the opposite quadrant, \(z<-|t|\), wherein a completely separate field construction parallel to (2) exists. The inverse of the Bogolubov formula (3) requires \(b_k\) operators from both quadrants. A Minkowski plane wave \(\phi_j\) is analytic in certain regions of a complexified coordinate space, and this requires its expansion in Rindler modes to involve \(\psi_k^*\) functions from the left quadrant, analytically continued into the right quadrant, as well as \(\psi_k\) functions from the right quadrant.

From either the analyticity argument or a direct calculation follows the fundamental conclusion that \(v_{kj}\) is not equal to zero. It follows that \[ \langle 0|b_k^\dagger b_k|0\rangle= \sum_j |v_{kj}|^2 \ne 0. \tag{4}\] Insofar as \(b_k^\dagger b_k\) can be regarded as the operator representing the observable "number of particles present in mode \(k\)", this equation states that the vacuum state of standard free field theory in Minkowski space contains "Rindler particles". Close inspection of \(v_{kj}\) reveals that, up to a normalization factor, \[\langle 0|b_k^\dagger b_k|0\rangle = (e^{2\pi \Omega_k/g} -1)^{-1},\tag{5}\] recognizable as a Planck factor. Thus the famous conclusion: The distribution of particle number corresponds to a temperature \[ T_\mathrm{Unruh} = \frac{g}{2\pi} =\frac{\hbar g}{2\pi c k_B} \tag{6}\] for observations in the neighborhood of \(\xi=1/g\). In (6) conventional units have been restored to emphasize the smallness of the effect in normal circumstances.

### Model detectors

One might very well suspect that the foregoing is merely a mathematical curiosity, and that the conclusion of "particles" and "temperature" in the vacuum is physically nonsense, a result of taking formula (2) too literally. W. G. Unruh (Unruh, 1976, 1977a) reasoned, however, that the response of a localized particle detector must be determined by the dependence of the quantum fields on the detector's proper time, not the time of a global coordinate system. He developed detector models (later simplified by DeWitt, 1979) to show that a uniformly accelerated detector (following a space-time path of constant \((x,y,\xi)\)) in the Minkowski vacuum \(|0\rangle\) does indeed "click" at a rate consistent with \(T_\mathrm{Unruh}\). Because the response is in one-one correspondence with the temperature, such a detector is also sometimes called an accelerated thermometer.

The amplitude for a transition in the internal state of the detector because of the coupling with the field is calculated by first-order perturbation theory. From the point of view of Rindler quantization (2) the detector is responding to the particles whose presence was calculated in (4). From the point of view of Minkowski quantization (1) the excitation of the detector is correlated with emission, not absorption, of particles (Unruh and Wald, 1984); thus a stationary (or inertial) observer "sees" the detector radiating, much like the classical radiation of photons by an accelerated electric charge (Pauri and Vallisneri, 1999), and regards the effect as a fairly routine quantum process having nothing necessarily to do with coordinate transformations. This point is discussed further in Sec. 3.

For the details of these calculations see Crispino et al., 2008, or the four original papers just cited. An important point is that they involve integration along (just) the detector's entire worldline. Thus, on the one hand, the effect is entirely determined by the state of the field at the detector, depending only indirectly on the global construction of the Rindler field (2) and its "vacuum" and \(n\)-particle states. On the other hand, this theory does not directly describe what happens to the detector during a finite time (Schlicht, 2004; Satz, 2007), and care must be exercised in extrapolating the conclusions to arbitrary coordinate systems or space-times that do not possess timelike Killing vectors (i.e., time-translation symmetries) whose orbits play the role of detector trajectories. (See, however, the next subsection.)

Besides showing that quantization in the accelerated
coordinate system has real physical significance,
the work of Unruh established the important point that
*what is important about Rindler space is not the Rindler vacuum*
(characterized by \(b_k|0\rangle_\mathrm{R}=0\)),
*but the ordinary Minkowski vacuum,* \(|0\rangle\),
*looked at from a Rindler point of view*.

### Periodicity of Green functions in imaginary time

The (Minkowski vacuum) two-point function,
\(\langle0|\phi(t,\mathbf{x}) \phi(t',\mathbf{x'})|0\rangle\),
which as \((t',\mathbf{x'})\to(t,\mathbf{x})\)
determines the expectation values of all local
observables at \((t,\mathbf{x})\), can be expressed
in terms of the Rindler coordinates.
The result is recognizable as the corresponding
function that could be computed directly for a
*thermal* state of the Rindler field (2)
at the Unruh temperature
(DeWitt, 1979; Sciama et al., 1981; Audretsch and Müller, 1994b;
Milonni, 1994 (Sec. 2.10).)
This results, of course, from the Planckian form of
\(v_{kj}\,\) (see (5)),
but it can also be understood as a manifestation of the
general relationship between temperature and imaginary time
in quantum statistical mechanics
(KMS theory).
When \(t\) and \(\tau\) are extended as complex variables,
\(i\tau\) is revealed as an angular variable in the
\((it,z)\) plane. The periodicity determined by \(g\), the
acceleration, is the
only one that makes functions of \(\tau\) be analytic in \(t\).
That period corresponds, under the KMS theory, to the
reciprocal of the Unruh temperature
(Dowker, 1978; Christensen and Duff, 1978; Sewell, 1982;
Bell et al., 1985; Fulling and Ruijsenaars, 1987).

In general, what an accelerated observer encounters in the vacuum is not exactly an ordinary flat-space Bose gas at finite temperature. The eigenfunctions and spectral density of \(A_\mathrm{Rind}\) are different from those of \(A_\mathrm{Mink}\). (The simplest model, a massless scalar field in two-dimensional space-time, is a misleading exception.) For example, in higher dimensions a mode of fixed, nonzero transverse momentum can have arbitrarily small Rindler energy; this becomes important in the discussion of bremsstrahlung in Sec. 3.

The analysis in terms of Green functions is truly local, showing that vacuum has thermal properties for acceleration through any space-time point. Global properties and constructions, including horizons, nontrivial topologies, Bogolubov transformations, and Killing trajectories, although prominent in the calculation and interpretation of numerous exactly solvable scenarios, appear irrelevant to the crux of the effect. Global conditions are pertinent, however, to the issue of what quantum states are likely to be produced in nature by normal processes.

The Unruh effects for quantized electromagnetic and fermion fields were perhaps first demonstrated by a Green function approach (Candelas and Deutsch, 1977, 1978).

### Persisting controversies

The foregoing is generally accepted by the pertinent research community, since a variety of lines of argument lead to the same conclusion and, as explained in Sec. 3, the Unruh effect is seen as a necessity to keep consistency between inertial and Rindler frame calculations. Nevertheless, some doubts continue to be expressed in the physics literature.

Some authors (Narozhny et al., 2002, 2004; Fedotov et al., 2009) deny that uniformly accelerated detectors show a universal thermal response to \(|0\rangle\); they attribute the calculated results to improperly imposed boundary conditions at the horizon. The response from the conventional viewpoint is that mathematical subtleties at the horizon do not affect physical observations inside a causally complete region inside Rindler space (Fulling and Unruh, 2004).

Others (Hu et al., 2004; Ford and O'Connell, 2006) question the conclusion that the accelerated detector is radiating as seen from an inertial frame. The question appears to be mired in some definitional disagreements, for example on how to draw the line between radiation and vacuum fluctuation, what initial state to consider, and whether and how to go beyond first order in perturbative calculations (Unruh, 1992; Audretsch and Müller, 1994a). Additional comments and references on this issue appear in the cited references and in Crispino et al., 2008.

## Rotational analog of the Unruh effect

Rotational motion involves acceleration, albeit acceleration perpendicular to the motion, which does not change the speed. Is there a corresponding Unruh effect? Answering this question has proved surprisingly complicated, mainly because of the ambiguity of the term "circularly accelerated coordinate system".

Because the original Unruh effect was most easily discovered and studied by doing calculations in the Rindler coordinates, it was natural to seek a coordinate system in which the lines of constant spatial coordinates are the worldlines of uniformly rotating observers. The most obvious construction is to modify ordinary cylindrical coordinates by \[ \phi' = \phi + \Omega t, \tag{7}\] where \(\Omega\) is the desired angular velocity. Development of quantum field theory in this frame (Denardo and Percacci, 1978; Letaw and Pfautsch, 1980) reveals that the normal modes are the same as those in the rest frame; there is no nontrivial Bogolubov transformation, hence no change of vacuum state. Nevertheless, Letaw and Pfautsch, 1980, analyzed a centripetally accelerating detector and found a nontrivial response.

The decision between the two seemingly contradictory conclusions of Letaw and Pfautsch was settled by Bell and Leinaas 1983, 1987, who pointed out that electrons in a circular accelerator can be regarded as Unruh-DeWitt detectors, because their two spin states have different energies in the surrounding magnetic field. Indeed, it had long been known that the electrons are polarized along the magnetic field, but not completely so. The remanent depolarization is an indication that the electrons see a nonzero effective temperature, because at zero temperature all electrons should come to equilibrium in the lower energy state. This work played a major role in convincing the physics community that the Unruh effect has real physical significance.

The coordinate system based on (7) is not a Lorentz frame, even in the infinitesimal neighborhood of a constant \(\phi\) observer. (It is more analogous to a "Galilean" frame for an observer with constant velocity.) Therefore, it is not surprising that routine mathematics in that frame does not automatically reproduce the correct physics. The problem is that a true instantaneous Lorentz frame cannot be extended to a coordinate system within which the canonical construction of a quantum field can be carried out. A local Lorentz transformation requires, at least locally, a transformation of time of the sort \[ t' = \gamma (t + \Omega r^2 \phi), \qquad \phi' = \gamma(\phi + \Omega t). \tag{8}\] The new time is thus constant on helical surfaces that are not suitable for prescribing initial data for the field. (More precisely, they do not provide Cauchy surfaces for any region of space-time that contains the entire rotating worldline.)

An important clarification was provided by Korsbakken and Leinaas, 2004, who transferred attention from the construction of a "rotating vacuum state" to the analysis of a single rotating detector in the usual vacuum state. (This is in accordance with the foregoing dictum that what is interesting about linear acceleration is not the Rindler vacuum, but the standard vacuum from an accelerated point of view.) For such analysis an extended coordinate system with Cauchy surfaces is not necessary; transformations into successive instantaneous local Lorentz frames are sufficient. The crucial fact is that in the rotating frame some states have negative energy; the Minkowski vacuum is no longer a ground state, and no ground state exists. The detector response is entirely due to radiation into the negative-energy modes, not absorption from a thermal gas of positive-energy particles.

Mathematically, this effect is related to the fact
that the time axis defined by (8) becomes
spacelike when \(r>c/\Omega\);
it is more closely related to the creation of particles
by rotating black holes
than to the Unruh (or the Hawking) effect.
The surface \(r=c/\Omega\) 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 \(c/\Omega\)
(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 \(e^{-(E_1-E_0)/k_B T}\). 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 \(1\, {\rm K}\) through the Unruh effect is of order \(10^{20} \, {\rm m/s^2}\), 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 \[ m_e + m_n - m_p \lesssim g \lesssim m_{\pi} + m_n - m_p \] is the weak-interaction one \[p^+ \stackrel{g}{\to} n^0 \; e^+ \; \nu, \tag{9} \] where \(m_e\), \(m_n\), \(m_p\), and \(m_\pi\) are the \(e^-\), \(n^0\), \(p^+\), and \(\pi^+\) masses, respectively. (For larger accelerations, the strong-interaction channel \(p^+ \stackrel{g}{\to} n^0 \; \pi^+ \) dominates, while for smaller ones the decay rate is negligible.) Clearly, the corresponding transition rate \(\Gamma_{W} = \Gamma_{W} (g)\) can be calculated from the point of view of inertial observers using standard quantum field theory.

Now, one may carry out an independent calculation of \(\Gamma_{W}\) 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): \begin{eqnarray} e^- \; p^+ &\stackrel{T_U}{\to}& n^0 \; \nu, \tag{10} \\ \bar \nu \; p^+ &\stackrel{T_U}{\to}& n^0 \; e^+, \tag{11} \\ e^- \; \bar \nu \; p^+ &\stackrel{T_U}{\to}& n^0, \tag{12} \end{eqnarray} where \(e^\pm\), \(\nu\), and \(\bar \nu\) in processes (10)-(12) are Rindler particles being absorbed from or emitted to the Unruh thermal bath at temperature \(T_\mathrm{Unruh} = g/2\pi\). 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 \(\Gamma_{W}\). (If the Unruh thermal bath were not present, the Rindler-frame calculation would lead to \(\Gamma_{W}=0\).) 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
\(m_n-m_p\). But the situation is different for the third example,
the electromagnetic process of photon
emission (**bremsstrahlung**) from uniformly accelerated protons:
\[
p^+ \stackrel{g}{\to} p^+ \; \gamma,
\tag{13}
\]
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
\(\Gamma_{E} = \Gamma_{E} (g)\)
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
\(\Gamma_{E}^{\bf k_\bot} \)
per fixed transverse momentum \({\bf k_\bot }\), which is finite.
In order to recover \(\Gamma_{E}^{\bf k_\bot} \) from the point
of view of Rindler observers, the Unruh thermal bath is crucial
again (Higuchi et al., 1992). For Rindler observers, process
(13) should be replaced by
\begin{eqnarray}
p^+ \; \gamma &\stackrel{T_U}{\to}& p^+,
\tag{14}
\\
p^+ &\stackrel{T_U}{\to}& p^+ \; \gamma,
\tag{15}
\end{eqnarray}
where \(\gamma\) 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
\({\bf k_\bot }\). Then, the interpretation goes as follows: Each
(finite-energy) photon with fixed \({\bf k_\bot }\) 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 \({\bf k_\bot }\) 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 \(\tau\) 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.

## References

J. Audretsch and R. Müller (1994a)
Radiation from a uniformly accelerated particle detector:
Energy, particles, and the quantum measurement process.
*Phys. Rev. A* 49:6566.

J. Audretsch and R. Müller (1994b)
Spontaneous excitation of an accelerated atom:
The contributions of vacuum fluctuations and radiation
reaction.
*Phys. Rev. A* 50:1755.

J. S. Bell, R. J. Hughes, and J. M. Leinaas (1984)
The Unruh effect in extended thermometers.
*Z. Phys. C* 28:75.

J. S. Bell and J. M. Leinaas (1983)
Electrons as accelerated thermometers.
*Nucl. Phys. B* 212:131.

J. S. Bell and J. M. Leinaas (1987)
The Unruh effect and quantum fluctuations of electrons
in storage rings.
*Nucl. Phys. B* 284:488.

N. D. Birrell and P. C. W. Davies (1982)
*Quantum fields in curved space*.
Cambridge University Press (Cambridge).

J. J. Bisognano and E. H. Wichmann (1975)
On the duality condition for a Hermitian scalar field.
*J. Math. Phys.* 16:985.

J. J. Bisognano and E. H. Wichmann (1976)
On the duality condition for quantum fields.
*J. Math. Phys.* 17:303.

G. Brodin, M. Marklund, R. Bingham, J. Collier, and
R. G. Evans (2008)
Laboratory soft x-ray emission due to the Hawking-Unruh
effect.
*Class. Quantum Grav.* 25:145005

P. Candelas and D. Deutsch (1977)
On the vacuum stress induced by uniform acceleration
or supporting the ether.
*Proc. Roy. Soc. London A* 354:79.

P. Candelas and D. Deutsch (1978)
Fermion fields in accelerated states.
*Proc. Roy. Soc. London A* 362:251.

P. Chen and T. Tajima (1999)
Testing Unruh radiation with ultraintense lasers.
*Phys. Rev. Lett.* 83:256.

S. M. Christensen and M. J. Duff (1978)
Flat space as a gravitational instanton.
*Nucl. Phys. B* 146:11.

L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas (2008)
The Unruh effect and its applications.
*Rev. Mod. Phys.* 80:787.

P. C. W. Davies (1975)
Scalar particle production in Schwarzschild and Rindler metrics.
*J. Phys. A* 8:609.

P. C. W. Davies, T. Dray, and C. A. Manogue (1996)
Detecting the rotating quantum vacuum.
*Phys. Rev. D* 53:4382.

P. C. Davies and S. A. Fulling (1977)
Radiation from moving mirrors and from black holes.
*Proc. Roy. Soc. London A* 356:237.

G. Denardo and R. Percacci (1978)
Quantum field theory for a rotating observer.
*Nuovo Cimento B* 48:81.

B. S. DeWitt (1975)
Quantum field theory in curved spacetime.
*Phys. Reports* 19:295.

B. S. DeWitt (1979)
Quantum gravity: The new synthesis.
*General Relativity: An Einstein Centenary Survey*,
ed. by S. W. Hawking and W. Israel,
Cambridge University Press (Cambridge).
pp. 680-745.

J. S. Dowker (1978)
Thermal properties of Green's functions in Rindler,
de Sitter, and Schwarzschild spaces.
*Phys. Rev. D* 18:1856.

A. M. Fedotov, N. B. Narozhny, V. D. Mur, and E. G. Gelfer
(2009)
On the incompleteness of the Unruh fermion modes in the Minkowski
space.
*JETP Lett.* 89:385
[*Pis. ZhETF* 89:449].

G. W. Ford and R. F. O'Connell (2006)
Is there Unruh radiation?
*Phys. Lett. A* 350:17.

S. A. Fulling (1973)
Nonuniqueness of canonical field quantization in Riemannian
space-time.
*Phys. Rev. D* 7:2850.

S. A. Fulling (1977)
"Radiation" and "vacuum polarization" near a black hole.
*Phys. Rev. D* 15:2411.

S. A. Fulling and P. C. W. Davies (1976)
Radiation from a moving mirror in two dimensional
space-time: Conformal anomaly.
*Proc. Roy. Soc. London A* 348:393.

S. A. Fulling and S. N. M. Ruijsenaars (1987)
Temperature, periodicity and horizons.
*Phys. Reports* 152:135.

S. A. Fulling and W. G. Unruh (2004)
Comment on "Boundary conditions in the
Unruh problem".
*Phys. Rev. D* 70:048701.

G. W. Gibbons and S. W. Hawking (1977)
Cosmological event horizons, thermodynamics,
and particle creation.
*Phys. Rev. D* 15:2738.

G. W. Gibbons and M. J. Perry (1978)
Black holes and thermal Green functions.
*Proc. Roy. Soc. London A* 358:467

W. Greiner, B. Müller, and J. Rafelski (1985)
*Quantum Electrodynamics of Strong Fields*.
Springer (Berlin).

S. W. Hawking (1975)
Particle creation by black holes.
*Commun. Math. Phys.* 43:199.

A. Higuchi, G. E. A. Matsas, and D. Sudarsky (1992)
Bremsstrahlung and zero-energy Rindler photons.
*Phys. Rev. D* 45:R3308.

B. L. Hu, A. Roura, and S. Shresta (2004)
Vacuum fluctuations and moving atoms/detectors:
From the Casimir-Polder to the
Unruh-Davies-DeWitt-Fulling effect.
*J. Optics B* 6:S698.

C. Itzykson and J.-B. Zuber (1987)
*Quantum Field Theory*.
McGraw--Hill (Singapore).

J. D. Jackson (1976)
On understanding spin-flip synchrotron radiation and
the transverse polarization of electrons in storage rings.
*Rev. Mod. Phys.* 48:417.

J. R. Johnson, R. Prepost, D. E. Wiser, J. J. Murray,
R. F. Schwitters, and C. K. Sinclair (1983)
Beam polarization measurements at the SPEAR storage ring.
*Nucl. Instr. Meth.* 204:261.

B. S. Kay and R. M. Wald (1991)
Theorems on the uniqueness and thermal properties of
stationary, nonsingular, quasifree states on spacetimes
with a bifurcate Killing horizon.
*Phys. Reports* 207:49.

P. Kok and U. Yurtsever (2003)
Gravitational decoherence.
*Phys. Rev. D* 68:085006.

J. I. Korsbakken and J. M. Leinaas (2004)
Fulling--Unruh effect in general stationary
accelerated frames.
*Phys. Rev. D* 70:084016

A. G. S. Landulfo and G. E. A. Matsas (2009)
Sudden death of entanglement and teleportation fidelity
loss via the Unruh effect.
*Phys. Rev. A* 80:032315.

J. R. Letaw and J. D. Pfautsch (1980)
Quantized scalar field in rotating coordinates.
*Phys. Rev. D* 22:1345.

J. R. Letaw and J. D. Pfautsch (1981)
Quantized scalar field in the stationary coordinate systems
of flat spacetime.
*Phys. Rev. D* 24:1491.

O. Levin, Y. Peleg, and A. Peres (1993)
Unruh effect for circular motion in a cavity.
*J. Phys. A* 26:3001.

P. W. Milonni (1994)
*The Quantum Vacuum*. Academic Press (San Diego).

G. T. Moore (1970)
Quantum theory of the electromagnetic field in a
variable-length one-dimensional cavity.
*J. Math. Phys.* 11:2679.

V. F. Mukhanov and S. Winitski (2007)
*Introduction to Quantum Effects in Gravity*.
Cambridge University Press (Cambridge).

R. Müller (1997)
Decay of accelerated particles
*Phys. Rev. D* 56:953

N. B. Narozhny, A. M. Fedotov, B. M. Karnakov, V. D. Mur,
and V. A. Belinskii (2002)
Boundary conditions in the Unruh problem.
*Phys. Rev. D* 65:025004.

N. B. Narozhny, A. M. Fedotov, B. M. Karnakov, V. D. Mur,
and V. A. Belinskii (2004)
Reply to "Comment on `Boundary conditions in the
Unruh problem'".
*Phys. Rev. D* 70:048702.

L. Parker (1969)
Quantized fields and particle creation in expanding
universes. I.
*Phys. Rev. D* 183:1057

M. Pauri and M. Vallisneri (1999)
Classical roots of the Unruh and Hawking effects.
*Found. Phys.* 29:1499.

I. Peña and D. Sudarsky (2014)
On the possibility of measuring the Unruh effect.
*Found. Phys.* 44:689.

A. Satz (2007)
Then again, how often does the Unruh-DeWitt
detector click if we switch it carefully?.
*Class. Quantum Grav.* 24:1719.

S. Schlicht (2004)
Considerations on the Unruh effect: causality and
regularization.
*Class. Quantum Grav.* 21:4647.

D. W. Sciama, P. Candelas, and D. Deutsch (1981)
Quantum field theory, horizons and thermodynamics.
*Adv. Phys.* 30:327.

M. O. Scully, V. V. Kocharovsky, A. Belyanin,
E. Fry, and F. Capasso (2003)
Enhancing acceleration radiation from ground-state atoms
via cavity quantum electrodynamics.
*Phys. Rev. Lett.* 91:243004.

G. L. Sewell (1982)
Quantum fields on manifolds: PCT and gravitationally
induced thermal states.
*Ann. Phys. (N.Y.)* 141:201.

A. A. Sokolov and I. M. Ternov (1963)
On polarization and spin effects in the theory of
synchrotron radiation.
*Sov. Phys. Dokl.* 8:1203
[*Dokl. Akad. Nauk SSSR* 153:1052].

A. A. Starobinsky (1973)
Amplification of waves during reflection from a rotating "black
hole".
*Soviet Phys. -- JETP* 37:28
[*Zh. Eksp. Teor. Fiz.* 64:48].

H. Suzuki and K. Yamada (2003)
Analytic evaluation of the decay rate for an accelerated
proton.
*Phys. Rev. D* 67:065002.

W. G. Unruh (1974)
Second quantization in the Kerr metric.
*Phys. Rev. D* 10:3194.

W. G. Unruh (1976) Notes on black-hole evaporation.
*Phys. Rev. D* 14:870

W. G. Unruh (1977a) Particle detectors and black holes.
*Proceedings of the First Marcel Grossmann Meeting*
on General Relativity*, ed. by R. Ruffini, North-Holland*
(Amsterdam) pp. 527-536.

W. G. Unruh (1977b)
Origin of the particles in black-hole evaporation.
*Phys. Rev. D* 15:365.

W. G. Unruh (1992)
Thermal bath and decoherence of Rindler spacetimes.
*Phys. Rev. D* 46:3271.

W. G. Unruh (1998)
Acceleration radiation for orbiting electrons.
*Phys. Reports* 307:163.

W. G. Unruh and R. M. Wald (1984)
What happens when an accelerating observer detects a Rindler
particle.
*Phys. Rev. D* 29:1047.

D. A. T. Vanzella and G. E. A. Matsas (2001)
Decay of accelerated protons and the existence of
the Fulling-Davies-Unruh effect.
*Phys. Rev. Lett.* 87:151301.

C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen,
J. R. Johansson, T. Duty, F. Nori, and P. Delsing (2011)
Observation of the dynamical Casimir effect in a
superconducting circuit.
*Nature* 479:376.

### Internal references

C. M. Becchi (2010) Second quantization. Scholarpedia, 5(6), 7902.

T. Creighton and R. H. Price (2008) Black holes. Scholarpedia, 3(1):4277.

I. Gladwell (2008) Boundary value problem. Scholarpedia, 3(1):2853.

S. K. Lamoreaux (2011) Casimir force. Scholarpedia, 6(10):9746.

R. Parentani and P. Spindel (2011) Hawking radiation. Scholarpedia, 6(12):6958.

W. Rindler (2011) Special relativity: kinematics. Scholarpedia, 6(2):8520.

R. F. Streater (2009) Wightman quantum field theory. Scholarpedia, 4(5):7123.

### Wikipedia references

Boltzmann distribution. Wikipedia

Kerr metric. Wikipedia

KMS state. Wikipedia

Lorentz transformation. Wikipedia

Natural units. Wikipedia

Planck's law. Wikipedia

## Further reading

Unruh effect. Wikipedia

P. Chen, ed. (1999)
*Quantum Aspects of Beam Physics: Advanced ICFA Beam Dynamics Workshop, Monterey, 1998*,
World Scientific (Singapore).

S. A. Fulling (2005)
Review of some recent work on acceleration radiation.
*J. Mod. Optics* 52:2207.

G. W. Gibbons and M. J. Perry (1976)
Black holes in thermal equilibrium.
*Phys. Rev. D* 16:985.

A. Higuchi, G. E. A. Matsas, and D. Sudarsky (1992b)
Bremsstrahlung and Fulling-Davies-Unruh thermal bath.
*Phys. Rev. D* 46:3450.

J. M. Leinaas (1991)
Hawking radiation, the Unruh effect and the polarization of electrons.
*Europhys. News* 22:78.

J. M. Leinaas (2002)
Unruh effect in storage rings.
*Quantum Aspects of Beam Physics: 18th Advanced ICFM Beam Dynamics Workshop, Capri, 2000*,
ed. P. Chen, World Scientific (Singapore) pp. 336-352.

J. R. Letaw (1981)
Stationary world lines and the vacuum excitation of noninertial
detectors.
*Phys. Rev. D* 23:1709.

J. R. Letaw and J. D. Pfautsch (1982)
The stationary coordinate systems in flat spacetime.
*J. Math. Phys.* 23:425.

E. Martín-Martínez and J. Louko (2014)
Particle detectors and the zero mode of a quantum field.
*Phys. Rev. D* 90:024015.

K. K. Ng, L. Hodgkinson, J. Louko, R. B. Mann, and E.
Martín-Martínez (2014)
Unruh-DeWitt detector response along static and
circular-geodesic trajectories for Schwarzschild-anti-de Sitter
black holes.
*Phys. Rev. D* 90:064003.