TheUnruh effect (also known as theFulling–Davies–Unruh effect) is a theoretical prediction inquantum field theory that an observer who is uniformly accelerating through empty space will perceive athermal bath. This means that even in the absence of any external heat sources, an accelerating observer will detect particles and experience a temperature. In contrast, aninertial observer in the same region of spacetime would observe no temperature.^[1]

In other words, the background appears to be warm from an acceleratingreference frame. In layman's terms, an accelerating thermometer in empty space (like one being waved around), without any other contribution to its temperature, will record a non-zero temperature, just from its acceleration. Heuristically, for a uniformly accelerating observer, theground state of an inertial observer is seen as a mixed state inthermodynamic equilibrium with a non-zero temperature bath.

The Unruh effect was first described byStephen Fulling in 1973,Paul Davies in 1975 andW. G. Unruh in 1976.^[2][3][4] It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are disputed. There is also some doubt about whether the Unruh effect implies the existence ofUnruh radiation.

TheUnruh temperature, sometimes called the Davies–Unruh temperature,^[5] was derived separately by Paul Davies^[3] and William Unruh^[4] and is the effective temperature experienced by a uniformly accelerating detector in avacuum field. It is given by^[6]

${\displaystyle T=\frac{\hslash a}{2\pi c{k}_{\mathrm{B}}}\approx 4.06\times {10}^{-21}\mathrm{K}\cdot {\mathrm{s}}^2\cdot {\mathrm{m}}^{-1}\times a,}$

whereħ is thereduced Planck constant,a is the proper uniform acceleration,c is thespeed of light, andk_B is theBoltzmann constant. Thus, for example, aproper acceleration of2.47×10²⁰ m⋅s⁻² corresponds approximately to a temperature of1 K. Conversely, an acceleration of1 m⋅s⁻² corresponds to a temperature of4.06×10⁻²¹ K.

The Unruh temperature has the same form as theHawking temperatureT_H =⁠ħg/2πck_B⁠ withg denoting the surface gravity of ablack hole, which was derived byStephen Hawking in 1974.^[7] In the light of theequivalence principle, it is, therefore, sometimes called the Hawking–Unruh temperature.^[8]

Solving the Unruh temperature for the uniform acceleration, it can be expressed as

${\displaystyle a=\frac{2\pi c{k}_{\mathrm{B}}}{\hslash }T=2\pi a_{\mathrm{P}}\frac{T}{T_{\mathrm{P}}}}$,

where${\displaystyle a_{\mathrm{P}}}$ isPlanck acceleration and${\displaystyle T_{\mathrm{P}}}$ isPlanck temperature.

Unruh demonstrated theoretically that the notion ofvacuum depends on the path of the observer throughspacetime. 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.^[9]

The Unruh effect would only appear to an accelerating observer. And although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the wordvacuum is interpreted in the following specific way. Inquantum field theory, the concept of "vacuum" is not the same as "empty space":Space is filled with the quantized fields that make up theuniverse. Vacuum is simply the lowestpossibleenergy state of these fields.

The energy states of any quantized field are defined by theHamiltonian, based on local conditions, including the time coordinate. According tospecial relativity, 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 fieldcanonical 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 (seeRindler spacetime). The existence of Unruh radiation could be linked to this apparentevent horizon, putting it in the same conceptual framework asHawking radiation. 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.

Thefree field needs to be decomposed into positive andnegative frequency components before defining thecreation andannihilation operators. This can only be done in spacetimes with atimelikeKilling vector field. This decomposition happens to be different inCartesian andRindler coordinates (although the two are related by aBogoliubov transformation). 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 ofblack holes andcosmological horizons. It is possible to rearrange the metric restricted to these regions to obtain the Rindler metric.^[10] The Unruh effect would then be the near-horizon form ofHawking radiation.

The Unruh effect is also expected to be present inde Sitter space.^[11]

It is worth stressing that the Unruh effect only says that, according to uniformly-accelerated observers, the vacuum state is a thermal state specified by its temperature, and one should resist reading too much into the thermal state or bath. Different thermal states or baths at the same temperature need not be equal, for they depend on the Hamiltonian describing the system. In particular, the thermal bath seen by accelerated observers in the vacuum state of a quantum field is not the same as a thermal state of the same field at the same temperature according to inertial observers. Furthermore, uniformly accelerated observers, static with respect to each other, can have different proper accelerationsa (depending on their separation), which is a direct consequence of relativistic red-shift effects. This makes the Unruh temperature spatially inhomogeneous across the uniformly accelerated frame.^[12]

Inspecial relativity, an observer moving with uniformproper accelerationa throughMinkowski spacetime is conveniently described withRindler coordinates, which are related to the standard (Cartesian)Minkowski coordinates by

Theline element in Rindler coordinates, i.e.Rindler space is

${\displaystyle \mathrm{d}{s}^2=-{\rho }^2\mathrm{d}{\sigma }^2+\mathrm{d}{\rho }^2,}$

whereρ =⁠1/a⁠, and whereσ is related to the observer's proper timeτ byσ =aτ (herec = 1).

An observer moving with fixedρ traces out ahyperbola in Minkowski space, therefore this type of motion is calledhyperbolic motion. The coordinate${\displaystyle \rho }$ is related to the Schwarzschild spherical coordinate${\displaystyle r_S}$ by the relation^[13]

${\displaystyle \rho ={\int }_{r_S}^r\frac{d{r}^{\mathrm{\prime }}}{\sqrt{1-r_S/r^{\mathrm{\prime }}}}.}$

An observer moving along a path of constantρ is uniformly accelerating, and is coupled to field modes which have a definite steady frequency as a function ofσ. These modes are constantlyDoppler 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 inσ is a symmetry of Minkowski space: it can be shown that it corresponds to aboost inx,t coordinate around the origin. Any time translation in quantum mechanics is generated by the Hamiltonian operator. For a detector coupled to modes with a definite frequency inσ, we can treatσ as "time" and the boost operator is then the corresponding Hamiltonian. In Euclidean field theory, where the minus sign in front of the time in the Rindler metric is changed to a plus sign by multiplying${\displaystyle i}$ to the Rindler time, i.e. aWick rotation or imaginary time, the Rindler metric is turned into a polar-coordinate-like metric. Therefore any rotations must close themselves after 2π in a Euclidean metric to avoid being singular. So

${\displaystyle e^{2\pi iH}=Id.}$

A path integral with real time coordinate is dual to a thermal partition function, related by aWick rotation. The periodicity${\displaystyle \beta }$ of imaginary time corresponds to a temperature of${\displaystyle \beta =1/T}$ inthermal quantum field theory. Note that the path integral for this Hamiltonian is closed with period 2π. This means that theH modes are thermally occupied with temperature⁠1/2π⁠. This is not an actual temperature, becauseH is dimensionless. It is conjugate to the timelike polar angleσ, which is also dimensionless. To restore the length dimension, note that a mode of fixed frequencyf inσ at positionρ has a frequency which is determined by the square root of the (absolute value of the) metric atρ, theredshift factor. This can be seen by transforming the time coordinate of a Rindler observer at fixedρ to an inertial, co-moving observer observing aproper time. From the Rindler-line-element given above, this is justρ. The actual inverse temperature at this point is therefore

${\displaystyle \beta =2\pi \rho .}$

It can be shown that the acceleration of a trajectory at constantρ in Rindler coordinates is equal to⁠1/ρ⁠, so the actual inverse temperature observed is

${\displaystyle \beta =\frac{2\pi }{a}.}$

Restoring units yields

${\displaystyle k_{\text{B}}T=\frac{\hslash a}{2\pi c}.}$

Thetemperature of the vacuum, seen by an isolated observer accelerating at the Earth's gravitational acceleration ofg =9.81 m·s⁻², is only4×10⁻²⁰ K. For an experimental test of the Unruh effect it is planned to use accelerations up to10²⁶ m·s⁻², which would give a temperature of about400000 K.^[14][15]

The Rindler derivation of the Unruh effect is unsatisfactory to some^[who?], since the detector's path issuper-deterministic. Unruh later developed theUnruh–DeWitt particle detector model to circumvent this objection.

The Unruh effect would also cause the decay rate of accelerating particles to differ from inertial particles. Stable particles like the electron could have nonzero transition rates to higher mass states when accelerating at a high enough rate.^[16][17][18]

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 is.^{[citation needed]} 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 asUnruh radiation.

The existence of Unruh radiation is not universally accepted. Smolyaninov claims that it has already been observed,^[19] while O'Connell and Ford claim that it is not emitted at all.^[20] While these skeptics accept that an accelerating object thermalizes 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.

Researchers claim experiments that successfully detected theSokolov–Ternov effect^[21] may also detect the Unruh effect under certain conditions.^[22]

Theoretical work in 2011 suggests that accelerating detectors could be used for the direct detection of the Unruh effect with current technology.^[23]

The Unruh effect may have been observed for the first time in 2019 in the high energychanneling radiation explored by theNA63 experiment at CERN.^[24]

• Dynamical Casimir effect
• Cosmic Background Radiation
• Hawking radiation
• Black hole thermodynamics
• Pair production
• Quantum information
• Superradiance
• Virtual particle

• Thorne, K. P. (1995). "Black holes evaporate".Black Holes and Time Warps (Reprint ed.).W. W. Norton & Company. Box 12.5, p. 444.ISBN 0-393-31276-3.
• Wald, R. M. (1994).Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics.University of Chicago Press. Ch. 5.ISBN 0-226-87027-8.
• Crispino, L. C. B.; Higuchi, A.; Matsas, G. E. A. (2008). "The Unruh effect and its applications".Reviews of Modern Physics.80 (3):787–838.arXiv:0710.5373.Bibcode:2008RvMP...80..787C.doi:10.1103/RevModPhys.80.787.S2CID 119223632.

• Stephen Fulling and George Matsas (ed.)."Unruh effect".Scholarpedia.

• Thermodynamics
• Quantum field theory
• Theory of relativity
• Acceleration
• Physical phenomena
• Hypothetical processes
• Paul Davies

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