# Unruh effect

The Unruh effect (or sometimes Fulling–Davies–Unruh effect) is the prediction that an accelerating observer will observe blackbody radiation where an inertial observer 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, subtracting any other contribution to its temperature, will record a non-zero temperature. For a uniformly accelerating observer, the ground state of an inertial observer is seen as in thermodynamic equilibrium with a non-zero temperature.

The Unruh effect was first described by Stephen Fulling in 1973, Paul Davies in 1975 and W. G. Unruh in 1976.[1][2][3] 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 of Unruh radiation.

## Temperature equation

The Unruh temperature, derived by William Unruh in 1976, is the effective temperature experienced by a uniformly accelerating detector in a vacuum field. It is given by[4]

${\displaystyle T={\frac {\hbar a}{2\pi ck_{\mathrm {B} }}},}$

where ħ is the reduced Planck constant, a is the local acceleration, c is the speed of light, and kB is the Boltzmann constant. Thus, for example, a proper acceleration of 2.47×1020 m·s-2 corresponds approximately to a temperature of 1 K. Conversely, an acceleration of 1 m·s-2 corresponds to a temperature of 4.06×10−21 K.

The Unruh temperature has the same form as the Hawking temperature TH = ħg/ckB of a black hole, which was derived (by Stephen Hawking) independently around the same time. It is, therefore, sometimes called the Hawking–Unruh temperature.[5]

## Explanation

Unruh demonstrated theoretically that the notion of vacuum depends on the path of the observer through spacetime. 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.[6]

Although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the word vacuum is interpreted in a specific way.

In modern terms, the concept of "vacuum" is not the same as "empty space": Space is filled with the quantized fields that make up the universe. Vacuum is simply the lowest possible energy state of these fields.

The energy states of any quantized field are defined by the Hamiltonian, based on local conditions, including the time coordinate. According to special 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 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, putting it in the same conceptual framework as Hawking 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.

The free field needs to be decomposed into positive and negative frequency 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 (although the two are related by a Bogoliubov 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 of black holes and cosmological horizons. The Unruh effect would then be the near-horizon form of Hawking radiation.

The Unruh effect is also expected to be present in de Sitter space.[7]

## Calculations

In special relativity, an observer moving with uniform proper acceleration a through Minkowski spacetime is conveniently described with Rindler coordinates, which are related to the standard (Cartesian) Minkowski coordinates by

{\displaystyle {\begin{aligned}x&=\rho \cosh(\sigma )\\t&=\rho \sinh(\sigma ).\end{aligned}}}

The line element in Rindler coordinates, i.e. Rindler space is

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

where ρ = 1/a, and where σ is related to the observer's proper time τ by σ = (here c = 1).

An observer moving with fixed ρ traces out a hyperbola in Minkowski space, therefore this type of motion is called hyperbolic motion.

An observer moving along a path of constant ρ is uniformly accelerating, and is coupled to field modes which have a definite steady frequency as a function of σ. 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 σ is a symmetry of Minkowski space: it can be shown that it corresponds to a boost in x, 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 in σ, we can treat σ as "time" and the boost operator is then the corresponding Hamiltonian. In the Euclidean field theory, where the minus sign in front of the time in Rindler metric is changed to a plus sign by multiplying ${\displaystyle i}$  to the Rindler time, i.e. a Wick rotation or the imaginary time, the Rindler Metric is turned into a polar-coordinate-like metric. Therefore the rotations must close itself after 2π in 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 a Wick rotation. The periodicity ${\displaystyle \beta }$  of imaginary time corresponds to a temperature of ${\displaystyle \beta =1/T}$  in thermal quantum field theory. Now that the path integral for this Hamiltonian is closed with period 2π. This means that the H modes are thermally occupied with temperature 1/2π. This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle σ, which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in σ at position ρ has a frequency which is determined by the square root of the (absolute value of the) metric at ρ, the redshift factor. This can be seen by transforming the time coordinated of an Rindler observer at fixed ρ to an inertial, co-moving observer observing a proper time. From the Rindler-line-element given above, this is just ρ. 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 constant ρ in Rindler coordinate is equal to 1/ρ, the actual inverse temperature observed is

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

Restoring units yields

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

The temperature of the vacuum, seen by an isolated observer accelerating at the Earth's gravitational acceleration of g = 9.81 m·s−2, is only 4×10−20 K. For an experimental test of the Unruh effect it is planned to use accelerations up to 1026 m·s−2, which would give a temperature of about 400000 K.[8][9]

The Rindler derivation of the Unruh effect is unsatisfactory to some, since the detector's path is super-deterministic. Unruh later developed the Unruh–DeWitt particle detector model to circumvent this objection.

## Other implications

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.[10][11][12]

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. 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. Smolyaninov claims that it has already been observed,[13] while O'Connell and Ford claim that it is not emitted at all.[14] 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.

## Experimental observation

Researchers claim experiments that successfully detected the Sokolov–Ternov effect[15] may also detect the Unruh effect under certain conditions.[16]

Theoretical work in 2011 suggests that accelerating detectors might be used for the direct detection of the Unruh effect with current technology.[17]

## References

1. ^ Fulling, S. A. (1973). "Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time". Physical Review D. 7 (10): 2850–2862. Bibcode:1973PhRvD...7.2850F. doi:10.1103/PhysRevD.7.2850.
2. ^ Davies, P. C. W. (1975). "Scalar production in Schwarzschild and Rindler metrics". Journal of Physics A. 8 (4): 609–616. Bibcode:1975JPhA....8..609D. doi:10.1088/0305-4470/8/4/022.
3. ^ Unruh, W. G. (1976). "Notes on black-hole evaporation". Physical Review D. 14 (4): 870–892. Bibcode:1976PhRvD..14..870U. doi:10.1103/PhysRevD.14.870.
4. ^ Unruh, W. G. (2001). "Black Holes, Dumb Holes, and Entropy". In Callender, C. (ed.). Physics meets Philosophy at the Planck Scale. Cambridge University Press. pp. 152–173, Eq. 7.6. ISBN 9780521664455.
5. ^ Alsing, P. M.; Milonni, P. W. (2004). "Simplified derivation of the Hawking–Unruh temperature for an accelerated observer in vacuum". American Journal of Physics. 72 (12): 1524–1529. arXiv:quant-ph/0401170. Bibcode:2004AmJPh..72.1524A. doi:10.1119/1.1761064.
6. ^ Bertlmann, R. A.; Zeilinger, A. (2002). Quantum (Un)Speakables: From Bell to Quantum Information. Springer. p. 401. ISBN 3-540-42756-2.
7. ^ Casadio, R., et al. "On the Unruh effect in de Sitter space." Modern Physics Letters A 26.28 (2011): 2149-2158.
8. ^ Visser, M. (2001). "Experimental Unruh radiation?". Matters of Gravity. 17: 4–5. arXiv:gr-qc/0102044. Bibcode:2001gr.qc.....2044P.
9. ^ Rosu, H. C. (2001). "Hawking-like effects and Unruh-like effects: Toward experiments?". Gravitation and Cosmology. 7: 1–17. arXiv:gr-qc/9406012. Bibcode:1994gr.qc.....6012R.
10. ^ Mueller, R. (1997). "Decay of accelerated particles". Physical Review D. 56 (2): 953–960. arXiv:hep-th/9706016. Bibcode:1997PhRvD..56..953M. doi:10.1103/PhysRevD.56.953.
11. ^ Vanzella, D. A. T.; Matsas, G. E. A. (2001). "Decay of accelerated protons and the existence of the Fulling-Davies-Unruh effect". Physical Review Letters. 87 (15): 151301. arXiv:gr-qc/0104030. Bibcode:2001PhRvL..87o1301V. doi:10.1103/PhysRevLett.87.151301. PMID 11580689.
12. ^ Suzuki, H.; Yamada, K. (2003). "Analytic Evaluation of the Decay Rate for Accelerated Proton". Physical Review D. 67 (6): 065002. arXiv:gr-qc/0211056. Bibcode:2003PhRvD..67f5002S. doi:10.1103/PhysRevD.67.065002.
13. ^ Smolyaninov, I. I. (2008). "Photoluminescence from a gold nanotip as an example of tabletop Unruh-Hawking radiation". Physics Letters A. 372 (47): 7043–7045. arXiv:cond-mat/0510743. Bibcode:2008PhLA..372.7043S. doi:10.1016/j.physleta.2008.10.061.
14. ^ Ford, G. W.; O'Connell, R. F. (2005). "Is there Unruh radiation?". Physics Letters A. 350 (1–2): 17–26. arXiv:quant-ph/0509151. Bibcode:2006PhLA..350...17F. doi:10.1016/j.physleta.2005.09.068.
15. ^ Bell, J. S.; Leinaas, J. M. (1983). "Electrons as accelerated thermometers". Nuclear Physics B. 212 (1): 131–150. Bibcode:1983NuPhB.212..131B. doi:10.1016/0550-3213(83)90601-6.
16. ^ Akhmedov, E. T.; Singleton, D. (2007). "On the physical meaning of the Unruh effect". JETP Letters. 86 (9): 615–619. arXiv:0705.2525. Bibcode:2007JETPL..86..615A. doi:10.1134/S0021364007210138.
17. ^ Martín Martínez, E.; Fuentes, I.; Mann, R. B. (2011). "Using Berry's Phase to Detect the Unruh Effect at Lower Accelerations". Physical Review Letters. 107 (13): 131301. arXiv:1012.2208. Bibcode:2011PhRvL.107m1301M. doi:10.1103/PhysRevLett.107.131301. PMID 22026837.