Unruh effect

The Unruh effect (or sometimes Fulling–Davies–Unruh effect), was first described by Stephen Fulling in 1973, Paul Davies in 1975 and Bill Unruh in 1976.^[1][2][3] It is the prediction that an accelerating observer will observe black-body 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 will record a non-zero temperature. The ground state for an inertial observer is seen as in thermodynamic equilibrium 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. It is given by:^[4]

$T = \frac{\hbar a}{2\pi c k}$

where:

a is the local acceleration

k is the Boltzmann constant

$\hbar$ is the reduced Planck's constant

c is the speed of light

Thus, for example, an acceleration of 10²²m / s² corresponds to a temperature of 1 K. The Unruh temperature has the same form as the Hawking temperature $T_H = \frac{\hbar g}{2\pi c k}$ 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 appropriately, as below.

Vacuum interpretation

In modern terms, the concept of "vacuum" is not the same as "empty space", as all of space is filled with the quantized fields that make up a universe. Vacuum is simply the lowest possible energy 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, 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 the Hawking radiation.

Calculations

The theory of the Unruh effect involves the Rindler coordinates ρ and τ, which have metric

$ds^2 = -\rho^2 d\tau^2 + d\rho^2 \,$

This is just ordinary Minkowski space in relativistic polar coordinates:

$x= \rho \cosh(\tau)\,$

$t= \rho \sinh(\tau)\,$

The orbit in (1+1) space-time is a regular hyperbola. The parametric equations are of the above form, but with ρ 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 τ. 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 ρ is uniformly accelerated, and is coupled to field modes which have a definite steady frequency as a function of τ. 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 τ is a symmetry of Minkowski space: It is a boost around the origin. For a detector coupled to modes with a definite frequency in τ, the boost operator is then the Hamiltonian. In the Euclidean field theory, these boosts analytically continue to rotations, and the rotations close after 2π. So

$e^{2\pi i H} = 1 \,$

The path integral for this Hamiltonian is closed with period 2π which guarantees that the H modes are thermally occupied with temperature $\scriptstyle (2\pi)^{-1}$. This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle τ which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in τ at position ρ has a frequency which is determined by the square root of the metric at ρ, the redshift factor. The actual inverse temperature at this point is therefore

$\beta= 2\pi \rho \,$

Since the acceleration of a trajectory at constant ρ is equal to 1 / a, the actual inverse temperature observed is:

$\beta = {2\pi \over a}$

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

$kT = \frac{\hbar a}{2\pi c}$

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 ρ₀, the thermal boundary condition for the field theory at ρ₀ is the temperature of the wall. By making the positive ρ side of the wall colder, the extension of the wall's state to ρ > ρ₀ 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^{[citation needed]}, because under these circumstances there is no Rindler horizon in the field of view.

The temperature of the vacuum, seen by an isolated observer accelerated at the Earth's gravitational acceleration of g = 9.81 m/s², is only 4×10⁻²⁰ K. 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.^[7][8]

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/√(3m_ekT) = 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 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.^[9][10][11]

Unruh radiation

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,^[12] while others claims that it is not emitted at all.^[13] 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^[14] 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 .^[15]

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 ^[16].

See also

• Pair production
• Virtual particle
• Superradiance
• Quantum information

References

1. ^ S.A. Fulling (1973). "Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time". Physical Review D 7 (10): 2850. Bibcode 1973PhRvD...7.2850F. doi:10.1103/PhysRevD.7.2850. 
2. ^ P.C.W. Davies (1975). "Scalar production in Schwarzschild and Rindler metrics". Journal of Physics A 8 (4): 609. Bibcode 1975JPhA....8..609D. doi:10.1088/0305-4470/8/4/022. 
3. ^ W.G. Unruh (1976). "Notes on black-hole evaporation". Physical Review D 14 (4): 870. Bibcode 1976PhRvD..14..870U. doi:10.1103/PhysRevD.14.870. 
4. ^ See equation 7.6 in W.G. Unruh (2001). "Black Holes, Dumb Holes, and Entropy". Physics meets Philosophy at the Planck Scale. Cambridge University Press. pp. 152–173. 
5. ^ P.M. Alsing, P.W. Milonni (2004). "Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum". American Journal of Physics 72 (12): 1524. arXiv:quant-ph/0401170v2. Bibcode 2004AmJPh..72.1524A. doi:10.1119/1.1761064. 
6. ^ Reinhold A. Bertlmann & Anton Zeilinger (2002). Quantum (un)speakables: From Bell to Quantum Information. Springer. p. 401 ff. ISBN 3540427562. http://books.google.com/?id=wiC0SEdQ454C&pg=PA483&dq=Unruh+%22Sokolov-Ternov+effect%22#PPA401,M1. 
7. ^ M. Visser (2001). "Experimental Unruh radiation?". Newsletter of the APS Topical Group on Gravitation 17: 2044. arXiv:gr-qc/0102044. Bibcode 2001gr.qc.....2044P. 
8. ^ H.C. Rosu (2001). "Hawking-like effects and Unruh-like effects: Toward experiments?". Gravitation and Cosmology 7: 1. arXiv:gr-qc/9406012. Bibcode 1994gr.qc.....6012R. 
9. ^ R. Mueller (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. 
10. ^ D.A.T. Vanzella, G.E.A. Matsas (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. 
11. ^ H. Suzuki, K. Yamada (2003). "Analytic Evaluation of the Decay Rate for Accelerated Proton". Physical Review D 67 (6): 065002. arXiv:arXiv:gr-qc/0211056 [[arXiv]]:[[arXiv:gr-qc/0211056|gr-qc/0211056]]. Bibcode 2003PhRvD..67f5002S. doi:10.1103/PhysRevD.67.065002. 
12. ^ I.I. Smolyaninov (2005). "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. 
13. ^ G.W. Ford, R.F. O'Connell (2005). "Is there Unruh radiation?". Physics Letters A 350: 17–26. arXiv:quant-ph/0509151. Bibcode 2006PhLA..350...17F. doi:10.1016/j.physleta.2005.09.068. 
14. ^ Bell, J. S.; Leinaas, J. M. (7 February 1983). "Electrons as accelerated thermometers". Nuclear Physics B 212 (1): 131–150. Bibcode 1983NuPhB.212..131B. doi:10.1016/0550-3213(83)90601-6. 
15. ^ E.T. Akhmedov, D. Singleton (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. 
16. ^ E. Martín-Martínez, I. Fuentes, R. B. Mann (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. 

• K.P. Thorne (1995). "Black holes evaporate". Black Holes and Time Warps (Reprint ed.). W. W. Norton & Company. ISBN 0-393-31276-3.  See especially box 12.5 on p. 444.

• R.M. Wald (1994). "Chapter 5". Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press. ISBN 0226870278. http://books.google.com/?id=Iud7eyDxT1AC&printsec=frontcover&dq=quantum+inauthor:wald#PPA105,M1. 

• L.C.B. Crispino, A. Higuchi, G.E.A. Matsas (2008). "The Unruh effect and its applications". Reviews of Modern Physics 80 (3): 787. arXiv:0710.5373. Bibcode 2008RvMP...80..787C. doi:10.1103/RevModPhys.80.787.