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| The hypothetical '''Unruh effect''' (or sometimes '''Fulling–Davies–Unruh effect''') is the prediction that an accelerating observer will observe [[black-body radiation]] where an [[Inertial frame of reference|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.
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| The Unruh effect was first described by [[Stephen Fulling]] in 1973, [[Paul Davies]] in 1975 and [[W. G. Unruh]] in 1976.<ref name="fdu">
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| {{cite journal
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| |author=S.A. Fulling
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| |year=1973
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| |title=Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time
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| |journal=[[Physical Review D]]
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| |volume=7 |pages=2850
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| |doi=10.1103/PhysRevD.7.2850
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| | bibcode = 1973PhRvD...7.2850F
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| |issue=10 }}</ref><ref>
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| {{cite journal
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| |author=P.C.W. Davies
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| |year=1975
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| |title=Scalar production in Schwarzschild and Rindler metrics
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| |journal=[[Journal of Physics A]]
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| |volume=8 |pages=609
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| |doi=10.1088/0305-4470/8/4/022
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| | bibcode = 1975JPhA....8..609D
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| |issue=4 }}</ref><ref>
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| {{cite journal
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| |author=W.G. Unruh
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| |year=1976
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| |title=Notes on black-hole evaporation
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| |journal=[[Physical Review D]]
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| |volume=14 |pages=870
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| |doi=10.1103/PhysRevD.14.870
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| | bibcode = 1976PhRvD..14..870U
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| |issue=4 }}</ref> 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'''.
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| == The equation ==
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| The '''Unruh temperature''', derived by William Unruh in 1976, is the effective temperature experienced by a uniformly accelerating detector in a [[vacuum state|vacuum field]]. It is given by<ref name=DUMB>See equation 7.6 in {{cite book
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| |author=W.G. Unruh
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| |year=2001
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| |chapter=Black Holes, Dumb Holes, and Entropy
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| |title=Physics meets Philosophy at the Planck Scale
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| |pages=152–173
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| |publisher=[[Cambridge University Press]]
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| }}</ref>
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| :<math>T = \frac{\hbar a}{2\pi c k_\text{B}},</math>
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| where <math>a</math> is the local acceleration, <math style="vertical-align:-17%;">k_\text{B}</math> is the [[Boltzmann constant]], <math style="vertical-align:0%;">\hbar</math> is the [[reduced Planck constant]], and <math style="vertical-align:0%;">c</math> is the [[speed of light]]. Thus, for example, a [[proper acceleration]] of 2.5 × 10<sup>20</sup> m s<sup>−2</sup> corresponds approximately to a temperature of 1 K.
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| The Unruh temperature has the same form as the [[Hawking temperature]] <math style="vertical-align:-23%;">T_\text{H} = \hbar g/(2\pi c k_\text{B})</math> of a [[black hole]], which was derived (by [[Stephen Hawking]]) independently around the same time. It is, therefore, sometimes called the Hawking–Unruh temperature.<ref name=SIMPLE>
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| {{cite journal
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| |author=P.M. Alsing, P.W. Milonni
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| |year=2004
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| |title=Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum
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| |journal=[[American Journal of Physics]]
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| |volume=72 |pages=1524
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| |doi=10.1119/1.1761064
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| | arxiv=quant-ph/0401170v2
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| | bibcode = 2004AmJPh..72.1524A
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| |issue=12 }}</ref>
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| == Explanation ==
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| 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.<ref name=Bertlmann>
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| {{cite book
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| | author=Reinhold A. Bertlmann & Anton Zeilinger
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| | year=2002
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| | title=Quantum (un)speakables: From Bell to Quantum Information
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| | url=http://books.google.com/?id=wiC0SEdQ454C&pg=PA483&dq=Unruh+%22Sokolov-Ternov+effect%22#PPA401,M1
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| | page=401 ''ff''
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| | publisher=[[Springer (publisher)|Springer]]
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| | isbn=3-540-42756-2
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| }}</ref>
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| Although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the word ''vacuum'' is interpreted in a specific way.
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| 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".
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| The energy states of any quantized field are defined by the [[Hamiltonian (quantum theory)|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.
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| 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.
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| 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.
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| The [[free field]] needs to be decomposed into positive and [[negative frequency]] components before defining the [[creation operator|creation]] and [[annihilation operator]]s. This can only be done in spacetimes with a [[timelike]] [[Killing vector]] field. This decomposition happens to be different in [[Cartesian coordinates|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. | |
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| 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 hole]]s and [[Observable universe#Cosmological horizon|cosmological horizon]]s. The Unruh effect would then be the near-horizon form of the [[Hawking radiation]].
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| == Calculations ==
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| In [[special relativity]], an observer moving with uniform [[proper acceleration]] ''a'' through [[Minkowski spacetime]] is conveniently described with [[Rindler coordinates]]. The [[line element]] in Rindler coordinates is
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| : <math>ds^2 = -\rho^2 d\sigma^2 + d\rho^2,</math>
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| where <math>\rho = 1/a</math>, and where <math>\sigma</math> is related to the observer's proper time <math>\tau</math> by <math> \sigma = g\tau</math> (here ''c'' = 1). Rindler coordinates are related to the standard (Cartesian) Minkowski coordinates by
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| : <math> x= \rho \cosh\sigma</math>
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| : <math> t= \rho \sinh\sigma.</math>
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| An observer moving with fixed <math>\rho</math> traces out a [[hyperbola]] in Minkowski space.
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| An observer moving along a path of constant <math>\rho</math> is uniformly accelerated, and is coupled to field modes which have a definite steady frequency as a function of <math>\sigma</math>. 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.
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| Translation in <math>\sigma</math> is a symmetry of Minkowski space: It is a [[Lorentz boost|boost]] around the origin. For a detector coupled to modes with a definite frequency in <math>\sigma</math>, the boost operator is then the Hamiltonian. In the Euclidean field theory, these boosts analytically continue to rotations, and the rotations close after <math>2\pi</math>. So
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| : <math>e^{2\pi i H} = 1.</math>
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| The path integral for this Hamiltonian is closed with period <math>2\pi</math> which guarantees that the H modes are thermally occupied with temperature <math>\scriptstyle (2\pi)^{-1}</math>. This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle <math>\sigma</math> which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in <math>\sigma</math> at position <math>\rho</math> has a frequency which is determined by the square root of the (absolute value of the) metric at <math>\rho</math>, the redshift factor. From the equation for the line element given above, it is easily seen that this is just <math>\rho</math>. The actual inverse temperature at this point is therefore
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| : <math>\beta= 2\pi \rho.</math>
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| Since the acceleration of a trajectory at constant <math>\rho</math> is equal to <math>1/a</math>, the actual inverse temperature observed is
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| :<math>\beta = {2\pi \over a}.</math>
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| Restoring units yields
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| : <math>k_\text{B}T = \frac{\hbar a}{2\pi c}.</math>
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| 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 <math>\rho_0</math>, the thermal boundary condition for the field theory at <math>\rho_0</math> is the temperature of the wall. By making the positive <math>\rho</math> side of the wall colder, the extension of the wall's state to <math>\rho>\rho_0</math> 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|date=April 2009}}, because under these circumstances there is no Rindler horizon in the field of view.
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| The [[temperature]] of the vacuum, seen by an isolated observer accelerated at the Earth's gravitational acceleration of [[standard gravity|''g'']] = 9.81 [[metre per second squared|m/s²]], is only 4×10<sup>−20</sup> [[Kelvin|K]]. For an experimental test of the Unruh effect it is planned to use accelerations up to 10<sup>26</sup> m/s², which would give a temperature of about 400,000 K.<ref>
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| {{cite journal
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| |author=M. Visser
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| |year=2001
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| |title=Experimental Unruh radiation?
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| |journal=Newsletter of the APS Topical Group on Gravitation
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| |volume=17
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| | arxiv=gr-qc/0102044
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| | bibcode = 2001gr.qc.....2044P
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| |pages=2044 }}</ref><ref>
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| {{cite journal
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| |author=H.C. Rosu
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| |year=2001
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| |title=Hawking-like effects and Unruh-like effects: Toward experiments?
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| |journal=[[Gravitation and Cosmology]]
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| |volume=7 |pages=1
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| |doi=
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| | arxiv=gr-qc/9406012
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| | bibcode = 1994gr.qc.....6012R }}</ref>
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| To put this in perspective, at a vacuum Unruh temperature of 3.978×10<sup>−20</sup> K, an electron would have a [[de Broglie Wavelength]] of ''h''/√(''3m''<sub>e</sub>''kT'') = 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.
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| 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<sup>−23</sup> 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.
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| == Other implications ==
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| 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.<ref name="muel">
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| {{cite journal
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| |author=R. Mueller
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| |year=1997
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| |title=Decay of accelerated particles
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| |journal=[[Physical Review D]]
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| |volume=56 |pages=953–960
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| |doi=10.1103/PhysRevD.56.953
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| | arxiv=hep-th/9706016
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| | bibcode = 1997PhRvD..56..953M
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| |issue=2 }}</ref><ref name="van">
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| {{cite journal
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| |author=D.A.T. Vanzella, G.E.A. Matsas
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| |year=2001
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| |title=Decay of accelerated protons and the existence of the Fulling-Davies-Unruh effect
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| |journal=[[Physical Review Letters]]
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| |volume=87 |pages=151301
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| |doi=10.1103/PhysRevLett.87.151301
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| | arxiv=gr-qc/0104030
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| | bibcode=2001PhRvL..87o1301V
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| |issue=15
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| }}</ref><ref name="suz">
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| {{cite journal
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| |author=H. Suzuki, K. Yamada
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| |year=2003
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| |title=Analytic Evaluation of the Decay Rate for Accelerated Proton
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| |journal=[[Physical Review D]]
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| |volume=67 |pages=065002
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| |doi=10.1103/PhysRevD.67.065002
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| |arxiv={{arxiv|gr-qc/0211056}}
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| | bibcode = 2003PhRvD..67f5002S
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| |issue=6 }}</ref>
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| == Unruh radiation ==
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| 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''.
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| The existence of Unruh radiation is not universally accepted. Some claim that it has already been observed,<ref>
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| {{cite journal
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| |author=I.I. Smolyaninov
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| |year=2005
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| |title=Photoluminescence from a gold nanotip as an example of tabletop Unruh-Hawking radiation
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| |doi=10.1016/j.physleta.2008.10.061
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| |journal=Physics Letters A
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| |volume=372
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| |issue=47
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| |pages=7043–7045
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| |arxiv=cond-mat/0510743
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| | bibcode = 2008PhLA..372.7043S }}</ref> while others claims that it is not emitted at all.<ref>
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| {{cite journal
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| |author=G.W. Ford, R.F. O'Connell
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| |year=2005
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| |title=Is there Unruh radiation?
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| |doi=10.1016/j.physleta.2005.09.068
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| |journal=Physics Letters A
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| |volume=350
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| |pages=17–26
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| |arxiv=quant-ph/0509151
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| | bibcode = 2006PhLA..350...17F }}</ref> 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.
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| == Experimental observation of the Unruh effect ==
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| Researchers claim experiments that successfully detected the [[Sokolov–Ternov effect]]<ref>
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| {{cite journal
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| |first1=J. S. |last1=Bell |first2=J. M. |last2=Leinaas
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| |date=7 February 1983
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| |title=Electrons as accelerated thermometers
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| |doi=10.1016/0550-3213(83)90601-6
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| |journal=Nuclear Physics B
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| |volume=212
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| |issue=1
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| |pages=131–150
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| |bibcode = 1983NuPhB.212..131B }}</ref>
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| may also detect the Unruh effect under certain conditions.<ref>
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| {{cite journal
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| |author=E.T. Akhmedov, D. Singleton
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| |year=2007
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| |title=On the physical meaning of the Unruh effect
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| |doi=10.1134/S0021364007210138
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| |journal=JETP Letters
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| |volume=86
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| |issue=9
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| |pages=615–619
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| |arxiv=0705.2525
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| | bibcode = 2007JETPL..86..615A }}</ref>
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| Theoretical work in 2011 suggests that accelerated detectors may be used for the direct detection of the Unruh effect with current technology.<ref>
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| {{cite journal
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| |author=E. Martín-Martínez, I. Fuentes, R. B. Mann
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| |year=2011
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| |title=Using Berry’s Phase to Detect the Unruh Effect at Lower Accelerations
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| |doi=10.1103/PhysRevLett.107.131301
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| |journal=Physical Review Letters
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| |volume=107
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| |issue=13
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| |pages=131301
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| |arxiv=1012.2208
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| |bibcode = 2011PhRvL.107m1301M }}</ref>
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| == See also ==
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| * [[Casimir effect#Dynamical Casimir effect|Dynamical Casimir effect]]
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| * [[Hawking radiation]]
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| * [[Pair production]]
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| * [[Quantum information]]
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| * [[Stochastic electrodynamics]]
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| * [[Superradiance]]
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| * [[Virtual particle]]
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| * [[Woodward effect]]
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| == References ==
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| {{reflist|30em}}
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| == Further reading ==
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| * {{cite book
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| |author=K.P. Thorne
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| |year=1995
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| |chapter=Black holes evaporate
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| |title=[[Black Holes and Time Warps]]
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| |publisher=[[W. W. Norton & Company]]
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| |edition=Reprint
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| |isbn=0-393-31276-3
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| }} See especially box 12.5 on p. 444.
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| * {{cite book
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| |author=R.M. Wald
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| |year=1994
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| |chapter=Chapter 5
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| |title=Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics
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| |publisher=[[University of Chicago Press]]
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| |url=http://books.google.com/?id=Iud7eyDxT1AC&printsec=frontcover&dq=quantum+inauthor:wald#PPA105,M1
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| |isbn=0-226-87027-8
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| }}
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| * {{cite journal
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| |author=L.C.B. Crispino, A. Higuchi, G.E.A. Matsas
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| |year=2008
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| |title=The Unruh effect and its applications
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| |journal=[[Reviews of Modern Physics]]
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| |volume=80 |pages=787
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| |doi=10.1103/RevModPhys.80.787
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| | arxiv=0710.5373
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| | bibcode=2008RvMP...80..787C
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| |issue=3
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| }}
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| {{DEFAULTSORT:Unruh Effect}}
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| [[Category:Thermodynamics]]
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| [[Category:Quantum field theory]]
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| [[Category:Theory of relativity]]
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