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Unruh effect
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== {{anchor|Unruh temperature}}Temperature equation == The '''Unruh temperature''', sometimes called the Davies–Unruh temperature,<ref name="takagi 1986">{{cite journal |last1=Takagi |first1=Shin |title=Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimensions |journal=Progress of Theoretical Physics Supplement |date=1986 |issue=88 |pages=1–142 |doi=10.1143/PTP.88.1 |doi-access=free }}</ref> was derived separately by Paul Davies<ref name="davies 1975" /> and William Unruh<ref name="unruh 1976" /> and is the effective temperature experienced by a uniformly accelerating detector in a [[vacuum state|vacuum field]]. It is given by<ref name=DUMB> {{cite book |last=Unruh |first=W. G. |year=2001 |chapter=Black Holes, Dumb Holes, and Entropy |editor-last=Callender |editor-first=C. |title=Physics meets Philosophy at the Planck Scale |pages=152–173, Eq. 7.6 |publisher=[[Cambridge University Press]] |isbn=9780521664455 }}</ref> :<math>T = \frac{\hbar a}{2\pi c k_\mathrm{B}}\approx 4.06\times 10^{-21}\,\mathrm{K{\cdot}s^2{\cdot}m^{-1}}\times a,</math> where {{mvar|ħ}} is the [[reduced Planck constant]], {{mvar|a}} is the proper uniform acceleration, {{mvar|c}} is the [[speed of light]], and {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]]. Thus, for example, a [[proper acceleration]] of {{val|2.47|e=20|u=m.s-2}} corresponds approximately to a temperature of {{val|1|u=K}}. Conversely, an acceleration of {{val|1|u=m.s-2}} corresponds to a temperature of {{val|4.06|e=-21|u=K}}. The Unruh temperature has the same form as the [[Hawking temperature]] {{math|''T''<sub>H</sub> {{=}} {{sfrac|''ħg''|2π''ck''<sub>B</sub>}}}} with {{mvar|g}} denoting the surface gravity of a [[black hole]], which was derived by [[Stephen Hawking]] in 1974.<ref name="Hawking 1974">{{cite journal |last1=Hawking |first1=S. W. |title=Black hole explosions? |journal=Nature |date=1974 |volume=248 |issue=5443 |pages=30–31 |doi=10.1038/248030a0 |bibcode=1974Natur.248...30H |s2cid=4290107 |url=https://www.nature.com/articles/248030a0}}</ref> In the light of the [[equivalence principle]], it is, therefore, sometimes called the Hawking–Unruh temperature.<ref name=SIMPLE> {{cite journal |last1=Alsing |first1=P. M. |last2=Milonni |first2=P. W. |author-link2=Peter W. Milonni |year=2004 |title=Simplified derivation of the Hawking–Unruh temperature for an accelerated observer in vacuum |journal=[[American Journal of Physics]] |volume=72 |issue=12 |pages=1524–1529 |arxiv=quant-ph/0401170 |bibcode=2004AmJPh..72.1524A |doi=10.1119/1.1761064 |s2cid=18194078 }}</ref> Solving the Unruh temperature for the uniform acceleration, it can be expressed as :<math>a = \frac{2\pi c k_\mathrm{B}}{\hbar}T = 2\pi a_\mathrm{P} \frac{T}{T_\mathrm{P}}</math>, where <math>a_\mathrm{P}</math> is [[Planck_units#Derived_units|Planck acceleration]] and <math>T_\mathrm{P}</math> is [[Planck_units#History_and_definition|Planck temperature]].
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