Editing Unruh effect (section)

== Calculations ==
In [[special relativity]], an observer moving with uniform [[proper acceleration]] {{mvar|a}} through [[Minkowski spacetime]] is conveniently described with [[Rindler coordinates]], which are related to the standard ([[Cartesian coordinates|Cartesian]]) [[Minkowski space|Minkowski coordinates]] by

: <math>\begin{align}
  x &= \rho \cosh(\sigma) \\
  t &= \rho \sinh(\sigma).
\end{align}</math>

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

: <math>\mathrm{d}s^2 = -\rho^2\, \mathrm{d}\sigma^2 + \mathrm{d}\rho^2,</math>

where {{math|''ρ'' {{=}} {{sfrac|1|''a''}}}}, and where {{mvar|σ}} is related to the observer's proper time {{mvar|τ}} by {{math|''σ'' {{=}} ''aτ''}} (here {{math|''c'' {{=}} 1}}).

An observer moving with fixed {{mvar|ρ}} traces out a [[hyperbola]] in Minkowski space, therefore this type of motion is called [[hyperbolic motion (relativity)|hyperbolic motion]]. The coordinate <math>\rho</math> is related to the Schwarzschild spherical coordinate <math>r_S</math> by the relation<ref>{{cite book |last1=Susskind |first1=Leonard |last2=Lindesay |first2=James |title=An introduction to black holes, information, and the string theory revolution: The holographic universe |date=2005 |publisher=World Scientific |location=Hackensack, NJ |isbn=978-9812561312 |page=8}}</ref> 
:<math> \rho = \int^r_{r_S}\frac{dr^\prime}{\sqrt{1-r_S/r^\prime}}.</math>

An observer moving along a path of constant {{mvar|ρ}} is uniformly accelerating, and is coupled to field modes which have a definite steady frequency as a function of {{mvar|σ}}. These modes are constantly [[Relativistic Doppler effect|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 {{mvar|σ}} is a symmetry of Minkowski space: it can be shown that it corresponds to a [[Lorentz boost|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 {{mvar|σ}}, we can treat {{mvar|σ}} 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 <math>i</math> to the Rindler time, i.e. a [[Wick rotation]] or imaginary time, the Rindler metric is turned into a polar-coordinate-like metric. Therefore any rotations must close themselves after 2{{pi}} in a Euclidean metric to avoid being singular. So

: <math>e^{2\pi i H} = Id.</math>

A path integral with real time coordinate is dual to a thermal partition function, related by a [[Wick rotation]]. The periodicity <math>\beta</math> of imaginary time corresponds to a temperature of <math>\beta = 1/T</math> in [[thermal quantum field theory]]. Note that the path integral for this Hamiltonian is closed with period 2{{pi}}. This means that the {{mvar|H}} modes are thermally occupied with temperature {{sfrac|1|2{{pi}}}}. This is not an actual temperature, because {{mvar|H}} is dimensionless. It is conjugate to the timelike polar angle {{mvar|σ}}, which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency {{mvar|f}} in {{mvar|σ}} at position {{mvar|ρ}} has a frequency which is determined by the square root of the (absolute value of the) metric at {{mvar|ρ}}, the [[redshift]] factor. This can be seen by transforming the time coordinate of a Rindler observer at fixed {{mvar|ρ}} to an inertial, co-moving observer observing a [[proper time]]. From the Rindler-line-element given above, this is just {{mvar|ρ}}. The actual inverse temperature at this point is therefore

: <math>\beta = 2\pi \rho.</math>

It can be shown that the acceleration of a trajectory at constant {{mvar|ρ}} in Rindler coordinates is equal to {{math|{{sfrac|1|''ρ''}}}}, so the actual inverse temperature observed is

:<math>\beta = \frac{2\pi}{a}.</math>

Restoring units yields

: <math>k_\text{B}T = \frac{\hbar a}{2\pi c}.</math>

The [[temperature]] of the vacuum, seen by an isolated observer accelerating at the Earth's gravitational acceleration of {{mvar|[[standard gravity|g]]}} = {{val|9.81|u=m·s{{sup|−2}}}}, is only {{val|4|e=-20|u=K}}. For an experimental test of the Unruh effect it is planned to use accelerations up to {{val|e=26|u=m·s{{sup|−2}}}}, which would give a temperature of about {{val|400000|u=K}}.<ref>
{{cite journal
 |last=Visser |first=M.
 |year=2001
 |title=Experimental Unruh radiation?
 |journal=[[Matters of Gravity (newsletter)|Matters of Gravity]]
 |volume=17 |pages=4–5
 |arxiv=gr-qc/0102044
 |bibcode=2001gr.qc.....2044P
}}</ref><ref>
{{cite journal
 |last=Rosu |first=H. C. 
 |year=2001
 |title=Hawking-like effects and Unruh-like effects: Toward experiments?
 |journal=[[Gravitation and Cosmology]]
 |volume=7 |pages=1–17
 |arxiv=gr-qc/9406012
 |bibcode=1994gr.qc.....6012R
}}</ref>

The Rindler derivation of the Unruh effect is unsatisfactory to some{{who|date=August 2020}}, since the detector's path is [[superdeterminism|super-deterministic]]. Unruh later developed the [[Unruh–DeWitt particle detector]] model to circumvent this objection.