LOG#144. Basic QFT in curved ST(IV): the Unruh effect.

What is the Unruh effect? Description: any accelerated observer in the traditional Minkovski state OBSERVE a thermal spectrum of particles. When we refer to the accelerated observer, usually it is called the Rindler observer in the literature as well. In fact, Unruh effect somehow “implies” the Hawking effect (despite the fact that the Unruh effect was discovered AFTER the seminal work by S.W. Hawking!).

Key points:

1st. Observers with different notions of positive and negative frequencies (or normal modes) will DISAGREE on particle content!

2nd. Uniformly accelerated observers (Rindler spacetimes) in Minkovski spacetime will move along time-like Killing vector orbits.

3rd. Inert vacuum is more like a “thermal state”.

This is certainly surprising but true! The mathematical background is the Rindler spacetime (accelerated spacetime from Minkovski spacetime):

ds^2=e^{2a\xi}(-d\eta^2+d\xi^2)

The particle number operator in Rindler spacetime reads

\langle N_R (k)\rangle =\langle 0_M\vert N_R^I(k)\vert 0_M\rangle=\langle 0_M\vert b_k^+b_k\vert 0_M\rangle=\dfrac{e^{-\pi \frac{\omega}{a}}}{2\sinh\left(\dfrac{\pi\omega}{a}\right)}

i.e., we have

\langle N_R (k)\rangle =\dfrac{1}{e^{2\pi\frac{\omega}{a}}-1}

But this is precisely the Planck blackbody distribution function, a pure thermal spectrum! If we identify the terms, we obtain that the so called Unruh temperature is

T_U=\dfrac{a}{2\pi}

or, equivalently, reintroducing the Boltzmann constant, the reduced Planck constant and the speed of light

k_BT_U=\dfrac{\hbar a}{2\pi c}\longleftrightarrow \boxed{T_U=\dfrac{\hbar a}{2\pi k_B c}}

This is one of my favourite formulae from Theoretical Physics! In fact, it is known that Feynman himself had this equation (and effects) on his thoughts before his death (in his blackboard, as far as I know…).

An interesting question is how can Rindler (accelerated) observers detect particles. After all, x \eta\sim constant implies that the energy-momentum tensor should be zero! The answer is subtle. When you accelerate a detector, it implies that E (energy) is NOT conserved. Work in necessary in order to keep it accelerating! The detector is excited form the energy used to keep it moving, and it is NOT caused by the background energy-momentum T_{\mu\nu}. Moreover, there are quantum fluctuations due to the spacetime curvature, and it is not captured by the background and we see it as an effect of geometry!

The Unruh effect plays also a key role in the entropic gravity approach launched by Verlinde in January, 2000. As I have reviewed here, Verlinde guessed the Newton’s law of inertia and the gravitational law of gravity from a pure “thermodynamical” or “statistical” setting. However, there is a big criticism on his approach, but it is a relatively new idea that deserver further attention. The last 3 years, this topic has exploded and produce new lines or research in Cosmology, holography, string theory, loop quantum cosmology and gravity.

Final remark (I):

\vert 0_M\rangle =\vert 0_I\rangle \otimes \vert 0_{II}\rangle

Final remark (II):

In general relativity, we get a redshift with a gravitational dilation factor and a gravitational field

g=\sqrt{1-\dfrac{2G_NM}{r}}

so we obtain

a=\dfrac{GM}{r^2\sqrt{1-\dfrac{2G_NM}{r}}}

Unruh’s original argument followed the next lines

a) \dfrac{T_2}{T_1}=\dfrac{V_1}{V_2}

and then

T_2=T_1\dfrac{V_1}{V_2}=\dfrac{V_1}{V_2}\dfrac{a}{2\pi}

b) At infinity,

V_2\rightarrow 1

and

V_1a_1\rightarrow \dfrac{G_NM}{r_1^2}=\dfrac{1}{4G_NM}=k=a

so if r_1\rightarrow 2G_NM, the Unruh temperature follows

(1)   \begin{equation*}\boxed{ T_U=\dfrac{a}{2\pi}=\dfrac{k}{2\pi}=T_H}\end{equation*}

where T_H is the Hawking temperature!

Please, don’t forget that the Unruh effect was discovered after the Hawking discovery of the quantum origin of the Black Hole thermodynamics (due to quantum effects!).

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