# 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):

The particle number operator in Rindler spacetime reads

i.e., we have

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

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

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, 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 . 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):

Final remark (II):

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

so we obtain

Unruh’s original argument followed the next lines

a)

and then

b) At infinity,

and

so if , the Unruh temperature follows

(1)

where 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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