# Is there somesort of "recipe" to check if an spacetime exibit the Unruh effect?

Firstly I don't know for sure if this question makes a solid sense, concerning the physical phenomena.

I) Unruh Effect

Now, we have then the Unruh effect which appears just by considering the Minkowski spacetime, of a (unifomely) accelerated observer; precisely the accelerated motion of a particle on Rindler Spacetime with a massless scalar field.

Unruh effect have a quantum nature, and the aspect of "thermal bath" is explained by both kinematics of the accelerated particle with the mean particle density given by $$[1]$$:

$$\langle 0_{M}|\hat{n}_{\Omega}| 0_{M}\rangle = \frac{1}{ e^{\frac{2\pi\Omega}{a}}-1} \tag{1}$$

And obeying the distribution (Bose-Einstein) with termal temperature$$[1]$$:

$$T = \frac{a}{2\pi} \tag{2}$$

II) Given a metric I can tell you the motion of a photon

Consider now, the standard algorithm procedure of finding the null-like geodesics of a given metric $$[2]$$:

1) You'll need a metric tensor $$g_{\mu\nu}$$

2) Construct "the lagrangian" for a affine paramter $$\lambda$$:

$$2L = g_{\mu\nu}\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}\lambda}$$

3) Solve the (null) geodesic equations:

$$\frac{\partial L}{\partial x^{\mu}} - \frac{\mathrm{d}}{\mathrm{d}\lambda}\Big(\frac{\partial L}{\partial (\mathrm{d}x^{\mu}/\mathrm{d}\lambda) }\Big) = 0$$

With this procedure we can explore more and define the causal structure (a.k.a. the light cones; the null-like paths) of Schwarzschild spacetime, and realize the basic features and phenomena of a black hole:

III) How can I tell, precisely, that a particle moving in a given background spacetime will exibit the Unruh phenomena?

Here comes, then, the doubt. Is there any computational procedure to answer if a spacetime will exibit Unruh effect? Because note that in II) the spirit is given 1), 2),3) we can see some new phenomena for this geometry. In I) it seems for me that the spirit is the same. Just like are exposed in $$[1]$$, we just give a metric and massless scalar field and then we can see new phenomena.

What I'm searching is:

1) Given a metric tensor $$g_{\mu\nu}$$

2) Given massless scalar field

3) For this background spacetime $$g_{\mu\nu}$$, there is (there isn't) Unruh effect.

$$* * *$$

$$[1]$$ ALBORNOZ.A.C.C.On Accelerating Observers and Evaporating Black Holes.MSc Tesis.

$$[2]$$ D'INVERNO.R. Introducing Einstein's Relativity.

• This may help scholarpedia.org/article/Unruh_effect#Observability Commented Oct 14, 2019 at 4:21
• What exactly do you mean by there is/isn't an Unruh effect for a given background? (I'm not sure there is a way of phrasing this that would make it a well defined question, but maybe there is.) Commented Oct 14, 2019 at 13:31

Several things to note:

1. Unruh effect is observed not in a spacetime in general but by an observer (or detector) moving along specific trajectory (non-inertial) in this spacetime.

2. Additionally, one has to specify a state of the scalar field in this spacetime. While quantum field in Minkowski spacetime has a unique vacuum state, in a curved spacetime there is generally a choice of Hadamard states that could play the same role as vacuum in Minkowski spacetime.

3. There is indeed a “recipe” called Unruh–deWitt detector which is a model of simple quantum system with two energy levels linearly coupled to a scalar field at the position of detector. Transitions between this levels (“detector clicking”) are interpreted as detection of scalar field quanta of corresponding energy. For more details see e.g. these papers:

• Louko, J., & Satz, A. (2008). Transition rate of the Unruh–DeWitt detector in curved spacetime. Classical and Quantum Gravity, 25(5), 055012, arXiv:0710.5671.

• Barbado, L. C., & Visser, M. (2012). Unruh-DeWitt detector event rate for trajectories with time-dependent acceleration. Physical Review D, 86(8), 084011, arXiv:1207.5525.

4. Whether a given pattern of quanta detection could be called “Unruh effect” is a matter of some interpretation. For example, detector moving along inertial trajectory in de Sitter spacetime would register thermal radiation (from cosmological horizon), this is an example of Gibbons–Hawking effect. But if we consider accelerated detector in such a spacetime then depending on acceleration parameter effective temperature would be smoothly interpolating from purely de Sitter temperature (for small accelerations) to Unruh temperature (for large accelerations). So, at what values we would call detector's behavior “Unruh effect”?