Doubts about the "periodicity trick" to compute temperature The "periodicity trick" is a mysterious way to compute some sort of temperature associated to a Rindler-like spacetime.
Suppose there exist coords $R\in(0,\infty), \eta\in(-\infty,\infty)$ covering a patch of spacetime such that the metric takes the form $$ds^2 = -R^2 d\eta^2+dR^2,\tag{$\star$}$$
where I'm suppressing any orthogonal flat directions. This is the metric of the right Rindler wedge, in Rindler coords. (Note: I've not included any constant factors like the surface gravity, so dimensions might look a bit odd throughout this question).
Defining $\theta=i\eta$ we find the Euclidean metric
$$ds_E^2 =  R^2 d\theta^2 + dR^2. \tag{$\dagger$}$$
The periodicity trick proceeds in three steps:
(1) We claim that $\theta$ should be taken to be a periodic coordinate, $\theta \sim \theta+c$ for some constant $c$.
(2) To avoid a conical singularity in the metric $(\dagger)$ at $R=0$, we must have $c=2\pi$.
(3) We wave our hands and mumble "periodicity in imaginary time = inverse temperature", therefore concluding that the Lorentzian spacetime $(\star)$ has inverse temperature $\beta = 2\pi$.
I have questions about each of these three steps:
(Q1) Why should $\theta$ be periodic at all? Why not have $\theta\in (-\infty,\infty)$?
(Q2) Why shouldn't there be a conical singularity at $R=0$? Usually something is said about the Einstein equations implying smoothness, but I don't see how the Einstein equations are relevant here.
(Q3) What state are we calculating the temperature of? Spacetimes do not have a temperature (as far as I understand). Temperature is a property of thermal density matrices. So if this calculation is to make any sense at all, it must refer to a specific state somewhere. (For example, perhaps the relevant density matrix is the one obtained by tracing out the pure vacuum state on a larger spacetime?)
Note: answers can assume that I am confident using Euclidean path integrals to prepare states/find transition amplitudes, and that I am familiar with the KMS condition $\langle \mathcal{O_1}(t_1,\mathbf{x_1})\mathcal{O_2}(t_2,\mathbf{x_2}) \rangle_\beta = \langle \mathcal{O_1}(t_1+i\beta,\mathbf{x_1})\mathcal{O_2}(t_2,\mathbf{x_2}) \rangle_\beta $.
 A: Amendment:
The answer below is not correct. See discussion thread. I am leaving it up in case somebody can point out exactly what is wrong with it or until somebody gives a correct answer.
Original Answer
This is an old post, but I was confused by this myself so perhaps this will be of some use to somebody. The periodicity trick is not supposed to give a derivation of anything, it is just supposed to be a trick to obtain the temperature of a thermal state that comes from reducing the density matrix of a state that is smooth across the Rindler horizon. As for why it works, here is one way to think of it:
Take a real scalar field on Minkowski space. Suppose we want to compute the the partition function of the thermal ensemble $e^{-\beta H}$ on a semi-infinite region of Minkowski space, which we can describe with the Rindler metric:
\begin{equation}
    ds^2 = dR^2 - R^2 d\eta^2
\end{equation}
Where  $H$ is the conserved charge associated with Lorentz boosts $H = Q[\partial_\eta]$. We can compute this partition function using a path integral with Euclidean action and periodic boundary conditions:
\begin{equation}
    Z(\beta) = \mathrm{Tr}(e^{-\beta H}) = \int \mathcal{D}\psi \mathcal{D}\chi \delta(\psi - \chi) \int_{\phi(0) = \psi}^{\phi(\beta) = \xi} \mathcal{D}\phi e^{-S_E[\phi]}
\end{equation}
Since $\eta$ is the time coordinate conjugate to $H$, we can evaluate this path integral by integrating over a Euclidean manifold obtained from analytically continuing $\eta$ to $\eta_E = i\eta$. This Euclidean manifold has metric:
\begin{equation}
    ds^2 = dR^2 + R^2 d\eta_E^2
\end{equation}
The periodic boundary conditions of the path integral are enforced by requiring that $\eta_E = \eta_E + \beta$. This turns the metric on the Euclidean manifold into a conical metric which has a conical singularity at $R = 0$. An integral over a manifold with a conical singularity is divergent, so we find that the partition function is ill-defined. This tells us that the state we started with was singular on the Rindler horizon. The only way to avoid this is if the inverse temperature $\beta = 2\pi$ in which case the metric becomes a smooth cylindrical metric. So any regular state (i.e. with regular energy-momentum tensor, two-point functions, etc.) which reduces to a thermal ensemble (with respect to $H = Q[\partial_\eta]$) on the right Rindler wedge must have inverse temperature $\beta = 2\pi$.
Of course, the Minkowski vacuum is regular in the above sense, and it reduces to a thermal ensemble on the right Rindler wedge, so the periodicity trick tells us the temperature of this ensemble. Equally, if you are interested in some other space-time, with a patch that can be described by the Rindler coordinates, then the same argument will apply. As long as you know already that the reduced density matrix of this state will be thermal on the patch described by Rindler coordinates, and that the state is smooth on the Rindler horizon, then the periodicity trick will automatically give you the temperature.
