((x - y)^\mu p_\mu)}} d^4p\end{aligned}$$ where $a > 0$ is some parameter, and $\epsilon$ denotes the sign function. To compare, the two-point function of a KMS-state with constant inverse temperature vector $\boldsymbol{\beta}'$ is given by $$\begin{aligned}
 \label{2pKMS}
w_{\boldsymbol{\beta}'}(x,y) &= \frac{1}{(2 \pi)^{3}} \int_{\mathbb{R}^4}
     {\rm e}^{- i (x - y)^\mu p_\mu} \epsilon(p_0) \delta(p^\mu p_\mu) 
     \frac{1}{1 - {\rm e}^{- \boldsymbol{\beta}'{}^\mu p_\mu}} d^4p\,.\end{aligned}$$ Upon comparison, one can see that the inverse temperature vector field of $\omega_{\rm HB}$ has an inverse temperature vector field $$\boldsymbol{\beta}(x) = 2 a x\,, \quad x \in V_+\,.$$ Thus, the temperature diverges at the boundary of $V_+$, with the thermal rest frame tilting lightlike; moreover, the temperature decreases away from the boundary with increasing coordinate time $x^0$. This behaviour is sketched in Figure 1.\

![image](hotbang.eps)\
[**Figure 1: Sketch of temperature distribution of the hot bang state.**]{}

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It also worth mentioning that the hot bang state provides an example of a state where the thermal stress-energy tensor $\varepsilon_{\mu\nu}(x) = -(1/4) \langle \eth_{\mu \nu} : \Phi_0^2:(x) \rangle_{\rm HB}$ deviates from the expectation values of the full stress energy tensor $\langle {\sf T}_{\mu\nu}(x) \rangle_{\rm HB}$. In fact, as discussed in [@BOR], for the stress energy tensor of the massless linear scalar field 