We revisit the definition of the temperature of a causal diamond for the case of a free massless scalar field. The stress is given to the intrinsic, direction-dependent character of this definition. Some important limits are also discussed.
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We restrict ourselves to the TT trajectories in the x0 − x1 plane. The explicit form for a more general trajectory will be more involved after a translation, but because the choice of the 1-direction is arbitrary, we do not loose any generality of the results.
Recall that we are considering only the TT trajectories in x0 − x1 plane. So the “corners” here are actually wedge-like regions for some neighbourhood in x2, x3 directions.
Here the TT trajectory that passes through z at some t, at t = 0 passes through x0(0) = 0 and r(0) = r.
