Temperature at horizon in de Sitter spacetime

– It is found that there is no period in the imaginary Beltrami-time of the de Sitter spacetime with Beltrami metric and that the 'surface-gravity' in view of inertial observers in de Sitter spacetime is zero! They show that the horizon might be at zero temperature in de Sitter spacetime and that the thermal property of the horizon in the de Sitter spacetime with a static metric should be analogous to that of the Rindler horizon in Minkowski spacetime. Introduction. – Recent observations show that our universe is dark and in accelerated expansion [1, 2]. It implies that our universe is probably asymptotically de Sitter (dS) with a tiny positive cosmological constant Λ. It is well known that dS-universe is empty and of constant curvature without intrinsic singularity but there is a cosmological horizon surrounding an observer at the spatial origin. The horizon is naturally associated with a temperature [3–5] and an entropy proportional to the area of the horizon [4]. The standard approaches in the framework of general relativity (GR) treat the thermodynamic aspects of the dS-universe and black holes in the same way [4–7]. This leads to the puzzles on dS-universe [8]: why dS-universe is like a black hole thermodynamically and what is the statistical origin of the cosmological horizon entropy? A lot of proposals have been presented to this puzzle so far, but there is not any satisfactory explanation to them yet [9]. As is well known, in Minkowski spacetime the Poincaré-invariant field theories are defined in inertial frames and are at zero temperature. While in the Rindler metric which describes a special non-inertial frame in which each observer moves at a constant 4-acceleration, there is a horizon with Hawking temperature. Thus, the Hawking temperature observed by Rindler observers should be aroused by non-inertial motions in the flat spacetime rather than gravity. In fact, it has been pointed out that the temperature is usually regarded as a kinetic effect,