Abstract
We discuss the field quantisation of a free massive Dirac fermion in the two causally disconnected static patches of the de Sitter spacetime, by using mode functions that are normalisable on the cosmological event horizon. Using this, we compute the entanglement entropy of the vacuum state corresponding to these two regions, for a given fermionic mode. Further extensions of this result to more general static spherically symmetric and stationary axisymmetric spacetimes are discussed. For the stationary axisymmetric Kerr-de Sitter spacetime in particular, the variations of the entanglement entropy with respect to various eigenvalues and spacetime parameters are depicted numerically. We also comment on such variations when instead we consider the non-extremal black hole event horizon of the same spacetime.
Highlights
The de Sitter spacetime is the simplest solution of the Einstein equation with a positive cosmological constant, Λ
Even though the exact nature/form of the dark energy is yet far from being well understood, it is reasonable to expect that the de Sitter spacetime would qualitatively model at least some salient features of any cosmological spacetime undergoing accelerated expansion
If we consider an “in” vacuum state in the cosmological de Sitter spacetime, due to the accelerated expansion, the state may evolve in the future to a different or “out” vacuum state, indicating particle pair production
Summary
The de Sitter spacetime is the simplest solution of the Einstein equation with a positive cosmological constant, Λ. If we consider an “in” vacuum state in the cosmological de Sitter spacetime, due to the accelerated expansion, the state may evolve in the future to a different or “out” vacuum state, indicating particle pair production Such pairs turn out to be entangled. Such similarity follows from the universal Rindler-like behavior of the t − r part of any (nonextremal) near–Killing horizon metric and the subsequent simplification of the mode functions Such universality and simplification allow us to extend our result further to (a) the de Sitter horizon of a general static and spherically symmetric spacetime, such as the Schwarzschild-de Sitter, Sec. IV, and to (b) stationary axisymmetric spacetime such as the Kerr-de Sitter, Sec. V. We shall work with the mostly negative signature of the metric in 3 þ 1 dimensions and will set c 1⁄4 G 1⁄4 ħ 1⁄4 1 throughout
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