Spacetime entanglement entropy of de Sitter and black hole horizons

Abstract

We calculate Sorkin's manifestly covariant, spacetime entanglement entropy (SSEE) for a massive and massless minimally coupled free Gaussian scalar field for the de Sitter horizon and Schwarzschild de Sitter horizons, respectively, in d > 2. In de Sitter spacetime we restrict the Bunch-Davies vacuum in the conformal patch to the static patch to obtain a mixed state. The finiteness of the spatial L2 norm in the static patch implies that the SSEE is well defined for each mode. We find that for this mixed state it is independent of the effective mass of the scalar field and matches results obtained by Higuchi and Yamamoto, where, a spatial density matrix was used to calculate the horizon entanglement entropy. Using a cut-off in the angular modes we show that the SSEE is proportional to the area of the de Sitter cosmological horizon. Our analysis can be carried over to the black hole and cosmological horizon in Schwarzschild de Sitter spacetime, which also has finite spatial L2 norm in the static regions. Although the explicit form of the modes is not known in this case, we use appropriate boundary conditions for a massless minimally coupled scalar field, to find the mode-wise SSEE for both the black hole and de Sitter cosmological horizons. As in the de Sitter calculation we see that SSEE is proportional to the horizon area in each case after taking a cut-off in the angular modes.

Spacetime entanglement entropy: covariance and discreteness

“…The modes we use in the static patch are normal modes found by Higuchi [29] which are also L 2 orthogonal. Following the calculations of [30] we find in [19] that which then leads to the mode-wise contribution to SSEE…”

“…We use the past boundary conditions for the static and the Kruskal modes, since this defines the Klein Gordon norm on the limiting initial null surface H − b ∪ H − c in Region I. As shown in [19] we find that where κ b is the surface gravity of the black hole horizon.…”

“…In Schwarzschild de Sitter spacetime we compute the SSEE of a massless scalar field restricted to one of the static patches (region I of Fig. 3) [19]. The field is assumed to be in the Kruskal vacuum state which is defined across the black hole horizon.…”

“…Sorkin's spacetime entanglement entropy (SSEE) formulation for Gaussian free scalar fields provides an alternative measure for entanglement in terms of spacetime correlators [16]. In the continuum the SSEE has been calculated for different types of horizons, in d ≥ 2 and shown to satisfy the expected area law behaviour [17,18,19]. On a causal set the SSEE can be calculated using the SJ vacuum, given the discrete Green's function.…”

“…In Section 2 we review the construction of the SSEE [16]. In Section 3 we discuss results in the the continuum which show that the SSEE gives the expected area laws for d > 2 de Sitter and Schwarzschild-de Sitter horizons [19]. For the d = 2 causal diamond in the cylinder spacetime the Calabrese-Cardy form for the EE is recovered, but the coefficients are not universal [18].…”