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Register a new userSpacetime Entanglement Entropy of de Sitter and Black Hole Horizons
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We calculate Sorkins manifestly covariant entanglement entropy $mathcal{S}$ 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 $mathcal{L}^2$ norm in the static patch implies that $mathcal{S}$ is well defined for each mode. We find that $mathcal{S}$ for this mixed state is independent of the effective mass of the scalar field, and matches that of 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 $mathcal{S} propto A_{c}$, where $A_c$ is 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 $mathcal{L}^2$ 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 $mathcal{S}_{b,c}$, where $b,c$ denote the black hole and de Sitter cosmological horizons, respectively. As in the de Sitter calculation we see that $mathcal{S}_{b,c} propto A_{b,c}$ after taking a cut-off in the angular modes.
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de Sitter cosmological horizons are known to exhibit thermodynamic properties similar to black hole horizons. In this work we study causal set de Sitter horizons, using Sorkins spacetime entanglement entropy (SSEE) formula, for a conformally coupled quantum scalar field. We calculate the causal set SSEE for the Rindler-like wedge of a symmetric slab of de Sitter spacetime in $d=2,4$ spacetime dimensions using the Sorkin-Johnston vacuum state. We find that the SSEE obeys an area law when the spectrum of the Pauli-Jordan operator is appropriately truncated in both the de Sitter slab as well as its restriction to the Rindler-like wedge. Without this truncation, the SSEE satisfies a volume law. This is in agreement with Sorkin and Yazdis calculations for the causal set SSEE for nested causal diamonds in $mathbb{M}^2$, where they showed that an area law is obtained only after truncating the Pauli-Jordan spectrum. In this work we explore different truncation schemes with the criterion that the SSEE so obtained obeys an area law.
Based on the consideration that the black hole horizon and the cosmological horizon of Kerr-de Sitter black hole are not independent each other, we conjecture the total entropy of the system should have an extra term contributed from the correlations between the two horizons, except for the sum of the two horizon entropies. By employing globally effective first law and effective thermodynamic quantities, we obtain the corrected total entropy and find that the region of stable state for kerr-de Sitter is related to the angular velocity parameter $a$, i.e., the region of stable state becomes bigger as the rotating parameters $a$ is increases.
The basic equations of the thermodynamic system give the relationship between the internal energy, entropy and volume of two neighboring equilibrium states. By using the functional relationship between the state parameters in the basic equation, we give the differential equation satisfied by the entropy of spacetime. We can obtain the expression of the entropy by solving the differential equationy. This expression is the sum of entropy corresponding to the two event horizons and the interaction term. The interaction term is a function of the ratio of the locations of the black hole horizon and the cosmological horizon. The entropic force, which is strikingly similar to the Lennard-Jones force between particles, varies with the ratio of the two event horizons. The discovery of this phenomenon makes us realize that the entropic force between the two horizons may be one of the candidates to promote the expansion of the universe.
Using the analytic extension method, we study Hawking radiation of an $(n + 4)$-dimensional Schwarzschild-de Sitter black hole. Under the condition that the total energy is conserved, taking the reaction of the radiation of particles to the spacetime into consideration and considering the relation between the black hole event horizon and cosmological horizon, we obtain the radiation spectrum of de Sitter spacetime. This radiation spectrum is no longer a strictly pure thermal spectrum. It is related to the change of the Bekenstein-Hawking(B-H) entropy corresponding the black hole event horizon and cosmological horizon. The result satisfies the unitary principle. At the same time, we also testify that the entropy of de Sitter spacetime is the sum of the entropy of black hole event horizon and the one of cosmological horizon.
The fundamental equation of the thermodynamic system gives the relation between internal energy, entropy and volume of two adjacent equilibrium states. Taking higher dimensional charged Gauss-Bonnet black hole in de Sitter space as a thermodynamic system, the state parameters have to meet the fundamental equation of thermodynamics. We introduce the effective thermodynamic quantities to describe the black hole in de Sitter space. Considering that in the lukewarm case the temperature of the black hole horizon is equal to that of the cosmological horizon, the effective temperature of spacetime is the same, we conjecture that the effective temperature has the same value. In this way, we can obtain the entropy formula of spacetime by solving the differential equation. We find that the total entropy contain an extra terms besides the sum of the entropies of the two horizons. The corrected terms of the entropy is a function of horizon radius ratio, and is independent of the charge of the spacetime.
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