2.11 An expression in terms of the determinant of the Laplacian on the surface

Even though the entanglement entropy is determined by the geometry of the surface Σ, in general, this can be not only its intrinsic geometry but also how the surface is embedded in the larger spacetime. The embedding is determined by the extrinsic curvature. The curvature of the larger spacetime enters through the Gauss–Cadazzi relations. But in some particularly simple cases the entropy can be given a purely intrinsic interpretation. To see this for the case when Σ is a plane we note that the entropy (28View Equation) or (30View Equation) originates from the surface term in the trace of the heat kernel (27View Equation) (or (31View Equation)). To leading order in (1 − α), the surface term in the case of a massive scalar field is
(1 − α) ⋅ 1-⋅ TrK Σ(s) , 6
--A-(Σ)-- −m2s TrK Σ(s) = (4πs )d−22⋅ e
can be interpreted as the trace of the heat kernel of operator − Δ (Σ) + m2, where Δ (Σ) is the intrinsic Laplace operator defined on the (d − 2)-plane Σ. The determinant of the operator − Δ (Σ ) + m2 is determined by
2 ∫ ∞ ds lndet (− Δ (Σ ) + m ) = − 2 ---TrK Σ(s). 𝜖 s
Thus, we obtain an interesting expression for the entanglement entropy
1 S = − ---ln det(− Δ (Σ ) + m2 ) (32 ) 12
in terms of geometric objects defined intrinsically on the surface Σ. A similar expression in the case of an ultra-extreme black hole was obtained in [172Jump To The Next Citation Point] and for a generic black hole with horizon approximated by a plane was obtained in [89Jump To The Next Citation Point].
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