2.14 Arbitrary surface in curved spacetime: general structure of UV divergences

The definition of the entanglement entropy and the procedure for its calculation generalize to curved spacetime. The surface Σ can then be any smooth closed co-dimension two surface2, which divides the space into two sub-regions. In Section 3 we will consider in detail the case where this surface is a black-hole horizon. Before proceeding to the black-hole case we would like to specify the general structure of UV divergent terms in the entanglement entropy. In d-dimensional curved spacetime, entanglement entropy is presented in the form of a Laurent series with respect to the UV cutoff 𝜖 (for d = 4 see [204Jump To The Next Citation Point])
s s s S = -dd−−22-+ -d−d−-44 + ..+ -dd−−22−−22nn-+ ..+ s0ln 𝜖 + s(g), (48 ) 𝜖 𝜖 𝜖
where sd−2 is proportional to the area of the surface Σ. All other terms in the expansion (48View Equation) can be presented as integrals over Σ of local quantities constructed in terms of the Riemann curvature of the spacetime and the extrinsic curvature of the surface Σ. Of course, the intrinsic curvature of the surface Σ can be expressed in terms of ℛ and k using the Gauss–Codazzi equations. Since nothing should depend on the direction of vectors normal to Σ, the integrands in expansion (48View Equation) should be even powers of extrinsic curvature. The general form of the sd−2−2n term can be symbolically presented in the form
∫ s = ∑ ℛl k2p , (49 ) d− 2− 2n Σ l+p=n
where ℛ stands for components of the Riemann tensor and their projections onto the sub-space orthogonal to Σ and k labels the components of the extrinsic curvature. Thus, since the integrands are even in derivatives, only terms 𝜖d− 2n− 2, n = 0,1,2,... appear in Eq. (48View Equation). If d is even, then there also may appear a logarithmic term s0. The term s(g) in Eq. (48View Equation) is a UV finite term, which may also depend on the geometry of the surface Σ, as well as on the geometry of the spacetime itself.


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