### 3.12 The statement on the renormalization of the entropy

As explained in Sections 3.9 and 3.10, the entanglement entropy is a UV divergent quantity. The other well-known quantity, which possesses UV divergences is the effective action. The standard way to handle the UV divergences in the action is to absorb them into a redefinition of the couplings, which appear in the gravitational action. In four dimensions the gravitational action should also include the terms quadratic in the Riemann curvature. The renormalization procedure is well studied and is described in textbooks (see for instance [22]). The idea now is that exactly the same procedure renormalizes the UV divergences in the entropy. In order to demonstrate this statement, consider a minimally-coupled scalar field. For simplicity suppose that the mass of the field vanishes. The bare (tree-level) gravitational action in four dimensions is the sum of the Einstein–Hilbert term and all possible combinations quadratic in the Riemann curvature,
where are the bare coupling constants in the gravitational action.

The UV divergences of the gravitational action are computed by the heat kernel method using the small expansion (69). For a minimal massless field ( in the scalar field equation) one finds

These divergences are removed by standard renormalization of the gravitational couplings in the bare gravitational action
where and are the renormalized couplings expressed in terms of the bare ones and the UV parameter ;

The tree-level entropy can be obtained by means of the same replica trick, considered in Sections 3.8, 3.9 and 3.10, upon introduction of the conical singularity with a small angle deficit , . The conical singularity at the horizon manifests itself in that a part of the Riemann tensor for such a manifold behaves as a distribution having support on the surface . Using formulas (56) – (59) one finds for the tree-level entropy

Thus, the Bekenstein–Hawking entropy is modified due to the presence of -terms in the action (112). It should be noted that Eq. (116) exactly coincides with the entropy computed by the Noether charge method of Wald [221, 144] (the relation between Wald’s method and the method of conical singularity is discussed in [140]).

The UV divergent part of the entanglement entropy of a black hole has already been calculated, see Eq. (82). For a minimal massless scalar, one has

The main point now is that the sum of the UV divergent part (117) of the entanglement entropy and the tree-level entropy (116)
takes again the tree-level form (116) if expressed in terms of the renormalized coupling constants defined in Eq. (115). Thus, the UV divergences in entanglement entropy can be handled by the standard renormalization of the gravitational couplings, so that no separate renormalization procedure for the entropy is required.

It should be noted that the proof of the renormalization statement is based on a nice property of the heat kernel coefficients  (68) on space with conical singularity. Namely, up to terms the exact coefficient on conical space is equal to the regular volume coefficient expressed in terms of the complete curvature, regular part plus a delta-like contribution, using relations (55)

The terms quadratic in are not well defined. However, these terms are proportional to and do not affect the entropy calculation. Thus, neglecting terms of order in the calculation of entropy, the renormalization of entropy (118) directly follows from the renormalization of the effective action (114).

That the leading divergence in the entropy can be handled by the standard renormalization of Newton’s constant has been suggested by Susskind and Uglum [213] and by Jacobson [141]. That one also has to renormalize the higher curvature couplings in the gravitational action in order to remove all divergences in the entropy of the Schwarzschild black hole was suggested by Solodukhin [196]. For a generic static black hole the renormalization statement was proven by Fursaev and Solodukhin in [112]. In a different approach based on ’t Hooft’s “brick-wall model” the renormalization was verified for the Reissner–Nordström black hole by Demers, Lafrance and Myers [62]. For the rotating black hole described by the Kerr–Newman metric the renormalization of the entropy was demonstrated by Mann and Solodukhin [170]. The non-equilibrium aspect (as defining the rate in a semiclassical decay of hot flat space by black hole nucleation) of the black hole entropy and the renormalization was discussed by Barbon and Emparan [12].