Mass and angular momentum

Any stationary, asymptotically-AdS4 black hole spacetime is equipped with at least two isometries, one associated with time translations and one associated with azimuthal rota-tions. We denote the corresponding Killing vectors that generate these isometries by K_(t) and K_(φ), respectively. There are conserved quantities associated with these isometries, namely the mass M and azimuthal angular momentum J, that play an important role in the thermodynamic properties of the black hole. In order to compute these quantities, we

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follow the formalism developed in [66] for computing conserved charges in asymptotically-AdS spacetimes via a Komar integral of the boundary stress-tensor. This method is the most natural in holographic settings, since it yields charges that are automatically free of divergences while also removing the ambiguities that can arise in other schemes, such as the Brown-York procedure [67].

Let Σ denote the conformal boundary of the spacetime at spatial infinity, where the unit normal to the boundary is n^µ, the induced boundary metric ishµν, the extrinsic curvature is Kµν (with K≡h^µνKµν), and the boundary Riemann tensor is R_µνρσ. We then denote by ∂Σ a constant time slice of the boundary. The conserved charge Q associated with a Killing vector K is computed by the Komar integral

Q[K] =^Z

∂Σ

d²x√

γ u^µK^ντµν, (5.15)

where γ is the induced metric on ∂Σ, u is the unit normal to ∂Σ, and τµν denotes the boundary stress-tensor, defined by

τµν ≡ √2

−h δL

δh^µν . (5.16)

Importantly, the boundary stress-tensor is computed by varying both the bulk action as well as the boundary counterterm action, since the counterterm action is required for a well-posed variational principle that ensures no derivatives of the metric fluctuation appear.

For the two-derivative minimal supergravity action and the Gauss-Bonnet action, the boundary stress-tensors are well-known in the literature (see e.g. [68, 69]), and we go through their computation extensively in appendix D. The results are:

τ_µν^(2∂) = 1 8πGN

Kµν−Khµν−LG_µν+ 2 Lhµν

, (5.17)

and

τ_µν^(GB)= 12J_µν−4Jhµν−8P_µρνσK^ρσ, (5.18) where G_µν ≡ R_µν − ¹₂hµνR is the boundary Einstein tensor, the boundary tensor J_µν is defined in (3.8), andP_µνρσ is the divergence-free part of the boundary Riemann tensor:

P_µνρσ ≡ R_µνρσ−2R_µ[ρh_σ]ν+ 2R_ν[ρh_σ]µ+Rh_µ[ρh_σ]ν . (5.19) With these boundary stress-tensors, we can now compute the mass M₀ and angular mo-mentumJ0 for any black hole in the two-derivative theory simply by utilizing the Komar integral (5.15) with the corresponding Killing vectors and the two-derivative boundary stress-tensor:

M₀ =^Z

∂Σ

d²x√

γ u^µK^ν_(t)τ_µν^(2∂), J0 =^Z

∂Σ

d²x√

γ u^µK^ν_(φ)τ_µν^(2∂) .

(5.20) For the Gauss-Bonnet boundary stress-tensor, on the other hand, the corresponding Ko-mar integrals vanish and thus the Gauss-Bonnet term yields no contribution to the mass

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and angular momentum. The particular sum of boundary tensors that show up in (5.18) conspire such that τµν^(GB) becomes very subleading in the radial coordinate, and so in the limit where the radial cutoff is sent to infinity the Komar integral (5.15) dies out:

Z

∂Σ

d²x√

γ u^µK^ν_(t)τ_µν^(GB)=^Z

∂Σ

d²x√

γ u^µK^ν_(φ)τ_µν^(GB) = 0. (5.21) Physically, this is simply a manifestation of the topological nature of the Gauss-Bonnet invariant in four dimensions, which forbids the Gauss-Bonnet term from affecting conserved charges [70]. In higher dimensions this is no longer the case, and the Gauss-Bonnet term can yield non-trivial corrections to thermodynamic properties of black holes.

We now have to determine the boundary stress-tensor for the W² component of the full HD action (5.4). This is in general a hard problem to tackle, as the variation of S_W2 with respect to the boundary metric is fairly complicated. However, since we are only interested in considering black holes that are solutions to the original two-derivative equations of motion, we can apply these equations to drastically simplify the variation. Additionally, the W² boundary counterterm S_W^CT2 is simply a linear combination of the two-derivative and Gauss-Bonnet countertermsS^CT_2∂ andS_GB^CT, and so their variations with respect to the boundary metric will be similarly related. Putting all of this together, we find that the W² boundary stress-tensor is simply given by a linear combination of τµν^(2∂) and τµν^(GB), i.e.

τ_µν^(W2)= 64πG_N

L² τ_µν^(2∂)+τ_µν^(GB) . (5.22) Again, we stress that this boundary stress-tensor will take a more complicated form when considering the four-derivative equations of motion as a whole, but when we restrict to only those solutions that satisfy the two-derivative equations of motion, it is forced to take the simpler form in (5.22).

Putting all this together, we find that the full boundary stress-tensor for our four-derivative supergravity theory is given by

τ_µν =1 +64πG_N(c₂−c₁) L²

τ_µν^(2∂)−c₁τ_µν^(GB) . (5.23) The mass M and angular momentum J in the four-derivative theory are then computed by inserting the boundary stress-tensor (5.23) into the Komar integral (5.15). Since the Gauss-Bonnet stress-tensor yields no contribution to the charges and the two-derivative stress-tensor in (5.23) is modified only by an overall rescaling, we find that

M =1 +64πG_N(c₂−c₁) L²

M0, J =1 +64πG_N(c₂−c₁) L²

J0 . (5.24) That is, the massM and angular momentumJ, as computed in the four-derivative theory, are related to the original massM₀ and angular momentumJ₀in the two-derivative theory by a constant rescaling. This relation holds for general stationary black hole solutions, so once the mass and angular momentum in the two-derivative theory are computed, we can immediately find their values in the four-derivative theory by simply using (5.24).

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The last point we would like to stress is that the W²counterterm is absolutely essential for deriving the results (5.24). Indeed, even though it evaluates to zero on-shell for all explicit solutions we have studied, its variation with respect to the metric is non-zero, which allows us to cancel bulk terms in the variation of S_W2 and end up with a well-defined boundary stress-tensor. The presence of this novel boundary term also explains the discrepancy between our results and those of [71], where it was argued that certain bulk four-derivative terms cannot affect the mass of the black hole. Our results therefore serve as an important reminder of the significance of finite boundary counterterms in holographic settings.