Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in $d\ge 5$ dimensions

TL;DR

This paper demonstrates that in five or more dimensions, quantum corrections can lead to violations of the Quantum Focusing Conjecture and covariant entropy bounds, challenging their universality in higher-dimensional gravity theories.

Contribution

It shows that Einstein-Hilbert-Gauss-Bonnet gravity in higher dimensions can violate the QFC and entropy bounds, revealing limitations of these conjectures beyond four dimensions.

Findings

Violations occur in $d extgreater 4$ dimensions with non-zero Gauss-Bonnet coupling.

Violations persist with arbitrary higher curvature terms and generic massive quantum fields.

The validity of QFC and entropy bounds in $d extless 4$ remains uncertain.

Abstract

We study the Quantum Focussing Conjecture (QFC) in curved spacetime. Noting that quantum corrections from integrating out massive fields generally induce a Gauss-Bonnet term, we study Einstein-Hilbert-Gauss-Bonnet gravity and show for $d\ge 5$ spacetime dimensions that weakly-curved solutions can violate the associated QFC for either sign of the Gauss-Bonnet coupling. The nature of the violation shows that -- so long as the Gauss-Bonnet coupling is non-zero -- it will continue to arise for local effective actions containing arbitrary further higher curvature terms, and when gravity is coupled to generic $d\ge 5$ theories of massive quantum fields. The argument also implies violations of a recently-conjectured form of the generalized covariant entropy bound. The possible validity of the QFC and covariant entropy bound in $d\le 4$ spacetime dimensions remains open.

Equations

$$\theta [\Sigma ,y]=\frac{4G}{\sqrt{\tilde{h}}}\frac{\delta S_{\mathrm{BH}}}{\delta \Sigma (y)},$$

$$S_{\text{gen}}=S_{\text{grav}}+S_{\text{out}}.$$

$$\frac{1}{\sqrt{\tilde{h}(y)}}\frac{\delta }{\delta \Sigma \left(y_2\right)}\Theta \left[\Sigma ;y_1\right]\le 0.$$

$$I=\frac{1}{16\pi G}\int d^{d}x\sqrt{-g}R+\gamma \int d^{d}x\sqrt{-g}\left(R_{abcd}{R}^{abcd}-4R_{ab}{R}^{ab}+R^2\right).$$

$$C_{abcd}=O\left(ϵ\right).$$

$$R_{ab}=\frac{16\pi G\gamma }{d-2}{C}_{cdef}{C}^{cdef}{g}_{ab}-32\pi G\gamma C_{acde}{C}_b^{cde}+O\left({ϵ}^3\right).$$

$$K_{ab}^{(k)}{|}_p:=({\tilde{h}}_a^c{\tilde{h}}_b^d{\nabla }_c{k}_d){|}_p=0,$$

$$g_{ab}={\tilde{h}}_{ab}-k_a{l}_b-l_a{k}_b,$$

$$\dot{\theta }{|}_p$$

$$=32\pi G\gamma \left(C_{k\alpha \beta \gamma }{C}_k^{\alpha \beta \gamma }-2C_{k\beta \alpha }^{\beta }{C}_{k\gamma }^{\alpha \gamma }-4C_{k\alpha k\beta }{C}_{kl}^{\alpha \beta }\right)+O\left({ϵ}^3\right).$$

$$S_{\text{GB}}=-8\pi \gamma {\int }_{\Sigma }{d}^{d-2}y\sqrt{\tilde{h}}\tilde{R},$$

$${\delta }_X{S}_{\text{GB}}$$

$$=-16\pi \gamma {\int }_{\Sigma }{d}^{d-2}y\sqrt{\tilde{h}}\left({\tilde{R}}^{ab}-\frac{1}{2}\tilde{R}{\tilde{h}}^{ab}\right)K_{ab}^{(X)}$$

$$=-16\pi \gamma \sqrt{\tilde{h}}\left({\tilde{R}}^{ab}-\frac{1}{2}\tilde{R}{\tilde{h}}^{ab}\right)K_{ab}^{(k)}.$$

$${\delta }_Z\left(\frac{1}{\sqrt{\tilde{h}}}{\delta }_X{S}_{\text{GB}}\right)=-16\pi \gamma \left({\tilde{R}}^{ab}-\frac{1}{2}\tilde{R}{\tilde{h}}^{ab}\right)({\delta }_Z{K}_{ab}^{(k)}){|}_p.$$

$$({\delta }_Z{K}_{ab}^{(k)}){|}_p=(-{\tilde{h}}_a^c{\tilde{h}}_b^d{Z}^e{k}^f{R}_{ecfd}){|}_p$$

$$S_{\text{GB}}^{\prime \prime }=16\pi \gamma \left({\tilde{R}}^{ab}-\frac{1}{2}\tilde{R}{\tilde{h}}^{ab}\right)R_{kakb}.$$

$$\frac{\delta }{\delta \Sigma \left(y_Z\right)}\Theta \left[\Sigma ;y_p\right]=\sqrt{\tilde{h}}Q\delta (y_p-y_Z)$$

$$Q=\dot{\theta }+4G{S}_{\text{GB}}^{\prime \prime }.$$

$$({\tilde{R}}_{abcd}){|}_p=({\tilde{h}}_a^e{\tilde{h}}_b^f{\tilde{h}}_c^g{\tilde{h}}_d^h{R}_{efgh}){|}_p,$$

$$S_{\text{GB}}^{\prime \prime }=16\pi \gamma \left(R_{cedf}{\tilde{h}}^{cd}{\tilde{h}}^{ae}{\tilde{h}}^{bf}-\frac{1}{2}{R}_{cedf}{\tilde{h}}^{cd}{\tilde{h}}^{ef}{\tilde{h}}^{ab}\right)R_{kakb}.$$

$$S_{\text{GB}}^{\prime \prime }$$

$$=32\pi \gamma C_{k\alpha k\beta }{C}_{kl}^{\alpha \beta }+O\left({ϵ}^3\right),$$

$$Q=32\pi G\gamma \left(C_{k\alpha \beta \gamma }{C}_k^{\alpha \beta \gamma }-2C_{k\beta \alpha }^{\beta }{C}_{k\gamma }^{\alpha \gamma }\right)+O\left({ϵ}^3\right).$$

$$C_{k\alpha \beta \gamma }={\tilde{h}}_{\alpha \beta }{v}_{\gamma }-{\tilde{h}}_{\alpha \gamma }{v}_{\beta }+{ϵ}_{\beta \gamma }^{\delta }{n}_{\delta \alpha }\text{, for }d=5,$$

$$Q=64\pi G\gamma \left(n_{\alpha \beta }{n}^{\alpha \beta }-2v_{\gamma }{v}^{\gamma }\right)+O\left({ϵ}^3\right)\text{, for }d=5.$$

$$\Delta S\le \Delta A/4G,$$

Full text

1 Introduction

The gravitational focussing theorem plays a key role in the modern understanding of General Relativity. This key results states (see e.g. Wald:1984rg ) that the expansion of null congruences cannot increase toward the future in any solution to Einstein-Hilbert gravity sourced by matter satisfying the Null Energy Condition (NEC). It leads to the second law of black hole thermodynamics Hawking:1971tu , singularity theorems Penrose:1964wq ; Hawking:1973uf , the chronology protection theorem Hawking:1991nk , topological censorship Friedman:1993ty , and other fundamental results. It also guarantees essential properties of holographic entanglement entropy Wall:2012uf ; Headrick:2014cta in the context of gauge/gravity duality.

However, the null energy condition is known to be violated by quantum effects Epstein:1965zza . This then raises the question of whether quantum corrections might enable fundamentally new and perhaps pathological gravitational phenomena. Indeed, it was recently established that traversable wormholes can be constructed in this way Gao:2016bin . On the other hand, the conjectured Generalized Second Law of thermodynamics (GSL) would both limit the utility of traversable wormholes and prohibit even more troubling exotic physics Wall:2013uza .

Motivated in part by the GSL, and also in part by the covariant entropy conjecture Bousso:1999xy , it was suggested in Bousso:2015mna that a generalization of the focussing theorem might continue to hold at the quantum level. Known as the Quantum Focussing Conjecture (QFC), it would imply both the GSL (for any causal horizon) and a form Bousso:2015mna of the covariant entropy bound of Bousso:1999xy related to the version discussed by Strominger and Thompson Strominger:2003br .

The QFC is formulated by noting that the expansion $\theta$ of any null congruence can be expressed as a first functional derivative of the area of cuts of the congruence, and that Einstein-Hilbert gravity associates a Bekenstein-Hawking entropy $S_{\rm BH}=A/4G$ with many surfaces of area $A$. In particular, given a region ${\cal R}$ with boundary $\Sigma=\partial{\cal R}$ in some Cauchy surface, and also given a null congruence $N$ orthogonal to $\Sigma$, we have

[TABLE]

where $y$ labels the space of null generators, $\delta\Sigma(y)$ is an infinitesimal displacement of the surface along the null generator $y$, and $\tilde{h}$ denotes the determinant of the transverse metric in the $y$-coordinate system on the null congruence $N$. For semi-classical gravity (and in particular where the metric itself may be treated classically), ref. Bousso:2015mna then defines the generalized expansion $\Theta[\sigma,y]$ by replacing $S_{\rm BH}=A/4G$ in (1) with the generalized entropy functional

[TABLE]

Here $S_{\text{grav}}$ is an appropriate gravitational entropy functional (say, from Dong:2013qoa ; Miao:2014nxa ; Dong:2015zba ; Wall:2015raa , which coincides with that of Jacobson:1993xs for the case studied here) and $S_{\text{out}}$ is a von Neumann entropy for quantum fields outside the null congruence.111$S_{\text{out}}$ presumably includes an appropriate set of boundary terms for gauge fields as in e.g. will_lattice ; will14 ; Ghosh:2015iwa ; Aoki:2015bsa ; Donnelly:2014fua ; Donnelly:2015hxa . Finally, the statement of the QFC is simply that $\Theta$ is semi-classically non-decreasing as we push the surface $\Sigma$ toward the future or, in other words, that a corresponding second derivative of $S_{\text{gen}}$ is negative or zero:

[TABLE]

While (3) is divergent for $y_{1}=y_{2}$, and in particular the contribution of the Einstein-Hilbert term to (3) is $\dot{\theta}\delta(y_{1}-y_{2})$ where $\dot{\theta}=k^{a}\nabla_{a}\theta$, the quantity (3) remains meaningful when treated as a distribution.

As evidence for the QFC, one may recall Bousso:2015mna that in Einstein-Hilbert gravity, taking a weakly-gravitating ($G\to 0$) limit implies quantum fields satisfy a so-called Quantum Null Energy Condition (QNEC) generalizing the classical NEC, and that this QNEC has now been established in a variety of contexts Bousso:2015wca ; Koeller:2015qmn . In such cases, an associated QFC follows immediately at first order in the coupling $G$ of such theories to Einstein-Hilbert gravity.

However, we argue here that for $d\geq 5$ spacetime dimensions the QFC generally fails. To do so, we recall that integrating out massive fields typically induces a Gauss-Bonnet term in the gravitational effective action; see e.g. Shapiro:2008sf . Classical Einstein-Hilbert-Gauss-Bonnet gravity is analyzed in section 2, and is shown to violate the QFC at weak curvature for $d\geq 5$.222Causality violations implying pathologies for non-stringy theories with large Gauss-Bonnet couplings were found in Camanho:2014apa . By contrast, we emphasize that the QFC violation found in this paper is present for the less restrictive class of theories containing even a small effective field theory Gauss-Bonnet term. The form of this violation shows that similar issues arise at the quantum level, and also in the presence of arbitrary higher derivative terms controlled by a single length scale so long as the coefficient of the Gauss-Bonnet term is non-zero. The QFC is thus violated in generic $d\geq 5$ theories of semi-classical gravity coupled to massive quantum fields, and presumably in the presence of massless quantum fields as well. Our example also leads in section 3 to violations of the generalized covariant entropy bound (also called the quantum Bousso bound) conjectured in Bousso:2015mna .333This conjecture is closely related to the Strominger-Thompson proposal Strominger:2003br . We close in section 4 with further discussion emphasizing future directions and the possibility that a reformulated QFC and quantum Bousso bound may nevertheless hold.

2 Violating the QFC in Gauss-Bonnet Gravity

Consider the the Einstein-Hilbert-Gauss-Bonnet action

[TABLE]

As noted above, we will first treat this theory classically and identify violations of the associated QFC (3). We will then note that explicit quantum corrections are sub-leading in a long-wavelength expansion so our classical violation extends directly to the quantum level.

We work in the weak curvature limit, taking the Weyl tensor to be first order in some small quantity $\epsilon$:

[TABLE]

In this limit, iteratively solving the equation of motion yields

[TABLE]

Note that since the right-hand side is non-zero only due to contributions to the equations of motion from the variation of the Gauss-Bonnet term, the Gauss-Bonnet theorem requires it to vanish for $d=4$. It also vanishes for $d<4$ where the Weyl tensor is identically zero.

Now consider a null hypersurface $N$ generated by a hypersurface-orthognal null normal vector field $k^{a}$. For simplicity, we choose both the expansion $\theta$ and the shear $\sigma_{ab}$ of $N$ to vanish at some point $p$, or equivalently that the extrinsic curvature along $k$ vanishes there for any cut $\Sigma$ of $N$ through $p$; i.e.,

[TABLE]

where $\tilde{h}_{a}^{~{}c}$ is the projector onto $\Sigma$. As in e.g. Cao:2010vj , we will use the notation $K^{(X)}_{ab}:=\tilde{h}_{a}^{~{}c}\tilde{h}_{b}^{~{}d}\nabla_{c}X_{d}$ below for any vector field $X_{d}$ orthogonal to $\Sigma$. Note that (7) does not restrict the spacetime at $p$ in any way; given any $p$ in any spacetime, we may choose $\Sigma$ and then define the orthogonal null congruence $N$ so that the above conditions are satisfied. We use indices $a$, $b$, $c$, $d$, …to denote coordinates in spacetime and indices $\alpha$, $\beta$, $\gamma$, $\delta$, …to denote coordinates on $\Sigma$.

It is convenient to also introduce an auxiliary null vector field $l^{a}$ orthogonal to $\Sigma$ and satisfying $g_{ab}k^{a}l^{b}=-1$. The spacetime metric can then be written

[TABLE]

where the transverse part $\tilde{h}_{ab}=\tilde{h}_{a}^{~{}c}g_{cb}$ is the induced metric on $\Sigma$. We will reserve $k$ and $l$ “indices” to denote contractions with $k^{a}$ and $l^{a}$, as in e.g. $A_{kl}:=A_{ab}k^{a}l^{b}$. Substituting (8) into equation (6) and noticing that $C_{klk\alpha}=-C_{k\beta\alpha}^{~{}~{}~{}\beta}$ for all $d$, the Raychaudhuri equation $\dot{\theta}=-\frac{\theta^{2}}{d-2}-\sigma^{ab}\sigma_{ab}-R_{ab}k^{a}k^{b}$ for hypersurface-orthogonal null congruences satisfying equation (7) yields

[TABLE]

As noted above, (9) vanishes for $d=4$. One may see this explicitly by using the $d=4$ identity $C_{k\alpha\beta\gamma}C_{k}^{~{}\alpha\beta\gamma}=2C_{k\beta\alpha}^{~{}~{}~{}\beta}C_{k\gamma}^{~{}~{}\alpha\gamma}$ from Coley:2009is so that the first two terms cancel in (9). To deal with the final term we again use the $d=4$ results from Coley:2009is to write $C_{k\alpha l\beta}$ as $C_{k\alpha l\beta}=-\frac{1}{4}A\tilde{h}_{\alpha\beta}+\frac{1}{2}B\epsilon_{\alpha\beta}$ where $\epsilon_{\alpha\beta}$ is the area element of $\Sigma$ and $A$ and $B$ are independent scalars; in particular, there is no traceless symmetric term. The final term in (9) then vanishes since $C_{k\alpha k~{}}^{~{}~{}~{}\alpha}=0=C_{k\alpha k\beta}\epsilon^{\alpha\beta}$ identically for all $d$.

To study the QFC, recall Dong:2013qoa ; Camps:2013zua that the entropy functional associated with the Gauss-Bonnet term is

[TABLE]

where $\tilde{R}$ is the scalar curvature of the induced metric $\tilde{h}_{\alpha\beta}$. Let us introduce a deformation vector field $X^{a}=fk^{a}$ on $N$, where $f$ is a scalar function of the null generators $y$. Taking $f=\delta(y-y_{p})$, when $\Sigma$ is is deformed along $X^{a}$ the first derivative of entropy (10) is

[TABLE]

Here, to obtain the second line, we used $\delta_{X}\tilde{h}_{ab}=2K^{(X)}_{ab}$ (i.e. equation (3.10) of Cao:2010vj ).

We now introduce another vector field $Z=\delta(y-y_{Z})k^{a}$. Recalling that $K^{(k)}_{ab}$ vanishes at $p$, we find the second derivative

[TABLE]

Since $K^{(k)}_{ab}|_{p}=0$ and $Z^{a}=\delta(y-y_{Z})k^{a}$, the derivative of $K^{(k)}_{ab}$ at $p$ takes the simple form Cao:2010vj

[TABLE]

and (12) becomes $\delta_{Z}\left(\frac{1}{\sqrt{\tilde{h}}}\delta_{X}S_{\text{GB}}\right)=\delta(y_{p}-y_{Z})S^{\prime\prime}_{\text{GB}}$ for

[TABLE]

Since we treat the theory classically, we save for the end of this section consideration of any explicit $S_{\rm out}$ term in equation (2) associated with the entropy of gravitons and thus find

[TABLE]

for

[TABLE]

Since $K^{(k)}_{ab}|_{p}=0$, the Gauss equation (i.e. equation (2.14) of Cao:2010vj ) at $p$ is simply

[TABLE]

and expression (14) becomes

[TABLE]

In the weak curvature limit, we may use (5) and (6) to further write

[TABLE]

where in the last step we have used $\tilde{h}^{ab}C_{kakb}=C_{kkkl}+C_{klkk}$ which vanishes since the Weyl tensor is anti-symmetric in pairs of indices ($C_{abcd}=-C_{bacd}=-C_{abdc}$). Combining (9) and (19) with the definition (16) yields

[TABLE]

As with (9), expression (20) vanishes for $d=4$. To show that it generally does not vanish for $d=5$, we use further results from Coley:2009is to write it in terms of independent components of the Weyl tensor; the Weyl tensor at a point is constrained by its symmetries, tracelessness, and the algebraic Bianchi identity. The block $C_{k\alpha\beta\gamma}$, which has boost weight $-1$, can be written in terms of $8$ independent components as

[TABLE]

where $\epsilon_{\alpha\beta\gamma}$ is the area element of $\Sigma$, $v_{\gamma}$ is a vector containing $3$ independent components and $n_{\delta\alpha}$ is a traceless symmetric matrix containing $5$ independent components. Thus,

[TABLE]

Furthermore, for $d>5$ we may again use Coley:2009is to take the block $C_{k\alpha\beta\gamma}$ to be of the form (21), although (21) is no longer the most general form for $C_{k\alpha\beta\gamma}$ and of course the number of components of each object above increases with the spacetime dimension $d$.

It is clear from (22) that (20) is generally non-zero for $d\geq 5$. Furthermore, while the QFC requires $Q$ to be non-positive, for $\gamma>0$ it can be made positive by setting $v_{\gamma}=0$ and taking $n_{\alpha\beta}\neq 0$, and for $\gamma<0$ we can make $Q$ positive by taking $n_{\alpha\beta}=0$ with $v_{\gamma}\neq 0$.

Violations of the QFC thus occur for either sign of the Gauss-Bonnet coupling $\gamma$ and the QFC generally fails for classical $d\geq 5$ Einstein-Hilbert-Gauss-Bonnet gravity. We may immediately extend this result to the quantum level by noting that graviton contributions to the $S_{\rm out}$ term of equation (2) are of order $G$ while our violation above is of order $G\gamma$. The key point here is that $\gamma$ has dimensions $(\textit{Length})^{-(d-4)}$ so that the $G\gamma$ term is more important at large length scales than the $G$ term in $S_{\text{out}}$. In other words, the classical contributions to (2) will dominate in the long-distance limit.

Let us now consider more general (perhaps, effective) theories of gravity with higher derivative terms. First, it is trivial to add a cosmological constant $\Lambda$ to the action (4). Noticing that $C_{k\alpha k}^{~{}~{}~{}\alpha}=0$ identically for all $d$, one finds no change to equation (19). Next, recall that at the four-derivative level, up to total derivatives there are only two further independent terms that we may add to the action, and we may choose to write both in terms of the square of the Ricci tensor (so that they do not depend on the Weyl tensor). Thus Ricci-flat metrics continue to solve the theory with $\gamma=0$, and there continue to be solutions of the form (6) in the presence of such terms, and in such cases we again find (20) (up to additional corrections that are also of order $\epsilon^{2}$ but involve additional derivatives and so remains smaller in the long-distance limit). Finally, so long as they are controlled by a common length scale, in a long-distance expansion any terms in the action with more than four derivatives can be ignored relative to those already discussed so that (6) continues to hold in that regime.

The key point, however, is the associated implication for generic quantum theories of massive fields when coupled to semi-classical gravity. Since integrating out massive fields gives an effective action of the above type, so long as the resulting Gauss-Bonnet coefficient444The final Gauss-Bonnet coefficient is of course formally the sum of the Gauss-Bonnet coefficient in the gravitational action and the coefficient induced by integrating out the matter. For $d\geq 5$ the latter is generally divergent, so the former must be as well if the effective action is to be finite. In this sense, as usual, there is generally no meaning to attempting to couple the massive field theory to Einstein-Hilbert gravity alone. is non-zero the theory will violate the associated QFC.

3 Violating the Generalized Covariant Entropy Bound

Bousso’s original covariant entropy bound Bousso:1999xy involved the concept of “entropy flux through a non-expanding null surface” and conjectured this to be bounded by $(\frac{1}{4G}$ times) the area of the largest cut. There has been much discussion of how this concept might be properly defined, with one seemingly-natural choice involving entropy defined directly on the null surface. This version was proven for free and interacting theories in the $G\to 0$ limit from the monotonicity property of the relative entropy Bousso:2014sda ; Bousso:2014uxa . Alternatively, Strominger and Thompson Strominger:2003br suggested focussing on the case where any cut of the null surface $N$ is closed and bounds a spacelike surface. One may then discuss the von Neumann entropy $S_{vN}$ of the region enclosed, and replace the “flux of entropy across $N$” with the change in $S_{vN}$ between the initial and final surfaces.

As noted in Bousso:2015mna , this choice gives rise to a putative (generalized) covariant entropy bound which is intrinsically finite and does not require renormalization. The conjecture of Bousso:2015mna states that if some set of null generators has non-positive quantum expansion ($\Theta\leq 0$) on some cut $C_{\text{initial}}$ of $N$, then any cut $C_{\text{final}}$ obtained by moving $C_{\text{initial}}$ to the future along these generators will have smaller or equal generalized entropy $S_{\text{gen}}$ so long as no caustic lies between $C_{\text{initial}}$ and $C_{\text{final}}$. The non-increase of $S_{\text{gen}}$ is equivalent to the claim

[TABLE]

which is a generalized covariant entropy bound of the form first discussed in Flanagan:1999jp . Note, however, that the condition $\Theta|_{C_{\text{initial}}}\leq 0$ under which this was conjectured in Bousso:2015mna differs from the assumption used in Bousso:1999xy ; Flanagan:1999jp which requires the classical expansion $\theta$ to be non-positive on all intermediate cuts. Furthermore, equation (23) follows directly from the QFC in cases where the latter is valid Bousso:2015mna .

However, it turns out the QFC violation constructed above is also a counterexample to the generalized covariant entropy bound (i.e. the quantum Bousso bound) of Bousso:2015mna . The key point is that the Gauss-Bonnet contribution (11) to the the quantum expansion vanishes at $p$ since $K_{ab}^{(X)}=0$. But since $\theta|_{p}=0$ as well, the full quantum expansion $\Theta$ also vanishes at $p$.

From here we need only note that we can then achieve $\Theta\leq 0$ near $p$ on $C_{\text{initial}}$ by taking the classical expansion $\theta$ sufficiently negative near $p$; i.e., by simply choosing $C_{\text{initial}}$ to have large enough extrinsic curvature of the appropriate sign. We then find that later cuts $C_{\text{final}}$ differing from $C_{\text{initial}}$ only very near $p$ and by small affine parameter distance along the QFC-violating generators must have larger generalized entropy $S_{\text{gen}}$, violating the conjecture of Bousso:2015mna . Indeed, in the appropriate limit the increase of $S_{\text{gen}}$ is determined by $\left(\pounds_{k}\Theta\right)|_{p}>0$.

4 Discussion

Using an explicit calculation for classical Einstein-Hilbert-Gauss-Bonnet gravity, we argued that the QFC of Bousso:2015mna is violated in generic $d\geq 5$ theories of gravity coupled to massive quantum fields. The key point is that integrating out the massive fields generically induces a Gauss-Bonnet term which, at least for a certain class of solutions, dominates in the long-distance limit. There we may use the explicit Einstein-Hilbert-Gauss-Bonnet calculation of section 2. We expect similar violations to continue to arise when massless quantum fields are included as well. Our construction also provides a counterexample to the generalized covariant entropy bound (i.e. the quantum Bousso bound) conjectured in Bousso:2015mna . It remains an open question whether the QFC and covariant entropy bound could hold for $d\leq 4$, and it would be interesting to investigate the affect of Ricci-squared terms in this context. As mentioned in the introduction, the QFC is closely related to the Quantum Null Energy Condition (QNEC). Indeed, when a matter theory satisfying the Quantum Null Energy Condition is coupled to Einstein-Hilbert gravity, the QFC will hold at least to first order in the gravitational coupling $G$. The reader may thus ask whether our results are in tension with the QNEC proofs in Bousso:2015wca and Koeller:2015qmn . The answer is no, as those results prove the QNEC only for congruences $N$ through $p$ that form bifurcate Killing horizons at $G=0$. And on a bifurcate Killing horizon components of the Weyl tensor with non-zero boost weight must vanish. This would then force $C_{k\alpha\beta\gamma}=0$ and thus $Q=0$ in (20), reproducing the expected result that the QFC hold at first order in $G\gamma$ for such cases.555Indeed, a result of Kolekar:2012tq shows that the QFC holds for any Lovelock theory of gravity (a class which includes the Einstein-Hilbert-Gauss-Bonnet gravity) when evaluated at first order in $G$ about a Killing horizon. This result was then generalized in Sarkar:2013swa and extended to arbitrary higher-derivative theories of gravity in Wall:2015raa .

Conversely, taking the limit $G\gamma\rightarrow 0$ of our results shows that for $d\geq 5$ the renormalized QNEC must generally fail666As will be discussed in more detail in ToAppear , the QNEC may still hold in some sense for appropriate bare quantities. But finite renormalized quantities cannot satisfy a QNEC-like bound. for surfaces $\Sigma$ defining null congruences $N$ that are only locally stationary at $p$; i.e., which satisfy $\theta=\sigma_{ab}=R_{ab}k^{a}k^{b}=0$ in the background spacetime. However, one may ask if the QNEC can hold at locally stationary points of null congruences for $d<5$ or where further conditions are satisfied. The forthcoming work ToAppear will provide results of this kind, including a proof for $d\leq 3$ holographic theories at locally stationary points.

It is natural to ask if our QFC violation also provides a perturbative counterexample to the GSL. While Einstein-Hilbert-Gauss-Bonnet gravity is known to violate the GSL at the non-perturbative level Jacobson:1993xs ; Liko:2007vi ; Sarkar:2010xp , these are of lesser interest as higher derivative theories of gravity are expected Camanho:2014apa to approximate UV-complete theories only when treated perturbatively as an effective field theory valid at lengths longer than some cutoff scale $\ell_{c}$. And indeed, as in section 3, one can certainly find cases where the generalized entropy inside the horizon increases and thus that outside decreases. But the GSL is naturally conjectured to hold at most for causal horizons (see e.g. Jacobson:2003wv , Wall:2011hj ), and determining whether a given null $N$ is a causal horizon requires understanding the very far future. Analyzing the constraints on $N$, thus requires going well beyond the local approximations used here, and thus beyond the scope of this work, though see Wall:2015raa ; Bhattacharjee:2015qaa ; Bhattacharyya:2016xfs for further work on the GSL for higher derivative gravity and more thorough reviews.

Finally, one may ask if some version of the QFC or quantum Bousso bound might yet be salvaged for general $d\geq 5$ theories. In particular, we recall again that higher derivative gravity should be treated as an effective field theory with a cutoff $\ell_{c}$. But the QFC, and in particular our construction of a counterexample, requires the choice of a null congruence $N$ that is taken to be arbitrarily well localized in the transverse directions. Furthermore, since the Gauss-Bonnet term should be treated as perturbatively small, correspondingly small changes in $N$ can make $\theta,\sigma_{ab}$ non zero at $p$ so that $\dot{\theta}=-\frac{\theta^{2}}{d-2}-\sigma^{ab}\sigma_{ab}-R_{ab}k^{a}k^{b}$ becomes sufficiently negative at $p$ that $Q<0$ for the new surface. In other words, perturbatively close to any compact QFC-violating null congruence $N$ lies a QFC-respecting null congruence $N^{\prime}$. If this can be interpreted as a distinction finer than the cutoff scale $\ell_{c}$, there is room for the formulation of an effective QFC valid only at larger scales.777We thank Aron Wall for this suggestion. But such an interpretation is not immediately clear as the above mentioned deformation from $N$ to $N^{\prime}$ involves adding extrinsic curvature of a particular sign; it is not just a transverse smearing of the surface. And while it is attractive from many perspectives to conjecture that a QFC-like inequality may hold in an appropriately cutoff sense, both the form that this effective QFC might take and how in practice it would be used to restrict possible pathologies of NEC-violating spacetimes remain open questions for future investigation.

Note added in v2.

After the appearance of our paper on the arXiv, it was pointed out in Leichenauer:2017bmc that the violation described above is removed by restricting the QFC to apply only to variations of the entropy defined by surfaces that are smooth on the scale set by $G\gamma$, and which is presumably associated with the cut-off that defines the effective theory. This emphasizes the importance of studying the effect of $R_{ab}R^{ab}$ terms in the action, which might contribute a different class of terms to the QFC.

Acknowledgements

It is a pleasure to thank Chris Akers, Raphael Bousso, Venkatesh Chandrasekaran, Netta Engelhardt, Zachary Fisher, Stefan Leichenauer, Adam Levine, Arvin Shahbazi Moghaddam, and Aron Wall for useful discussions. ZF and DM were supported in part by the Simons Foundation and by funds from the University of California. JK was supported in part by the Berkeley Center for Theoretical Physics, by the National Science Foundation (award numbers 1521446, and 1316783), by FQXi, and by the US Department of Energy under contract DE-AC02-05CH11231.