On black holes in higher-derivative gravities

Most of the material presented in sections 2 and 3 can be encapsulated in two main results (which we prove) and two conjectures (in favor of which we provide strong evidence). In order to formulate them, let us start with a couple of definitions. First, let $L_{N,\, f}$ be the effective Lagrangian resulting from the evaluation of $\newcommand{\sq}{\sqrt{-g}\,} \newcommand{\m}{{\rm min}} \sqrt{\vert g\vert }\mathcal{L}$ in the general static and spherically symmetric ansatz

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle \label{Nf} {\rm d}s^2_{N,f}=-N^2(r)\,f(r){\rm d}t^2+\frac{{\rm d}r^2}{f(r)}+r^2 {\rm d}\Omega_{(D-2)}^2\,. \nonumber \end{align} \tag{ 1.5 }$$

More precisely, we define

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\la}{\lambda} \displaystyle \label{LNf} L_{N,\,f}(r,f(r), N(r),f^\prime(r), N^\prime(r), \ldots)\equiv N(r)r^{D-2}\mathcal{L}\big\vert _{g^{ab}=g^{ab}_{N,f}}\,. \nonumber \end{align} \tag{ 1.6 }$$

Analogously, we will denote by L_f the expression resulting from setting $N=1$ in $L_{N,\, f}$, which of course corresponds to the effective Lagrangian for $f(r)$ resulting from the evaluation of $\newcommand{\sq}{\sqrt{-g}\,} \newcommand{\m}{{\rm min}} \sqrt{\vert g\vert }\mathcal{L}$ in the single-function ansatz (1.3). With these definitions at hand, we are ready to enumerate our results:

Result 1. Given a $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ gravity of the form (1.2) such that: 1) It allows for static and spherically symmetric solutions characterized by a single function (1.3); 2) These solutions represent the exterior gravitational field of a spherically symmetric mass distribution, then:

• 1.  
  The theory only propagates a traceless and massless graviton on the vacuum⁶.

Result 2. Given a higher-derivative gravity Lagrangian $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd}, \nabla_{e}R_{abcd}, \ldots)$ of the form (1.2) involving terms with up to n covariant derivatives of the Riemann tensor. If the Euler–Lagrange equation of L_f vanishes identically, i.e. if

$$\begin{align} \newcommand{\q}{a} \newcommand{\la}{\lambda} \displaystyle \label{EELL} \frac{\delta L_f}{\delta f}\equiv \frac{\partial L_f}{\partial f}-\frac{\rm d}{{\rm d}r}\frac{\partial L_f}{\partial f^\prime}+\frac{{\rm d}^2}{{\rm d}r^2}\frac{\partial L_f}{\partial f^{\prime\prime}}-\ldots= 0\quad \forall \, f(r)\,, \nonumber \end{align} \tag{ 1.7 }$$

then:

• 1.  
  The theory allows for solutions of the form (1.3), where $f(r)$ satisfies a differential equation of order $\leqslant 2n+2$.
• 2.  
  The former solutions coincide with the exterior gravitational field of a spherically symmetric body, and they are fully characterized by the total mass M.
• 3.  
  At least in the $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ case, the theory only propagates a traceless and massless graviton on the vacuum.

Conjecture 1. For all theories fulfilling the hypothesis of result 2:

• 1.  
  (No-hair) There is a unique black hole solution of the form (1.3) solely characterized by its mass M.
• 2.  
  The thermodynamic properties (e.g. T, $\newcommand{\m}{{\rm min}} \mathsf{S}$) of this black hole can be determined by solving a system of algebraic equations without free parameters.

Conjecture 2. Given a $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory fulfilling the hypothesis of result 2, if $L_{N,\, f}$ can be written as

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle \label{siss} L_{N,\,f}=N L_f+ N^\prime F_1+N^{\prime\prime} F_2\,, \nonumber \end{align} \tag{ 1.8 }$$

where $F_{1, 2}$ are functions of $f(r)$ and its derivatives, then:

• 1.  
  The equation determining $f(r)$ is algebraic.

Some comments are in order.

• Result 1, which we prove in section 2.1, explains the previously noticed [17, 30, 49–53] (but so far unexplained) fact, that certain higher-order gravities admitting simple black hole solutions (in particular, of the form (1.3)), have the interesting property of sharing the linearized spectrum of Einstein gravity. A crucial hypothesis in our result is the assumption that the corresponding solutions describe the exterior field of spherically symmetric mass distributions. The apparent contradiction of result 1 with the fact that certain theories which propagate extra modes at the linearized level do admit black hole solutions of the form 2 is not such. The reason is that, in those cases, the corresponding solutions cannot correspond to the exterior gravitational field of generic spherical distributions, which makes them somewhat 'unnatural'. We illustrate this very explicitly for the well-known Schwarzschild–(A)dS solution in $f(R)$ gravity in section 2.2.
• While result 1 provides a necessary condition for a theory to allow for solutions characterized by a single function and representing the exterior field of a spherical body, result 2, which we prove in section 3.1, yields a sufficient condition for a theory to satisfy this requirement. Interestingly, the proof of the first item of result 2, provides a very efficient method for identifying higher-derivative gravities allowing for solutions of the form (1.3) as well as for obtaining the differential equation determining $f(r)$ in each case. We present this method in the form of a simple recipe in section 3.2.
• Note that the reason for considering the subclass of theories $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ (which do not include terms involving covariant derivatives of the Riemann tensor), in result 1 and in epigraph 3 of result 2 is that the spectrum and Newtonian limit of these theories has been exhaustively classified in [53], which we use to prove our results. We are not aware of an analogous general classification in the general higher-derivative case. However, in light of some related recent works [54, 55], we strongly believe those results apply in the general case as well.
• Observe that result 2 does not really make reference to whether the solutions described by (1.3) in each case can correspond, in particular, to black holes. However, the great amount of evidence accumulated so far [17, 30, 41–52], along with our results, provide strong support for the validity of conjecture 1.
• For a $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory satisfying the hypothesis of result 1, the order of the differential equation determining $f(r)$ is usually 2—see e.g. [30, 50, 52]. However, in some well-known cases [17, 41–49], the equation is algebraic instead, which represents a considerable simplification. Conjecture 2, which we motivate in sections 3.2.1 and 3.2.2, provides a straightforward guiding principle for identifying such class of theories from a given larger set.
• Finally, note that, even though we focus on the spherically symmetric case, the solutions which can be constructed using our method can be straightforwardly generalized to the hyperbolic and flat transverse geometry cases. In particular, the recipe explained in section 3.2 can equally be applied to those cases. We illustrate this in section 4, where we obtain new five-dimensional asymptotically AdS planar black hole solutions of the recently constructed generalized quasitopological gravity [52]—we also comment on this theory in section 3.2.2. This also serves to further support conjecture 1 and is of course motivated by holography, where planar black hole solutions to various higher-order gravities have proven to be remarkably useful for different purposes—see e.g. [8, 15, 16, 19, 46].

In this section we will prove that, in fact, given a general $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory, if the exterior gravitational field of a spherically symmetric body is given by a metric of the form (1.3)⁷, then the theory only propagates a massless graviton at the linearized level. In other words, only theories sharing the linearized spectrum of Einstein gravity are susceptible of admitting single-function (black-hole) solutions corresponding to the exterior field of a spherically symmetric body.

In order to prove this statement, let us consider the linearization of a general $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory on a maximally symmetric background. In [53], we proved that the Newtonian metric of a point-particle of mass M in any such theory in D-dimensions is given by

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle {\rm d}s_N^2=-(1+2U(\rho)){\rm d}t^2+(1-2V(\rho))({\rm d}\rho^2+\rho^2 {\rm d}\Omega_{(D-2)}^2), \label{Newtonian} \nonumber \end{align} \tag{ 2.1 }$$

where $U(\rho)$ is the generalized Newton potential and $V(\rho)=\gamma(\rho) U(\rho)$ where $\gamma(\rho)$ is one of the so-called parametrized Post-Newtonian parameters. In four dimensions, these functions are explicitly given by⁸

$$\begin{align} \newcommand{\q}{a} \newcommand{\la}{\lambda} \displaystyle U (\rho)=-\frac{G_{\rm{eff}}M}{\rho}\left[1-\frac{4}{3}{\rm e}^{-m_g \rho}+\frac{1}{3}{\rm e}^{-m_s \rho}\right],\quad V (\rho)=-\frac{G_{\rm{eff}}M}{\rho}\left[1-\frac{2}{3}{\rm e}^{-m_g \rho}-\frac{1}{3}{\rm e}^{-m_s \rho}\right]. \label{potentials} \nonumber \end{align} \tag{ 2.4 }$$

In these expressions, m_g and m_s are, respectively, the masses of the additional spin-2 and spin-0 modes propagated by the metric perturbation for a generic $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory. These can be easily computed for a given theory using the method developed in [51, 53]⁹.

The Newtonian metric (2.1) is written in isotropic coordinates. In order to express it in Schwarzschild-like coordinates, we perform the change of variable $r^2=\rho^2(1-2V(\rho))$. We stay at linear order in V and U, which suffices for our purposes. Note that $\newcommand{\m}{{\rm min}} V(\rho)=V(r)+\mathcal{O}(V^2)$ and the same is true for U. Then, we obtain

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle \label{tte} {\rm d}s_N^2=-(1+2 U(r)){\rm d}t^2+(1+2r V^\prime(r)){\rm d}r^2+r^2 {\rm d}\Omega_{(D-2)}^2. \nonumber \end{align} \tag{ 2.5 }$$

Now we make use of our hypothesis and assume that this is indeed the linearized limit of a full non-linear solution of the form (1.3). This means that $g_{tt}g_{rr}=-1$, which, when applied to (2.5), imposes the following condition on U and V,

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle U(r)+ r V^\prime(r)=0\,. \label{condition1} \nonumber \end{align} \tag{ 2.6 }$$

From the expressions in (2.4), it follows that this condition holds only when $m_g^2=m_s^2=+\infty$. In other words, the single-function condition on the non-linear solution implies the absence of the massive graviton and the scalar field, so that the only mode which is propagated on the vacuum is the massless graviton. The argument extends straightforwardly to general dimensions using (2.2) and (2.3). This simple argument shows that if a higher-order gravity allows for a solution of the form (1.3) representing the exterior field of a spherical body, the theory only propagates a massless graviton on the vacuum. Indeed, when $m_g^2=m_s^2=+\infty$, the linearized equations of the corresponding theory on a maximally symmetric background reduce to [51, 53]

$$\begin{align} \newcommand{\ka}{\kappa} \displaystyle G_{ab}^{L}= \kappa_{\rm eff } T_{ab}\,, \nonumber \end{align} \tag{ 2.7 }$$

where $G_{ab}^{L}$ is the linearized Einstein tensor, $T_{ab}$ is some putative matter energy momentum tensor, and $\newcommand{\ka}{\kappa} \kappa_{\rm eff }$ is the effective Einstein constant.

Note that, strictly speaking, the Newtonian metric (2.1) only applies in the asymptotically flat case. In that situation, only terms up to quadratic order in curvature contribute to the masses m_g and m_s, so our argument does not immediately go through beyond the quadratic level. In order to extend it to terms of arbitrary order in curvature, we must generalize this argument to include asymptotically (A)dS solutions, since in that case all terms do contribute to the masses of the modes. For a background of curvature $\newcommand{\m}{{\rm min}} \newcommand{\La}{\mathcal{L}} \Lambda$, the Newtonian solution (2.1) is also a good approximation as long as $\newcommand{\m}{{\rm min}} \newcommand{\La}{\mathcal{L}} \rho<<\vert \Lambda\vert ^{-1/2}$. Hence, the same analysis can be applied and the same conclusions are reached, namely, $m_g^2=m_s^2=+\infty$, where now the masses contain information about terms at every order in curvature. Alternatively, one can compute explicitly the Newtonian potential in the (A)dS case and repeat the analysis, but the conclusions remain unchanged.

As we have seen, the results in this section rely on the analysis performed in [51, 53] for the Newtonian metric of a general $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory. Hence, these do not include the more general higher-derivative case $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd}, \nabla_{e}R_{abcd}, \ldots)$. However, in view of some results available in the literature for the Newtonian potential in some of these theories [54, 55], we are confident that result 1 extends as well to that case. The Newtonian metric in the general case seems to have the same structure as in (2.1), with exponential terms including the masses of all the additional modes. Then, we would expect that the condition $g_{tt}g_{rr}=-1$ would only hold if all the masses are infinite, similarly to what we found in our analysis. This suggests that the result 1 would also hold in the general higher-derivative case. A rigorous analysis would require the systematic classification of the spectrum of these theories.

Naively, the result found in the previous section seems to be incompatible with the fact that certain theories which propagate extra modes at the linearized level do also admit single-function black hole solutions of the form (1.3). This apparent contradiction is not such. The reason is that, as we have stressed, our result holds only whenever (1.3) describes the gravitational field of a spherical mass distribution.

In order to illustrate this point, let us consider the case of $f(R)$ gravity. It is well-known that this theory allows for the Schwarzschild–(A)dS solution in the absence of matter—e.g. [31]. Hence, it possesses black hole solutions characterized by a single function. However, we also know that this theory propagates a scalar mode along with the massless graviton on the vacuum. Hence, our result implies that, even though the Schwarzschild–(A)dS metric is a vacuum solution of $f(R)$ gravity, it does not describe the external field of a generic spherically symmetric mass distribution for this theory. Let us verify this statement explicitly. The $f(R)$ field equations coupled to matter read

$$\begin{align} \newcommand{\ka}{\kappa} \newcommand{\la}{\lambda} \newcommand{\B}{\mathcal{B}} \newcommand{\m}{{\rm min}} \displaystyle \label{fr} f^\prime(R) R_{ab}-\frac{1}{2}f(R)g_{ab}+\left(g_{ab}\Box-\nabla_{a}\nabla_{b}\right)\,f^\prime(R)=\kappa T_{ab}\,, \nonumber \end{align} \tag{ 2.8 }$$

where κ is proportional to Newton's constant. Now, let us consider a static and spherically symmetric configuration with an energy-momentum tensor $T_{ab}$ such that $T_{ab}(r)=0$ if $r> r_0$, for certain r₀. Further, let us assume this situation to be compatible with an exterior metric of constant scalar curvature, i.e. satisfying $R=\bar R$, where the constant $\bar R$ would be obtained from the algebraic equation $2\bar R f^\prime(\bar R)-Df(\bar R)=0$. If that was the case, (2.8) would imply $R_{ab}\propto g_{ab}$, and we would obtain Schwarzschild–(A)dS in the exterior region.

However, we will show that no constant-R solution compatible with the above assumptions can exist in the outside region. The trace of the field equations reads

$$\begin{align} \newcommand{\ka}{\kappa} \newcommand{\la}{\lambda} \newcommand{\B}{\mathcal{B}} \newcommand{\m}{{\rm min}} \displaystyle (D-1)\Box f^\prime(R)+R f^\prime(R)-\frac{D}{2}f(R)=\kappa T\,. \label{Requation} \nonumber \end{align} \tag{ 2.9 }$$

This can be thought of as an equation for R (or $f^\prime(R)$). Since we are considering a spherically symmetric situation, we can assume $R=R(r)$, which reduces (2.9) to an ordinary second-order differential equation for $R(r)$. A solution to this equation is then determined by specifying the values of R and $dR/{\rm d}r$ at some r. Now, in the transition point r₀ we must demand continuity and differentiability (otherwise there is no solution). Taking into account our assumptions for the exterior solution, this fixes the following boundary conditions for the internal one:

$$\begin{align} \newcommand{\q}{a} \newcommand{\la}{\lambda} \displaystyle \label{bdc} R(r_0)=\bar R\,, \quad \frac{{\rm d}R}{{\rm d}r}(r_0)=0\,. \nonumber \end{align} \tag{ 2.10 }$$

The solution for $r<r_0$ is then completely specified. However, let us now consider a $(D-2)$-sphere of radius $r_s>r_0$ and unit normal $n^{a}$ at some time slice. Then, using the spherical symmetry of the problem and Stokes' theorem, it is straightforward to prove the following identities

$$\begin{align} \newcommand{\sq}{\sqrt{-g}\,} \newcommand{\B}{\mathcal{B}} \newcommand{\m}{{\rm min}} \displaystyle \Omega_{(D-2)} r_s^{D-2} \frac{{\rm d} f^\prime(R)}{{\rm d}r}(r_s)=\oint_{S_{r_s}} {\rm d}^{D-2}S\, n^{a}\nabla_{a}f^\prime(R)=\int_{r<r_s} {\rm d}^{D-1}x\sqrt{\vert g\vert }\Box f^\prime(R)\,. \nonumber \end{align} \tag{ 2.11 }$$

Finally, taking into account that $f^\prime(R)$ is constant for $r>r_0$ and using equation (2.9), which holds in the $r<r_0$ region, we get

$$\begin{align} \newcommand{\ka}{\kappa} \newcommand{\sq}{\sqrt{-g}\,} \newcommand{\la}{\lambda} \displaystyle \Omega_{(D-2)} r_s^{D-2} \frac{{\rm d} f^\prime(R)}{{\rm d}r}(r_s)=\frac{1}{D-1}\int_{r<r_0} {\rm d}^{D-1}x\sqrt{\vert g\vert }\left(\kappa T-R f^\prime(R)+\frac{D}{2}f(R)\right). \label{contradiction} \nonumber \end{align} \tag{ 2.12 }$$

Now it is immediate to see that there is a problem here. Indeed, while the left-hand side (lhs) is zero, the rhs is non-vanishing in general. The reason is that the interior solution is already completely specified by the boundary conditions (2.10) but those conditions do not depend on the particular form of the matter distribution, while the integral in the rhs does. Hence, that integral will be in general non-vanishing, which leads to a contradiction when compared to the lhs. This implies that no constant-R solutions can describe the gravitational field in the outer region of a spherically symmetric matter distribution for general $f(R)$ theories¹⁰.

Observe that in the Einstein gravity case, i.e. when $\newcommand{\m}{{\rm min}} \newcommand{\La}{\mathcal{L}} f(R)=R-2\Lambda_0$, the contradiction in (2.12) disappears, as the rhs vanishes in that case by virtue of Einstein's equation. This is naturally related to the absence of terms involving covariant derivatives of $f^\prime(R)$—such as $\newcommand{\m}{{\rm min}} \newcommand{\B}{\mathcal{B}} \Box f^\prime(R)$—in the equations of motion, which are in fact ultimately responsible for the appearance of the extra spin-0 mode in the linearized spectrum in the general $f(R)$ case.

In this section we consider the general action (1.2), i.e. with respect to the previous section, we also allow for an arbitrary dependence on terms involving covariant derivatives of the Riemann tensor. Then, as explained before, if the Lagrangian contains up to n derivatives of the Riemann tensor, the field equations generally involve $2n+4$ derivatives of the metric. Such equations can be equivalently studied, for a general static and spherically symmetric ansatz of the form (1.5), by considering the action functional $S[N, f]$, where all the metrics in the gravitational Lagrangian are evaluated on that ansatz, i.e.

$$\begin{align} \displaystyle S[N,f]=\Omega_{(D-2)}\int {\rm d}t \int {\rm d}r L_{N,\,f}\,, \nonumber \end{align} \tag{ 3.1 }$$

where $L_{N, f}$ was defined in (1.6). More explicitly, the variations of $S[N, f]$ with respect to N and f are related to the tt and rr components of the corresponding field equations $\newcommand{\sq}{\sqrt{-g}\,} \newcommand{\m}{{\rm min}} \mathcal{E}_{ab}\equiv \frac{1}{\sqrt{\vert g\vert }}\frac{\delta S}{\delta g^{ab}}$ through [30]

$$\begin{align} \newcommand{\q}{a} \newcommand{\m}{{\rm min}} \displaystyle \frac{1}{\Omega_{(D-2)}r^{D-2}}\frac{\delta S[N,f]}{\delta N}=\frac{2\mathcal{E}_{tt}}{f N^2}\,, \quad \frac{1}{\Omega_{(D-2)}r^{D-2}}\frac{\delta S[N,f]}{\delta f}=\frac{\mathcal{E}_{tt}}{N f^2}+N \mathcal{E}_{rr}\,. \nonumber \end{align} \tag{ 3.2 }$$

Hence, imposing the Euler–Lagrange equations of N and f to hold is equivalent to imposing $\newcommand{\m}{{\rm min}} \mathcal{E}_{tt}=\mathcal{E}_{rr}=0$. Finally, the Bianchi identity $\newcommand{\m}{{\rm min}} \nabla^a \mathcal{E}_{ab}=0$ automatically makes the angular components vanish whenever $\newcommand{\m}{{\rm min}} \mathcal{E}_{tt}=\mathcal{E}_{rr}=0$.

Observe now that the constant rescaling $N\rightarrow N\alpha$, for an arbitrary α, is equivalent to the time rescaling $t\rightarrow t\, \alpha$, which leads to the following identities

$$\begin{align} \displaystyle S[\alpha N, f]=\Omega_{(D-2)}\int {\rm d}t \int {\rm d}r L_{\alpha N,f}=\Omega_{(D-2)}\int d(\alpha t) \int {\rm d}r L_{N,f} =\alpha S[N,f]\,. \nonumber \end{align} \tag{ 3.3 }$$

This implies that both $S[N, f]$ and $L_{N, f}$ are homogeneous of degree 1 in N. Moreover, the Lagrangian is formed by products, quotients and derivatives of the metric components, and the only homogeneous monomials of degree 1 which can be formed in this way with N and its derivatives are of the form $\newcommand{\hs}{h^{\rm cs}} \newcommand{\ren}{R\'enyi} \renewcommand{\c}{$c$} N^{i_1}N^{\prime{i_2}}N^{{\prime\prime}{i_3}}\cdots(N^{(n+2)}){\hspace{0pt}}^{i_{n+3}}$, with i_k integers such that $i_1+\ldots +i_{n+3}=1$. Also, we must have $i_k\geqslant0$ for $k>1$ because the derivatives cannot appear in the denominator. Taking this into account, we observe that the Lagrangian can always be expanded in the following way¹¹,

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\la}{\lambda} \displaystyle L_{N,f}=N L_{f}+\sum_{i=1}^{n+2}N^{(i)}F_i+\mathcal{O}(N^{\prime2/N})\,, \label{homogeneus} \nonumber \end{align} \tag{ 3.4 }$$

where $L_f(r, f, f^\prime, f^{\prime\prime}, \ldots)\equiv L_{N=1, f}$ is the effective Lagrangian resulting from the evaluation of the gravitational Lagrangian in the single-function ansatz (1.3), and the $F_i=F_i(r, f, f^\prime, f^{\prime\prime}, \ldots, f^{(n+2)})$ are functions of f and its derivatives. Finally, $\newcommand{\m}{{\rm min}} \mathcal{O}(N^{\prime2/N})$ denotes all the terms which are at least quadratic in derivatives of N,

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\la}{\lambda} \displaystyle \label{ooo} \mathcal{O}(N^{\prime2/N})\equiv \sum_{i,j=1}^{n+2}\frac{N^{(i)}N^{(\,j)}}{N}F_{ij}+\sum_{i,j,k=1}^{n+2}\frac{N^{(i)}N^{(\,j)}N^{(k)}}{N^2}F_{ijk}+\ldots\,, \nonumber \end{align} \tag{ 3.5 }$$

where, again, $F_{ij}$, $F_{ijk}$, etc only depend on f and its derivatives.

The analysis so far is completely general. Let us now make use of the hypothesis of result 2: we assume that (1.7) holds, i.e. that the Euler–Lagrange equation of $f(r)$ for the Lagrangian L_f vanishes identically. Of course, this is equivalent to the assumption that L_f is a total derivative, this is, that there exists a function $F_0(r, f, f^\prime, \ldots, f^{(n+1)})$ such that

$$\begin{align} \displaystyle L_f=F_0^\prime, \nonumber \end{align} \tag{ 3.6 }$$

where again the prime denotes a total derivative with respect to r. Using this result and the expansion (3.4), we can express the action functional as

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\la}{\lambda} \displaystyle \label{acc} S[N,f]=\Omega_{(D-2)}\int {\rm d}t \int {\rm d}r \left[ N \left(F_0+\sum_{i=1}^{n+1}(-1)^iF_ i^{(i-1)}\right)^\prime+\mathcal{O}(N^{\prime2/N})\right]\,, \nonumber \end{align} \tag{ 3.7 }$$

where we have integrated by parts several times. Now we are ready to compute the Euler–Lagrange equations for N and f. First, it is immediate to see that the equation $\delta_f S=0$ is trivially satisfied whenever $N^\prime=0$. Hence, we can just set N to a constant value, which we choose to be one. On the other hand, variation with respect to N and evaluation at $N=1$ yields

$$\begin{align} \displaystyle \delta_N S=\Omega_{(D-2)}\int {\rm d}t \int {\rm d}r \delta N \left(F_0+\sum_{i=1}^{n+1}(-1)^iF_ i^{(i-1)}\right)^\prime=0\,. \nonumber \end{align} \tag{ 3.8 }$$

Hence, integrating once the equation, we get

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle F_0+\sum_{i=1}^{n+1}(-1)^iF_ i^{(i-1)}=C\,, \label{fequation} \nonumber \end{align} \tag{ 3.9 }$$

for some integration constant C. This is the differential equation which determines $f(r)$ in each case. In order to determine its order, let us consider the Bianchi identity, $\newcommand{\m}{{\rm min}} \nabla_{a}\mathcal{E}^{ab}=0$. The $\nu=r$ component reads

$$\begin{align} \newcommand{\m}{{\rm min}} \displaystyle \frac{{\rm d} \mathcal{E}^{rr}}{{\rm d}r}+\left(\frac{2}{r}-\frac{1}{2}f^{-1}f^\prime\right)\mathcal{E}^{rr}+\frac{1}{2}f f^\prime\mathcal{E}^{tt}-r f\mathcal{E}^{\theta\theta}-\sin^2(\theta) r f\mathcal{E}^{\varphi\varphi}=0\,. \nonumber \end{align} \tag{ 3.10 }$$

Since all the components of $\newcommand{\m}{{\rm min}} \mathcal{E}^{ab}$ contain derivatives up to order $2n+4$ and this identity relates the derivative of $\newcommand{\m}{{\rm min}} \mathcal{E}^{rr}$ to the rest of components (without derivatives), we must conclude that in fact $\newcommand{\m}{{\rm min}} \mathcal{E}^{rr}$ contains derivatives up to order $2n+3$. Now, in (3.9) we have integrated the equation once, so the order of the equation is reduced yet another order. Therefore, (3.9) is in general of order $2n+2$, which means two orders less than the equations determining $N(r)$ and $f(r)$ in the general case. Naturally, in $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theories, for which $n=0$, (3.9) becomes a differential equation of order 2 or less—see (3.23) below. This completes the proof of the first part of result 2.

Let us now proceed with the second. In order to do so, let us start by adding some minimally coupled matter to the gravity action (1.2), $S\rightarrow S + S_{\rm matter}$, where $\newcommand{\sq}{\sqrt{-g}\,} S_{\rm matter}= \int {\rm d}^Dx \sqrt{\vert g\vert }\, L_{\rm matter}$. The field equations would read now $\newcommand{\m}{{\rm min}} \mathcal{E}_{ab}=\frac{1}{2}T_{ab}$, where the matter stress-energy tensor is defined as usual, $\newcommand{\sq}{\sqrt{-g}\,} T_{ab}=-\frac{2}{\sqrt{\vert g\vert }}\frac{\delta S_{\rm matter}}{\delta g^{ab}}.$ Then, it can be shown that the equations for f and N corresponding to the general ansatz (1.5) read

$$\begin{align} \newcommand{\q}{a} \newcommand{\m}{{\rm min}} \newcommand{\la}{\lambda} \displaystyle \left(F_0+\sum_{i=1}^{n+1}(-1)^iF_ i^{(i-1)}\right)^\prime+\mathcal{O}(N^\prime)=r^{D-2} f N^2T^{tt}, \quad \frac{\delta S[N,f]}{\delta f}=\Omega_{(D-2)}\frac{r^{D-2}}{2}\left(N^3T^{tt}+\frac{N}{f^2}T^{rr}\right)\,. \label{matterequations} \nonumber \end{align} \tag{ 3.11 }$$

We are interested in a compact, spherically symmetric and static source of radius r₀, so that $T_{ab}(r)=0$ if $r>r_0$. Let us begin by constructing the interior solution, $r<r_0$. In order to find it, we need to impose some boundary conditions at $r=r_0$. In particular, since we want an exterior solution with $N=1$, we set $N(r_0)=1$, $N^\prime(r_0)=0, \ldots, N^{(n_{\rm{max}}-1)}(r_0)=0$, where $n_{\rm{max}}$ is the highest derivative of N in the equations. On the other hand, we allow the values of f and its derivatives at $r=r_0$ to be arbitrary, but observe that once they are specified, the interior solution is fully determined. The idea is now to find the exterior solution $r>r_0$ which corresponds to this internal configuration. In order to do so, note that the second equation in (3.11) is solved outside the source for $N=1$, and this choice guarantees the continuity of N and its $n_{\rm{max}}-1$ first derivatives at $r=r_0$. On the other hand, applying Stokes' theorem on the first equation in (3.11) yields

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\sq}{\sqrt{-g}\,} \newcommand{\la}{\lambda} \displaystyle F_0(r)+\sum_{i=1}^{n+1}(-1)^iF_ i^{(i-1)}(r)=\frac{1}{\Omega_{(D-2)}}\int_{r<r_0}{\rm d}^{D-1}x\sqrt{\vert g\vert }\left[\,f N T^{tt}-\frac{\mathcal{O}(N^\prime)}{N r^{D-2}}\right]\,, \label{FequationT} \nonumber \end{align} \tag{ 3.12 }$$

where the lhs is evaluated on arbitrary $r>r_0$ while the rhs is an integral over the matter distribution, and thus, it is independent of r. Therefore, (3.9) holds outside the source but now the constant C is not arbitrary, but determined by the mass distribution¹². Even though it seems that C could depend on the density profile or on the radius of the source, as we show next, studying the asymptotic behavior of the solution one finds that it is always proportional to the total mass. Indeed, assuming that our higher-derivative theory has a well-defined Einstein gravity limit as in (1.2), then the asymptotic expansion of (3.9) becomes

$$\begin{align} \newcommand{\La}{\mathcal{L}} \newcommand{\m}{{\rm min}} \displaystyle \frac{1}{16\pi G}\left[-\frac{2}{(D-1)}\Lambda_0 r^{D-1}-(D-2)(\,f-1)r^{D-3}+\ldots\right]=C. \nonumber \end{align} \tag{ 3.13 }$$

In the asymptotically flat case, $\newcommand{\m}{{\rm min}} \newcommand{\La}{\mathcal{L}} \Lambda_0=0$, this equation implies

$$\begin{align} \newcommand{\m}{{\rm min}} \displaystyle f(r)=1-\frac{16 \pi G C}{(D-2) r^{D-3}}+\mathcal{O}(r^{2-D})\,. \nonumber \end{align} \tag{ 3.14 }$$

Now, for an asymptotically flat spacetime, no matter the higher-derivative corrections, the total mass can be computed according to ADM formula [25–27, 65], which yields

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle M=\frac{(D-2)\Omega_{(D-2)}}{16\pi G}\lim_{r\rightarrow\infty} r^{D-3}\left(\frac{1}{f(r)}-1\right)\,. \label{massC} \nonumber \end{align} \tag{ 3.15 }$$

Then, we obtain

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle C=\frac{M}{\Omega_{(D-2)}}\,. \label{Cmass} \nonumber \end{align} \tag{ 3.16 }$$

In the case of a non-vanishing cosmological constant, this relation still holds. Indeed, in the asymptotically (A)dS case, the mass is given by a generalized version of (3.15) [65, 66], but the identity above remains unchanged. In sum, the equation for f can always be written as

$$\begin{align} \displaystyle F_0-F_1+F_2^\prime-\ldots=\frac{M }{\Omega_{(D-2)}}\,. \nonumber \end{align} \tag{ 3.17 }$$

Returning to the interior solution, remember that we intentionally left undetermined the values of f and their first derivatives at $r=r_0$. Then, these values are chosen so that the interior solution is well glued with the exterior one (i.e. there is continuity and differentiability).

This completes the proof that the vacuum solutions of the form (1.3) do describe the exterior gravitational field of spherically symmetric matter distributions and that they are characterized by the total mass. There is no unicity though, since the vacuum equation (3.9) could have more than one solution even for a fixed M. In other words, the solutions which represent the exterior field of a source could have 'hair'. When we impose the solution to be a black hole, this is no longer the case, as the existence of a horizon imposes a natural boundary condition which kills the possible hair, as we explicitly showed for $D=4$ Einsteinian cubic gravity in [30]—see also section 4 for a new five-dimensional example¹³. This result is captured in conjecture 1.

The third part of result 2 follows from the first two along with result 1, which we proved in the previous section.

Let us conclude this section by stressing that Lovelock theories, quasitopological gravity (and its higher-order generalizations), four-dimensional Einsteinian cubic gravity and generalized quasitopological gravity satisfy the hypothesis of result 2. On the one hand, our results explain why they all have the same linearized equations as Einstein gravity (up to a redefinition of Newton's constant [53]). On the other, they imply that their static and spherically symmetric black hole solutions of the form (1.3) are the natural generalizations of Schwarzschild–(A)dS, as they represent the exterior field of spherically symmetric mass distributions.

The results obtained in the previous section provide a very simple and efficient method for identifying higher-derivative gravities with simple black hole solutions of the form (1.3), and for characterizing those solutions. For the same price, the solutions constructed in this way correspond to the exterior field of a spherically symmetric mass distribution, and the corresponding theories are automatically equivalent to Einstein gravity at the linearized level. Our method is a refinement of an often utilized procedure, e.g. in [17, 30, 48, 49, 52, 67, 68], consisting in evaluating $L_{N,\, f}$ and performing a repeated integration by parts in the aim of bringing it to the form (3.7) for a particular combination of couplings.

Let us now present our method in the form of a recipe ready to be applied to any higher-derivative gravity. From a computational perspective, our procedure is considerably faster than the one just described. It involves two trivial on-shell evaluations of the higher-derivative action, computing the Euler–Lagrange equation of a one-dimensional Lagrangian, writing an expression as a total derivative of another function (which is guaranteed to be possible), and computing some trivial derivatives. Here is the recipe:

• 1.  
  Evaluate the gravity Lagrangian on the single-function ansatz (1.3), namely, $\newcommand{\m}{{\rm min}} L_f(r, f^\prime, f^{\prime\prime}, \ldots)\equiv r^{D-2}\mathcal{L}\vert _{g^{ab}=g^{ab}_f}$.
• 2.  
  Compute the Euler–Lagrange equation of $f(r)$ for the effective Lagrangian $L_f(r, f^\prime, f^{\prime\prime}, \ldots)$.
• 3.  
  Fix the higher-derivative couplings in a way such that this equation is identically satisfied, i.e. impose $\delta L_f/\delta f=0$ for all $f(r)$.
• 4.  
  Find F₀, namely, the function of $f(r)$ satisfying $L_f=F_0^\prime$.
• 5.  
  Substitute the general ansatz (1.5) in the corresponding gravity Lagrangian: $L_{N, f}(r, f, N, f^\prime, N^\prime, \ldots)$ $\newcommand{\m}{{\rm min}} \equiv N(r)r^{D-2}\mathcal{L}\big\vert _{g^{ab}=g^{ab}_{N, f}}$. The result should take the form (3.4), where now $L_f=F_0^\prime$.
• 6.  
  Identify the functions F_i by inspection.
• 7.  
  Plug F₀ and the (corresponding derivatives of the) F_i in (3.9). This is the equation that determines $f(r)$.

The first three steps select, from all the possible theories considered originally, the ones which allow for single-function solutions of the form (1.3). The last four allow one to determine the differential (or algebraic) equation which needs to be solved in order to determine $f(r)$ for the corresponding theory.

3.2.1. Quadratic gravities.

In order to illustrate this method, let us apply it to the D-dimensional quadratic theory

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\La}{\mathcal{L}} \displaystyle \mathcal{L}_{\rm quadratic}=\frac{1}{16\pi G}\left[-2\Lambda_0+R+ \alpha_1 R^2+\alpha_2 R_{ab}R^{ab}+\alpha_3 R_{abcd}R^{abcd} \right]\,. \nonumber \end{align} \tag{ 3.18 }$$

Evaluating the Lagrangian on the single-function metric ansatz, we obtain the effective Lagrangian

$$\begin{align} \newcommand{\La}{\mathcal{L}} \newcommand{\m}{{\rm min}} \newcommand{\B}{\mathcal{B}} \displaystyle L_f=&\frac{1}{16\pi G}\Big[-2\Lambda_0 r^{D-2}+(D-2)(D-3) r^{D-4} (\,f-1)+2(D-2)r^{D-3}f^\prime+r^{D-2}f^{\prime\prime}\nonumber \\ &+\alpha_1 r^{D-6}\left((D-2)(D-3)(\,f-1)+2(D-2)rf^\prime+r^2f^{\prime\prime}\right)^2\nonumber \\ &+\alpha_2r^{D-6}\left((D-2)((D-3)(\,f-1)+rf^\prime)+((D-2)rf^\prime+r^2f^{\prime\prime})^2/2\right)\nonumber \\ &+\alpha_3r^{D-6}\left(2(D-2)(D-3)(\,f-1)^2+2(D-2)r^2f^{\prime2}+r^4 f^{{\prime\prime}4}\right)\Big]. \nonumber \end{align} \tag{ 3.19 }$$

From this, it is straightforward to compute the Euler–Lagrange derivative, which yields

$$\begin{align} \displaystyle \frac{\delta L_f}{\delta f}=&\frac{(D-2)}{16\pi G}\left[ (3\alpha_1+\alpha_2+\alpha_3)\left(4(D-3)(\,f-1)-2r^2f^{\prime\prime}\right)\right.\nonumber \\ &\left.+(2\alpha_1+\alpha_2+2\alpha_3)\left((D-4)r^2 f^{\prime\prime}+2r^2f^{(3)}+r^4 f^{(4)}\right)\right]. \nonumber \end{align} \tag{ 3.20 }$$

Then, applying the third step of the recipe, we find that imposing $\delta L_f/ \delta f= 0$ $\forall f(r)$ fixes $\alpha_1=\alpha_3=-\alpha_2/4=\alpha$ which, unsurprisingly, leads to the usual Gauss–Bonnet combination. Having fixed these couplings, we can compute F₀ for this theory, which turns out to read

$$\begin{align} \newcommand{\La}{\mathcal{L}} \newcommand{\m}{{\rm min}} \displaystyle 16\pi GF_0=\,\,&(D-2)r^{D-3}(1-2r^2\Lambda_0/((D-2)(D-1))-f)+f^\prime(2(D-3)(D-2) r^{D-4}(\,f-1)\alpha-r^{D-2}) \nonumber \\ &+(D-4)(D-3)(D-2)r^{D-5} (\,f-1)^2\alpha\,. \nonumber \end{align} \tag{ 3.21 }$$

The next step is to evaluate the Lagrangian in the general metric ansatz with two functions. Amusingly, the effective Lagrangian $L_{N,\, f}$, which in general takes the form (3.4), does not contain any $\newcommand{\m}{{\rm min}} \mathcal{O}(N^{\prime2/N})$ term in this case, and is simply given by (1.8), where

$$\begin{align} \displaystyle 16\pi GF_1&=r^{D-5}(-3f^\prime r^3+2(D-3)(D-2)(5f-3)\,f^\prime r \alpha-2(D-2)\,f (r^2-2(D-4)(D-3)(\,f-1)\alpha))\,,\nonumber \\ 16\pi GF_2&=-2r^{D-4}f(r^2+2(D-3)(D-2)\alpha(1-f))\,. \nonumber \end{align} \tag{ 3.22 }$$

This is all we need to determine the equation of $f(r)$, (1.3), which in this case (and for any $\newcommand{\m}{{\rm min}} \mathcal{L}(g^{ab}, R_{abcd})$ theory satisfying the hypothesis of result 2) reads:

$$\begin{align} \newcommand{\la}{\lambda} \displaystyle \label{fff} F_0-F_1+F_2^\prime=C. \nonumber \end{align} \tag{ 3.23 }$$

Explicitly, one finds

$$\begin{align} \newcommand{\la}{\lambda} \newcommand{\La}{\mathcal{L}} \newcommand{\m}{{\rm min}} \displaystyle \label{GB} r^{D-3}\left((D-2)(1-f)-\frac{2r^2\Lambda_0}{(D-1)}\right)+\alpha (D-4)(D-3)(D-2) r^{D-5}(\,f-1)^2=\frac{16\pi G M}{\Omega_{(D-2)}}, \nonumber \end{align} \tag{ 3.24 }$$

where we have taken into account (3.16). This equation can be solved for $f(r)$ to yield

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\sq}{\sqrt{-g}\,} \newcommand{\La}{\mathcal{L}} \displaystyle f(r)=1+\frac{r^2}{2(D-3)(D-4)\alpha}\left[1\mp \sqrt{1+\frac{8\alpha \Lambda_0 (D-4)(D-3)}{(D-2)(D-1)}+\frac{64 \alpha\pi G M(D-3)(D-4)}{(D-2)\Omega_{(D-2)}r^{D-1}}} \right]. \nonumber \end{align} \tag{ 3.25 }$$

This is naturally the blackening factor of the usual D-dimensional static and spherically-symmetric Gauss–Bonnet black hole¹⁴.

Notice that the fact that (3.24) is algebraic instead of a second-order differential equation is a non-generic feature which also occurs for general Lovelock [41–46] and quasitopological theories [17, 47–49], but not in other cases like Einsteinian cubic gravity [30] or the recently constructed generalized quasitopological gravity [52]—see below. Interestingly, the algebraicity of (3.23) (which is a result of the non-trivial cancellation of the different terms in F₀, F₁ and $F_2^\prime$ involving derivatives of $f(r)$) appears to be related to the absence of $\newcommand{\m}{{\rm min}} \mathcal{O}(N^{\prime2/N})$ terms in $L_{N, \,f}$. This is, very likely, a general feature which we encapsule in conjecture 2.

In addition, let us comment that by using the solution above and the results in [65] it is easy to check explicitly that M is in fact the total mass in any dimension and for any asymptotic behavior.

3.2.2. Cubic gravities.

Let us be less detailed with the cubic case. If the analysis is repeated at this order in curvature, one is led to the recently constructed generalized quasitopological gravity [52], whose action is given by

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\la}{\lambda} \newcommand{\La}{\mathcal{L}} \displaystyle \label{gQuasi2} \mathcal{L}_{\rm cubic}=\frac{1}{16\pi G} \left[-2\Lambda_0+R+\alpha \mathcal{X}_4+ \gamma \mathcal{X}_6+\mu \mathcal{Z}_D+\xi \mathcal{S}_D\right]\,. \nonumber \end{align} \tag{ 3.26 }$$

In this expression, $\newcommand{\m}{{\rm min}} \mathcal{X}_6$ is the cubic Lovelock density [56, 57], $\newcommand{\m}{{\rm min}} \mathcal{Z}_D$ is the D-dimensional cubic quasitopological term [17, 47], and $\newcommand{\m}{{\rm min}} \mathcal{S}_D$ is the new term [52] which generalizes Einsteinian cubic gravity [51] to higher-dimensions¹⁵. This term is explicitly given by

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\hs}{h^{\rm cs}} \newcommand{\la}{\lambda} \displaystyle \label{SD} \mathcal{S}_D=&14{R}{_{a}^{c}{\hspace{0pt}}_{b}^{d}}{R}{_{c}^{e}{\hspace{0pt}}_{d}^{\,f}}{R}{_{e}^{a}{\hspace{0pt}}_{f}^{b}} +2{R}{_{abcd}}{R}{^{abc}_{e}}R^{de}-\frac{(38-29 D+4 D^2)}{4(D-2)(2D-1)}{R}{_{abcd}}{R}{^{abcd}}R\nonumber \\ &-\frac{2(-30+9D+4D^2)}{(D-2)(2D-1)}{R}{_{abcd}}{R}{^{ac}}{R}{^{bd}} -\frac{4(66-35D+2D^2))}{3(D-2)(2D-1)} R^{b}_{a} R_{b}^{c} R_{c}^{a}\nonumber \\ &+\frac{(34-21D+4D^2)}{(D-2)(2D-1)}R_{ab}R^{ab} R-\frac{(30-13D+4D^2)}{12(D-2)(2D-1)} R^3. \nonumber \end{align} \tag{ 3.27 }$$

When restricted to four dimensions, one finds that [52]

$$\begin{align} \newcommand{\m}{{\rm min}} \displaystyle \mathcal{S}_4=\mathcal{P}-\frac{1}{4}\mathcal{X}_6+4\mathcal{C}\,, \nonumber \end{align} \tag{ 3.28 }$$

where

$$\begin{align} \newcommand{\m}{{\rm min}} \newcommand{\hs}{h^{\rm cs}} \newcommand{\la}{\lambda} \displaystyle \label{ecg} \mathcal{P}=12 {R}{_{a}^{c}{\hspace{0pt}}_{b}^{d}}{R}{_{c}^{e}{\hspace{0pt}}_{d}^{\,f}}{R}{_{e}^{a}{\hspace{0pt}}_{f}^{b}}+{R}{_{ab}^{cd}} {R}{_{cd}^{ef}}{R}{_{ef}^{ab}}-12{R}{_{abcd}}{R}{^{ac}}{R}{^{bd}}+8R^{b}_{a} R_{b}^{c} R_{c}^{a}\,, \nonumber \end{align} \tag{ 3.29 }$$

is the usual Einsteinian cubic gravity term [51], $\newcommand{\m}{{\rm min}} \mathcal{X}_6$ identically vanishes (for any metric in $D\leqslant 5$), and $\newcommand{\m}{{\rm min}} \mathcal{C}$ is a cubic term which does not contribute to the equations of motion for a static and spherically symmetric ansatz (1.5). It is in this sense that $\newcommand{\m}{{\rm min}} \mathcal{S}_D$ provides a D-dimensional generalization of four-dimensional Einsteinian cubic gravity, as explained in [52]. As emphasized in that paper, the linearized spectrum of (3.26) coincides with Einstein's, which of course can be now understood as a consequence of this theory satisfying the hypothesis of result 2. This also implies that the black hole solutions of (3.26) constructed solving the corresponding equation for $f(r)$ [52] describe the exterior gravitational field of generic spherically symmetric distributions of mass for the theory (3.26) minimally coupled to matter.

We also point out that in the absence of $\newcommand{\m}{{\rm min}} \mathcal{S}_D$ (or, equivalently, $\newcommand{\m}{{\rm min}} \mathcal{P}$ in $D=4$) the equation that determines $f(r)$ is algebraic, and no $\newcommand{\m}{{\rm min}} \mathcal{O}(N^{\prime2/N})$ terms of the form (3.5) appear in $L_{N, \, f}$. This is no longer the case when $\newcommand{\m}{{\rm min}} \mathcal{S}_D$ is included. These observations strongly support conjecture 2. Finally, let us also mention that for all the different theories contained in (3.26), conjecture 1 holds.