Extensions of Einstein gravity with higher-order derivative terms arise in string theory and other effective theories, as well as being of interest in their own right. In this paper we study static black-hole solutions in the example of Einstein gravity with additional quadratic curvature terms. A Lichnerowicz-type theorem simplifies the analysis by establishing that they must have vanishing Ricci scalar curvature. By numerical methods we then demonstrate the existence of further blackhole solutions over and above the Schwarzschild solution. We discuss some of their thermodynamic properties, and show that they obey the first law of thermodynamics.The well-known problem of the non-renormalisability of Einstein gravity has given rise to many attempts to view it as an effective low-energy theory that will receive higher-order corrections that become important as the energy scale increases (see, for example, [1]). In string theory, the Einstein-Hilbert action is just the first term in an infinite series of gravitational corrections built from powers of the curvature tensor and its derivatives. In other approaches, only a finite number of additional terms might be added. It was shown in [2] that if one adds all possible quadratic curvature invariants to the usual Einstein-Hilbert action one obtains a renormalisable theory, albeit at the price of introducing ghost-like modes. Arguments have been given for why these might not be fatal to the theory (for example, see [3] for a recent discussion). In any case, it is worthwhile to study in detail the properties of the theory of Einstein gravity with added quadratic curvature terms, in order shed light on the question of whether it has irredeemable pathologies or whether they can be controlled in some manner.Black holes are the most fundamental objects in a theory of gravity, and they provide powerful probes for studying some of the more subtle global aspects of the theory. It is therefore of considerable interest to investigate the structure of black-hole solutions in theories of gravity with higher-order curvature terms. In this paper, we report on some investigations of the static, spherically-symmetric black-hole solutions in four-dimensional Einstein-Hilbert gravity with added quadratic curvature terms, for which the most general action can be taken to bewhere α, β and γ are constants and C µνρσ is the Weyl tensor. We shall work in units where we set γ = 1, and the equations of motion following from (1) are thenwhereC µρνσ is the Bach tensor, which is tracefree.In general, the theory describes a system with a massive spin-2 mode with mass-squared m 2 2 = 1/(2α) and a massive spin-0 mode with mass-squared m 2 0 = 1/(6β), in addition to the massless spin-2 graviton. These massive modes will be associated with rising and falling Yukawa type behaviour in the metric modes near infinity [4], of the form 1 r e ±m2r and 1 r e ±m0r . In particular, one can expect that if generic initial data is set at some small distance, the rising exponentials will eventually dominate, leadi...

Extensions of Einstein gravity with quadratic curvature terms in the action arise in most effective theories of quantised gravity, including string theory. This article explores the set of static, spherically symmetric and asymptotically flat solutions of this class of theories. An important element in the analysis is the careful treatment of a Lichnerowicz-type 'no-hair' theorem. From a Frobenius analysis of the asymptotic small-radius behaviour, the solution space is found to split into three asymptotic families, one of which contains the classic Schwarzschild solution. These three families are carefully analysed to determine the corresponding numbers of free parameters in each. One solution family is capable of arising from coupling to a distributional shell of matter near the origin; this family can then match on to an asymptotically flat solution at spatial infinity without encountering a horizon. Another family, with horizons, contains the Schwarzschild solution but includes also non-Schwarzschild black holes. The third family of solutions obtained from the Frobenius analysis is nonsingular and corresponds to 'vacuum' solutions. In addition to the three families identified from near-origin behaviour, there are solutions that may be identified as 'wormholes', which can match symmetrically on to another sheet of spacetime at finite radius.1

A new branch of black hole solutions occurs along with the standard Schwarzschild branch in n-dimensional extensions of general relativity including terms quadratic in the Ricci tensor. The standard and new branches cross at a point determined by a static negative-eigenvalue eigenfunction of the Lichnerowicz operator, analogous to the GrossPerry-Yaffe eigenfunction for the Schwarzschild solution in standard n = 4 dimensional general relativity. This static eigenfunction has two rôles: both as a perturbation away from Schwarzschild along the new black-hole branch and also as a threshold unstable mode lying at the edge of a domain of Gregory-Laflamme-type instability of the Schwarzschild solution for small-radius black holes. A thermodynamic analogy with the Gubser and Mitra conjecture on the relation between quantum thermodynamic and classical dynamical instabilities leads to a suggestion that there may be a switch of stability properties between the old and new black-hole branches for small black holes with radii below the branch crossing point.

Extensions of Einstein gravity with higher-order derivative terms are natural generalizations of Einstein’s theory of gravity. They may arise in string theory and other effective theories, as well as being of interest in their own right. In this paper we study static black-hole solutions in the example of Einstein gravity with additional quadratic curvature terms in four dimensions. A Lichnerowicz-type theorem simplifies the analysis by establishing that they must have vanishing Ricci scalar curvature. By numerical methods we then demonstrate the existence of further black-hole solutions over and above the Schwarzschild solution. We discuss some of their thermodynamic properties, and show that they obey the first law of thermodynamics.

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