Recently, a perturbative duality between gauge and gravity theories (the double copy) has been discovered, that is believed to hold to all loop orders. In this paper, we examine the relationship between classical solutions of non-Abelian gauge theory and gravity. We propose a general class of gauge theory solutions that double copy to gravity, namely those involving stationary Kerr-Schild metrics. The Schwarzschild and Kerr black holes (plus their higher-dimensional equivalents) emerge as special cases. We also discuss plane wave solutions. Furthermore, a recently examined double copy between the self-dual sectors of Yang-Mills theory and gravity can be reinterpreted using a momentum-space generalisation of the Kerr-Schild framework.

The scattering equations on the Riemann sphere give rise to remarkable formulae for tree-level gauge theory and gravity amplitudes. Adamo, Casali and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann sphere. What emerges is a framework for loop integrands on the Riemann sphere that promises to have wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulae, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.

We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.

The double copy is a much-studied relationship between gauge theory and gravity amplitudes. Recently, this was generalised to an infinite family of classical solutions to Einstein's equations, namely stationary Kerr-Schild geometries. In this paper, we extend this to the Taub-NUT solution in gravity, which has a double Kerr-Schild form. The single copy of this solution is a dyon, whose electric and magnetic charges are related to the mass and NUT charge in the gravity theory. Finally, we find hints that the classical double copy extends to curved background geometries. 1 a.luna-godoy.1@research.gla.ac.uk 2 monteiro@maths.ox.ac.uk 3

Abstract:The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.

We present the first example of a linearized gravitational instability of an asymptotically flat vacuum black hole. We study perturbations of a Myers-Perry black hole with equal angular momenta in an odd number of dimensions. We find no evidence of any instability in five or seven dimensions, but in nine dimensions, for sufficiently rapid rotation, we find perturbations that grow exponentially in time. The onset of instability is associated with the appearance of time-independent perturbations which generically break all but one of the rotational symmetries. This is interpreted as evidence for the existence of a new 70-parameter family of black hole solutions with only a single rotational symmetry. We also present results for the Gregory-Laflamme instability of rotating black strings, demonstrating that rotation makes black strings more unstable.

We study the double-copy relation between classical solutions in gauge theory and gravity, focusing on four-dimensional vacuum metrics of algebraic type D, a class that includes several important solutions. We present a double copy of curvatures that applies to all spacetimes of this type -the Weyl double copy -relating the curvature of the spacetime to an electromagnetic field strength. We show that the Weyl double copy is consistent with the previously known Kerr-Schild double copy, and in fact resolves certain ambiguities of the latter. The most interesting new example of the classical double copy presented here is that of the C-metric. This well-known solution, which represents a pair of uniformly accelerated black holes, is mapped to the Liénard-Wiechert potential for a pair of uniformly accelerated charges. We also present a new double-copy interpretation of the Eguchi-Hanson instanton. arXiv:1810.08183v2 [hep-th] 1 Apr 2019 -1 -an antisymmetric tensor. The focus of this article will be on the double copy in classical field theory. Moreover, we will restrict to pure Einstein gravity, so that the dilaton and the antisymmetric tensor are not present; we will comment on these fields in section 6.The KLT relations have the advantage that an underlying reason for the relation between gauge theory and gravity is clear: by joining two open strings, you get one closed string. But they have the disadvantage that the relations themselves become quite complicated for high multiplicity. More recently, Bern, Carrasco and Johansson (BCJ) discovered a new and simple form for the double copy [2] which also leads to a natural formulation of the double copy at loop level [3]. This form of the double copy has been extensively studied at tree level, leading to various proofs [4][5][6][7][8][9][10][11][12][13][14][15][16]. At loop level, the double copy has been extensively studied [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], but to date it is still a conjecture [27]. Nevertheless, the BCJ double copy is a powerful tool in the theory of scattering amplitudes, which has led to rich new insights into the structure of supergravity, e.g. [32][33][34][35][36][37][38][39][40]. A celebrated recent example is the detailed computation of the UV structure of maximal supergravity at five loops [41]. The double copy is reviewed in, for example, [42][43][44].The success of the double copy for scattering amplitudes motivated the investigation of its manifestation for solutions of the classical field equations, with early steps given in [45][46][47]. Since the principles of the double copy, as currently understood, are perturbative in nature, it is remarkable that exact relations between solutions can be found, as discovered in [48]. In the same way that solutions to the Maxwell equations provide a class of linear solutions to the Yang-Mills equations (with trivial colour dependence), there is a class of solutions that linearise the Einstein equations. Kerr-Schild metrics belong to this class, and so do certain multi-Kerr-Sch...

Advances in our understanding of perturbation theory suggest the existence of a correspondence between classical general relativity and Yang-Mills theory. A concrete example of this correspondence, which is known as the double copy, was recently introduced for the case of stationary Kerr-Schild spacetimes. Building on this foundation, we examine the simple time-dependent case of an accelerating, radiating point source. The gravitational solution, which generalises the Schwarzschild solution, includes a non-trivial stress-energy tensor. This stress-energy tensor corresponds to a gauge theoretic current in the double copy. We interpret both of these sources as representing the radiative part of the field. Furthermore, in the simple example of Bremsstrahlung, we determine a scattering amplitude describing the radiation, maintaining the double copy throughout. Our results provide the strongest evidence yet that the classical double copy is directly related to the BCJ double copy for scattering amplitudes.