The Weyl double copy in vacuum spacetimes with a cosmological constant

Published for SISSA by Springer Received: June 1, 2022 Revised: September 19, 2022 Accepted: September 20, 2022 Published: September 28, 2022 Shanzhong Han The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark E-mail: Abstract: We examine the Weyl double copy relation for vacuum solutions of the Einstein equations with a cosmological constant using the approach we previously described, in which the spin-1/2 massless free-field spinors (Dirac-Weyl fields) are regarded as basic units. Based on the exact non-twisting vacuum type N and vacuum type D solutions, the finding explicitly shows that the single and zeroth copies fulfill conformally invariant field equations in conformally flat spacetime. In addition, irrespective of the presence of a cosmological constant, we demonstrate that the zeroth copy connects Dirac-Weyl fields with the degenerate electromagnetic fields in the curved spacetime in addition to connecting gravity fields with the single copy in conformally flat spacetime. Moreover, the study also demonstrates the critical significance the zeroth copy plays in time-dependent radiation solutions. In particular, for Robinson-Trautman (Λ) gravitational waves, unlike the single copy, we find that the zeroth copy carries additional information to specify whether the sources of associated gravitational waves are time-like, null, or space-like, at least in the weak field limit. Keywords: Black Holes, Classical Theories of Gravity, Gauge-Gravity Correspondence, Black Holes in String Theory ArXiv ePrint: 2205.08654 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP09(2022)238 JHEP09(2022)238 The Weyl double copy in vacuum spacetimes with a cosmological constant Contents 1 2 Massless free-fields in spinor formalism 2 3 The Weyl double copy in curved spacetimes 3.1 The case of non-twisting vacuum type N solutions 3.1.1 The Kundt(Λ) class 3.1.2 The Robinson-Trautman(Λ) class 3.2 The case of vacuum type D solutions 3.2.1 Kerr-(A)dS black holes 3.2.2 The most general vacuum type D solutions 6 7 7 9 10 10 12 4 Discussion and conclusions 15 1 Introduction The double copy originates from the study of perturbative scattering amplitudes [1–3], which brings forth a fascinating connection between gauge amplitudes and gravity amplitudes. Moreover, this idea has been extended to the classical context. In Kerr-Schild coordinate system, a map between gravity theory and gauge theory was proposed, called Kerr-Schild double copy [4]. A wide array of such classes of spacetimes has been studied [5–19]. Inspired by this, a new type of double copy relation called Weyl double copy is drawing more attention [20–28]. This prescription is represented by ΨABCD = Φ(AB ΦCD) , S (1.1) where ΨABCD is a Weyl spinor describing vacuum gravity fields, ΦAB is an electromagnetic spinor referring to a Maxwell field in Minkowski spacetime — the simplest solution of the gauge theory, and S is an auxiliary scalar field satisfying the wave equation in Minkowski spacetime. The last two fields are called single copy and zeroth copy, respectively. Starting from the gravity fields, the Weyl double copy relation leads to a gauge field that is completely independent of the gravity theory. As a result, it is thought that, the Weyl double copy relation could serve as a link between gravity theory and gauge theory. Luna et al. proposed for the first time the Weyl double copy relation for the case of vacuum type D solutions [20]. Then, in spinor language, this relation was extended to non-twisting vacuum type N solutions by Godazgar et al. [25]. Making use of the peeling property [29, 30] of the Weyl tensor, they further showed that the Weyl double copy relation also holds asymptotically for algebraically general solutions [27]. In addition, at the linearised lever, the Weyl double copy relation was shown to hold for arbitrary Petrov type –1– JHEP09(2022)238 1 Introduction 2 Massless free-fields in spinor formalism In this section, we will briefly review how to construct electromagnetic spinors in order to verify the Weyl double copy relation using the methodology of the previous work [31]. –2– JHEP09(2022)238 solutions using the twistor formalism [22, 23]. An extended Weyl double copy prescription was also proposed recently for non-vacuum solutions, whose Weyl spinor is decomposed into a sum of source terms [28]. Very recently, regarding the Dirac-Weyl (DW) spinors (spin-1/2 massless free-field spinors) as the basic units of other higher spin massless free-field spinors, we systematically revisited the Weyl double copy relation for non-twisting vacuum type N and vacuum type D solutions [31]. We further found a map similar to the Weyl double copy prescription for non-twisting vacuum type III spacetimes. However, the Weyl double copy relation for the exact vacuum solutions with a cosmological constant has not yet been investigated. This is the primary objective of the current effort. In fact, since 1998, by the observations of supernovae of Ia type [32, 33], studies have shown that the expansion of our universe is accelerating, which strongly supports the condition that the cosmological constant Λ is nonzero and positive. On the other hand, although Anti-de Sitter (AdS) spacetime does not appear to have direct cosmological applications, it plays a crucial role in AdS/CFT correspondence. Therefore, investigating the Weyl double copy relation in the presence of a cosmological constant would be of interest. Currently, there are two possible research directions: one is to interpret the cosmological constant as a source of the single and zeroth copies in the flat spacetime; the other is to consider the (A)dS spacetime to be the background of the single and zeroth copies. The former idea was proposed for the first time in Kerr-Schild double copy in Taub-NUT spacetime [8] and it would be natural in the direct investigation of the relationship between gravity theory and gauge theory. On the other hand, the latter can be viewed as a precursor to the former. Moreover, it is also advantageous for extending the remit of the Weyl double copy, including cosmological applications and perturbation theory. This has been done in ref. [15] for Kerr-Schild(Λ) double copy, which shows that the single and zeroth copies satisfy different equations for time-dependent and time-independent solutions. These outcomes encourage us to study whether or not the Weyl double copy relation shares this property. In this paper, we shall give an explicit demonstration to show that, different from the Kerr-Schild(Λ) double copy, the single and zeroth copies in the Weyl double copy prescription all satisfy conformal invariant field equations in conformally flat spacetime, both for time-independent solutions and time-dependent solutions. Our finding coincides with the statement of ref. [22] in the twistorial version. Some interesting relations between th (...truncated)