PHYSICAL REVIEW D 102, 024079 (2020) Black hole gravitational waves in the effective field theory of gravity † ‡ Claudia de Rham,1,2,* J´er´emie Francfort ,3, and Jun Zhang 1, 1Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom 2CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA 3D´epartement de Physique Th´eorique, Universit´e de Gen`eve, 24 quai Ansermet, CH1211 Gen`eve 4, Switzerland (Received 19 June 2020; accepted 16 July 2020; published 27 July 2020) We investigate the propagation of gravitational waves on a black hole background within the low-energy effective field theory of gravity, where effects from heavy fields are captured by higher-dimensional curvature operators. Depending on the spin of the particles integrated out, the speed of gravitational waves at low energy can be either superluminal or subluminal as compared to the causal structure observed by other species. Interestingly, however, gravitational waves are always exactly luminal at the black hole horizon, implying that the horizon is identically defined for all species. We further compute the corrections on quasinormal frequencies caused by the higher-dimensional curvature operators and highlight the corrections arising from the low-energy effective field. DOI: 10.1103/PhysRevD.102.024079 I. INTRODUCTION charged particle mass, the low-energy effective field theory contains operators that can lead to a superluminal group The detection of gravitational waves (GWs) opens up a and phase velocity on certain backgrounds [4,5]. Yet this brand new window of opportunity to test gravity. The low-energy superluminal group and phase velocity is not in observation of GW170817 [1] together with its gamma-ray – counterpart GRB170817A [2] constrains the speed differ- conflict with causality as discussed in [6 16]. ence between GWs and photons propagating on a cosmo- By analogy, the same is expected for GWs. When −15 accounting for the interactions between gravity and the logical background down to 10 [3]. In this new era of other heavy fields, one would expect the speed of GWs to GW astronomy, it has become more important than ever to understand how GWs propagate especially in the strong naturally depart from unity at low energy in backgrounds gravity regime and get a handle on the types of corrections that spontaneously break Lorentz invariance, while recov- that are expected to arise in the effective field theory of ering a luminal speed at high energy [17]. gravity. In order to remain general and agnostic on the precise Lorentz invariance dictates that in the vacuum any high-energy completion of gravity (i.e., on the precise massless particle propagates at the speed of light, but in spectrum of the heavy fields considered), we work here media that (spontaneously) break Lorentz invariance, we within the low-energy effective field theory (EFT) of are used to expect a frequency-dependent deviation from gravity framework, where the classical and quantum effects luminal propagation, as is well known for light propagating of heavy fields is captured by the inclusion of higher- – through glass or water. This effect emerges naturally from dimensional curvature operators [18 21]. Indeed, we the interactions between light and the medium it propagates expect these operators to naturally arise from an arbitrary through. The speed of photons can also be modified in a underlying UV complete gravity theory, such as string curved background due to loop corrections from charged theory [22–24], although we do not need to commit to any particles (e.g., electrons). At energy scales well below the particular realization in what follows. Within this low- energy EFT of gravity, it was shown in [25] that GWs propagating in a Friedmann-Lemaître-Robertson-Walker *[email protected] † [email protected] background do not generically propagate exactly luminally ‡ [email protected] at low energy. In this work, we push this investigation further by analyzing the speed of GWs propagating on a Published by the American Physical Society under the terms of Schwarzschild-like background. This will represent an the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to interesting situation where the GWs are propagating in the author(s) and the published article’s title, journal citation, the vacuum but the presence of a black hole spontaneously and DOI. Funded by SCOAP3. breaks Poincar´e invariance. This implies that the 2470-0010=2020=102(2)=024079(14) 024079-1 Published by the American Physical Society DE RHAM, FRANCFORT, and ZHANG PHYS. REV. D 102, 024079 (2020) low-energy speed of GWs can (and indeed does) differ operators are extremely suppressed and determined in from the speed of other massless minimally coupled terms of only two of the dimension-six EFT operators. particles. Since the speed of various species is not invariant The rest of this paper is organized as follows. In Sec. II, under a change of frames, we qualify our statement and we introduce the low-energy EFT of gravity, including the make the impact on the causal structure manifest by dimension-six operators. We study their perturbative effects working in the Jordan frame, where all the matter fields on the black hole solution and derive the modified Regge- (including light) are minimally coupled to gravity, ensuring Wheeler-Zerilli equations for the metric perturbations. This that electromagnetic waves travel at a luminal speed with allows us to investigate the speed of GWs and the causal respect to the background metric. In this frame, we consider structure in Sec. III, and compute the corrections on the the low-energy EFT of gravity by including the local and black hole quasinormal frequencies in Sec. IV. Section V is covariant higher-order curvature operators present in the devoted to discussions and outlook. Technical details and low-energy EFT. These can emerge from weakly coupled some expressions are given in the Appendixes. We work UV completions after integrating out fields of higher spin at with the ð−; þ; þ; þÞ signature, and in units where tree level, or they can emerge from integrating out loops of ℏ ¼ c ¼ 1. particles of all spins, including standard model particles (see Ref. [25] for a detailed discussion). Once again, for the II. BLACK HOLES IN THE LOW-ENERGY most part of this work, we shall remain agnostic on the EFFECTIVE FIELD THEORY OF GRAVITY precise realization. A. EFT of gravity If treated nonperturbatively, the higher-dimensional cur- vature operators may lead to interesting features as pointed We consider the low-energy EFT of gravity including out for instance in [26]. In what follows we shall, however, curvature operators up to dimension-six. The Lagrangian of take an EFT approach considering the higher-dimensional the theory is given by [22] operators to represent only the leading contributions in an ffiffiffiffiffiffi M2 infinite low-energy expansion. Since the low-energy EFT is L p− Pl L L L ψ ¼ g 2 R þ D4 þ D6 þ light matter fieldsðg; Þ only meaningful at energy scales well below the cutoff, all higher-dimensional curvature operators should be under- Riemann4 þ O ; ð2:1Þ stood as being treated perturbatively, and this is indeed the M4 approach we shall take in what follows. Working pertur- batively implies that the dimension-four (curvature- where ψ designates symbolically all the light fields squared) operators do not affect the evolution of GWs (including the photon) that are explicitly included within when we restrict ourselves to a background perturbatively the low-energy EFT. The dynamics of these fields will not connected to the GR Schwarzschild background.1 To be relevant for this study as we shall be interested in determine the leading order corrections on the propaga- vacuum solutions. The higher-dimensional operators are tion of GWs, we therefore have to consider curvature given by dimension-six (curvature-cubed) operators. Including the pffiffiffiffiffiffi 2 2 2 L − 2 2 perturbative contribution from these operators, we extract D4 ¼ g½cR R þ cW Wμναβ þ cGBRGBð2:2Þ the effective metric seen by the metric perturbations and identify the speed of GWs. We find that the speed indeed and deviates from the speed of photons in general. As expected, 1 the deviation caused by the higher-dimensional operators is pffiffiffiffiffiffi μν 3 2 L 6 ¼ −g½d1R□R þ d2Rμν□R þ d3R þ d4RRμν highly suppressed. However, any departure from unity D M2 2 3 μν αβ is significant in itself, as it reflects the causal structure þ d5RRμναβ þ d6Rμν þ d7R R Rμναβ of the theory. In particular, the deviation vanishes as one μν αβγ αβ γσ μν approaches the horizon. This remarkable feature implies þ d8R RμαβγRν þ d9Rμν Rαβ Rγσ α β γ σ μ ν that while GWs and photons see a different causal structure þ d10Rμ ν Rα β Rγ σ ; ð2:3Þ almost everywhere, they still experience the horizon at the precise same location. We argue that this has to always be 2 2 − 4 2 2 where RGB ¼ Rμναβ Rμν þ R is the Gauss-Bonnet the case. For completeness we also compute the corrections term, and Wμναβ is the Weyl tensor. In four dimensions, on the quasinormal frequencies of the black holes in the the Gauss-Bonnet term is topological, which allows EFT of gravity. As expected, the corrections from the EFT us to rewrite the dimension-four curvature operator Lagrangian as 1 ffiffiffiffiffiffi In principle, there could be other branches of solutions in a L p− 2 μν theory with higher-dimensionaal operators, but those solutions D4 ¼ g½c1R þ c2RμνR ; ð2:4Þ rely on exciting the higher-dimensional operators beyond the regime of validity of the low-energy EFT.