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Inverting no-hair theorems: How requiring general relativity solutions restricts scalar-tensor theories

Hajime Kobayashi1, Shinji Mukohyama1,2, Johannes Noller3,4, Sergi Sirera1,3,4,*, Kazufumi Takahashi1, and Vicharit Yingcharoenrat2,5

  • 1Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan
  • 2Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
  • 3Department of Physics and Astronomy, University College London, London, WC1E 6BT, United Kingdom
  • 4Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom
  • 5High Energy Physics Research Unit, Department of Physics, Faculty of Science, Chulalongkorn University, Pathumwan, Bangkok 10330, Thailand

  • *Contact author: sergi.sirera-lahoz@port.ac.uk
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Phys. Rev. D 111, 124022 – Published 16 June, 2025

DOI: https://doi.org/10.1103/75sw-4f7n

Abstract

Black hole solutions in general scalar-tensor theories are known to permit hair, i.e., nontrivial scalar profiles and/or metric solutions different from the ones of general relativity (GR). Imposing that some such solutions—e.g., Schwarzschild or de Sitter solutions motivated in the context of black hole physics or cosmology—should exist, the space of scalar-tensor theories is strongly restricted. Here we investigate precisely what these restrictions are within general quadratic/cubic higher-order scalar-tensor theories for stealth solutions, whose metric is given by that in GR, supporting time-dependent scalar hair with a constant kinetic term. We derive, in a fully covariant approach, the conditions under which the Euler-Lagrange equations admit all (or a specific set of) exact GR solutions, as the first step toward our understanding of a wider class of theories that admit approximately stealth solutions. Focusing on static and spherically symmetric black hole spacetimes, we study the dynamics of linear odd-parity perturbations and discuss possible deviations from GR. Importantly, we find that requiring the existence of all stealth solutions prevents any deviations from GR in the odd-parity sector. In less restrictive scenarios, in particular, for theories only requiring the existence of Schwarzschild(–de Sitter) black holes, we identify allowed deviations from GR, derive the stability conditions for the odd modes, and investigate the generic deviation of a nontrivial speed of gravitational waves. All calculations performed in this paper are reproducible via companion Mathematica notebooks.

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Phys. Rev. D 111, 124022– Published 16 June, 2025
  • Received 18 March 2025
  • Accepted 27 May 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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