Kip S. Thorne's Nobel Lecture "LIGO and gravitational waves III" (Reviews of Modern Physics 90, 040503, 2018) closes with a set of open questions for gravitational-wave astronomy. The sharpest, stated as a major goal of the LISA mission, is whether the spacetime geometry of the massive compact objects at the centers of galaxies is the Kerr geometry predicted by general relativity, with all multipole moments fixed by the no-hair relation, or whether observations will reveal deviations: multipolar deviations, horizonless or hairy compact objects, firewall-modified near-horizon geometry, or naked singularities. Extreme-mass-ratio inspirals (EMRIs), which encode a high-precision map of the central object's multipole structure over thousands to hundreds of thousands of gravitational-wave cycles in the strong-field regime, are the designated probe, via the LISA space interferometer (adopted by ESA in January 2024, launch planned 2035). The problem is open: as of 2026 no EMRI has been observed, all EMRI Kerr-test literature is prospective, and current ground-based tests of the Kerr question reach only percent-level precision.

Background

In September 2015 the LIGO interferometers detected gravitational waves — propagating ripples of spacetime curvature — from the merger of two stellar-mass black holes, opening gravitational-wave astronomy. Kip S. Thorne's Nobel Lecture (Reviews of Modern Physics 90, 040503 (2018)) recounts that achievement and ends with an outlook of open questions for the field's future.

According to general relativity, an isolated, stationary, spinning black hole in vacuum is described by the Kerr geometry, a solution fixed by only two parameters: the mass MM and the angular momentum (spin) J=MaJ = Ma, where aa is the spin parameter. In particular, all of the hole's multipole moments — the coefficients MℓM_\ell (mass) and SℓS_\ell (current) of the far-field gravitational potential, analogous to the multipoles of a charge distribution in electrostatics — obey the "no-hair" relation Mℓ+iSℓ=M(ia)ℓM_\ell + iS_\ell = M(ia)^\ell. This is a strong and testable statement: for example, the quadrupole moment must be exactly M2=−Ma2M_2 = -Ma^2. Any measured deviation from this relation would mean the central object is not a Kerr black hole. Candidate alternatives discussed in the literature include boson stars and other horizonless "exotic compact objects", black holes carrying additional fields ("hair"), and — motivated by the quantum-mechanical debate about information loss in black holes — "firewalls": hypothetical high-energy structure at the would-be horizon that could modify the exterior geometry near it (Almheiri, Marolf, Polchinski, and Sully, 2013). A stationary, axisymmetric vacuum spacetime that is not Kerr could even contain a naked singularity, a curvature singularity not hidden behind a horizon.

The observational route that the lecture singles out is extreme-mass-ratio inspirals (EMRIs): a stellar-mass or intermediate-mass compact object (the secondary) spiraling into a massive black hole of roughly 10410^4 to 10710^7 solar masses, radiating away orbital energy and angular momentum as gravitational waves (a process called radiation reaction, computed within black-hole perturbation theory as a "self-force"). Over the years before plunge, the secondary completes 10310^3 to 10510^5 orbital cycles in the strong-field region at radii of a few gravitational radii, and the emitted millihertz gravitational waves phase-encode the central geometry. Fintan Ryan showed in 1995 that the sequence of orbital frequencies of a body inspiraling into an axisymmetric body with arbitrary multipole moments reveals those multipoles one after another; this is the basis of "spacetime mapping" — using the inspiral as a ruler for the central object's multipole structure.

Detecting EMRIs requires a space-based interferometer in the millihertz band, below the seismic and gravity-gradient noise floor of ground-based detectors: LISA (Laser Interferometer Space Antenna), a constellation of three drag-free spacecraft tracked by laser beams over millions of kilometers, which was adopted by the European Space Agency in January 2024 with launch planned for 2035. The LISA Definition Study Report codifies the question as a formal science investigation: "Are the massive objects observed at the centres of galaxies consistent with the rotating Kerr [massive black holes] predicted by [general relativity]?", with high-signal-to-noise "golden" EMRIs expected to constrain deviations of the quadrupole moment from Kerr down to the 10−410^{-4} level, the spin to 10−410^{-4}, and the mass to 10−510^{-5}.

Current ground-based detectors already test the Kerr question, but only weakly: black-hole spectroscopy — checking that the "ringdown" of a merger remnant, a superposition of damped oscillations called quasinormal modes, matches the Kerr spectrum — and inspiral measurements of the spin-induced multipoles find consistency with Kerr at the percent to tens-of-percent level (e.g., the LIGO–Virgo–KAGRA ringdown analysis of GW250114 in 2025, and the quadrupole/octupole test of GW241011 in 2026). They cannot reach the 10−410^{-4} multipole precision of a high-signal-to-noise EMRI, nor probe near-horizon structure, and they probe remnants of stellar-mass mergers rather than the massive objects at galactic centers. As of 2026, no EMRI has been observed; all EMRI Kerr-test literature consists of waveform frameworks and forecasting studies. The open problem is the determination itself, together with the scientific apparatus needed to make it sound.

Problem Statement

Determine whether the spacetime geometry of the massive compact objects observed at the centers of galaxies is the Kerr geometry of general relativity — the multipole structure fixed by the no-hair relation Mℓ+iSℓ=M(ia)ℓM_\ell + iS_\ell = M(ia)^\ell — or whether it deviates from Kerr, using the high-precision mapping of the geometry encoded in extreme-mass-ratio-inspiral (EMRI) gravitational waves. This is the question posed as a major goal of the LISA mission in Thorne's Nobel Lecture and codified in the LISA science case, and it comprises the jointly linked components needed to answer it:

(1) Waveform models for EMRIs into parameterized non-Kerr geometries (generic multipole structure, and possibly non-Kerr near-horizon behavior), with radiation-reaction-driven orbital evolution, phase-accurate at the sub-radian level across the 10310^3 to 10510^5 in-band cycles — the standard accuracy convention for EMRI parameter estimation — so that genuine geometric deviations are not confused with waveform-modeling error.

(2) A measurement framework that converts EMRI observations into constraints on the deviation parameters (for example, fractional quadrupole deviations at the 10−410^{-4} level and the corresponding higher moments), with a controlled treatment of the systematics that could mimic deviations: environmental matter around the massive object, the secondary's spin and possible charge, and instrumental noise.

(3) The observational determination itself, once LISA data exist (launch planned 2035): per-event and population-level statements of consistency with, or deviation from, the Kerr geometry, including the targeted search for near-horizon structures such as firewall-modified geometry, and for massive compact objects that are not black holes at all, such as horizonless bodies or naked singularities orbited by smaller bodies.

The determination rests, at the precision EMRI mapping affords, on the sound scientific apparatus of components (1) and (2), with the data-driven determination of component (3) deferred until LISA data exist; the apparatus and the level of Kerr-deviation constraint it makes reachable are therefore themselves the subject of the question.

References

  1. Kip S. Thorne (2018), "Nobel Lecture: LIGO and gravitational waves III", Reviews of Modern Physics 90, 040503. DOI: 10.1103/RevModPhys.90.040503, https://doi.org/10.1103/RevModPhys.90.040503
  2. Fintan D. Ryan (1995), "Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments", Physical Review D 52, 5707. DOI: 10.1103/PhysRevD.52.5707, https://doi.org/10.1103/PhysRevD.52.5707
  3. Monica Colpi, Karsten Danzmann, Martin Hewitson, Kelly Holley-Bockelmann, Philippe Jetzer, et al. (LISA Consortium) (2024), "LISA Definition Study Report" (Red Book), arXiv:2402.07571, https://arxiv.org/abs/2402.07571 (Sec. 3.5.2: EMRI spacetime mapping and the Kerr question)
  4. NASA LISA Project Office (2024–2026), "LISA: Laser Interferometer Space Antenna" (mission website; ESA's Science Programme Committee formally adopted LISA as a flight project on 25 January 2024, moving it into the implementation phase, and launch is planned for 2035), https://lisa.nasa.gov/
  5. LIGO Scientific Collaboration, Virgo Collaboration, KAGRA Collaboration (2025), "Black Hole Spectroscopy and Tests of General Relativity with GW250114", arXiv:2509.08099, https://arxiv.org/abs/2509.08099
  6. Rimo Das, N. V. Krishnendu, M. Saleem, Chandra Kant Mishra, K. G. Arun (2026), "Testing the Kerr hypothesis beyond the quadrupole with GW241011", arXiv:2604.09828, https://arxiv.org/abs/2604.09828
  7. Lorenzo Speri, Francisco Duque, Susanna Barsanti, Alessandro Santini, Shubham Kejriwal, et al. (2026), "Quantifying the Scientific Potential of Intermediate and Extreme Mass Ratio Inspirals with the Laser Interferometer Space Antenna", arXiv:2603.17072, https://arxiv.org/abs/2603.17072
  8. Pablo F. Muguruza, Carlos F. Sopuerta (2026), "Probing Kerr Symmetry Breaking with LISA Extreme-Mass-Ratio Inspirals", arXiv:2604.06053, https://arxiv.org/abs/2604.06053
  9. Ahmed Almheiri, Donald Marolf, Joseph Polchinski, James Sully (2013), "Black holes: complementarity or firewalls?", Journal of High Energy Physics 2013, 062. DOI: 10.1007/JHEP02(2013)062, https://doi.org/10.1007/JHEP02(2013)062