Hees et al. (Phys. Rev. Lett. 124, 081101 (2020)) performed the first search for a variation of the fine-structure constant around a black hole by comparing absorption lines of different α\alpha-sensitivity in spectra of five late-type stars orbiting the supermassive black hole Sgr A*, obtaining Δα/α=(1.0±5.8)×10−6\Delta\alpha/\alpha=(1.0\pm5.8)\times10^{-6} and a coupling to gravitational potential βα=3.6±12.0\beta_\alpha=3.6\pm12.0 (68% C.L.) — still eight orders of magnitude less constraining than atomic-clock limits and one order less than the white-dwarf result, and limited by spectral resolution (R~5000–20000), few usable lines, and astrometric uncertainties in the gravitational potential. The open problem is to determine by improved high-resolution near-infrared spectroscopy of S-cluster stars whether α\alpha depends on gravitational potential in the black-hole regime: improve the βα\beta_\alpha limit by about one order of magnitude with dedicated observations of S0-6-class stars, by up to four orders of magnitude with next-generation adaptive-optics spectrographs on fainter stars closer to Sgr A*, control the potential-estimate uncertainties, and constrain scalar-field enhancement (black-hole scalarization) around the SMBH. The problem is open.

Background

The fine-structure constant α\alpha is the dimensionless coupling that sets the strength of electromagnetic interactions, and it enters atomic transition frequencies in a way that differs from line to line. Many unified theories and models of dark matter and dark energy predict that constants such as α\alpha may vary slowly in space-time; any such variation would violate Local Position Invariance, the part of the Einstein Equivalence Principle stating that the outcomes of non-gravitational experiments are independent of where and in what gravitational potential they are performed. Searches therefore compare atomic frequencies measured in different environments. The comparison is made quantitative through the sensitivity coefficient qq of each transition, defined by ω=ω0+qx\omega=\omega_0+qx with x=(α/α0)2−1x=(\alpha/\alpha_0)^2-1: lines with different qq shift by different fractional amounts if α\alpha differs between the source and the laboratory, so the offsets between lines within one spectrum yield Δα/α\Delta\alpha/\alpha directly, largely breaking the degeneracy with an ordinary Doppler redshift. When the environments sit at different Newtonian gravitational potentials UU, results are conventionally parametrized as Δα/α=βα ΔU/c2\Delta\alpha/\alpha=\beta_\alpha\,\Delta U/c^2, where βα\beta_\alpha is a dimensionless coupling fixed by the underlying theory and cc is the speed of light. Prior regimes constrain this coupling at very different depths of potential: long-term comparisons of atomic transition frequencies in the laboratory (Ashby, Parker and Patla 2018) bound βα\beta_\alpha at roughly the 10−710^{-7} level but only sample the shallow solar-system potential (ΔU/c2∼10−8\Delta U/c^2\sim10^{-8}), while spectra of white-dwarf photospheres (Berengut et al. 2013) probe deeper stellar potentials. Neither regime reaches the environment of a black hole, where several extensions of general relativity predict qualitatively new behavior: in scalar–Gauss–Bonnet gravity, vacuum black-hole solutions can undergo spontaneous growth of a scalar field (black-hole scalarization; Doneva and Yazadjiev 2018, Antoniou, Bakopoulos and Kanti 2018, Silva et al. 2018), and a scalar field coupled to the electromagnetic sector would make α\alpha vary with the local field value — an effect that could be enhanced near a black hole and is essentially unconstrained elsewhere. The Galactic Center hosts the supermassive black hole Sgr A* (mass ≈4×106\approx4\times10^6 solar masses) surrounded by the S-cluster of stars, including cool late-type giants whose atmospheres show many narrow metal absorption lines usable for the qq-coefficient method. Hees et al. (2020) combined spectra of five such stars (S0-6, S0-12, S0-13, S1-5, S1-23) from the NIFS instrument at Gemini (resolution R5000), IRCS at Subaru, and NIRSPEC at Keck (R20,000), all fed by adaptive optics (the technique that corrects atmospheric blur to reach diffraction-limited images in the crowded, dust-extincted Galactic Center field), with 25 years of astrometry used to estimate the gravitational potential U/c2∼10−6U/c^2\sim10^{-6} at each star. They obtained Δα/α=(1.0±5.8)×10−6\Delta\alpha/\alpha=(1.0\pm5.8)\times10^{-6} between the Galactic Center and Earth and βα=3.6±12.0\beta_\alpha=3.6\pm12.0, the first such measurement around a black hole and around so massive an object. The authors identify the limiting factors: spectral resolution and the small number of usable atomic lines; uncertainties in the gravitational potential inferred from astrometry and the black-hole mass model; and sensitivity far from the atomic-clock and white-dwarf benchmarks. They note that a dedicated NIRSPEC observation of S0-6 could improve the βα\beta_\alpha limit by about one order of magnitude, and that future adaptive-optics-fed spectrographs such as KAPA at Keck could observe fainter late-type stars orbiting closer to the Galactic Center (e.g., S0-38/S38), improving the constraint by up to four orders of magnitude and probing a higher gravitational potential, precisely the regime relevant to black-hole scalarization.

Problem Statement

Determine, by high-resolution near-infrared spectroscopy of late-type stars in the S-cluster of the Galactic Center, whether the fine-structure constant α\alpha differs between the strong gravitational environment of the supermassive black hole Sgr A* and Earth, and constrain the coupling of α\alpha to gravitational potential. Specifically: (1) measure Δα/α\Delta\alpha/\alpha from the relative offsets of atomic absorption lines with different sensitivity coefficients qq in spectra of Galactic Center late-type stars, improving on the first search's constraint of Δα/α=(1.0±5.8)×10−6\Delta\alpha/\alpha=(1.0\pm5.8)\times10^{-6} and βα=3.6±12.0\beta_\alpha=3.6\pm12.0 (68% C.L.) obtained from five stars at R~5000–20000 — a dedicated high-resolution observation of a star such as S0-6 is expected to improve the βα\beta_\alpha limit by about one order of magnitude, and adaptive-optics-fed next-generation spectrographs observing fainter late-type stars closer to Sgr A* (such as S0-38/S38) by up to four orders of magnitude, probing a higher gravitational potential; (2) determine the gravitational potential at each target star's location accurately enough that astrometric and black-hole-mass uncertainties do not dominate the βα\beta_\alpha error budget; (3) translate the measurement into bounds on scalar-field enhancement around the black hole, including black-hole scalarization scenarios. These three components are all required parts of one objective: a statistically defensible determination, at least one order of magnitude tighter in βα\beta_\alpha than Hees et al. (2020), of the dependence — or its absence — of the fine-structure constant on gravitational potential in the black-hole regime.

References

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