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ABSTRACT Ultra-massive white dwarf stars are currently being discovered at a considerable rate, thanks to surveys such as the Gaia space mission. These dense and compact stellar remnants likely play a major role in Type Ia supernova explosions. It is possible to probe the interiors of ultra-massive white dwarfs through asteroseismology. In the case of the most massive white dwarfs, general relativity could affect their structure and pulsations substantially. In this work, we present results of relativistic pulsation calculations employing relativistic ultra-massive ONe-core white dwarf models with hydrogen-rich atmospheres and masses ranging from 1.29 to $$1.369 \ \mathrm{M}_{\odot }$$ with the aim of assessing the impact of general relativity on the adiabatic gravity (g)-mode period spectrum of very high mass ZZ Ceti stars. Employing the relativistic Cowling approximation for the pulsation analysis, we find that the critical buoyancy (Brunt–Väisälä) and acoustic (Lamb) frequencies are larger for the relativistic case, compared to the Newtonian case, due to the relativistic white dwarf models having smaller radii and higher gravities for a fixed stellar mass. In addition, the g-mode periods are shorter in the relativistic case than those in the Newtonian computations, with relative differences of up to ∼$50$ per cent for the highest mass models ($$1.369 \ \mathrm{M}_{\odot }$$) and for effective temperatures typical of the ZZ Ceti instability strip. Hence, the effects of general relativity on the structure, evolution, and pulsations of white dwarfs with masses larger than ∼$$1.29 \ \mathrm{M}_{\odot }$$ cannot be ignored in the asteroseismological analysis of ultra-massive ZZ Ceti stars. 

ABSTRACT G 29 − 38 (TIC 422526868) is one of the brightest (V = 13.1) and closest (d = 17.51 pc) pulsating white dwarfs with a hydrogen-rich atmosphere (DAV/ZZ Ceti class). It was observed by the TESS spacecraft in sectors 42 and 56. The atmosphere of G 29 − 38 is polluted by heavy elements that are expected to sink out of visible layers on short time-scales. The photometric TESS data set spans ∼51 d in total, and from this, we identified 56 significant pulsation frequencies, that include rotational frequency multiplets. In addition, we identified 30 combination frequencies in each sector. The oscillation frequencies that we found are associated with g-mode pulsations, with periods spanning from ∼ 260 to ∼ 1400 s. We identified rotational frequency triplets with a mean separation δνℓ = 1 of 4.67 μHz and a quintuplet with a mean separation δνℓ = 2 of 6.67 μHz, from which we estimated a rotation period of about 1.35 ± 0.1 d. We determined a constant period spacing of 41.20 s for ℓ = 1 modes and 22.58 s for ℓ = 2 modes. We performed period-to-period fit analyses and found an asteroseismological model with M⋆/M⊙ = 0.632 ± 0.03, $$T_{\rm eff}=11\, 635\pm 178$$ K, and log g = 8.048 ± 0.005 (with a hydrogen envelope mass of MH ∼ 5.6 × 10−5M⋆), in good agreement with the values derived from spectroscopy. We obtained an asteroseismic distance of 17.54 pc, which is in excellent agreement with that provided by Gaia (17.51 pc).