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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. 

We derive new terms in the post-Newtonian (PN) expansion of the generalized redshift invariant hu tiτ for a small body in eccentric, equatorial orbit about a massive Kerr black hole. The series is computed analytically using the Teukolsky formalism for first-order black hole perturbation theory, along with the Chrzanowski, Cohen, Kegeles method for metric reconstruction using the Hertz potential in ingoing radiation gauge. Modal contributions with small values of l are derived via the semianalytic solution of Mano-Suzuki-Takasugi, while the remaining values of l to infinity are determined via direct expansion of the Teukolsky equation. Each PN order is calculated as a series in eccentricity e but kept exact in the primary black hole’s spin parameter a. In total, the PN terms are expanded to e16 through 6PN relative order, and separately to e10 through 8PN relative order. Upon grouping eccentricity coefficients by spin dependence, we find that many resulting component terms can be simplified to closed-form functions of eccentricity, in close analogy to corresponding terms derived previously in the Schwarzschild limit. We use numerical calculations to compare convergence of the full series to its Schwarzschild counterpart and discuss implications for gravitational wave analysis. 

We calculate the high-order post-Newtonian (PN) expansion of the energy and angular momentum fluxes onto the horizon of a nonspinning black hole primary in eccentric-orbit extreme-mass-ratio inspirals. The first-order black hole perturbation theory calculation uses Mathematica and makes an analytic expansion of the Regge-Wheeler-Zerilli functions using the Mano-Suzuki-Takasugi formalism. The horizon absorption, or tidal heating and torquing, is calculated to 18PN relative to the leading horizon flux (i.e., 22PN order relative to the leading quadrupole flux at infinity). Each PN term is a function of eccentricity e and is calculated as a series to e10. A second expansion, to 10PN horizon-relative order (or 14PN relative to the flux at infinity), is computed deeper in eccentricity to e20. A number of resummed closed-form functions are found for the low PN terms in the series. Using a separate Teukolsky perturbation code, numerical comparisons are made to test how accurate the PN expansion is when extended to a close p =10 orbit. We find that the horizon absorption expansion is not as convergent as a previously computed infinity-side flux expansion. However, given that the horizon absorption is suppressed by 4PN, useful results can be obtained even with an orbit as tight as this for e ≲1 /2 . Combining the present results with our earlier expansion of the fluxes to infinity makes the knowledge of the total dissipation known to 19PN for eccentric-orbit nonspinning extreme-mass-ratio inspirals. 

With the precision now afforded by modern space-based photometric observations from the retired K2 and current TESS missions, the effects of general relativity (GR) may be detectable in the light curves of pulsating white dwarfs (WDs). Almost all WD models are calculated using a Newtonian description of gravity and hydrodynamics. To determine if the inclusion of GR leads to observable effects, we used idealized models of compact stars and made side-by-side comparisons of mode periods computed using a: (i) Newtonian and (ii) GR description of the equilibrium structure and nonradial pulsations. For application to WDs, it is only necessary to include the first post- Newtonian (1PN) approximation to GR. The mathematical nature of the linear nonradial pulsation problem is then qualitatively unchanged and the GR corrections can be written as extensions of the classic Dziembowski equations. As such, GR effects might easily be included in existing asteroseismology codes. The idealized stellar models are (i) 1PN relativistic polytropes and (ii) stars with a cold degenerate electron equation of state featuring a near-surface chemical transition from μe = 2 to μe = 1, simulating a surface hydrogen layer. A comparison of Newtonian and 1PN normal mode periods reveals fractional differences in the order of the surface gravitational redshift z. For a typical WD, this fractional difference is ∼10−4 and is greater than the period uncertainty σΠ/Π of many WD pulsation modes observed by TESS. Consistent theoretical modeling of periods observed in these stars should, in principle, include GR effects to 1PN order.