Relativistic Tidal Dissipation and the Gravitational-wave Signal of a White Dwarf Orbiting an Intermediate-Mass Black Hole

Abstract

Finding intermediate-mass black holes (IMBHs) and measuring their masses and spins are key to understanding massive black hole formation. White dwarf (WD)-IMBH binaries provide a unique probe because they emit both electromagnetic radiation and gravitational waves (GWs), thereby conveying richer information. However, such multi-messenger sources often enter the regime of strong gravity, where existing models fail to capture their relativistic dynamics. Here, we develop a fully relativistic model for the tidal response of a WD close to an IMBH and use it to study the secular orbital evolution as well as the GW signal. We find that for IMBHs more massive than 10^5 solar masses, tidal interaction becomes relativistic and sensitive to IMBH spin. The interaction generally dissipates binary orbital energy and angular momentum, but due to relativistic frame rotation, which reduces phase coherence across pericenter passages, the orbit-averaged tidal dissipation rate can be suppressed by up to about 50% relative to Newtonian predictions. Including tidal dissipation leads to more rapid damping of the orbital eccentricity, to the extent that the pericenter distance may even increase over time, potentially explaining quasi-periodic eruptions and secular orbital period growth. Such tidal effects accumulate into measurable phase and amplitude deviations in the GW signal. For typical space-based observations, the GW waveform mismatch can reach values of order 0.1 within 6 months. Our results indicate that relativistic tidal dissipation is both dynamically important and observationally essential for reliably predicting the multi-messenger signals of WD-IMBH systems.

Figures (9)

• Figure 1: Ratio of the Newtonian TDE radius of a WD, $r_{t}$, to the gravitational radius of an IMBH, $r_{g}$, shown in the $M_{\star}$-$M_{\bullet}$ parameter space. The white lines indicate constant constant SNRs ($=10$, $20$, $40$) for LISA observation, assuming a pericenter distance of $r_p = 2\,r_t$, a redshift of $z=0.02$ ($d_{L} \simeq 85$Mpc), and an observation period of $4$ years, whereas the solid and dotted line styles correspond to orbital eccentricities of $e=0.95$ and $e=0.99$, respectively. The region below the black dot-dashed line in the lower-right corner corresponds to $R_{\star}/r_{t} > 5\%$, where the assumption of quadrupole tidal field becomes invalid. The stars mark the inferred locations of a sample of QPEs.
• Figure 2: Minimum allowed pericenter distance for a WD–IMBH system due to either relativistic plunge or tidal disruption, shown in the parameter space of IMBH spin $a$ and orbital eccentricity $e$. The red line marks the points at which the critical plunge radius $r_{\rm plunge}(a,e)$ equals the critical TDE radius $r_{\rm TDE}(M_{\star},M_{\bullet},a,e)$. Therefore, in the lower-left region, $r_{\rm plunge}>r_{\rm TDE}$, and a WD will plunge into IMBH before tidal disruption. On the contrary, a WD in the upper-right region gets tidally disrupted before plunging. Here we have assumed $(M_{\star},M_{\bullet}) = (0.6\,M_{\odot},10^{5}\,M_{\odot})$.
• Figure 3: Orientation of the Fermi normal coordinates for a WD on (a) an eccentric Newtonian orbit, (b) an eccentric orbit around a Schwarzschild IMBH, (c) a prograde circular orbit in the equatorial plane of a Kerr IMBH, and (d) a retrograde circular orbit in the equatorial plane of a Kerr IMBH. In each panel, the blue arrow represents the tetrad vector $\mathbf{\Lambda}^{(1)}$. The azimuthal angle $\phi$ in the Boyer-Lindquist coordinates and the rotational angle $\Psi$ associated with the Fermi-normal coordinates are also marked.
• Figure 4: Orbital evolution streamlines in the $e-r_{p}$ parameter space for a WD orbiting a non-spinning IMBH under the combined effects of gravitational radiation and TD. Solid curves show orbital trajectories when both mechanisms are included, while dashed curves correspond to evolution driven solely by gravitational radiation. The color map encodes the pericenter evolution rate $\log_{10}\left ( \left | \dot r_{p,\mathrm{GW}}+\dot r_{p,\mathrm{TD}} \right | \right )$ under two effects, where $\dot r_{ p}= ({\rm d}r_{ p}/{\rm d}t)/c$. The region below the red curve indicates the plunge region, defined by $r_{p} < r_{\rm plunge}$, where the orbit directly falls into the IMBH event horizon. The purple strip adjacent to the red curve denotes the TDE region, defined by $r_{\mathrm{plunge}} <r_{p} < r_{t}$. Yellow dots mark the locations along the streamlines where the pericenter distance $r_{p}$ reaches a maximum. In this plot, we assume $M_{\star} = 0.6\,M_\odot$ and $M_{\bullet} = 10^{5}\,M_\odot$.
• Figure 5: Comparison of the TD-induced angular-momentum (upper panels) and energy (lower panels) loss rates computed in the Newtonian and full GR cases. The color maps show the ratios of the rates obtained in the two cases, and the purple curves mark the loci where the ratio equals unity. The left and right panels correspond to IMBH spins $a=-0.9M_{\bullet}$ and $a=0.9M_{\bullet}$, respectively. All other parameters are the same as those in FIG. \ref{['fig:drpdt_dedt_combined']}.