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Relativistic and Dynamical Love

Abhishek Hegade K. R., K. J. Kwon, Tejaswi Venumadhav, Hang Yu, Nicolás Yunes

Abstract

Gravitational waves emitted in the late inspiral of binary neutron stars are affected by their tidal deformation. We study the tidal dynamics in full general relativity through matched-asymptotic expansions and prove that the dynamical tidal response can be expanded in a complete set of modes. We further prove that the mode amplitudes satisfy an effective, forced harmonic oscillator equation, which generalizes the overlap-integral formulation of Newtonian gravity. Our relativistic treatment of dynamical tides will avoid systematic biases in future gravitational-wave parameter estimation.

Relativistic and Dynamical Love

Abstract

Gravitational waves emitted in the late inspiral of binary neutron stars are affected by their tidal deformation. We study the tidal dynamics in full general relativity through matched-asymptotic expansions and prove that the dynamical tidal response can be expanded in a complete set of modes. We further prove that the mode amplitudes satisfy an effective, forced harmonic oscillator equation, which generalizes the overlap-integral formulation of Newtonian gravity. Our relativistic treatment of dynamical tides will avoid systematic biases in future gravitational-wave parameter estimation.

Paper Structure

This paper contains 11 sections, 73 equations, 1 figure.

Table of Contents

  1. Outline of the proof of self-adjointness of the operator equation
  2. Background Einstein and stress energy conservation equations
  3. Linearized Einstein-Euler system
  4. 3+1 split of the operator equation
  5. Proof in harmonic gauge
  6. Comments on the proof and comparison to Newtonian theory
  7. Validity of the 1PN boundary conditions and extension to higher PN order
  8. Tidal perturbations and the dynamical tidal response function
  9. Expression for the tidal contributions to the metric perturbation in a PN expansion
  10. Derivation of the formula for the dynamical tidal response function
  11. Expressions for the tidal force, tidal stress-energy tensor and the dimensionless overlap integrals

Figures (1)

  • Figure 1: Cartoon (not to scale) of the near zone (full figure) in a tidally interacting system, consisting of a star (pink disk) of finite radius (dashed white circle), and a tidal source (green disk) located at a characteristic distance $L\sim L_{\mathrm{source}}$ from the center of mass of the star. We divide the near zone into different regions: inner and outer body zones, a buffer zone and a PN zone. The gravitational field in the inner (pink disk) and outer (red annulus) body zones is strong, while that in the PN zone (blue) is weak. The outer body and PN zone solutions are matched asymptotically in the buffer zone (purple annulus), which has a mean radius $d$ from the center of mass of the star.