Where is Love? Tidal deformability in the black hole compactness limit

Abstract

One of the macroscopically measurable effects of gravity is the tidal deformability of astrophysical objects, which can be quantified by their tidal Love numbers. For planets and stars, these numbers measure the resistance of their material against the tidal forces, and the resulting contribution to their gravitational multipole moments. According to general relativity, deformed black holes, instead, show no addition to their gravitational multipole moments, and all of their Love numbers are zero. In this paper we explore different configurations of nonrotating compact and ultracompact stars to bridge the compactness gap between black holes and neutron stars and calculate their Love number k 2 . We calculate k 2 for the first time for uniform density ultracompact stars with mass M and radius R beyond the Buchdahl limit (compactness C = M/R > 4/9), and we find that k 2 → 0 + as C → 1/2, i.e., the Schwarzschild black hole limit. Our results provide insight on the zero tidal deformability limit and we use current constraints on the binary tidal deformability Λ from GW170817 (and future upper limits from binary black hole mergers) to propose tests of alternative models. ‡ It is also a case of bad scientific work eventually causing a big headache for its author [2].

Tidal deformability of ultracompact Schwarzschild stars and their approach to the black hole limit

“…In this paper, we will review the main results of the tidal deformability of ultracompact Schwarzschild stars presented by Chirenti et al (2020). As an addition to those results, we will show that the tidal Love number k 2 for ultracompact Schwarzschild stars do not follow the 1/ log ξ proposed by (Cardoso et al, 2017), instead, we found that k 2 decays exponentially as a function of the compactness.…”

“…The tidal Love number for constant-density stars, or Schwarzschild stars, was studied by (Damour and Nagar, 2009;Postnikov et al, 2010;Chan et al, 2015) and more recently by (Chirenti et al, 2020). The details of the computation were discussed in these papers so we refer the reader to those works.…”

“…The interior Schwarzschild solution describes a configuration of constant energy density . For convenience, it can be written in terms of the auxiliary function y (Chandrasekhar and Miller, 1974;Posada and Chirenti, 2019;Chirenti et al, 2020) defined as…”

“…( 14), has a regular singular point at x 0 ≡ −κ. The careful analysis of this singularity, for the computation of the Love number, was done in (Chirenti et al, 2020) so we refer the reader to that paper for more details.…”

“…On the other hand, it has been found that the Love numbers of general ECOs scale as k ∼ 1/ log ξ, where ξ is a parameter which measures how much the object deviates from the BH geometry (Cardoso et al, 2017). Recently Chirenti et al (2020) studied the tidal Love number of ultracompact Schwarzschild stars, below and beyond the Buchdahl limit. These authors found that the Love number of these configurations tends to zero as the compactness approaches the BH limit.…”