# Tidal Love numbers Love numbers tell us about the deformability or rigidity of an object, so studying them, e.g. for a neutron star or a black hole, is of astrophysical interest. In general, each Love number has a real part and an imaginary part:$k_{\ell m}(\omega)=\kappa_{\ell m}(\omega)+i \nu_{\ell m}(\omega).$For the Kerr black hole in four spacetime dimensions (including Schwarzschild), the *real* parts of the *static* ($\omega=0$) Love numbers are exactly zero. This suggests that such a black hole is in a sense perfectly rigid. It has been explained by the so-called Love symmetry. More generally, Love numbers do not vanish, e.g. in higher dimensions, in AdS, or in theories beyond vacuum Einstein. ## Love symmetry vs other known symmetries The Love symmetry is related to the near-horizon symmetries of extremal black holes, though this relation is not always trivial. For RN black holes,$\lim _{Q \rightarrow M} S L(2, \mathbb{R})_{\text {Love }}=S L(2, \mathbb{R})_{\mathrm{NH}}.$For Kerr black holes,$\lim _{a \rightarrow M} S L(2, \mathbb{R})_{\text {Love }} \ltimes \hat{U}(1) \supset S L(2, \mathbb{R})_{\mathrm{NH}}.$ The Kerr symmetry $SL(2,\mathbb{R})_L\times SL(2,R)_\mathbb{R}$ as studied in [[0520 Kerr-CFT correspondence|Kerr/CFT correspondence]] does not have a smooth Schwarzschild limit, while the Love symmetry does. ## Refs - original work: - [[1909#Love]] - original work for black holes in relativity: - [[2009#Damour, Nagar]] - [[2009#Binnington, Poisson]] (defines Love numbers in GR in a way similar to AdS/CFT) - higher dimensions (than 3+1): - [[2011#Kol, Smolkin]] - [[2019#Cardoso, Gualtieri, Moore]] - [[2024#Charalambous]] - rotating - [[2023#Rodriguez, Santoni, Solomon, Temoche]] (higher dimensions) - dynamical - [[2023#Perry, Rodriguez]]: dynamical Love numbers for Kerr are generally non-zero - [[2023#Chakraborty, Maggio, Silvestrini, Pani]]: Kerr-like objects - in [[0001 AdS-CFT|AdS/CFT]]: - [[2017#Emparan, Fernandez-Pique, Luna]] (planar black hole) - with [[0006 Higher-derivative gravity|higher-curvature]] corrections: - [[2018#Cardoso, Kimura, Maselli, Senatore]] - other theories beyond Einstein gravity: - [[2017#Cardoso, Franzin, Maselli, Pani, Raposo]] - in [[0582 Worldline EFT|worldline EFT]]: - [[2016#Porto]] - dimensionally reduced geometry and AdS$_2$/CFT$_1$: - [[2022#Kehagias, Perrone, Riotto]] - symmetry - [[2022#Charalambous, Dubovsky, Ivanov]]: Love symmetry - $SL(2,R)\times U(1)$ - [[2022#Hui, Joyce, Penco, Santoni, Solomon]]: Starobinsky symmetry - $SO(4,2)$ - [[2023#Perry, Rodriguez]]: $SL(2,R)\times SL(2,R)$ - [[2024#Sharma, Ghosh, Sarkar]]: ladder symmetry (for scalar perturbations) - [[2024#Gray, Keeler, Kubiznak, Martin]]: higher dimensions - dual CFT via [[0520 Kerr-CFT correspondence|Kerr/CFT]] - [[2023#Perry, Rodriguez]] - relation to Green's function - [[2024#De Luca, Garoffolo, Khoury, Trodden]]