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Rotating Stars in Relativity Living Rev Relativ (2017) 20:7 https://doi.org/10.1007/s41114-017-0008-x REVIEW ARTICLE Rotating stars in relativity Vasileios Paschalidis1,2 · Nikolaos Stergioulas3 Received: 8 December 2016 / Accepted: 3 October 2017 / Published online: 29 November 2017 © The Author(s) 2017. This article is an open access publication Abstract Rotating relativistic stars have been studied extensively in recent years, both theoretically and observationally, because of the information they might yield about the equation of state of matter at extremely high densities and because they are considered to be promising sources of gravitational waves. The latest theoretical understanding of rotating stars in relativity is reviewed in this updated article. The sec- tions on equilibrium properties and on nonaxisymmetric oscillations and instabilities in f -modes and r-modes have been updated. Several new sections have been added on equilibria in modified theories of gravity, approximate universal relationships, the one-arm spiral instability, on analytic solutions for the exterior spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating stars in numerical relativity including both hydrodynamic and magnetohydrodynamic studies of these objects. Keywords Relativistic stars · Rotation · Stability · Oscillations · Magnetic fields · Numerical relativity This article is a revised version of https://doi.org/10.12942/lrr-2003-3. Change summary: Major revision, updated and expanded. Change details: The article has been substantially revised and updated. Expanded all sections, and moved former Subsect. 2.10 to Sect. 3. The number of references (now 863) has more than doubled. B Nikolaos Stergioulas [email protected] Vasileios Paschalidis [email protected] 1 Theoretical Astrophysics Program, Departments of Astronomy and Physics, University of Arizona, Tucson, AZ 85721, USA 2 Department of Physics, Princeton University, Princeton, NJ 08544, USA 3 Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece 123 7 Page 2 of 169 V. Paschalidis, N. Stergioulas Contents 1 Introduction ............................................... 2 The equilibrium structure of rotating relativistic stars ......................... 2.1 Assumptions ............................................ 2.2 Geometry of spacetime ...................................... 2.3 The rotating fluid ......................................... 2.4 Equations of structure ....................................... 2.5 Rotation law and equilibrium quantities ............................. 2.6 Equations of state ......................................... 2.6.1 Relativistic polytropes ................................... 2.6.2 Hadronic equations of state ................................ 2.6.3 Strange quark equations of state .............................. 2.7 Numerical schemes ........................................ 2.7.1 Hartle ........................................... 2.7.2 Butterworth and Ipser (BI) ................................ 2.7.3 Komatsu, Eriguchi, and Hachisu (KEH) ......................... 2.7.4 Bonazzola et al. (BGSM) ................................. 2.7.5 LORENE/rotstar ...................................... 2.7.6 Ansorg et al. (AKM) .................................... 2.7.7 IWM-CFC approximation ................................. 2.7.8 The virial identities .................................... 2.7.9 Direct comparison of numerical codes .......................... 2.8 Analytic approximations to the exterior spacetime ........................ 2.9 Properties of equilibrium models ................................. 2.9.1 Bulk properties and sequences of equilibrium models .................. 2.9.2 Multipole moments .................................... 2.9.3 Mass-shedding limit and the empirical formula ...................... 2.9.4 Upper limits on mass and rotation: theory versus observation .............. 2.9.5 Maximum mass set by causality .............................. 2.9.6 Minimum period set by causality ............................. 2.9.7 Moment of inertia and ellipticity ............................. 2.9.8 Rotating strange quark stars ................................ 2.9.9 Rotating magnetized neutron stars ............................ 2.10 Approximate universal relationships ............................... 2.11 Rapidly rotating equilibrium configurations in modified theories of gravitation ......... 2.12 Differentially rotating neutron stars ................................ 2.13 Proto-neutron stars ........................................ 3 Rotating relativistic stars in LMXBs .................................. 3.1 Particle orbits and kHz quasi-periodic oscillations ........................ 3.2 Angular momentum conservation during burst oscillations ................... 4 Oscillations and stability ........................................ 4.1 Quasi-normal modes of oscillation ................................ 4.2 Effect of rotation on quasi-normal modes ............................. 4.3 Axisymmetric perturbations .................................... 4.3.1 Secular and dynamical axisymmetric instability ..................... 4.3.2 Axisymmetric pulsation modes .............................. 4.4 Nonaxisymmetric perturbations .................................. 4.4.1 Nonrotating limit ..................................... 4.4.2 Slow rotation approximation ............................... 4.4.3 Post-Newtonian approximation .............................. 4.4.4 Cowling approximation .................................. 4.5 Nonaxisymmetric instabilities .................................. 4.5.1 Introduction ........................................ 4.5.2 CFS instability of polar modes .............................. 4.5.3 CFS instability of axial modes .............................. 4.5.4 Effect of viscosity on the CFS instability ......................... 123 Rotating stars in relativity Page 3 of 169 7 4.5.5 Gravitational radiation from CFS instability ....................... 4.5.6 Viscosity-driven instability ................................ 4.5.7 One-arm (spiral) instability ................................ 5 Dynamical simulations of rotating stars in numerical relativity .................... 5.1 Numerical evolution of equilibrium models ........................... 5.1.1 Stable equilibrium ..................................... 5.1.2 Instability to collapse ................................... 5.1.3 Dynamical bar-mode instability .............................. 5.2 Pulsations of rotating stars .................................... 5.3 Rotating core collapse ....................................... 5.3.1 Collapse to a rotating black hole ............................. 5.4 Formation of rotating neutron stars ................................ 5.4.1 Stellar collapse to a rotating neutron star ......................... 5.4.2 Binary neutron star mergers ................................ 5.4.3 One-arm instability .................................... 5.5 Evolution of magnetized, rotating neutron stars ......................... 5.5.1 Magnetic instabilities ................................... 5.5.2 Magnetically driven outflows ............................... 5.5.3 Magnetospheric studies .................................. References .................................................. Publisher’s Note ....................................... 1 Introduction Rotating relativistic stars are of fundamental interest in physics. Their bulk properties constrain the proposed equations of state for densities greater than the nuclear satu- ration density. Accreted matter in their gravitational fields undergoes high-frequency oscillations that could become a sensitive probe for general relativistic effects. Tem- poral changes in the rotational period of millisecond pulsars can also reveal a wealth of information about important physical processes inside the stars or of cosmological relevance. In addition, rotational instabilities can result in the generation of copious amounts of gravitational waves the detection of which would initiate a new field of observational asteroseismology of relativistic stars. The latter is of particular impor- tance because with the first direct detections of gravitational waves by the LIGO and VIRGO collaborations (Abbott et al. 2016a, b) the era of gravitational wave astronomy has arrived. There exist several independent numerical codes for obtaining accurate models of rotating neutron stars in full general relativity, including two that are publicly avail- able. The uncertainty in the high-density equation of state still allows numerically constructed maximum mass models to differ by more than 50% in mass, radius and angular velocity, and by a larger factor in the moment of inertia. Given these uncer- tainties, an absolute upper limit on the rotation of relativistic stars can be obtained by imposing causality as the only requirement on the equation of state. It then follows that gravitationally bound stars cannot rotate faster than 0.41 ms. In rotating stars, nonaxisymmetric oscillations have been studied in various approxi- mations (the Newtonian limit and the post-Newtonian approximation, the slow rotation limit, the Cowling approximation, the spatial conformal flatness approximation) as an eigenvalue problem. Normal modes in full general relativity have been obtained through numerical simulations only. Time evolutions of the linearized equations have 123 7 Page 4 of 169 V. Paschalidis, N. Stergioulas improved our understanding of the spectrum of axial and hybrid modes
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