Rotating Stars in Relativity

Noname manuscript No. (will be inserted by the editor) Rotating Stars in Relativity Vasileios Paschalidis Nikolaos Stergioulas · Received: date / Accepted: date 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 re- viewed in this updated article. The sections 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 modi- fied 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 Vasileios Paschalidis Theoretical Astrophysics Program, Departments of Astronomy and Physics, University of Arizona, Tucson, AZ 85721, USA Department of Physics, Princeton University Princeton, NJ 08544, USA E-mail: [email protected] http://physics.princeton.edu/~vp16/ Nikolaos Stergioulas Department of Physics, Aristotle University of Thessaloniki Thessaloniki, 54124, Greece E-mail: [email protected] http://www.astro.auth.gr/~niksterg arXiv:1612.03050v2 [astro-ph.HE] 30 Nov 2017 2 Vasileios Paschalidis, Nikolaos Stergioulas The article has been substantially revised and updated. New Section 5 and various Subsections (2.3 { 2.5, 2.7.5 { 2.7.7, 2.8, 2.10-2.11,4.5.7) have been added. The number of references (877) has more than doubled. 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 saturation density. Accreted matter in their gravitational fields undergoes high-frequency oscillations that could become a sensitive probe for general relativistic effects. Temporal 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 importance because with the first direct detections of gravitational waves by the LIGO and VIRGO collaborations [6,5] 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 available. 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 uncertainties, 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 approximations (the Newtonian limit and the post-Newtonian approximation, the slow rotation limit, the Cowling approximation, the spatial conformal flat- ness approximation) as an eigenvalue problem. Normal modes in full general relativity have been obtained through numerical simulations only. Time evo- lutions of the linearized equations have improved our understanding of the spectrum of axial and hybrid modes in relativistic stars. Nonaxisymmetric instabilities in rotating stars can be driven by the emission of gravitational waves [Chandrasekhar-Friedman-Schutz (CFS) instability] or by viscosity. Relativity strengthens the former, but weakens the latter. Nascent neutron stars can be subject to the l = 2 bar mode CFS instability, which would turn them into a strong gravitational wave source. Axial fluid modes in rotating stars (r-modes) have received considerable attention since the discovery that they are generically unstable to the emission of gravitational waves. The r-mode instability could slow down newly-born relativistic stars and limit their spin during accretion-induced spin-up, which would explain the absence of millisecond pulsars with rotational periods less than ∼ 1.5 ms. Rotating Stars in Relativity 3 Gravitational waves from the r-mode instability could become detectable if the amplitude of r-modes is sufficiently large, however, nonlinear effects seem to set a small saturation amplitude on long timescales. Nevertheless, if the sig- nal persists for a long time, even a small amplitude could become detectable. Highly differentially rotating neutron stars are also subject to the development of a one-arm (m = 1) instability, as well as to the development of a dynamical bar-mode (m = 2) instability which both act as emitters of potentially detectable gravitational waves. Recent advances in numerical relativity have enabled the long-term dynamical evolution of rotating stars. Several interesting phenomena, such as dynamical instabilities, pulsation modes, and neutron star and black hole formation in rotating collapse have now been studied in full general relativity. The latest studies include realistic equations of state and also magnetic fields. The aim of this article is to present a summary of theoretical and numerical methods that are used to describe the equilibrium properties of rotating relativistic stars, their oscillations and dynamical evolution. The focus is on the most recently available publications in the field, in order to rapidly com- municate new methods and results. At the end of some sections, the reader is directed to papers that could not be presented in detail here, or to other review articles. As new developments in the field occur, updated versions of this article will appear. Another review on rotating relativistic stars has appeared by Gourgoulhon [318] while monographs appeared by Meinel et al. [544], and Friedman and Stergioulas [285]. In several sections, our Living Review article updates and extends previous versions [766,767] using also abridged discus- sions of topics from [285]. Notation and conventions. Throughout the article, gravitational units, where G = c = 1 (these units are also referred to as geometrized), will be adopted in writing the equations governing stellar structure and dynamics, while numerical properties of stellar models will be listed in cgs units, unless otherwise noted. We use the conventions of Misner, Thorne, and Wheeler (1973) for the signature of the spacetime metric (−+++) and for signs of the curvature ten- sor and its contractions. Spacetime indices will be denoted by Greek letters, α, β; : : :, while Latin a; b; : : : characters will be reserved to denote spatial indices. (Readers familiar with abstract indices can regard indices early in the alpha- bet as abstract, while indices µ, ν; λ and i; j; k will be concrete, taking values µ = 0; 1; 2; 3, i = 1; 2; 3.) Components of a vector uα in an orthonormal frame, 0^ 3^ fe0^; :::; e3^g, will be written as fu ; :::; u g. Parentheses enclosing a set of indices indicate symmetrization, while square brackets indicate anti-symmetrization. Numbers that rely on physical constants are based on the values c = 2:9979 × 1010 cm s−1, G = 6:670 × 10−8g−1cm3s−2, ~ = 1:0545 × 10−27g 2 −1 −24 33 cm s , baryon mass mB = 1:659×10 g, and M = 1:989×10 g = 1:477 km. 4 Vasileios Paschalidis, Nikolaos Stergioulas 2 The Equilibrium Structure of Rotating Relativistic Stars 2.1 Assumptions A relativistic star can have a complicated structure (such as a solid crust, magnetic field, possible superfluid interior, possible quark core, etc.). Depend- ing on which phase in the lifetime of the star one wants to study, a number of physical effects can be ignored, so that the description becomes significantly simplified. In the following, we will take the case of a cold, uniformly rotating relativistic star as a reference case and mention additional assumptions for other cases, where necessary. The matter can be modeled as a perfect fluid because observations of pulsar glitches are consistent with departures from a perfect fluid equilibrium (due to the presence of a solid crust) of order 10−5 (see [276]). The temperature of a cold neutron star has a negligible affect on its bulk properties and can be assumed to be 0 K, because its thermal energy ( 1 MeV ∼ 1010 K) is much smaller than Fermi energies of the interior (> 60 MeV). One can then use a one-parameter, barotropic equation of state (EOS) to describe the matter: " = "(P ); (1) where " is the energy density and P is the pressure. At birth, a neutron star is expected to be rotating differentially, but as the neutron star cools, several mechanisms can act to enforce uniform rotation. Kinematical shear viscosity is acting against differential rotation on a timescale that has been estimated to be [263,264,187] ρ −5=4 T 2 R τ ∼ 18 yr; (2) 1015 g cm−3 109 K 106 cm where ρ, T and R are the central density, temperature, and radius of the star. It has also been suggested that convective and turbulent motions may enforce uniform rotation on a timescale of the order of days [366]. Shapiro [705] suggested that magnetic braking of differential rotation by Alfvénwaves could be the most effective damping mechanism, acting on short timescales, possibly of the order of minutes. Within a short time after its formation, the temperature of a neutron star becomes less than 1010 K (due to neutrino emission). When the temperature drops further, below roughly 109 K, its outer core is expected to become superfluid (see [547] and references therein).

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