Binary Neutron Stars in General Relativity John Friedman & Koji Uryu I. Formalism and Analytic Results University of Wisconsin-Milwaukee Center for Gravitation and Cosmology

I.EINSTEIN EULER SYSTEM II.HELICAL SYMMETRY III.STATIONARY AND QUASI-STATIONARY EQUILIBRIA FOR ROTATING STARS IV.1 st LAW OF THERMODYNAMICS FOR BINARY SYSTEMS V.TURNING-POINT CRITERION AND LOCATION OF INNERMOST STABLE CIRCULAR ORBIT (ISCO)

I. EINSTEIN-EULER SYSTEM: PERFECT FLUID SPACETIMES A perfect fluid is defined by its local isotropy: At each point of the matter there is a timelike direction u  for which T  is invariant under rotations in the 3-space orthogonal to u  ; that is, a comoving observer sees no anisotropic stresses. Denoting by q  = g  + u  the projection ? u , one can decompose T  into a scalar T  u   u  a spatial vector q   T  and a symmetric tensor q   q   T .

is rotationally invariant only if it is proportional to the 3-metric q  : Because any nonzero spatial vector picks out a direction, q   T  = 0 for a perfect fluid. Similarly, the 3-dimensional tensor Then

For equilibria, these are the main corrections. For dynamical evolutions -- oscillations, instabilities, collapse, and binary inspiral, one must worry about the microphysics governing, for example viscosity, heat flow, magnetic fields, superfluid modes (2-fluid flow), and turbulence.

Binary NS inspiral is modeled by a perfect-fluid spacetime, a spacetime M,g whose whose metric satisfies with T  a perfect-fluid energy-momentum tensor. The Bianchi identities imply and this equation, together with an equation of state, determines the motion of the fluid.

3+1 DECOMPOSITION FOR FLUID MOTION The projection of along u  expresses conservation of energy, while the projection orthogonal to u  is the relativistic Euler equation:

Interpretation of the equation: The fractional change in fluid volume V in time d  is implying Then means the energy of a fluid element of volume V decreases by the work p dV in proper time d .

The projection of perpendicular to u  is the RELATIVISTIC EULER EQUATION : Newtonian limit:

Barotropic flows: enthalpy and injection energy A fluid with a one-parameter EOS is called barotropic. Neutron star matter is accurately described by a one-parameter EOS because it is approximately isentropic: Neutron stars rapidly cool far below the Fermi energy (10 13 K » m p ), effectively to zero temperature and entropy. (There is, however, a composition gradient in neutron stars, with the density of protons and electrons ordinarily increasing outward, and this dominates a departure from a barotropic equation of state in stellar oscillations).

1-PARAMETER EOS Then the Euler eqn becomes As you have seen in Nick Stergioulas’ first talk, introducing h allows one to find a first integral of the equation of hydrostatic equilibrium for a uniformly rotating star. We’ll derive the equation in a broader context including stationary binaries.

COMPACT BINARIES: QUASISTATIONARY EQUILIBRIA In the Newtonian limit, because a binary system does not radiate, it is stationary in a rotating frame. Because radiation appears only in the 2 1/2 post- Newtonian order -- to order (v/c) 5 £ Newtonian theory, one computes radiation for most of the inspiral from a stationary post-Newtonian orbit. Time translations in a rotating frame are generated by a helical Killing vector k .

In the Newtonian limit and in the curved spacetime of a rotating star, k  has the form where t  and   are timelike and rotational Killing vectors. For a stationary binary system in GR, one can choose t  and  coordinates for which k  has this form with t   t  and    . (One can define a helical KV by its helical structure in spacetime: There is a unique period T for which each point P is timelike separated from the corresponding point parameter distance T later along the orbit.)

k  is timelike near the fluid

k  is spacelike outside the light cylinder at  = 1

Although k  is spacelike outside a large cylinder, one can, as usual, introduce a 3+1 split associated with a spacelike hypersurface . Evolution along k  can again be expressed in terms a lapse and shift, n  is the future pointing unit normal to    a vector on  In a chart t, x i, for which  is a t = constant surface, the metric is

Equation of hydrostatic equilibrium We found that the relativistic Euler equation is When the spacetime has a Killing vector k   with u  along k  , we have

Proof of :

The equation of hydrostatic equilibrium now takes the form with first integral where is a constant ( is the injection energy per unit baryon mass needed to bring baryons at infinity to the same internal state as that in the star, lower them, give them the speed of the baryons in a fluid element, open a space to put them, and inject them into the star).

For irrotational flow, a good approximation at late stages of inspiral, u is not along a Killing vector, and the proof fails. It is nonetheless still possible to recover an equation of hydrostatic equilibrium.

Quasiequilibrium models are based on helically symmetric spacetimes in which a set of field equations are solved for the independent metric potentials and the fluid density. From, e.g., the angular velocity and multipole moments of a model, one can compute the energy radiated and construct a quasiequilibrium sequence. Until recently, these sequences (and initial data for NS binaries) were restricted to spatially conformally flat metrics, the IWM (Isenberg-Wilson-Mathews) approximation.

ISENBERG-WILSON-MATHEWS ANSATZ: SPATIALLY CONFORMALLY FLAT METRIC f ab flat. Five metric potentials , ,  i are found from five components of the Einstein equation: 1 Hamiltonian constraint 3 components of the Momentum constraint Spatial trace of Einstein eq:   (G  -8  T  )

IWM solutions have 5, not 6, metric functions and satisfy only 5 of the 6 independent components of the Einstein equation. An IWM spacetime agrees with an exact solution only to 1 st post-Newtonian order. Initial data then has some spurious radiation and cannot accurately enforce the  (r) relation. Orbits from the data can be elliptical. One improvement is obtained by adding the asymptotic equality between Komar and ADM mass. To do better, we need the remaining metric degree of freedom.

Shibata and Uryu have recently solved the full set of equations, in a waveless approximation in which time derivatives of the extrinsic curvature are artificially dropped.

Two advantanges of helically symmetric solutions to the full set of Einstein equations:  Energy in gravitational radiation is controlled (smaller than that of the outgoing solution, for practical grids).  By satisfying a full set of Einstein-Euler equations, one enforces a circular orbit. Because data obtained by solving the initial value equations alone or from an (spatially conformally flat) approximation satisfy a truncated set of field equations, they yield elliptical orbits.

An exact helically symmmetric solution is not asymptotically flat, because the energy radiated at all past times is present on a spacelike hypersurface. At a distance of a few wavelengths (larger than the present grid size) the energy is dominated by the mass of the binary system, and the solution appears to be asymptotically flat. Only at distances larger than about 10 4 M is the energy in the radiation field comparable to the mass of the binary system.

Work in progress by Price, Beetle, Bromley; Shinichirou Yoshida, Uryu, JF seeks to solve the full equations without truncation for a helically symmetric spacetime. In a helically symmetric spacetime, the constraints remain elliptic, but the dynamical equations have a mixed elliptic-hyperbolic character: elliptic where k  is spacelike and hyperbolic where k  is timelike:

Flat space wave equation with helical symmetry: In general,

Problem is not intrinsically elliptic, BUT after spherical harmonic decomposition, have coupled system of elliptic equations (Helmholtz eqs), each of form Uryu has a full code (a modification of the waveless code). No convergence yet for the full problem, but toy problems with nonlinear wave equation and two orbiting point scalar charges as source converge:

Remarkably, iteration converges for  of order unity. In example below (due to Shin Yoshida),  Curt- This figure was too large – I’ll send it separately.

For a change that locally preserves vorticity, baryon number and entropy,  M =  J

TURNING POINT STABILITY AND LOCATION OF THE ISCO The first law allows one to apply a turning point theorem (Sorkin 1981) to sequence of binary equilibria. The theorem shows that on one side of a turning point in M at fixed J or in J and fixed baryon mass M 0, the sequence is unstable. The side on which M is smaller is more tightly bound - the stable side. The other side is unstable. (The argument for the instability does not imply that one can reach the lower energy state by a dynamical evolution; the angular momentum distribution might need to be redistributed by viscosity to allow transition to the lower energy state. But the fact that there is a lower energy state with the same vorticity suggests that we may have identified the point of dynamical instability.)

Note: The theorem gives a sufficient condition for secular instability. It does not imply that the low-mass side is stable – there could be other nearby configurations, not on the sequence, that have lower mass.

Lai, Shapiro, Shibata, Uryu, Meudon,... Baumgarte, Cook, Scheel, Shapiro, Teukolsky: In the figure thin solid lines of constant J are plotted on a graph of rest mass vs central density. A red curve marks the onset of orbital instability, the set of models with maximum rest mass M 0 (and maximum mass M along a sequence of models with constant rest mass M 0. Blue segment is sequence of corotating stars, ISCO at intersection.

At much larger central density, the stars are widely separated, and another set of mass- maxima along the constant J curves (e.g., point B in the inset) marks the the onset of instability to collapse of the individual stars. The turning point method identifies both instabilities.

Gravitational mass vs frequency at the ISCO (Oechslin, Uryu, Poghosyan, Thielemann 2003) for a hadronic EOS and an EOS with a quark core