Binary Neutron Stars in General Relativity John Friedman

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Binary Neutron Stars in General Relativity John
Friedman Koji Uryu I. Formalism and Analytic
Results
University of Wisconsin-MilwaukeeCenter for
Gravitation and Cosmology
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• EINSTEIN EULER SYSTEM
• HELICAL SYMMETRY
• STATIONARY AND QUASI-STATIONARY EQUILIBRIA FOR
  ROTATING STARS
• 1st LAW OF THERMODYNAMICS FOR BINARY SYSTEMS
• TURNING-POINT CRITERION AND LOCATION OF INNERMOST
  STABLE CIRCULAR ORBIT (ISCO)

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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 ua for which Tab is
invariant under rotations in the 3-space
orthogonal to ua that is, a comoving observer
sees no anisotropic stresses.Denoting by qab
gab ub the projection ? ua , one can decompose
Tab into a scalar Tab ua ub a spatial vector qag
Tag and a symmetric tensor qag qbd Tgd .
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Because any nonzero spatial vector picks out a
direction, qag Tag 0 for a perfect
fluid.Similarly, the 3-dimensional tensor
is rotationally invariant only if it is
proportional to the 3-metric qab
Then
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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.
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Binary NS inspiral is modeled by a perfect-fluid
spacetime, a spacetime M,g whose whose metric
satisfieswith Tab 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.
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31 DECOMPOSITION FOR FLUID MOTION The projection
of along ua expresses
conservation of energy, while the projection
orthogonal to ua is the relativistic Euler
equation
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Interpretation of the equationThe fractional
change in fluid volume V in time dt isimplying
Then means the
energy of a fluid element of volume V decreases
by the work p dV in proper time dt .
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The projection of
perpendicular to ua is the
RELATIVISTIC EULER EQUATION
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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 (1013K mp),
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).
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1-PARAMETER EOSThen the Euler
eqnbecomesAs 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. Well derive the equation in a broader
context including stationary binaries.
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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 ka.
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In the Newtonian limit and in the curved
spacetime of a rotating star, ka has the
formwhere ta and fa are timelike and rotational
Killing vectors. For a stationary binary system
in GR, one can choose t and f coordinates for
which ka has this form with tat and faf. .
(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.)
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ka is timelike near the fluid
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ka is spacelike outside the light cylinder at v W
1
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Although ka is spacelike outside a large
cylinder, one can, as usual, introduce a 31
split associated with a spacelike hypersurface S.
Evolution along ka can again be expressed in
terms a lapse and shift,na is the future
pointing unit normal to S ba a vector on S In
a chart t, xi , for which S is a t constant
surface,the metric is
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Equation of hydrostatic equilibrium We found
that the relativistic Euler equation is When
the spacetime has a Killing vector ka with ua
along ka , we have
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Proof of
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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).
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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.
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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.
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ISENBERG-WILSON-MATHEWS ANSATZSPATIALLY
CONFORMALLY FLAT METRIC fab flat. Five
metric potentials y, a, b i are found from five
components of the Einstein equation1
Hamiltonian constraint 3 components of the
Momentum constraintSpatial trace of Einstein
eq gab(Gab-8p Tab)
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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 1st post-Newtonian
order.Initial data then has some spurious
radiation and cannot accurately enforce the W (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.
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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.
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• 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.

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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 104 M is the energy in the radiation
field comparable to the mass of the binary system.
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Work in progress by Price, Beetle, Bromley
Shinichirou Yoshida, Uryu, JFseeks 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 ka is spacelike and hyperbolic where ka is
timelike
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Flat space wave equation with helical
symmetry In general,
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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
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Remarkably, iteration converges for l of order
unity.In example below (due to Shin Yoshida),
l1. Curt- This figure was too large Ill send
it separately.
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For a change that locally preserves
vorticity, baryon number and entropy, d M
W d J
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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 M0, 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.)
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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.
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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 M0 (and maximum mass M along a sequence of
models with constant rest mass M0. Blue segment
is sequence of corotating stars, ISCO at
intersection.
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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.
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Gravitational mass vs frequency at the
ISCO(Oechslin, Uryu, Poghosyan, Thielemann
2003)for a hadronic EOS and an EOS with a quark
core