4.2 Isolated Neutron Stars4 Neutron-Star Initial Data4 Neutron-Star Initial Data

4.1 Hydrostatic Equilibrium 

For a neutron star to be in true equilibrium, the spacetime must be stationary as discussed in Section  2.4 . This means that the spacetime possesses both ``temporal'' and ``angular'' Killing vectors (cf. Ref. [35Jump To The Next Citation Point In The Article]). If the matter is also to be in equilibrium, then the 4-velocity of the matter tex2html_wrap_inline3601 must be a linear combination of these two Killing vectors. If we use coordinates as defined in (53Popup Equation) with the angular Killing vector in the tex2html_wrap_inline3603 direction, then

  equation1469

Here, tex2html_wrap_inline3605 and tex2html_wrap_inline3607 are functions of r and tex2html_wrap_inline3305 only. tex2html_wrap_inline3607 is the angular velocity of the matter as measured at infinity.

It is common to define v as the relative velocity between the matter and a normal observer (often called a zero angular momentum observer) so that

  equation1472

The velocity v is then fixed by the normalization condition tex2html_wrap_inline3619 .

If we assume that the matter source is a perfect fluid, then the stress-energy tensor is given by

  equation1476

where tex2html_wrap_inline3621 and P are the total energy density and pressure, respectively, as measured in the rest frame of the fluid. The vanishing of the divergence of the stress-energy tensor yields the equation of hydrostatic equilibrium (often referred to as the relativistic Bernoulli equation). In differential form, this is

  equation1481

If the fluid is barytropic Popup Footnote, then we can define the relativistic enthalpy as

  equation1489

and rewrite the relativistic Bernoulli equation as

  equation1494

The constants tex2html_wrap_inline3629, tex2html_wrap_inline3631, and tex2html_wrap_inline3633 are the values their respective quantities have at some reference point, often taken to be the surface of the neutron star at the axis of rotation. When uniform rotation is assumed (tex2html_wrap_inline3635), Eq. (118Popup Equation) is rather easy to solve. The case of differential rotation is somewhat more complicated. An integrability condition of (116Popup Equation) requires that tex2html_wrap_inline3637 be expressible as a function of tex2html_wrap_inline3607, so

  equation1501

tex2html_wrap_inline3641 is a specifiable function of tex2html_wrap_inline3607 which determines the rotation law that the neutron star must obey [35Jump To The Next Citation Point In The Article].



4.2 Isolated Neutron Stars4 Neutron-Star Initial Data4 Neutron-Star Initial Data

image Initial Data for Numerical Relativity
Gregory B. Cook
http://www.livingreviews.org/lrr-2000-5
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