### 6.1 Spherical nonrotating neutron stars

A static (nonrotating) neutron star, built of a perfect fluid, has spherical symmetry. The spacetime is also spherically symmetric, with the metric (see, for instance, Landau & Lifshitz [248]; here we use the notation of Haensel, Potekhin and Yakovlev [184])
where is a time coordinate, is a radial coordinate called “circumferential radius” (see below), and and are polar and azimuthal angular coordinates. The dimensionless metric function and have to be determined from Einstein’s equations. For a flat (Minkowski) spacetime we would have . One can show that is determined by the mass-energy contained within radius , divided by , to be denoted by  [248],

Let us limit ourselves to the static case . Fixing , , and then integrating over from zero to , we find that the proper length of the equator of the star, i.e., its circumference, as measured by a local observer, is equal to . This is why is called the circumferential radius. Notice that Equation (90) implies that the infinitesimal proper radial distance (corresponding to the infinitesimal difference of radial coordinates ) is given by .

From Einstein’s equations, we get the (relativistic) equations of hydrostatic equilibrium for a static spherically-symmetric star

Equation (92) is the famous Tolman–Oppenheimer–Volkoff equation of hydrostatic equilibrium [407313]. Equation (93) enables one to calculate within a radius . Finally, Equation (94) determines the metric function . Equations (92)–(94) have to be supplemented with an equation of state (EoS) .

Let us consider the differential Equations (92) and (93), which determine the global structure of a neutron star. They are integrated from the star center, , with the boundary conditions [] and . It is clear from Equation (92), that pressure is strictly decreasing with increasing . The integration is continued until , which corresponds to the surface of the star, with radial coordinate , usually called the star radius.

The gravitational mass of the star is defined by . The mass is the source of the gravitational field outside the star (), and creates an outer spacetime described by the Schwarzschild metric,

The crust corresponds to the layer , where determines the crust-core interface. The depth below the stellar surface, , is defined as the proper radial distance between the star surface and a given surface of radius . It is given by

The structure of the crust depends on its EoS, stellar mass, and the EoS of its liquid core. In Figures 37 and 38 we present the structure of the crust of a star, for two EoSs of the neutron star interior.

An accreted crust has a different composition and thus a different EoS (stiffer) than the ground-state crust (see Sections 4 and 5). For the comparison to be meaningful, however, these two EoSs have to be calculated from the same nuclear Hamiltonian. We satisfied this by using in both cases the same compressible liquid drop model of Mackie & Baym [277]. The plots of for the ground state and accreted crust of a neutron star are shown in Figure 39.