### 6.3 Crust in rotating neutron stars

We consider stationary neutron-star rotation and assume, for the time being, perfect axial symmetry.
Consequently, the spacetime metric is axially symmetric too. Using coordinates , , , and , we
can write the spacetime metric in the form (we use the notation of Haensel, Potekhin and Yakovlev [184])
where the metric functions , , , and depend on and . The metric function
has the asymptotic behavior . It is usually referred to as the “angular velocity of the local
inertial frames”. Einstein’s equations for a stationary rigid rotation of perfect fluid stars can be reduced to a
set of 2-D coupled partial differential equations [53, 389]. Their numerical solution can now be obtained
using publicly available domain codes, for example, the code rotstar from the LORENE library [270] and
the code RNS [388].
As can be seen on Figure 40, the outer crust is the most strongly deformed by the centrifugal
force. The spin frequency of 1200 Hz, significantly larger than the highest detected up to now,
716 Hz, was chosen to make the effect more spectacular. Let us nevertheless point out that some
evidence suggesting the existence of a rapidly-rotating neutron star has recently been found in
XTE J1739–285 [227] with a spin frequency of about 1122 Hz. Let us consider, for example,
the difference between the equatorial (maximal) and polar (minimal) thickness of the crust,
. For a , which is not too close to the mass-shedding limit,
. Therefore, .

The crustal baryon mass (not to be confused with the gravitational mass) of a neutron star
rotating at angular frequency , is larger than the crustal baryon mass of the
static star (with the same total baryon mass). The baryon mass (also called the rest mass) of a
star is equal to the number of baryons it contains times an assumed baryon mass .
One may take or . We take . For not too close to
, we have . Due to the radiation of electromagnetic waves
and particles, a pulsar spins down, so that . Consequently, the baryon mass of the
pulsar crust decreases in time, . Nucleons pass from the crust to the liquid
core, releasing some heat. As shown in Figure 40, the crust is decompressed near the equator
and compressed near the pole. These deformations trigger various nuclear reactions involving
electrons, neutrons, and nuclei. These reactions tend to drive the deformed crust towards its
equilibrium shape and release heat, which influences the cooling of a spinning down pulsar [205].
Additional heating results from crust cracking when local shear strain exceeds the maximal
one.