### 2.4 Equations of structure

Having specified an equation of state of the form , the structure of the star is determined by
solving four components of Einstein’s gravitational field equations
(where is the Ricci tensor and ) and the equation of hydrostationary equilibrium. Setting
, one common choice for the gravitational field equations is [55]
supplemented by a first order differential equation for (see [55]). Above, is the 3-dimensional
derivative operator in a flat 3-space with spherical polar coordinates , , .
Thus, three of the four gravitational field equations are elliptic, while the fourth equation is a first order
partial differential equation, relating only metric functions. The remaining nonzero components of the
gravitational field equations yield two more elliptic equations and one first order partial differential
equation, which are consistent with the above set of four equations.

The equation of hydrostationary equilibrium follows from the projection of the conservation of
the stress-energy tensor normal to the 4-velocity , and is written as

where a comma denotes partial differentiation and . When the equation of state is barotropic
then the hydrostationary equilibrium equation has a first integral of motion
where is some specifiable function of only, and is the angular velocity on the
symmetry axis. In the Newtonian limit, the assumption of a barotropic equation of state implies that the
differential rotation is necessarily constant on cylinders, and the existence of the integral of
motion (20) is a direct consequence of the Poincaré–Wavre theorem (which implies that when
the rotation is constant on cylinders, the effective gravity can be derived from a potential;
see [301]).