2.5 Rotation law and equilibrium 2 The Equilibrium Structure of 2.3 The rotating fluid

2.4 Equations of structure

Having specified an equation of state of the form tex2html_wrap_inline3820, the structure of the star is determined by solving four components of Einstein's gravitational field equations

equation136

(where tex2html_wrap_inline3822 is the Ricci tensor and tex2html_wrap_inline3824) and the equation of hydrostationary equilibrium. Setting tex2html_wrap_inline3826, one common choice for the gravitational field equations is [55Jump To The Next Citation Point In The Article]

eqnarray146

supplemented by a first order differential equation for tex2html_wrap_inline3828 (see [55Jump To The Next Citation Point In The Article]). Above, tex2html_wrap_inline3830 is the 3-dimensional derivative operator in a flat 3-space with spherical polar coordinates r, tex2html_wrap_inline3712, tex2html_wrap_inline3836 .

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 tex2html_wrap_inline3838, and is written as

  equation162

where a comma denotes partial differentiation and tex2html_wrap_inline3840 . When the equation of state is barotropic then the hydrostationary equilibrium equation has a first integral of motion

  equation172

where tex2html_wrap_inline3842 is some specifiable function of tex2html_wrap_inline3844 only, and tex2html_wrap_inline3596 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 (20Popup Equation) 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 [302]).



2.5 Rotation law and equilibrium 2 The Equilibrium Structure of 2.3 The rotating fluid

image Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr-2003-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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