Rotating and Magnetized Stars

Rotation and magnetic fields make a compact star genuinely multidimensional. They deform the material surface, source off-diagonal geometry, redistribute support, and open instabilities that do not exist in a spherical TOV model. The organizing question is not merely how much extra mass can be supported, but which symmetries, rotation law, current prescription, EOS, and timescale make a claimed equilibrium meaningful. This chapter moves from the exact stationary framework to numerical boundaries, exterior multipoles, and the frontier between equilibrium and nonlinear survival.

Enter this chapter

Read the six pages in the following sequence:

1. Stationary Axisymmetric Stellar Equilibrium derives the circular metric, frame-dragging velocity, fluid first integral, and free-surface conditions.
2. Uniform Rotation and Mass Shedding turns that system into controlled sequences and identifies the equatorial geodesic condition at the Kepler boundary.
3. Differential Rotation and Hypermassive Stars introduces rotation-law dependence, transient support, and the distinction between equilibrium and evolution.
4. Magnetic Equilibria and Deformation couples current functions and flux surfaces to Einstein–Maxwell stress and the stellar quadrupole.
5. Relativistic Stellar Multipoles defines invariant exterior mass and current moments, the moment of inertia, and the restricted domain of approximate universal relations.
6. Stability of Rotating Magnetized Configurations compares bar, corotation-shear, magnetorotational, Tayler, and Parker channels under a dated evidence boundary.

Pages 1–2 form the uniform-rotation construction. Page 3 changes the material flow assumption; page 4 changes the stress and exterior boundary problem. Page 5 extracts invariant consequences, and page 6 asks whether any of these stationary configurations survives the relevant perturbations and timescales.

Symmetry makes equilibrium tractable

Stationarity and axisymmetry provide commuting Killing fields $t^\mu$ and $\phi^\mu$. For a circular barotropic fluid,

$$u^\mu=u^t(t^\mu+\Omega\phi^\mu),$$

and the Euler equation admits

$$H+\ln N-\ln\Gamma +\int^\Omega F(\Omega')\,d\Omega'=C.$$

Here $H=\int_0^p dp'/(\varepsilon+p')$, $N$ is the lapse, and $\Gamma$ is the Lorentz factor measured by a locally nonrotating observer. Uniform rotation absorbs the last term into the constant; differential rotation requires the integrability function $F(\Omega)=u^tu_\phi$. The surface is the level set where the log enthalpy vanishes. The metric and fluid must be solved together because frame dragging, deformation, and support all respond to the same stress tensor. The complete stationary framework and its boundary conditions are reviewed in Paschalidis and Stergioulas 2017, §§2–3.

The main limiting checks are immediate. Zero rotation must recover a spherical TOV solution. Slow uniform rotation must overlap perturbation theory. At mass shedding the equatorial surface element follows a circular geodesic in the solved exterior. These checks address implementation consistency; they do not prove stability or select an astrophysical formation channel.

Support depends on the family

Uniform rotation creates supramassive equilibria above the cold nonrotating ceiling, but a constant-baryon-mass spin-down track eventually meets a stability boundary. Differential rotation can support still larger hypermassive configurations by placing angular momentum unevenly. The amount is conditional on the rotation law, topology, thermal profile, and composition. Magnetic and viscous transport erase shear, so a stationary differentially rotating model is best interpreted as an instantaneous reference state.

Numerical sequence claims require an EOS checksum and interpolation, target family, gauge and surface representation, resolution sequence, virial identities, and independent limits. Solver failure is not a physical endpoint. A surface cusp, branch change, EOS boundary, or poorly conditioned target can all halt an iteration. The physical distinction among mass shedding, axisymmetric collapse, and nonaxisymmetric instability must be diagnosed rather than inferred from nonconvergence.

Magnetic stress and exterior structure

Axisymmetric ideal-MHD equilibria use a flux function $\Psi=A_\phi$ together with current functions and an exterior boundary condition. Purely poloidal, purely toroidal, and mixed configurations are families of current choices, not complete models. Bocquet and collaborators demonstrate self-consistent Einstein–Maxwell rotating equilibria and their numerical checks in Bocquet et al. 1995, §§2–5 and Tables 1–2.

For weak fields, magnetic deformation scales as $\epsilon_B\sim C_{\rm geom}E_B/\lvert W\rvert$. The coefficient and sign depend on topology: common poloidal models are oblate and common toroidal models prolate. In the strong-field regime, one must recompute the density, surface, spacetime, and exterior field rather than extrapolate this scaling.

Invariant asymptotic moments separate exterior information from coordinates. $M_0=M$, $S_1=J$, and $M_2=Q$ lead to $\chi=J/(GM^2)$, $\bar I=I/(G^2M^3)$, and $\bar Q=-Q/(G^2M^3\chi^2)$. Hartle’s slow-rotation construction provides the overlap calculation for $I$ and $Q$ in Hartle 1967, §IV, Eqs. (46)–(49), and §VIII, Eqs. (138)–(140). Kerr has $\bar Q=1$; a material star generally retains EOS and structural information in its higher moments.

Stability is a separate calculation

A classical high-$T/\lvert W\rvert$ bar mode, a low-$T/\lvert W\rvert$ corotation mode, MRI, and magnetic kink or buoyancy modes tap different free energy. A threshold for one does not rule out the others. Dimensionality, symmetry, stratification, magnetic topology, numerical resistivity, scale resolution, and evolution duration must accompany any stability statement.

This chapter stops at isolated stationary stars and their local or nonlinear stability. Observable magnetospheric torques continue in Pulsars and Magnetospheres. Mode spectra and gravitational-radiation instabilities continue in Stellar Oscillations and Instabilities. Formation of hot, differentially rotating remnants belongs to later collapse and binary chapters, while production evolution methods belong in Computational Relativity.

Review the chapter

A uniformly rotating barotrope has constant $\Omega$. A proposed second model keeps the same metric and enthalpy but imposes a radius-dependent $\Omega$ without changing any other term in the first integral. Is it an equilibrium?

Solution

Not in general. Uniform rotation allowed the rotation integral to be absorbed into the constant. With differential rotation, Euler integrability requires a specified relation $F(\Omega)$ and contributes $\int F(\Omega')d\Omega'$ spatially. Changing $\Omega$ also changes the four-velocity, stress-energy, and hence the metric. A consistent model must resolve the fluid and field equations with the new rotation law; repainting a profile on the old solution does not satisfy force balance.

References

• Bocquet, Marc, Silvano Bonazzola, Eric Gourgoulhon, and Jérôme Novak. “Rotating Neutron Star Models with Magnetic Field.” Astronomy and Astrophysics 301 (1995): 757–775. arXiv:gr-qc/9503044.
• Hartle, James B. “Slowly Rotating Relativistic Stars. I. Equations of Structure.” The Astrophysical Journal 150 (1967): 1005–1029. DOI.
• Paschalidis, Vasileios, and Nikolaos Stergioulas. “Rotating Stars in Relativity.” Living Reviews in Relativity 20 (2017): 7. DOI.