Stationary Axisymmetric Stellar Equilibrium

A stationary rotating star is an equilibrium in spacetime, not a spherical star with a centrifugal term appended to the TOV equation. Rotation deforms the surface, drags inertial frames, and makes the metric depend on radius and polar angle. Under circularity, barotropy, and purely azimuthal flow, the relativistic Euler equation nevertheless has a first integral. That integral is the practical bridge between two commuting Killing symmetries and a two-dimensional stellar model; it also makes clear what fails when meridional flow, composition gradients, or noncircular stresses are admitted.

Required background. Killing Fields and Isometry Groups supplies the stationary and axial Killing fields and their conserved contractions.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies stress-energy conventions; The Einstein Equation supplies the field equations; Relativistic Matter and Compact Objects supplies the fluid closure; Formulations of General Relativity supplies gauge and constraint language; Computational Relativity supplies elliptic-solver checks.

Circular stationary geometry

Let $t^\mu$ generate time translations and $\phi^\mu$ generate rotations, with $[t,\phi]^\mu=0$. Stationarity and axisymmetry mean that metric and matter fields have zero Lie derivative along both vectors. Circularity adds the stronger statement that the two-planes orthogonal to the Killing orbits are integrable. With no meridional fluid current, a useful quasi-isotropic form is

$$ds^2=-N^2dt^2+A^2\left(dr^2+r^2d\theta^2\right) +B^2r^2\sin^2\theta\left(d\phi-\omega_{\rm fd}dt\right)^2.$$

The lapse $N$, conformal factors $A,B$, and frame-dragging angular velocity $\omega_{\rm fd}$ depend on $(r,\theta)$. The subscript distinguishes frame dragging from the mode frequency $\omega$ used in stability calculations. This ansatz is justified for a perfect fluid in purely circular motion; meridional circulation or a stress tensor that violates the circularity conditions requires a more general metric. The geometric hypotheses and reduction are set out in Gourgoulhon 2011, §2.2, Eq. (2.32), and §3.2.1, Eqs. (3.14)–(3.17).

Regularity requires even angular behavior on the symmetry axis and elementary flatness there, in particular $A=B$ on the axis so circumference divided by proper distance from the axis tends to $2\pi$. At the center, scalar potentials have vanishing first radial derivatives. Asymptotic flatness requires $N,A,B\to1$ and $\omega_{\rm fd}\to0$. Across the material surface, the interior geometry and appropriate normal derivatives match a vacuum exterior; the surface itself is part of the solution rather than a prescribed coordinate sphere.

Circular motion and frame dragging

Purely azimuthal flow has four-velocity

$$u^\mu=u^t\left(t^\mu+\Omega\phi^\mu\right), \qquad \Omega=\frac{u^\phi}{u^t}.$$

The physical speed relative to a locally nonrotating observer is

$$U=\frac{Br\sin\theta}{N}(\Omega-\omega_{\rm fd}), \qquad \Gamma=(1-U^2)^{-1/2}.$$

Thus $\Omega$ is a coordinate angular velocity, whereas $\Omega-\omega_{\rm fd}$ is the rotation relative to dragged inertial frames. Timelikeness requires $U^2\lt1$. On the axis $U$ vanishes geometrically even if $\Omega$ is finite. In the nonrotating limit $\Omega=\omega_{\rm fd}=0$, $A=B$ in isotropic coordinates and the spherical stellar equations are recovered.

The Einstein equation supplies coupled elliptic equations for the metric potentials, sourced by energy density, pressure, momentum density, and stress. Their explicit form depends on variables and gauge. The invariant content is the field equation plus regularity, surface, and asymptotic conditions; a particular potential should not be compared across codes without translating definitions.

The fluid first integral

For a zero-temperature barotrope define the log enthalpy

$$H(p)=\int_0^p\frac{dp'}{\varepsilon(p')+p'}.$$

Normalization gives $u^t=\Gamma/N$. For $i=r,\theta$, the spatial Euler equation can therefore be written

$$\partial_i H-\partial_i\ln u^t +u^tu_\phi\,\partial_i\Omega=0.$$

The first two terms already form a gradient. Taking the antisymmetric derivative shows why differential rotation needs an extra restriction: $d(u^tu_\phi)\wedge d\Omega=0$. Locally, where $d\Omega\ne0$, this means $u^tu_\phi=F(\Omega)$. Uniform rotation is a separate integrable case and does not require $u^tu_\phi$ to be spatially constant. The first integral is

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

Uniform rotation sets $d\Omega=0$, so the last term is absorbed into the constant. The surface of an ordinary fluid is the level set $H=0$. The relation simultaneously determines that surface and updates the matter source after the metric is solved. The complete derivation is in Gourgoulhon 2011, §3.4.3, Eqs. (3.90)–(3.104) and Paschalidis and Stergioulas 2017, §2.4, Eqs. (30)–(41).

For uniform rotation, the weak-field limit gives $H\simeq h_N=\int dp/\rho$, $\ln N\simeq\Phi_N$, and $\ln\Gamma\simeq\Omega^2\varpi^2/2$, where $\varpi=r\sin\theta$. Thus the first integral reduces to

$$h_N+\Phi_N-\frac12\Omega^2\varpi^2=C_N.$$

The minus sign is the centrifugal potential in the rotating frame. Differentiating this expression recovers outward centrifugal support; a plus sign would instead strengthen gravity.

The integrability assumptions matter. A nonbarotropic fluid has entropy or composition gradients, so $dp/(\varepsilon+p)$ need not be an exact differential of one scalar. Meridional circulation adds velocity components outside the Killing plane. Magnetic forces introduce current functions and their own integrability conditions. In each case, writing the same first integral without the additional fields would discard part of the Euler equation.

Construction and independent checks

An equilibrium iteration alternates between the elliptic metric equations and the matter first integral, adjusting a central enthalpy and rotation parameter until the desired global quantities are reached. Useful independent checks are asymptotic mass and angular momentum, the relativistic virial identities, surface-force balance, axis regularity, and convergence with radial and angular resolution. Zero rotation must recover a TOV sequence in the same EOS and unit conventions. Slow rotation must overlap perturbative results before a rapid-rotation branch is trusted.

The endpoint can be mass shedding, an axisymmetric stability boundary, a causality or EOS-table boundary, or solver failure. Mass shedding and an instability are physical boundaries; an acausal EOS or the edge of an EOS table instead limits the chosen material model. Each requires a different diagnostic. An iteration that ceases to converge does not by itself locate the maximum mass or Kepler limit.

Check your understanding

Take a nonrotating barotropic star, so $\Gamma=1$ and the rotation term vanishes. If the lapse is $N_c=0.70$ at the center and $N_s=0.80$ at the surface where $H_s=0$, what central log enthalpy follows from the first integral?

Solution

The nonrotating first integral is $H+\ln N=C$. At the surface, $C=\ln N_s$, hence

$$H_c=\ln\frac{N_s}{N_c} =\ln\frac{0.80}{0.70}\simeq0.134.$$

The positive value is consistent with positive central pressure. The numbers are only a first-integral check; they do not ensure that the chosen lapse also solves the Einstein equation.

References

• Gourgoulhon, Éric. “An Introduction to the Theory of Rotating Relativistic Stars.” COMPSTAR 2010 lecture notes, revised 20 September 2011. arXiv:1003.5015v2.
• Paschalidis, Vasileios, and Nikolaos Stergioulas. “Rotating Stars in Relativity.” Living Reviews in Relativity 20 (2017): 7. DOI.