Relativistic Stellar Multipoles

Relativistic multipole moments compress a stationary star’s exterior gravity into invariant asymptotic data. The mass, angular momentum, and quadrupole are not arbitrary coefficients read from any convenient coordinate expansion: beyond the leading terms, gauge choices mix coefficients. With an accepted asymptotic definition, the moment of inertia and spin-induced quadrupole connect interior matter to orbital and timing observables. Unlike a Kerr black hole, a material star is not determined by mass and spin alone.

Required background. Killing Fields and Isometry Groups supplies the stationary and axial symmetries and their conserved charges.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies the source multipoles; The Einstein Equation supplies the exterior geometry; Relativistic Matter and Compact Objects supplies the stellar EOS; Formulations of General Relativity supplies asymptotic and gauge language; Computational Relativity supplies extraction and convergence checks.

Mass and current moments

For an asymptotically flat, stationary, axisymmetric vacuum exterior, the Geroch–Hansen construction defines mass moments $M_\ell$ and current moments $S_\ell$ from the conformally completed orbit space. The first moments are

$$M_0=M, \qquad S_1=J, \qquad M_2=Q.$$

Equatorial reflection symmetry sets odd mass moments and even current moments to zero. The orbit-space construction, its extension to electrovacuum, and the normalization of the first moments are developed in Fodor, Costa Filho, and Hartmann 2021, §§II.B–II.C and III.C, Eqs. (13)–(17) and (43). Their complex moment $P_\ell$ corresponds to $G(M_\ell+iS_\ell)$ here; the reflection property is summarized in their §I. An exterior electromagnetic field is not vacuum: magnetized stars require an electrovacuum generalization or a stated alternative extraction prescription. The vacuum construction must not be applied unchanged through a current-carrying magnetosphere.

Equivalent extraction in asymptotically Cartesian, mass-centered coordinates is useful numerically, but the coordinate and normalization translation must be explicit.

The sign convention here takes $Q\lt0$ for an oblate axisymmetric source. Some stellar papers define a positive oblateness parameter instead, so a numerical comparison must translate the sign before discussing agreement. Magnetic stress can generate a quadrupole even at zero spin, and differential rotation can add structure not captured by a one-parameter slow-rotation expansion.

Moment of inertia and slow rotation

Along a family of uniformly rotating stars, the linear-response moment of inertia is

$$I=\left.\frac{\partial J}{\partial\Omega}\right|_{\Omega=0},$$

and $I=J/\Omega+O(\Omega^2)$ for sufficiently slow rotation. It is a global response, not the Newtonian integral $(2/3)\int r^2dm$ evaluated on an undeformed density profile. Frame dragging changes the local angular velocity relative to inertial frames, and relativistic pressure contributes to the source.

Hartle expands the metric and fluid through first order for $J$ and second order for the deformation, matching the interior to a vacuum exterior in Hartle 1967, §IV, Eqs. (46)–(49), and §VIII, Eqs. (138)–(140). The method requires uniform slow rotation and a barotropic background. It is an excellent overlap benchmark for a two-dimensional code, but it does not control a mass-shedding model or a strongly magnetized noncircular equilibrium.

With $c=1$ and $G$ explicit, useful dimensionless quantities are

$$\chi=\frac{J}{GM^2}, \qquad \bar I=\frac{I}{G^2M^3}, \qquad \bar Q=-\frac{Q}{G^2M^3\chi^2}.$$

These definitions remove leading mass and spin scaling. For the usual I–Love–Q comparison, $Q$ means the quadrupole induced by rotation of an otherwise spherical, unmagnetized star. If a magnetic distortion remains at $\Omega=0$, inserting its total quadrupole into $\bar Q$ produces a $\chi^{-2}$ divergence; it is not the same spin-induced relation. The definitions do not remove EOS, temperature, rotation-law, or magnetic dependence outside that calibrated problem.

For completeness, the tidal variable in that comparison is $\Lambda=\lambda/(G^4M^5)=2k_2/(3C^5)$, where $Q_{ij}^{\rm tidal}=-\lambda\mathcal E_{ij}$ defines the linear quadrupolar response to an external electric tidal field, $C=GM/R$, and $k_2$ is the quadrupolar Love number. This externally induced quadrupole differs from the isolated spin-induced $Q$ above; see Tidal Deformability and Love Response.

Stellar moments versus Kerr moments

Let $\mathcal M_\ell=G M_\ell$ and $\mathcal S_\ell=G S_\ell$ denote geometrized moments. Kerr satisfies

$$\mathcal M_\ell+i\mathcal S_\ell =(GM)\left(i\frac{J}{M}\right)^\ell.$$

In particular, $Q_{\rm Kerr}=-J^2/M$ and $\bar Q_{\rm Kerr}=1$. A rotating material star generally has a larger magnitude of the oblate quadrupole at the same $M$ and $J$, because its surface and density distribution respond to rotation. Higher multipoles likewise retain information about internal structure rather than following a no-hair sequence.

Cold, slowly and uniformly rotating neutron-star models exhibit approximate EOS-insensitive relations among $\bar I$, tidal deformability, and $\bar Q$. The numerical evidence and residual EOS spread are displayed in Yagi and Yunes 2013, Fig. 1 and Eq. (1). “Universal” means weakly sensitive over the tested family, not exact. Hot remnants, rapid or differential rotation, anisotropy, and strong magnetic fields can move a model outside that calibration domain. Strong phase transitions require renewed residual checks; their presence does not by itself prove that an approximate relation fails.

Extraction and consistency checks

A multipole calculation must specify the asymptotic definition, coordinate map, fit region, radial order, and treatment of electromagnetic stress outside the star. Extracting $Q$ at several outer radii and expansion orders tests contamination by neglected higher terms. A coordinate coefficient that moves under an allowed radial transformation is not yet an invariant multipole.

The zero-rotation limit should give $J\propto\Omega$ and a spin-induced $Q\propto\Omega^2$, leaving finite $I$. Reflection-forbidden moments should converge to zero. The asymptotic $M$ and $J$ should agree with independent surface or volume integrals, and a two-dimensional solution should overlap the Hartle expansion at small $\chi$. For a magnetized model, subtracting the unmagnetized sequence at matched baryon mass isolates a magnetic deformation only after exterior field energy is treated consistently.

Check your understanding

A slowly rotating star has $Q=-4G^2M^3\chi^2$ in the sign convention above. What is $\bar Q$, and how does its quadrupole compare with Kerr at the same $M$ and $J$?

Solution

Substitution gives

$$\bar Q=-\frac{-4G^2M^3\chi^2}{G^2M^3\chi^2}=4.$$

Kerr has $\bar Q=1$, so this star has four times the magnitude of the oblate quadrupole at fixed mass and angular momentum. The result says nothing about higher moments unless an additional relation has been established.

References

• Fodor, Gyula, Etevaldo dos Santos Costa Filho, and Betti Hartmann. “Calculation of Multipole Moments of Axistationary Electrovacuum Spacetimes.” Physical Review D 104 (2021): 064012. DOI. Open preprint.
• Hartle, James B. “Slowly Rotating Relativistic Stars. I. Equations of Structure.” The Astrophysical Journal 150 (1967): 1005–1029. DOI.
• Yagi, Kent, and Nicolás Yunes. “I-Love-Q.” Science 341 (2013): 365–368. DOI.