---
title: Spin-Induced Quadrupole Moments
url: https://www.emergentmind.com/topics/spin-induced-quadrupole-moments
type: topic
---

# Spin-Induced Quadrupole Moments

Searching arXiv for recent and foundational papers on spin-induced quadrupole moments to ground the article.
Spin-induced quadrupole moments are higher-multipole deformations generated by spin. In gravitational-wave theory for compact binaries, the standard parametrization is
\[
Q_i=-\kappa_i\,m_i^3\,\chi_i^2,
\]
with \(m_i\) the mass, \(\chi_i\equiv |S_i|/m_i^2\) the dimensionless spin, and \(\kappa_i\) a dimensionless quadrupole parameter; for a Kerr black hole, general relativity fixes \(\kappa_i=1\) exactly, while \(\kappa_i\neq 1\) indicates non-Kerr structure [2111.04135]. Closely related forms, such as \(Q_i=-\kappa_i S_i^2/m_i\), are also used [1811.00317]. The same phrase also appears in operator-based settings, where quadrupole tensors quantify anisotropic spin distributions in composite hadrons, nuclei, magnets, and quantum spin liquids [1810.03281; 2209.04070; 2203.09819; 2603.23145]. Across these contexts, spin-induced quadrupole moments encode how spin degrees of freedom depart from spherical symmetry.

## 1. Kerr normalization, deformation parameters, and object classification

For compact objects, the central benchmark is the Kerr relation implied by the no-hair theorem: a Kerr black hole has \(\kappa=1\), independent of mass or spin [2111.04135]. In the Hansen-Geroch-Thorne multipole language, the Kerr multipoles satisfy
\[
\mathcal{M}_\ell+i\mathcal{S}_\ell=M(ia)^\ell,
\]
so that the mass quadrupole is \(Q_{\rm Kerr}=-a^2M=-\chi^2 M^3\) [2401.12066]. A convenient deviation parameter is \(\delta\kappa\), defined by
\[
Q=Q_{\rm Kerr}(1+\delta\kappa)=-\chi^2 M^3(1+\delta\kappa),
\]
with \(\delta\kappa=0\) recovering the Kerr result [2401.12066].

The expected values of \(\kappa\) are object-dependent. Neutron stars have \(\kappa_{\rm NS}\) ranging roughly from \(2\) to \(14\), depending on the equation of state and rotation rate [2111.04135]. Boson stars can have \(\kappa\sim 10\!-\!150\), while gravastars can sit near \(\kappa\sim 1\) with \(O(0.1\!-\!1)\) deviations in thin-shell models; some gravastar constructions even permit negative \(\kappa\) [1811.00317; 2111.04135]. This object dependence is what makes the parameter a direct probe of compact-object nature.

Binary analyses often replace \((\kappa_1,\kappa_2)\) by symmetric and antisymmetric combinations,
\[
\kappa_s=\frac{\kappa_1+\kappa_2}{2},\qquad
\kappa_a=\frac{\kappa_1-\kappa_2}{2},
\]
or, equivalently, \(\delta\kappa_s\) and \(\delta\kappa_a\) defined relative to the Kerr value [1811.00317; 1908.02247]. In the equal-deformation limit \(\kappa_1=\kappa_2\), only \(\kappa_s\) survives [1811.00317]. This simplification is widely used because measuring the individual coefficients is strongly degenerate with other intrinsic parameters [1908.02247].

A common misconception is that non-Kerr structure necessarily corresponds to a large positive quadrupole deformation. The literature does not support that simplification. Some models predict very large positive \(\kappa\), some allow negative \(\kappa\), and fully back-reacted string-theory constructions can even produce a positive four-dimensional mass quadrupole \(M_2>0\), opposite in sign to Kerr [2111.04135; 2510.05217].

## 2. Post-Newtonian entry, precession dynamics, and waveform imprint

In the post-Newtonian expansion for compact binaries, the leading correction to the conservative dynamics from the rotation of each component appears at \(2\)PN order and is entirely captured by the second mass moment [2111.04135]. In frequency-domain inspiral waveforms, the spin-induced quadrupole enters through the phasing. One representative expression is
\[
\Psi(f)\supset -\frac{75}{128\eta}\,\chi_i^2\,\kappa_i\,m_i^2\,(\pi M f)^{-1/3},
\]
plus symmetric combinations of the two bodies’ contributions, with \(M=m_1+m_2\) and \(\eta=m_1m_2/(m_1+m_2)^2\) [2111.04135]. In the aligned-spin decomposition used for third-generation forecasts, the waveform phase contains a \(2\)PN spin-spin term carrying the quadrupole dependence, while higher spin-spin corrections enter at \(3\)PN and the leading cubic-in-spin octupole term at \(3.5\)PN [1811.00317].

The same dependence can be expressed in terms of \(\kappa_s\) and \(\kappa_a\). At \(2\)PN,
\[
\kappa_1\alpha\,\chi_1^2+\kappa_2\beta\,\chi_2^2
=\kappa_s[\alpha\chi_1^2+\beta\chi_2^2]+\kappa_a[\alpha\chi_1^2-\beta\chi_2^2],
\]
so analyses that set \(\kappa_a=0\) measure only the symmetric combination [1811.00317]. In TaylorF2-like parametrizations for large quadrupolar deviations, the \(2\)PN coefficient \(S_4\) explicitly contains \(\kappa_s\) and \(\kappa_a\), while additional quadrupole terms appear in the \(3\)PN and \(3.5\)PN coefficients \(S_6\) and \(S_7\) [2211.00039].

For generic precessing binaries, spin-induced quadrupole moments also modify the precession dynamics. The precession-frequency vectors \(\Omega_i\) include spin-orbit, spin-spin, and quadrupole-curvature couplings up to \(2\)PN, and each object’s own \(\kappa_i\) appears multiplying its spin \(S_i\) inside the precession equations [2308.09032]. This changes the Euler angles used in twisting-up procedures and therefore modulates higher-mode amplitudes and phases beyond the non-precessing \((2,2)\) carrier [2308.09032]. The same work distinguishes the aligned-spin phase correction (“AI” effect) from the precession-induced amplitude and phase modulation (“PI” effect), and shows that for highly precessing systems about \(20\!-\!30\%\) of random orientations yield PI-effect mismatches larger than the AI effect [2308.09032].

The extended-body formulation gives the same physics in a different language. In the Mathisson-Papapetrou-Dixon system, a spin-induced quadrupole is introduced through a quadrupole tensor \(J^{\alpha\beta\gamma\delta}\) built from the STF part of the spin bilinear, with a coefficient \(C_Q\) equal to unity for a Kerr black hole and of order \(4\!-\!8\) for neutron-star equations of state [1611.07602]. This formulation is used for orbital dynamics in Kerr spacetime rather than directly for comparable-mass waveform inference, but it encodes the same idea: spin induces a quadrupolar deformation that couples to curvature.

## 3. Current-detector inference: single events, hierarchical methods, and population tests

With second-generation detectors, single-event measurements are generally weak. The \(\delta\kappa_s\) parameter is weakly constrained unless the event has moderately high spins, \(\chi_{\rm eff}\gtrsim 0.15\), and good inspiral SNR, \({\rm SNR}_{\rm insp}\gtrsim 20\); otherwise \(\delta\kappa_s\) is effectively unmeasured, which dilutes population inference [2111.04135]. In dedicated simulations for Advanced LIGO-Virgo design sensitivity, \(90\%\) credible intervals remain very broad for low-spin systems, tighten to \(|\delta\kappa_s|\lesssim 30\) for moderate effective spin, and to \(|\delta\kappa_s|\lesssim 10\) for high \(|\chi_{\rm eff}|\gtrsim 0.7\) [1908.02247].

A Bayesian single-event framework measures the symmetric deformation parameter under the assumption \(\delta\kappa_a=0\), inserts the inspiral corrections into IMRPhenomPv2, and truncates the waveform to avoid unmodeled post-inspiral effects [1908.02247]. Applied to GW151226 and GW170608, the posteriors remain consistent with the binary-black-hole hypothesis. The reported \(90\%\) intervals and Bayes factors were:

| Event | \(90\%\) interval on \(\delta\kappa_s\) | \(\log_{10} B_{\rm non/BH}\) |
|---|---:|---:|
| GW151226 | \([-191.8,\;13.5]\) | \(-0.94\) |
| GW170608 | \([-177.4,\;122.9]\) | \(-0.15\) |

These results place \(\delta\kappa_s=0\) well inside the inferred intervals and do not favor the non-BH model [1908.02247].

Population analyses were introduced to overcome the weakness of single-event constraints. One approach posits that the true \(\delta\kappa_s\) values follow a Gaussian hyperdistribution \(N(\mu,\sigma^2)\), and combines event-level posteriors through
\[
L(d_1\dots d_N\mid \mu,\sigma)=\prod_{j=1}^N \int p_j(\delta\kappa_s\mid d_j)\,N(\delta\kappa_s\mid \mu,\sigma^2)\,d\delta\kappa_s.
\]
A narrow \(\sigma\to 0\), \(\mu\to 0\) is consistent with an all-Kerr population [2111.04135]. The second approach is a hierarchical mixture likelihood in which each event is either a BBH with \(\kappa=1\) exactly or a non-BBH with \(\kappa\neq 1\), with mixture fraction \(f_{\rm nbh}\). The single-event likelihood is
\[
L_j(d_j\mid f)=(1-f)\,Z_j^{\rm BH}+f\,Z_j^{\rm nBH},
\]
and the \(N\)-event likelihood is the product over events [2111.04135].

Both methods are equally effective to hint at inhomogeneous populations, but the mixture-likelihood approach was found to be more natural for mixture populations comprising compact objects of diverse classes [2111.04135]. On simulated high-quality injections satisfying \(\chi_{\rm eff}\ge 0.15\) and inspiral SNR \(\ge 20\), the mixture method recovered \(f_{\rm nbh}\sim 1\) for a pure non-BH set and \(f_{\rm nbh}\sim 1/3\) for a \(1{:}2\) mixture [2111.04135]. Applied to the LIGO-Virgo detections published in GWTC-2, the posterior on \(f_{\rm nbh}\) peaks at zero, \(f_{\rm nbh}=1\) is strongly disfavored, and the hierarchical-Gaussian analysis likewise finds \(\mu\approx 0\) and \(\sigma\approx 0\), all consistent with a pure BH population within present statistical precision [2111.04135].

## 4. Systematics, degeneracies, and reduced waveform constructions

The main limitations of present-day inference are not only statistical. Strong degeneracies exist between \(\delta\kappa_s\), effective spin, and mass ratio, and overlap calculations show that for a given injected \(\delta\kappa_s\) there can exist black-hole waveform parameters \((\chi_{\rm eff},q)\) yielding overlaps above \(0.995\) [1908.02247]. This explains skewed posteriors and the slow growth of evidence for non-Kerr structure in second-generation data.

Prior choices are also important. In one population study the prior range on \(\delta\kappa_s\) in single-event runs was uniform in \([-500,500]\); unless the true non-BBH \(\kappa\) lie in similarly wide ranges, Occam’s penalty disfavors non-BH interpretations [2111.04135]. The same analysis imposed \(\kappa_1=\kappa_2\), i.e. \(\delta\kappa_a=0\), so events with mismatched \(\kappa_1,\kappa_2\) would be mis-modeled and could pull \(f_{\rm nbh}\) downward [2111.04135]. Selection biases as a function of \(\kappa\) were not included there and were identified as a necessary ingredient for a fully astrophysical analysis [2111.04135].

Waveform-model dependence introduces an additional systematic. Re-analyses with IMRPhenomXPHM plus the new precession module showed that IMRPhenomPv2 can artificially tighten \(\kappa\) bounds when its internal phase differences relative to the true signal are larger than the true-signal phase shifts due to \(\kappa\neq 1\) [2308.09032]. In those injected-signal studies, mismatch-versus-\(\kappa\) curves confirmed that IMRPhenomPv2 overestimates constraining power [2308.09032].

Large deviations from Kerr require dedicated search models. A six-dimensional post-Newtonian waveform with \(\{m_1,m_2,\chi_1,\chi_2,\kappa_1,\kappa_2\}\) dependence was reduced to five- and four-dimensional models by reparameterizing the dominant quadrupole terms into effective parameters and truncating subdominant \(\chi_a\)-dependent contributions [2211.00039]. The same work showed that large positive \(\kappa_i\chi_i^2\) lowers the minimum-binding-energy frequency substantially, providing a natural inspiral cutoff; the analytic approximation to that cutoff reproduces the true PN ISCO to about \(10\%\) even for \(\kappa_i\) up to \(10^3\) [2211.00039]. In injection studies drawn from \(m_{1,2}\in[1,5]\,M_\odot\), \(\chi_{1,2}\in[-0.6,0.6]\), and \(\kappa_{1,2}\in[1,10^3]\), both reduced models achieved mean effectualness \(\varepsilon\approx 0.999\), more than \(80\%\) of injections had \(\varepsilon>0.999\), and even the worst cases remained above \(0.97\); by contrast, a standard BBH template bank had mean \(\varepsilon\sim 0.58\) and failed for \(\kappa_i\gg 1\) [2211.00039].

## 5. Measurement prospects with third-generation and space-borne detectors

Third-generation ground-based detectors substantially improve the measurement of spin-induced quadrupoles. For a single optimally oriented binary at \(D_L=400\) Mpc, mass ratio \(q=1.2\), \(\chi_1=0.9\), and \(\chi_2=0.8\), forecasts for Cosmic Explorer and Einstein Telescope ET-D give
\[
\sigma(\kappa_s)\simeq 1.0-1.2
\]
across total masses \(M\in[20,100]\,M_\odot\), with the tightest bound reaching \(\sigma(\kappa_s)\approx 1.0-1.1\) [1811.00317]. The error improves for larger total mass, higher mass ratio, and spins aligned with \(\hat L\); relative to Advanced LIGO, the error shrinks by about \(40\times\) for CE [1811.00317]. For simulated populations, roughly \(50\%\) of detected binaries satisfy \(\sigma(\kappa_s)\lesssim 5\), and about \(30\%\) satisfy \(\sigma(\kappa_s)\lesssim 2\) [1811.00317].

Precession adds further information for mass-gap systems. In representative A#-network studies at \(200\) Mpc, a GW190814-like configuration with \(\chi_1\approx 0.07\), \(\chi_2\approx 0.7\), and injected \(\kappa_2=2\) yielded
\[
\kappa_2=2.0_{-2.0}^{+1.9}\quad (90\%)
\]
for A#, while Cosmic Explorer gave
\[
\kappa_2=1.95_{-0.21}^{+0.13}.
\]
The same analysis found that, in a high-SNR O4 injection with moderate precession, including the PI effect shrinks \(\delta\kappa_s\) upper limits by about \(20\!-\!50\%\) relative to AI-only recovery [2308.09032].

Space-borne detectors target a different regime. For a canonical massive black hole binary with \(\mathcal{M}=1.24\times10^6\,M_\odot\), \(\eta=2/9\), true \(\delta\kappa=0\), and \(D_L=1\) Gpc, TianQin forecasts a total one-year SNR of about \(6600\), with the final day alone carrying \(99.9\%\) of that SNR [2401.12066]. The corresponding Fisher errors are \(\sigma_{\delta\kappa}\approx 0.17\) for a full year or one month of data and \(\sigma_{\delta\kappa}\approx 0.33\) for one day, indicating constraints at the \(\sim 0.1\) level [2401.12066]. The optimal mass range is around \(M_{\rm tot}\sim 10^{5.5}M_\odot\), larger mass ratios improve the constraints, and including higher modes tightens \(\sigma_{\delta\kappa}\) by roughly a factor of \(3\) in the Bayesian study [2401.12066].

These forecasted accuracies imply progressively different tests. Third-generation ground networks can directly probe the inspiral quadrupole coefficient \(\kappa_s\) at the level needed to separate Kerr binaries from neutron-star or boson-star mimickers in favorable events [1811.00317]. Space-borne observations of massive black hole inspirals can test the Kerr relation \(Q=-\chi^2M^3\) to about \(10\%\) and use Bayes factors to distinguish black-hole and boson-star-like injections decisively when higher modes are included [2401.12066].

## 6. Extended-body dynamics, sign structure, and broader spin-system realizations

In extreme-mass-ratio motion, spin-induced quadrupoles can be incorporated directly in the Mathisson-Papapetrou-Dixon equations. Numerical studies of circular and generic Kerr orbits found that the orbital-frequency shift scales as \(\Delta\Omega_\phi/\Omega_\phi\propto S^2 C_Q\), peaks at an intermediate radius, and is only about \(10^{-6}\!-\!10^{-8}\) for typical EMRI parameters \(S\lesssim 10^{-2}\), \(C_Q\lesssim 8\), and \(r\gtrsim 6M\) [1611.07602]. The same analysis concluded that one may safely neglect spin-induced quadrupoles in EMRI waveform templates for space-based detectors, although periodic variations in the kinematical spin \(s=S/m\) at the level of \(10^{-6}\!-\!10^{-7}\) could in principle be relevant for pulsar timing around intermediate-mass black holes [1611.07602].

The sign of the quadrupole is not universal. In the Kerr family one has \(Q_{\rm Kerr}=-J^2/M\), or equivalently \(R\equiv M_2 M/J^2=-1\) [2510.05217]. By contrast, singularity-free Running-Kerr-Taub-Bolt solutions descending from eleven-dimensional supergravity can have strictly positive \(M_2\) for a range of charges. At \(\gamma=0\), the explicit formula given for \(G_4 M_2\) is positive, and the sign changes as the bolt’s running speed increases [2510.05217]. The paper interprets positive \(M_2\) as elongation along the spin axis rather than Kerr-like pancaking [2510.05217]. A plausible implication is that future measurements sensitive to the sign structure of the spin-induced quadrupole would test not only the magnitude of non-Kerr deviations but also the class of microphysics supporting the compact object.

Outside relativistic compact-object astrophysics, spin-induced quadrupole moments appear as operator-valued observables. In spin-1 magnets, the on-site traceless symmetric tensor
\[
Q_i^{\alpha\beta}\equiv \tfrac12(S_i^\alpha S_i^\beta+S_i^\beta S_i^\alpha)-\tfrac23\delta^{\alpha\beta}[S_i\!\cdot\! S_i]
\]
quantifies quadrupolar or spin-nematic order, and the classical limit of spin-1 is not a single \(O(3)\) vector but a point on \(U(3)/U(2)\simeq \mathbb{C}P^2\) carrying both dipole and quadrupole components [2203.09819]. In magnetic materials with transition metals, the quadrupole tensor of the spin density contributes to magnetocrystalline anisotropy through the spin-flip term
\[
E_{\rm MCA}^Q\approx \frac{21}{8}\sum_I \frac{\xi_I^2}{\Delta_{\rm exc}}\,m_T^I,
\]
with \(m_T^I\equiv -\langle \hat Q_{zz}\hat S_z\rangle_I\) the intra-atomic magnetic dipole or spin-quadrupole moment [2209.04070]. In relativistic two-body \(S\)-wave spin-1 composites, a nonzero static quadrupole moment arises even when \(l=0\), because Wigner (Melosh) rotations break the non-relativistic cancellation [1810.03281]. In the Kitaev spin liquid with spin-\(3/2\) impurities, the impurity quadrupole moment exhibits discontinuous jumps at flux-sector transitions and thus acts as a local probe of the underlying \(Z_2\) gauge sector [2603.23145].

Across these realizations, the shared structure is precise but not identical. In compact-binary relativity, spin-induced quadrupole moments are usually effective multipole coefficients in orbital dynamics and waveform phasing. In composite and many-body spin systems, they are often expectation values of traceless rank-2 operators that diagnose anisotropic spin structure. The common theme is that spin generates quadrupolar information beyond the dipole level; the operational meaning depends on the dynamical framework in which the quadrupole is defined.

Source: https://www.emergentmind.com/topics/spin-induced-quadrupole-moments