Mass Quadrupole Moment Parameter

• Mass quadrupole moment parameter is a dimensionless measure defined via Q2 = M³q, capturing how a body’s external spacetime deviates from spherical symmetry.
• The associated 2- and 3-parameter metrics allow arbitrary q values, enabling realistic modeling of deformed, spinning astrophysical objects beyond traditional Kerr models.
• This parameter underpins no-hair theorem tests, gravitational-wave template development, and multipole extraction, enhancing our understanding of strong-field gravity.

The mass quadrupole moment parameter quantifies the leading-order deviation of a gravitating object's external spacetime from spherical symmetry, corresponding to the $l=2$ term in the multipole expansion of the gravitational potential or curvature. In general relativity, exact solutions for isolated bodies—whether black holes, neutron stars, composite systems, or even theoretical models—characterize the quadrupole through a dimensionless parameter, often denoted $q$, that encodes the magnitude (and sometimes sign) of the deviation from sphericity. This parameter serves as an essential diagnostic for distinguishing the spacetime of realistic astrophysical objects from idealized black holes and for probing the influence of internal structure, rotation, and equation of state.

1. Multipole Structure and Precise Definition

The multipolar structure of stationary, axisymmetric vacuum spacetimes is classified by the Geroch–Hansen formalism, which yields a hierarchy of mass $(M_\ell)$ and current $(S_\ell)$ moments. The mass quadrupole moment $Q$, or $Q_2$, corresponds to the $l=2$ mass moment. For a general solution with total mass $M$, the dimensionless quadrupole parameter $q$ is often defined so that

$Q_2 = M^3 q,$

where $q$ encodes the degree of oblateness ($q > 0$) or prolateness ($q < 0$) relative to a reference spherically symmetric solution (Mejía et al., 2019). For the Kerr metric, the quadrupole is fixed by the mass $M$ and dimensionless spin $a/M$ as $Q_\text{Kerr} = -a^2 M$; deviations from this value signal either non-Kerr structure or the influence of an additional degree of freedom, as in the generalized metrics described below.

2. The Simplest 2- and 3-Parameter Quadrupolar Metrics

Mía, Manko, and Ruiz introduced a remarkably streamlined family of static and stationary Weyl solutions parameterized directly by $M$ and $q$ (static case), or $M$, $q$, and $a$ (stationary case) (Mejía et al., 2019):

• Static Solution:

$$ds^2 = f^{-1} \left[e^{2\gamma}(d\rho^2 + dz^2) + \rho^2 d\varphi^2 \right] - f dt^2$$

where the metric potentials $f$ and $e^{2\gamma}$ depend algebraically on $M$ and $q$. The axis expansion yields the Geroch–Hansen quadrupole $Q_2 = M^3 q$.

• Stationary Extension: By specifying the Ernst potential on the axis as

$e(z) = \frac{z - M - Mq - iMj}{z + M - Mq + iMj},$

with $j = J/M^2 = a/M$, the full 3-parameter family is constructed. The Fodor–Hoenselaers–Perjés expansion explicitly gives

$m_{2k} = M q^k, \qquad m_{2k+1} = iM q^k j$

(for $k=0,1,\ldots$), so the independent mass quadrupole remains $Q_2 = M^3 q$, decoupled from the value fixed by the Kerr relation.

This construction admits arbitrary values of $q \in \mathbb{R}$, thus allowing for both large oblate and prolate deformations, unlike other classic solutions.

3. Comparison to the Zipoy–Voorhees and Other Quadrupolar Metrics

The Zipoy–Voorhees (ZV) $\delta$-metric, another well-known static axially symmetric solution, encodes deviations from sphericity via a $\delta$ parameter (or, in Quevedo’s notation, $p$). However, the range of attainable dimensionless quadrupoles

$q = -\frac{p(2+p)}{(1+p)^2}$

is restricted to $-1/3 < q < 0$ for real $p \geq 0$, so the ZV metric cannot realize arbitrarily large negative (oblate) values of $q$ (Mejía et al., 2019). This limitation distinguishes the new 2-parameter model, which encompasses the full range of physically and mathematically admissible quadrupole values.

4. Multipole Extraction and Physical Interpretation

The complete multipolar content of these solutions can be extracted from the asymptotic expansion of the Ernst potential or, equivalently, by evaluating the large-$z$ expansion on the axis: $\xi(z) = \sum_{n=0}^\infty m_n z^{-n-1}$ with the first three moments corresponding to mass ($M$), angular momentum ($J$), and mass quadrupole ($Q_2 = M^3 q$). In the stationary generalization, $q$ determines the principal even mass moments, $j$ the odd current moments.

• Static case: $q < 0$ yields an oblate mass distribution, $q > 0$ a prolate one.
• Stationary case: $q$ continues to control the quadrupolar and higher mass multipoles, while spin $a$ (or $j$) determines the current moments.

Because the metric potentials $f,\,\gamma,\,\omega$ (including ergoregions and light-ring structure) directly encode $q$, this parameter governs the qualitative and quantitative properties of the spacetime geometry.

5. Astrophysical and Mathematical Applications

Arbitrary quadrupole parameters are essential for modeling isolated bodies with realistic internal structure, as in non-Kerr neutron stars, deformed compact objects, or exotics. In contrast to restricted models such as ZV, the new metrics enable:

• Representation of a broad class of isolated, deformed static or spinning bodies in general relativity.
• Systematic investigation of the interplay between mass, spin, and quadrupole effects (e.g., on test-particle motion, gravitational lensing, or orbit precession).
• Construction of interior–exterior matched models: The explicit form and generality of $q$ facilitate the matching of these exteriors to physically reasonable perfect-fluid interiors with prescribed multipole structure, a requirement for astrophysical relevance.

6. Limit Cases and Relevance for Relativistic Theory

Setting $q = 0$ in the static metric recovers the (single-object) Schwarzschild spacetime (trivial quadrupole, spherical symmetry). Introducing spin $a$, but holding $q = 0$, reduces the solution to the axisymmetric Kerr geometry, where quadrupolar structure is not independent but determined by $a$ and $M$ via $Q_\text{Kerr} = -a^2 M$.

By promoting $q$ to an independent parameter, one relaxes the constraints of the Kerr family and provides a testbed for no-hair theorem tests, gravitational-wave template development, and analytic studies of strong-field phenomena.

---

Key references:

• Mía, Manko & Ruiz, "On the simplest static and stationary vacuum quadrupolar metrics" (Mejía et al., 2019)
• For classic metrics with restricted $q$: Zipoy–Voorhees solution
• For formalism and multipole moments: Fodor–Hoenselaers–Perjés and Geroch–Hansen

---