Quadrupole formula

The quadrupole formula is a cornerstone of gravitational wave theory in general relativity, providing the leading-order approximation for the emission of gravitational radiation from an isolated, slowly evolving system characterized by a time-varying mass quadrupole moment. In the weak-field limit, it expresses the transverse-traceless (TT) component of the metric perturbation, or strain, as h_{ij}^{\rm TT}(t, \mathbf{x}) = \frac{2G}{c^4 r} \ddot{\bar{I}}_{ij}^{\rm TT}(t - r/c), where G is the gravitational constant, c is the speed of light, r is the distance from the source, \bar{I}_{ij} is the reduced quadrupole moment tensor defined by \bar{I}_{ij} = \int d^3x' \, \rho(t, \mathbf{x}') (x'_i x'_j - \frac{1}{3} \delta_{ij} x'^2), and the double dot denotes the second time derivative evaluated at retarded time.^[1] This formula captures the dominant quadrupolar (l=2) mode of radiation, analogous to electric dipole radiation being absent in neutral systems, and applies to non-spherical mass distributions undergoing acceleration, such as orbiting binaries or rotating asymmetric bodies.^[1] The formula also yields the luminosity, or average power radiated in gravitational waves, as \langle P \rangle = \frac{G}{5 c^5} \left\langle \dddot{\bar{I}}_{ij} \dddot{\bar{I}}^{ij} \right\rangle, where the triple dot indicates the third time derivative and the angle brackets denote a time average over several cycles.^[2] This expression highlights the inefficiency of gravitational radiation, scaling with the fifth power of velocity over c in typical astrophysical sources, and has been instrumental in predicting energy loss mechanisms, such as the inspiral of compact binary systems due to "quadrupole drag."^[1] Originally derived by Albert Einstein in 1918 through the linearized Einstein field equations in the post-Newtonian approximation, the formula corrected an earlier error in his 1916 work and laid the foundation for understanding radiative degrees of freedom in gravity.^[1] Beyond its classical form, the quadrupole formula serves as the starting point for higher-order post-Newtonian expansions and numerical relativity simulations, enabling precise waveform modeling for detectors like LIGO and Virgo.^[1] Its predictions were first observationally validated through the orbital decay of the Hulse-Taylor binary pulsar, discovered in 1974, with measurements confirming general relativity to within 0.2% accuracy by 1978, and continue to underpin analyses of direct gravitational wave detections from events like binary black hole mergers since 2015.^[1] Extensions to cosmological backgrounds, such as de Sitter spacetime with a positive cosmological constant, have confirmed the formula's robustness up to first order in the Hubble parameter in studies since 2017, broadening its applicability to universe-scale phenomena.^[3]

Background

Gravitational Waves Overview

Gravitational waves are ripples in the curvature of spacetime generated by the acceleration of massive objects, as predicted by Albert Einstein's general theory of relativity. These perturbations propagate outward from their sources, representing dynamic disturbances in the gravitational field analogous to electromagnetic waves but arising from the tensor nature of gravity. Einstein first derived their existence in 1916, demonstrating that they emerge as solutions to the linearized Einstein field equations in the weak-field limit.^[4]^[5] Unlike electromagnetic radiation, which includes both monopole and dipole contributions, gravitational waves exhibit a predominantly quadrupolar character because conservation laws in general relativity forbid net monopole and dipole radiation. Specifically, the conservation of total mass-energy eliminates monopole emission, while the conservation of linear momentum prevents dipole gravitational radiation from isolated systems. This quadrupolar nature implies that only asymmetric, non-spherical mass distributions, such as orbiting binaries, can efficiently produce detectable waves.^[6] Gravitational waves travel at the speed of light through vacuum, carrying energy, linear momentum, and angular momentum away from their sources, which leads to observable effects like orbital decay in emitting systems. Upon passing through a region, they induce tidal distortions, alternately stretching and compressing spacetime in a transverse pattern, which can be detected as minute relative displacements in free test masses. In Einstein's 1916 analysis, these waves were treated within the framework of weak-field approximations, where the amplitude is small compared to the background curvature.^[4]^[7]

Multipole Expansion in General Relativity

In the weak-field approximation of general relativity, the multipole expansion provides a Taylor series-like decomposition of the metric perturbation h_{\mu\nu} generated by a localized, slowly moving source, analogous to the multipole expansion in electrostatics where the potential is expanded in powers of distance from the source. In electrostatics, the monopole term corresponds to the total charge, the dipole to the first moment of the charge distribution, and the quadrupole to the second moment capturing asymmetries. Similarly, in gravity, the monopole term represents the total mass (or energy), which is conserved and does not radiate; the dipole relates to the center-of-mass motion; and the quadrupole encodes the asymmetry in the mass distribution, becoming the leading contributor to gravitational radiation for non-spherical sources. Dipole radiation is absent in general relativity for isolated systems due to the conservation of total linear momentum, which implies that the mass dipole moment—essentially the position of the center of mass—remains constant in the center-of-mass frame, yielding zero second time derivative and thus no dipole contribution to the emitted waves. This contrasts with electromagnetism, where charge is conserved but dipole radiation from oscillating dipoles is allowed, highlighting the structural differences arising from the spin-2 nature of the gravitational field. The general form of the multipole expansion for the gravitational wave amplitude h_{\mu\nu} in the transverse-traceless (TT) gauge, valid far from the source in the wave zone, expresses h_{\mu\nu}^{\rm TT} as a sum over multipole moments starting from the quadrupole order (l \geq 2):

h_{\mu\nu}^{\rm TT}(t, \mathbf{x}) = \sum_{l=2}^\infty \sum_{m=-l}^l \left[ \frac{(-1)^l}{r} \frac{\partial^l I_{lm}}{\partial t^l}(t - r) \, {}^{\rm E}T_{\mu\nu lm}(\mathbf{n}) + \frac{(-1)^{l+1}}{r} \frac{\partial^l J_{lm}}{\partial t^l}(t - r) \, {}^{\rm B}T_{\mu\nu lm}(\mathbf{n}) \right],

where I_{lm} and J_{lm} are the mass and current multipole moments, respectively, r is the distance to the observer, \mathbf{n} is the unit vector in the propagation direction, and {}^{\rm E}T_{\mu\nu lm}, {}^{\rm B}T_{\mu\nu lm} are electric- and magnetic-type tensor spherical harmonics that ensure the TT properties (transverse to \mathbf{n} and traceless). Higher-order multipoles radiate more weakly than the quadrupole because the radiated power scales with the (2l + 2)-th time derivative of the source moments, suppressing contributions from higher l in the low-frequency, slow-motion regime typical of astrophysical sources, where the quadrupole term dominates as the primary carrier of gravitational wave energy.

Mathematical Formulation

Definition of the Quadrupole Moment

The mass quadrupole moment tensor provides a mathematical description of the second-order moments of a mass distribution, capturing its spatial asymmetry beyond the monopole (total mass) and dipole (center-of-mass motion) terms. In the context of general relativity, it is defined for a continuous mass distribution with density \rho(\mathbf{x}) as the integral

I_{ij} = \int \rho(\mathbf{x}) \, x_i x_j \, d^3x,

where x_i and x_j are Cartesian coordinates relative to the center of mass, and the integration extends over the source volume.^[8] This tensor is symmetric by construction, I_{ij} = I_{ji}, reflecting the commutative nature of the coordinate products.^[9] For time-varying sources relevant to gravitational wave emission, the reduced quadrupole moment tensor \bar{I}_{ij} is commonly employed, which subtracts the trace to ensure tracelessness:

\bar{I}_{ij} = I_{ij} - \frac{1}{3} I_{kk} \delta_{ij},

where \delta_{ij} is the Kronecker delta and repeated indices imply summation (Einstein notation).^[9] The trace I_{kk} = \int \rho(\mathbf{x}) r^2 \, d^3x, with r^2 = x_k x_k, represents a scalar moment akin to the moment of inertia. The resulting \bar{I}_{ij} satisfies \bar{I}_{kk} = 0 and remains symmetric, making it suitable for the transverse-traceless gauge used in wave propagation.^[8] Physically, the quadrupole moment quantifies deviations from spherical symmetry in the mass distribution, as a perfectly spherical configuration yields \bar{I}_{ij} = 0.^[10] For instance, in a binary system of point masses m_1 and m_2 separated by \mathbf{r}, treating them as delta functions in the density, the reduced quadrupole moment simplifies to \bar{I}_{ij} = \mu (r_i r_j - \frac{1}{3} r^2 \delta_{ij}), where \mu = m_1 m_2 / (m_1 + m_2) is the reduced mass and r = |\mathbf{r}|.^[11] This encodes the orbital asymmetry that drives gravitational radiation in such systems. It forms the leading-order contribution in the multipole expansion for weak-field gravitational wave sources.^[8]

Gravitational Wave Strain

The gravitational wave strain in the quadrupole approximation describes the metric perturbation caused by the time-varying mass quadrupole moment of a compact source, observable as tidal distortions in the far field at distances much larger than the source size or wavelength. The explicit formula for the trace-reversed strain tensor is

\bar{h}_{ij}(t, \mathbf{r}) = \frac{2G}{c^4 r} \ddot{I}_{ij}^{\rm TT}\left(t - \frac{r}{c}\right),

where G is the Newtonian gravitational constant, c is the speed of light, r = |\mathbf{r}| is the luminosity distance to the observer, the double dots denote the second time derivative, and the argument t - r/c is the retarded time that enforces causality by evaluating the source at the moment the signal was emitted.^[8] This expression holds in the weak-field, slow-motion (v \ll c) regime, where higher-order multipoles and relativistic corrections are neglected, providing the dominant contribution to the waveform for most astrophysical sources. The transverse-traceless (TT) gauge simplifies the strain for a wave propagating in direction \mathbf{n} = \mathbf{r}/r, ensuring h_{ij}^{\rm TT} is spatial, traceless (h_{kk}^{\rm TT} = 0), transverse (n^j h_{ij}^{\rm TT} = 0), and satisfies the Lorenz gauge condition. The TT quadrupole moment I_{ij}^{\rm TT} is obtained by projecting the source's mass quadrupole moment I_{ij} (a brief reference to its definition as the second mass moment) onto the plane perpendicular to \mathbf{n} using the projector

I_{ij}^{\rm TT} = \left( P_{i k} P_{j l} - \frac{1}{2} P_{i j} P_{k l} \right) \ddot{I}_{k l},

with P_{i j} = \delta_{i j} - n_i n_j isolating the transverse components, leaving two independent polarizations: the plus (h_+) mode, which stretches one transverse axis while compressing the orthogonal one, and the cross (h_\times) mode, which stretches at 45 degrees to the plus axes.^[8] The observed strain amplitude and polarization depend on the source's orientation relative to the line of sight, parameterized by the inclination angle \theta (angle between the source's angular momentum and \mathbf{n}) and azimuthal angle \phi. For typical sources like circular binaries, the plus polarization amplitude scales with the angular factor (1 + \cos^2 \theta)/2, while the cross polarization scales with \cos \theta, modulating the signal strength and introducing a phase shift related to the polarization angle; these factors arise from the TT projection and determine the beamed emission pattern, with maximum amplitude for face-on (\theta = 0) views.^[12]

Energy Loss Formula

The energy loss due to gravitational wave emission in the quadrupole approximation quantifies the rate at which a source system dissipates energy through the radiation of quadrupole modes, which dominate in the weak-field, slow-motion limit of general relativity. This process is described by the time-averaged power radiated, which arises from the far-field propagation of waves generated by the third time derivative of the system's mass quadrupole moment tensor. The quadrupole moment I_{ij} is defined as I_{ij} = \int d^3x \, \rho(t, \mathbf{x}) (x_i x_j - \frac{1}{3} \delta_{ij} r^2), where \rho is the mass density and the traceless part ensures the transverse nature of the radiation.^[13] The averaged power formula is

\left\langle \frac{dE}{dt} \right\rangle = \frac{G}{5 c^5} \left\langle \dddot{I}_{ij}^\mathrm{T} \dddot{I}^{T ij} \right\rangle,

where G is the gravitational constant, c is the speed of light, the superscript T denotes the traceless part, overdots indicate time derivatives, and the angle brackets denote a time average over several cycles of the source motion. This expression is obtained by integrating the energy flux over a distant sphere surrounding the source, using the linearized theory of gravitational waves. The factor of $1/5 emerges from angular integration over all directions of propagation.^[13]^[14] To define the energy flux carried by the waves, the Isaacson effective stress-energy tensor is employed, which averages the pseudotensor contributions from the metric perturbations over wavelengths much shorter than the scale of background curvature. This tensor, t_{\mu\nu} = \frac{c^4}{32\pi G} \left\langle \partial_\mu h_{ij}^\mathrm{TT} \partial^\nu h^{TT ij} \right\rangle (in the transverse-traceless gauge), captures the energy-momentum flux of high-frequency gravitational waves as an effective matter source in the Einstein field equations. The total luminosity is then the surface integral of the radial component of this flux over a large sphere, yielding the quadrupole power formula after substituting the wave strain h_{ij}^\mathrm{TT} from the quadrupole approximation.^[14] For non-relativistic sources with characteristic mass M and velocity v \ll c, the radiated power scales as \left\langle dE/dt \right\rangle \propto \frac{G M^2 v^6}{c^5 R^2}, where R is the system size; the sixth power of v/c reflects the involvement of three time derivatives acting on the quadrupole moment, which itself scales with v^2. For periodic sources with angular frequency \omega, the scaling becomes \left\langle dE/dt \right\rangle \propto G \omega^6 I^2 / c^5, emphasizing the strong frequency dependence that makes high-frequency systems efficient radiators. This energy drain leads to secular changes in the source dynamics, such as the tightening and circularization of orbits in bound systems due to the preferential loss of energy in non-spherical modes.^[13]

Derivation

From the Einstein Field Equations

The derivation of the quadrupole formula begins with the Einstein field equations (EFEs) in the context of general relativity, which describe the curvature of spacetime due to mass and energy: R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, where R_{\mu\nu} is the Ricci tensor, R is the Ricci scalar, g_{\mu\nu} is the metric tensor, G is the gravitational constant, c is the speed of light, and T_{\mu\nu} is the stress-energy tensor.^[15] To address gravitational radiation from isolated sources, the weak-field approximation is employed, assuming the metric is a small perturbation around the flat Minkowski background: g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, with |h_{\mu\nu}| \ll 1, where \eta_{\mu\nu} is the Minkowski metric.^[16] Linearizing the EFEs by expanding to first order in h_{\mu\nu} yields the simplified form \Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}, where \Box = \eta^{\alpha\beta} \partial_\alpha \partial_\beta is the d'Alembertian operator, and \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} h is the trace-reversed perturbation, with h = \eta^{\mu\nu} h_{\mu\nu}.^[15]^[16] This wave equation resembles the form for electromagnetic waves but sourced by the stress-energy tensor, indicating that gravitational disturbances propagate at the speed of light. To solve it explicitly, the harmonic (or de Donder) gauge is imposed: \partial^\mu \bar{h}_{\mu\nu} = 0, which simplifies the equations further and ensures a unique propagation form analogous to the Lorenz gauge in electromagnetism.^[15] In this gauge, the linearized EFEs reduce to a sourced wave equation, facilitating the identification of radiative solutions.^[16] For far-field radiation from an isolated source, appropriate boundary conditions are applied, assuming no incoming waves from infinity and that the fields vanish at spatial infinity, which selects the retarded Green's function solution for outgoing waves.^[15] In the slow-motion limit relevant to non-relativistic sources, such as binary systems, the stress-energy tensor is dominated by the mass-energy component: T_{00} \approx \rho c^2, where \rho is the mass density, while other components like momentum density are negligible compared to T_{00}.^[16] This approximation sets the stage for expanding the source in multipoles via the Green's function solution.^[15]

Post-Minkowskian Expansion

The post-Minkowskian (PM) expansion solves Einstein's field equations iteratively in powers of Newton's constant G, beginning with the flat Minkowski metric and incorporating successive gravitational self-interactions, providing a framework that extends beyond the leading linearized approximation while maintaining validity in the weak-field regime. At the first post-Minkowskian (1PM) order, this coincides with the linearized theory, yielding the dominant quadrupole contribution to gravitational radiation without assuming slow velocities, though the slow-motion limit v ≪ c is typically imposed for isolated sources. This iterative approach facilitates the extraction of radiation terms by treating the metric perturbation as a series h_{μν} = h_{μν}^{(1)} + h_{μν}^{(2)} + ⋯, where each term satisfies the field equations sourced by the previous ones. The 1PM solution for the metric perturbation in the Lorenz-de Donder gauge is given by the retarded Green's function integral over the stress-energy tensor: \begin{equation} \bar{h}{\mu\nu}(t, \mathbf{x}) = \frac{4G}{c^4} \int \frac{T{\mu\nu}\left(t - \frac{|\mathbf{x} - \mathbf{x}'|}{c}, \mathbf{x}'\right)}{|\mathbf{x} - \mathbf{x}'|} , d^3\mathbf{x}', \end{equation} which propagates causal influences from the source at retarded times, ensuring no incoming radiation from spatial infinity. This expression serves as the starting point for higher PM orders, where subsequent iterations include stress-energy pseudotensors accounting for nonlinearities. In the far-field limit, where the observation distance r ≫ source size and r ≫ wavelength, the integral is expanded by Taylor series in the source coordinates x' relative to the observer position x, approximating 1/|x - x'| ≈ 1/r + (n · x')/r^2 + ⋯, with n the unit propagation vector. The leading 1/r radiation term emerges from the quadrupolar contribution, involving the second time derivative of the source's mass quadrupole moment, as monopole and dipole terms vanish due to conservation laws and the transverse nature of waves in the slow-motion regime. Lower-order terms contribute only to near-field Coulomb-like potentials, while the quadrupole term dominates the propagating wave amplitude. To isolate the physical degrees of freedom, the spatial components are projected into the transverse-traceless (TT) gauge, which removes gauge artifacts, the trace, and longitudinal modes aligned with n. The TT projector operator is \begin{equation} P_{ijkl} = \left(\delta_{ik} - n_i n_k\right)\left(\delta_{jl} - n_j n_l\right) - \frac{1}{2} \left(\delta_{ij} - n_i n_j\right)\left(\delta_{kl} - n_k n_l\right), \end{equation} applied to the trace-reversed perturbation \bar{h}{ij} = h{ij} - \frac{1}{2} \delta_{ij} h_{kk}, yielding the strain \begin{equation} \bar{h}{ij}^{\rm TT}(t, \mathbf{x}) = P{ijkl} \frac{2G}{c^4 r} \partial_t^2 Q_{kl}\left(t - \frac{r}{c}\right), \end{equation} where Q_{kl} = \frac{1}{c^2} \int (x'_k x'_l - \frac{1}{3} \delta_{kl} x'^m x'_m) T_{00} \, d^3\mathbf{x}' is the mass quadrupole moment.^[17] This projection ensures the wave is transverse and traceless, encoding the two polarization modes. The PM formalism incorporates the slow-motion approximation v ≪ c through a matched asymptotic expansion, dividing spacetime into the near zone (r ≲ c^2/GM, strong internal fields, post-Newtonian expanded in v^2/c^2), the radiation zone (r ≳ source size, PM expanded in G), and an intermediate buffer region where both expansions converge term by term. Matching equates the near-zone expansion of the external PM solution to the post-Newtonian interior solution, ensuring global consistency and allowing the quadrupole moment to be computed from near-zone dynamics while propagating it accurately to infinity. This procedure has been implemented to arbitrary PM order in the multipolar-post-Minkowskian extension, validating the leading quadrupole term against higher corrections.

Applications and Observations

In Binary Star Systems

In compact binary star systems, such as those consisting of neutron stars or black holes, the quadrupole formula is essential for modeling the emission of gravitational waves due to the accelerating masses in orbit, which leads to a time-varying mass quadrupole moment and consequent orbital inspiral. These systems are particularly efficient sources because their small sizes and high densities result in strong gravitational fields, amplifying the quadrupole radiation relative to less compact binaries. For a circular binary orbit, the mass quadrupole moment tensor takes the form I_{ij} \propto \mu a^2, where \mu = \frac{m_1 m_2}{m_1 + m_2} is the reduced mass and a is the orbital separation; this moment oscillates at twice the orbital angular frequency due to the quadrupolar nature of the gravitational wave emission. The time variation arises from the relative motion of the two bodies, with the projection of their positions contributing to the off-diagonal components of I_{ij} in the transverse-traceless gauge. The average power radiated via gravitational waves in such systems is described by the Peters-Mathews formula:

\langle P \rangle = \frac{32}{5} \frac{G^4 \mu^2 M^3}{c^5 a^5},

where M = m_1 + m_2 is the total mass; this expression captures the leading-order energy flux from the quadrupole approximation for Keplerian orbits. This power loss extracts orbital energy, causing the binary to shrink over time. The resulting orbital decay rate is given by

\frac{da}{dt} \propto -\frac{G^3 \mu M^2}{c^5 a^3},

which drives the inspiral phase as the separation decreases, accelerating the wave emission in a runaway process until merger. For typical compact binaries, this evolution occurs over cosmological timescales for wide orbits but rapidly intensifies at small separations. A seminal observational confirmation comes from the Hulse-Taylor binary pulsar PSR B1913+16, a neutron star pair where the measured orbital period decrease agrees with the quadrupole formula's prediction for gravitational radiation energy loss to within 0.2%.^[18] This match provided early indirect evidence for the existence of gravitational waves, validating the quadrupole mechanism in a real astrophysical context.

Modern Detections with Gravitational Wave Observatories

The first direct detection of gravitational waves, announced by the LIGO Scientific Collaboration and Virgo Collaboration, was the event GW150914 on September 14, 2015. This signal arose from the inspiral, merger, and ringdown of a binary black hole system with component masses of approximately 36 M_⊙ and 29 M_⊙, resulting in a final black hole of about 62 M_⊙. The observed strain amplitude reached around 10^{-21}, with the signal spanning frequencies from 35 Hz to 250 Hz in the sensitive band of the Advanced LIGO detectors. The waveform's inspiral phase closely matched predictions from general relativity's quadrupole formula, confirming the quadrupole nature of the emitted radiation without significant deviations. Subsequent analyses of GW150914 and similar events rely on advanced waveform modeling to extract source parameters and test theoretical predictions. The inspiral is approximated using post-Newtonian expansions rooted in the quadrupole formula, which provide accurate descriptions up to the last few orbits before merger. For the full signal, including merger and ringdown, numerical relativity simulations are employed to generate template waveforms that incorporate strong-field general relativity effects. These models, when compared to data via matched filtering, yield high-fidelity agreement, with the quadrupole approximation serving as the foundational low-frequency limit for binary black hole systems. A landmark multi-messenger observation came with GW170817 on August 17, 2017, detected jointly by Advanced LIGO and Advanced Virgo. This event stemmed from the merger of two neutron stars with masses around 1.4 M_⊙ each, producing a strain amplitude of roughly 10^{-21} and frequencies from 30 Hz to over 400 Hz. The gravitational wave signal was accompanied by an electromagnetic counterpart, including a short gamma-ray burst observed by Fermi and INTEGRAL, followed by a kilonova visible in optical and infrared wavelengths. These observations independently measured the event's distance at approximately 40 Mpc and verified that gravitational waves propagate at the speed of light to within 10^{-15}, aligning with quadrupole formula expectations for the emission and propagation from binary neutron star inspirals. Precision tests of the quadrupole formula have been conducted using data from the observing runs of LIGO and Virgo, culminating in the GWTC-4.0 catalog with 218 confident detections as of August 2025.^[19] These tests parameterize potential deviations in the spin-induced quadrupole moments of compact objects, where general relativity predicts a specific relation tied to mass and spin (κ = 1 for Kerr black holes). No significant deviations are found, with combined bounds from GWTC-4 events constraining the deviation parameter δκ to within approximately ±5% at 90% credibility for the most informative events, affirming the quadrupole formula's validity in strong-field regimes.^[20]