Fully Gravitational Wave Function

• Fully gravitational wave function is a concept that treats gravity as fully dynamical by using the full geometric data rather than relying on fixed background metrics or frozen boundary conditions.
• It unifies diverse approaches, including curvature-based detector responses, second-order full-metric perturbation theory, and representation-theoretic quantum constructions in settings like AdS and JT supergravity.
• The operational perspective extends to waveform modeling for fully precessing eccentric binaries, enabling accurate simulations even in regimes with high eccentricity and complex dynamics.

Fully gravitational wave function denotes a family of constructions in which gravitational radiation or gravitational states are formulated using fully dynamical geometric data rather than a fixed background, a special gauge choice, or externally frozen boundary data. The literature uses the expression, and closely related ones, in several technically distinct settings: curvature-based detector response in classical general relativity, second-order nonlinear wave equations built from the full metric, Hartle–Hawking states with dynamical boundaries in AdS, representation-theoretic bulk states in JT supergravity, and operational waveform models for fully-precessing eccentric binaries (Koop et al., 2013, Hayward et al., 2020, Anikeeva et al., 13 May 2026, Belaey et al., 2024, Arredondo et al., 2024). This suggests that the unifying idea is not a single canonical object but a programmatic requirement: gravity is to be treated as fully dynamical at the level where the wave description is defined.

1. Terminological scope and common theme

In the detector-response literature, the fully gravitational viewpoint replaces gauge-dependent metric perturbations by the Riemann curvature tensor and measured worldline data, so that the response is “separately gauge invariant and has a clear physical interpretation” (Koop et al., 2013). In nonlinear perturbation theory, the same phraseology points to a description in which $h_{\mu\nu}$ is not treated as a field on a passive Minkowski background, but as part of the geometry itself, with indices raised and lowered by the full metric and the harmonic gauge imposed consistently to second order (Hayward et al., 2020).

In quantum gravity, the phrase becomes more literal. For the Hartle–Hawking wave function in AdS, a fully gravitational construction is one in which the extra spacetime boundary induced by an open spatial slice is itself dynamical and integrated over; the contrasting partially frozen construction fixes that boundary as in AdS/CFT (Anikeeva et al., 13 May 2026). In JT supergravity, “gravitational wavefunctions” are specific mixed parabolic matrix elements of a positive semigroup version of the relevant supergroup, chosen so that asymptotic AdS boundary conditions are enforced directly at the quantum level (Belaey et al., 2024).

A further, operationally distinct usage appears in waveform modeling. The EFPE and EFPE$_{\rm ME}$ families provide efficient frequency-domain inspiral waveforms for compact binaries that are both spin-precessing and eccentric, and the synthesis of that work describes the result as a “fully gravitational-wave function” in the sense of a waveform model that retains both effects simultaneously (Arredondo et al., 2024). The coexistence of these usages indicates that the term has no single universal definition across gravitational physics.

2. Curvature-based classical description

A central classical formulation holds that gravitational-wave detectors do not fundamentally measure metric perturbations $h_{ab}$; they measure the effect of spacetime curvature on light propagation (Koop et al., 2013). The detector class considered is broad: any detector that measures the phase of a remote clock by means of an electromagnetic signal propagating along null geodesics. In that framework the observable is written in terms of intrinsic clock evolution, the accelerations of clock and receiver, a kinematic light-time term, and a curvature term containing the null-path integral

$$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$

In the short-wavelength or high-frequency limit, the Riemann tensor separates into a background curvature contribution and a wave curvature contribution. The perturbation $R^{(1)}_{abcd}$ is then gauge invariant up to terms suppressed by $\delta^2$, with $\delta=(\omega L)^{-1}\ll 1$, and the gravitational-wave response reduces to the integral of the wave curvature along the unperturbed null geodesic: $$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$ This applies, according to the paper, to ground-based interferometers, LISA-like detectors, spacecraft Doppler tracking, and pulsar timing arrays (Koop et al., 2013).

A related fully gravitational characterization of radiation in full nonlinear GR is formulated at null infinity rather than in the bulk. In asymptotically Minkowskian spacetimes, the radiative degrees of freedom are not encoded in the universal structure $(q_{ab},n^a)$ of $\scri$, but in the derivative operator $_{\rm ME}$0 induced on $_{\rm ME}$1. The gauge-invariant radiative data are carried by the trace-free shear $_{\rm ME}$2, and the Bondi news satisfies

$_{\rm ME}$3

while the radiative Weyl scalar obeys

$_{\rm ME}$4

In the linearized limit this becomes

$_{\rm ME}$5

so the familiar strain polarizations arise as the asymptotic shadow of the nonlinear shear (D'Ambrosio et al., 2022).

A recurrent misconception addressed by these formulations is that a detector’s operation is fundamentally explained by a gauge-fixed metric perturbation, for example in TT gauge. The curvature-based account instead treats TT-gauge formulas as special cases and relocates the invariant content to curvature transport and null-infinity radiative data (Koop et al., 2013, D'Ambrosio et al., 2022).

3. Nonlinear wave descriptions based on the full metric

A second use of the fully gravitational idea appears in second-order perturbation theory. The key change is to write

$_{\rm ME}$6

but to raise and lower indices with the full inverse metric expanded consistently to second order,

$_{\rm ME}$7

and only then discard terms of order $_{\rm ME}$8 and above (Hayward et al., 2020).

The motivation is that once nonlinear curvature terms are retained, the perturbation contributes to the geometry, so traces, contractions, and gauge restrictions must be defined at the same perturbative order. The Ricci tensor is expanded to $_{\rm ME}$9 using Christoffel symbols built from the full metric, vacuum is imposed through $h_{ab}$0, and the full harmonic gauge condition

$h_{ab}$1

is also expanded consistently to second order. The resulting equation is a nonlinear vacuum wave equation of schematic form

$h_{ab}$2

with self-couplings and derivative couplings that are absent in the usual linear plane-wave treatment (Hayward et al., 2020).

Within that framework, the perturbation ansatz

$h_{ab}$3

leads to the claim that non-trivial solutions are necessarily non-plane-wave modes with non-zero trace when the amplitude varies. The paper further argues that allowed nonlinear solutions generically contain both longitudinal and transverse components, and that the second-order harmonic gauge does not preserve the standard linear TT reduction. These claims are specific to that second-order full-metric construction and depart from the familiar transverse-traceless plane-wave picture (Hayward et al., 2020).

This creates an important distinction within the literature. Null-infinity treatments identify radiative content through shear, news, and peeling in full GR (D'Ambrosio et al., 2022), whereas the second-order full-metric perturbative program emphasizes nonlinear trace structure and longitudinal-transverse mixing (Hayward et al., 2020). The two are not formulated at the same level of description, and conflating them obscures what each is designed to capture.

4. Single-equation formulations on curved backgrounds

A different route toward a fully gravitational description seeks a single master equation for perturbations on curved backgrounds. In type D spacetimes, one starts from the Newman–Penrose formalism, where the radiative gravitational variables are the extreme Weyl scalars $h_{ab}$4 and $h_{ab}$5, and Teukolsky’s equations provide two decoupled equations for them (Li, 2020).

The proposed unification introduces a transformation function $h_{ab}$6 and a perturbation variable $h_{ab}$7, allowing the spin-$h_{ab}$8 equations to be rewritten in a common form. The geometrical content is then packaged into the “spin-coefficient connection”

$h_{ab}$9

Using NP identities, the unified perturbation equation becomes

$$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$0

with source-free form

$$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$1

In this formulation, the kinetic operator describes propagation with a built-in spin connection, the term $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$2 encodes the background tidal field in type D spacetimes, and the Ricci scalar term captures scalar-curvature effects. The same structural equation is claimed to govern massless fields of nonzero spin $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$3, including Weyl, Maxwell, and Rarita–Schwinger fields, which the paper interprets as analogue models of gravitational waves (Li, 2020).

This program is not a quantum-mechanical wave function in the Hartle–Hawking or JT sense. Rather, it is a unified classical field equation whose “fully gravitational” character lies in making the background geometry explicit through $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$4, $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$5, and $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$6, and in avoiding dependence on a specific coordinate system or a specific metric form within the type D class (Li, 2020).

5. Fully gravitational Hartle–Hawking constructions in AdS

For the Hartle–Hawking wave function in AdS, open spatial slices require an additional spacetime boundary $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$7 meeting the slice $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$8 at a corner $$\int_0^{\lambda_R} R(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda .$$9. This produces two inequivalent ensembles. The fully gravitational wave function integrates over the geometry of $R^{(1)}_{abcd}$0,

$R^{(1)}_{abcd}$1

or equivalently fixes the metric on $R^{(1)}_{abcd}$2 while summing over bulk geometries with $R^{(1)}_{abcd}$3. The partially frozen version instead holds $R^{(1)}_{abcd}$4 fixed, as in an AdS/CFT Dirichlet ensemble (Anikeeva et al., 13 May 2026).

In AdS$R^{(1)}_{abcd}$5 Einstein gravity, the fully gravitational action includes the bulk Einstein–Hilbert term, Gibbons–Hawking–York terms on $R^{(1)}_{abcd}$6 and $R^{(1)}_{abcd}$7, a tension term on $R^{(1)}_{abcd}$8, and a Hayward corner term at $R^{(1)}_{abcd}$9. Varying the metric on $\delta^2$0 yields a Neumann-type condition that reduces in $\delta^2$1 to

$\delta^2$2

so the boundary geometry is not fixed externally but determined by the saddle-point equations. Explicit minisuperspace analyses are carried out in AdS$\delta^2$3 Einstein gravity and AdS$\delta^2$4 Jackiw–Teitelboim gravity, and in both cases real saddles exist only below a critical boundary size, beyond which the relevant saddle becomes complex (Anikeeva et al., 13 May 2026).

The one-loop distinction is especially sharp. For the hyperbolic-ball partition function with a dynamical boundary, the trace operator on the boundary sphere has $\delta^2$5 negative modes, giving

$\delta^2$6

whereas with Dirichlet boundary conditions the one-loop answer remains real and positive: $\delta^2$7 A converse de Sitter calculation with a fixed equator also yields a real positive answer after nontrivial phase cancellation. The paper’s conclusion is that the phase problem is controlled by whether the path integral is fully dynamical or partially frozen, not by AdS versus dS alone (Anikeeva et al., 13 May 2026).

6. Gravitational wavefunctions in JT supergravity

In supersymmetric JT gravity, gravitational wavefunctions are constructed representation-theoretically rather than by solving a minisuperspace differential equation in isolation. For $\delta^2$8 JT supergravity, the relevant structure is the positive semigroup

$\delta^2$9

retaining only the hyperbolic conjugacy class so as to match smooth asymptotically AdS Euclidean bulk geometries (Belaey et al., 2024).

The basic objects are mixed parabolic matrix elements

$\delta=(\omega L)^{-1}\ll 1$0

where the left and right boundary states are Whittaker-type eigenvectors of the parabolic generators. Because the fermionic generators anticommute, the construction uses Majorana/Clifford variables $\delta=(\omega L)^{-1}\ll 1$1 satisfying

$\delta=(\omega L)^{-1}\ll 1$2

so that the fermionic eigenvalue problem is encoded in a Clifford algebra. After fixing the parabolic data, the bulk dependence reduces to the Cartan variables, and the nontrivial wavefunction is the Whittaker function

$\delta=(\omega L)^{-1}\ll 1$3

which the paper computes explicitly in terms of modified Bessel functions $\delta=(\omega L)^{-1}\ll 1$4, $\delta=(\omega L)^{-1}\ll 1$5, and $\delta=(\omega L)^{-1}\ll 1$6 (Belaey et al., 2024).

After imposing the Brown–Henneaux choice $\delta=(\omega L)^{-1}\ll 1$7 with $\delta=(\omega L)^{-1}\ll 1$8, the resulting wavefunction matches the $\delta=(\omega L)^{-1}\ll 1$9 Liouville minisuperspace Hamiltonian derived from the quadratic Casimir,

$$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$0

with $$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$1 and $$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$2. The representation-theoretic method also fixes the normalization of the fermionic multiplet and yields the density of states

$$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$3

The thermofield-double or Hartle–Hawking state exhibits the strongest “fully gravitational” feature in this setting: the no-boundary condition does not permit an arbitrary fermionic multiplet component, but selects a unique linear combination,

$$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$4

with a $$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$5 holonomy fixing the relative phase when present. Here the geometry, boundary condition, and fermionic structure are tied together at the level of group representation theory (Belaey et al., 2024).

7. Operational waveform constructions for fully-precessing eccentric binaries

A more operational usage appears in gravitational-wave modeling for compact binary inspirals. The Efficient Fully-Precessing Eccentric family, EFPE, is a fast frequency-domain model for spin-precessing, eccentric compact-binary inspirals; EFPE$$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$6 extends it to moderate eccentricities by retaining the same precession dynamics and Fourier-transform framework but replacing the small-$$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$7 amplitude expansion with a full harmonic-series representation (Arredondo et al., 2024).

The model includes inspiral dynamics only, with no merger or ringdown, spin-induced orbital precession, orbital eccentricity described by the time eccentricity $$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$8, and orbit-averaged PN radiation reaction. The source-frame geometry is organized using Euler angles $$\left(\frac{d^2\phi}{d\tau^2}\right)^{(1)} = \frac{d\phi/dt}{\mathbf U\cdot\left(\frac{d\phi}{d\tau}\right)^2} \int_0^{\lambda_R} R^{(1)}(\mathbf k,\mathbf U,\mathbf k,\mathbf U)\, d\lambda +\mathcal O(\delta).$$9, $(q_{ab},n^a)$0, and $(q_{ab},n^a)$1, and the waveform is built by twisting orbital-frame modes into an inertial frame: $(q_{ab},n^a)$2 The orbital-frame modes are decomposed into harmonics of the mean anomaly,

$(q_{ab},n^a)$3

and the Fourier transform is computed with shifted uniform asymptotics rather than plain stationary phase, because precession introduces stationary-phase pathologies (Arredondo et al., 2024).

The distinction between EFPE and EFPE$(q_{ab},n^a)$4 lies in the amplitude sector. Earlier EFPE amplitudes used a post-circular expansion truncated at about $(q_{ab},n^a)$5, giving a regime of validity around $(q_{ab},n^a)$6 at the reference time four years before merger. EFPE$(q_{ab},n^a)$7 instead constructs the amplitudes as full Fourier series in the mean anomaly and evaluates enough harmonics to meet a target error tolerance, remaining accurate up to $(q_{ab},n^a)$8. In the LISA band, the comparison shows no significant difference between EFPE and EFPE$(q_{ab},n^a)$9 for $\scri$0, but significant deviations for $\scri$1, especially for total masses below about $\scri$2. The extension is, however, computationally expensive: at $\scri$3, EFPE$\scri$4 can be about 1000 times slower than EFPE (Arredondo et al., 2024).

This usage is conceptually different from curvature-based detector response, null-infinity radiative data, or Hartle–Hawking/JT states. It nevertheless shares the same methodological tendency: the waveform is constructed so that the relevant gravitational structure—here simultaneous precession and eccentricity—is retained rather than frozen or linearized away. Across the literature, that is the strongest common thread linking the otherwise disparate meanings of “fully gravitational wave function.”