Gravitational On-Shell Action

Updated 4 August 2025

Gravitational On-Shell Action is defined by evaluating the action on field configurations that satisfy the equations of motion, yielding gauge-independent physical insights.
It achieves gauge invariance through on-shell projection, metric decomposition, and ghost cancellation, ensuring that only physical degrees of freedom contribute.
This approach is pivotal in studies of asymptotic safety, black hole thermodynamics, and holography, enabling precise analysis of fixed points and observable quantities.

The gravitational on-shell action is the value of the gravitational action evaluated for field configurations that solve the equations of motion. In general relativity and its extensions, the on-shell action encodes physical information relevant to thermodynamic properties of spacetimes, the structure of quantum amplitudes, and the renormalization group flow. Unlike the off-shell action, which depends on gauge and parametrization choices, the on-shell gravitational action is insensitive to pure-gauge artifacts and carries direct physical significance—manifesting, for example, as the partition function in semiclassical gravity and as a generator of symmetry charges in asymptotically flat contexts. This article surveys the technical formulation, operational significance, and research applications of gravitational on-shell actions, drawing on techniques in both perturbative and nonperturbative regimes, their role in asymptotic safety, black hole thermodynamics, functional renormalization, and on-shell amplitudes.

1. Definition and Context of the On-Shell Gravitational Action

The gravitational on-shell action is defined by evaluating the action functional, typically based on the Einstein–Hilbert or extended Lagrangians, on solutions to the equations of motion (i.e., on-shell field configurations). For general relativity in $d$ dimensions, the standard action is

$S = \frac{1}{16\pi G} \int d^d x \sqrt{-g} (R - 2\Lambda) + S_{\text{boundary}},$

where $S_{\text{boundary}}$ consists of terms (such as the Gibbons–Hawking–York term) needed for a well-posed variational problem. For more general theories (e.g., Ricci-based gravity (Mora-Pérez et al., 13 Jun 2024) or higher-derivative models), $S$ is a non-linear function of curvature invariants and possibly independent connections.

The on-shell value of $S$ is crucial for several reasons:

It directly yields thermodynamic quantities (free energy, entropy) in black hole physics and gravitational thermodynamics (Halmagyi et al., 2017, Suh, 2018).
It serves as the semiclassical weight $e^{-S}$ in the path integral, controlling saddle-point contributions to the quantum gravitational partition function, and thus yields generating functionals of $S$ -matrix elements or conformal field theory correlators via AdS/CFT (Chakrabarti et al., 2022).
It encodes asymptotic and boundary symmetries, making it central in studies of soft theorems, BMS charges, and gravitational memory effects (He et al., 2 Aug 2024, Upadhyay, 13 Jan 2025).

In the context of quantum field theory, "on-shell" also refers to quantities that are free of gauge and parametrization ambiguities, as all terms vanishing by the equations of motion cancel.

2. Gauge (In)dependence and the On-Shell Projection

A major technical challenge in gravitational functional integrals and effective actions is the gauge and parametrization dependence of the off-shell effective action, which arises from choices in gauge fixing and metric decomposition (Benedetti, 2011). The on-shell gravitational action, by contrast, is constructed to be invariant under infinitesimal gauge transformations (diffeomorphisms) that preserve the solution. This is achieved by projecting the action onto solutions of the Einstein equations or their analogs in modified gravity.

In "Asymptotic safety goes on shell" (Benedetti, 2011), this projection is realized by expanding around backgrounds that satisfy the background equations of motion, specifically imposing

$R = 2A_k,$

where $A_k$ is a scale-dependent cosmological constant. This procedure removes terms which are proportional to the equations of motion (and thus vanish on physical solutions), leading to

RG flows for couplings that are free of explicit gauge artifacts,
beta functions for the essential couplings (e.g., cosmological constant) that are manifestly gauge independent.

Similarly, in the functional renormalization group equation (FRGE) framework, the on-shell projection isolates the physical content relevant for observable quantities, distinguishing it from unphysical, gauge-dependent off-shell flows.

3. Technical Methodologies for Isolating the On-Shell Sector

Achieving a gauge-invariant, physically meaningful on-shell action in gravity requires addressing decomposition, regularization, and ghost sector issues:

Metric Fluctuation Decomposition: The fluctuation field is decomposed into transverse-traceless (physical) and pure-gauge components; only the former survive the on-shell projection [(Benedetti, 2011), eqs. (3.8)-(3.9)].
Ghost Sector Treatment: Modifications to the Faddeev–Popov ghost action, including "squared" determinants and auxiliary field constructions, ensure that ghost contributions cancel gauge-variant fluctuation contributions exactly on shell [(Benedetti, 2011), eqs. (4.2)-(4.4)].
On-Shell Cutoff Schemes: Regulators are applied only to components of the Hessian that remain nonvanishing on shell, substantially simplifying the cancellation of gauge artifacts [(Benedetti, 2011), eq. (5.1)], and allowing spectral summation over physical degrees of freedom.

The net effect is to generate flow equations and physical couplings (e.g., cosmological constant, Newton's constant) whose running is gauge independent in the essential, observable combinations.

4. Physical Consequences and Fixed Points in Asymptotic Safety

The on-shell effective action framework leads to physically meaningful renormalization group flows and fixed points. An important result is the existence of a nontrivial, gauge-independent fixed point for the dimensionless cosmological constant $A_k$ within the Einstein–Hilbert truncation (Benedetti, 2011): $A_k^* = 0.2612.$ The fixed point for Newton's constant $G_k^* \approx 1.458$ does retain gauge dependence, but this dependence affects only inessential, non-observable parameterizations; the on-shell cosmological constant, entering physical observables (e.g., S-matrix elements), does not.

This property supports the broader scenario of asymptotic safety in gravity: universal, physically measurable observables are encoded in on-shell quantities whose universality is robust against gauge choices and parametrization ambiguities—a central requirement for the predictive power of any purported UV completion of quantum gravity.

5. Mathematical Formulations and Functional Traces

The formal structure of on-shell gravitational actions is encoded in several key equations:

Flow Equation (FRGE):

$\partial_t \Gamma_k = \frac{1}{2} \text{STr} \left\{ (\Gamma_k^{(2)} + R_k)^{-1} \partial_t R_k \right\}$

evaluated such that $\Gamma_k^{(2)}$ and $R_k$ act nontrivially only on physical, on-shell degrees of freedom [(Benedetti, 2011), eq. (2.1)].

Einstein–Hilbert Truncation and Essential Coupling:

$\Gamma_k[g] = \frac{1}{16\pi G_k} \int d^n x \sqrt{g} (2A_k - R)$

with the essential coupling $T_k = G_k^{-1} - 2A_k$ .

On-Shell Condition (Background Equations):

$R = 2A_k$

ensuring expansion about physical backgrounds.

Spectral Sums:

$\text{Tr}_s W(\Delta) = \sum_{n} D_{n, s} W(\lambda_{n, s})$

where $W$ is a test function, $\lambda_{n, s}$ are eigenvalues, and $D_{n,s}$ are the corresponding degeneracies.

These structures allow for explicit computation of the RG flow, fixed points, and physical coupling constants in a gauge-invariant (on-shell) manner.

6. Broader Role in Quantum Gravity and Observable Quantities

In practical applications, the on-shell gravitational action enters into various physically significant constructs:

Black Hole Free Energy and Entropy: The on-shell action evaluated on black hole backgrounds yields the free energy, from which the entropy and thermodynamics follow via standard thermodynamic relations (Halmagyi et al., 2017, Suh, 2018).
Gauge/Gravity Duality (AdS/CFT): In the semiclassical limit, the exponential of (minus) the renormalized on-shell action gives the leading-order boundary conformal field theory partition function (Chakrabarti et al., 2022).
S-Matrix and Soft Theorems: On-shell boundary actions at null infinity underpin the generation of tree-level amplitudes, the derivation of soft graviton theorems, and encode the infinite tower of symmetry charges in asymptotically flat gravity (He et al., 2 Aug 2024, Upadhyay, 13 Jan 2025).
Functional Renormalization and Asymptotic Safety: The explicit construction of gauge-independent beta functions and fixed points in the on-shell sector strengthens the theoretical underpinnings of nonperturbative approaches to quantum gravity (Benedetti, 2011).

7. Connections to Extended Frameworks and Future Directions

Recent developments demonstrate that the on-shell gravitational action plays a central role beyond general relativity:

In Ricci-based gravity and related extended theories, careful analysis of the independent connection, boundary terms, and 3+1 decomposition preserve the ADM energy structure and ensure the physical content remains encoded in the on-shell action (Mora-Pérez et al., 13 Jun 2024).
In semiclassical and quantum corrections, the implementation of on-shell renormalization schemes ensures that quantum-corrected propagators retain physical normalization, removing ambiguities in the definition of Newton's constant and yielding modified observable potentials with direct experimental implications (Jimu et al., 2 Oct 2024).
In the context of boundary terms at null infinity, the explicit resolution of corner ambiguities ensures that on-shell boundary actions obey the requisite symmetries and match to scattering amplitudes and soft factor insertions (Lehner et al., 2016, Upadhyay, 13 Jan 2025).

These developments point toward the universality and centrality of the gravitational on-shell action as a vehicle for connecting classical, semiclassical, and quantum gravity.

Table: Key Distinctions between Off-shell and On-shell Effective Actions in Gravity

Property	Off-shell Effective Action	On-shell Effective Action
Gauge dependence	Explicit, depends on gauge/parametrization	Eliminated: only physical, gauge-invariant content
Includes unphysical modes	Yes	No, only physical solutions contribute
Observable predictions	Not directly; must project further	Directly contains S-matrix, charges, thermodynamic data
Functional RG flow	Gauge-dependent beta functions	Beta functions for observables are manifestly gauge-free
Role in black hole entropy	Indirect	Computes entropy/free energy on solution
Boundary terms	Model dependent, may be ambiguous	Fixed to ensure variational principle on physical space