Non-Local Gravitational Self-Energy

• Non-local gravitational self-energy is defined as field-theoretic and geometric effects where gravitational interactions are expressed via integral kernels that connect spacetime-separated events.
• It arises from quantum corrections and modifications to the Einstein–Hilbert action, yielding finite energy densities, regular black holes, and altered singularity structures.
• These effects enable ultraviolet regularization, influence dark energy estimates, and support phenomena like quantum decoherence through nonlocal memory in gravitational fields.

Non-local gravitational self-energy refers to the class of field-theoretic and geometric effects in gravity where the energy associated with the gravitational field—and its self-interaction—cannot be expressed as a local function of the metric and its derivatives alone, but instead involves integral operators or kernels relating spacetime-separated events. This non-locality emerges both in quantum field theory through loop corrections, as well as in classical or modified gravity frameworks that respect covariance but extend the Einstein–Hilbert action by non-local functional dependence on curvature invariants. Non-local self-energy terms regulate ultraviolet divergences, alter the structure of singularities, and have profound implications for phenomena ranging from quantum decoherence, regular black holes, and wormhole geometries, to potential explanations for dark matter and dark energy.

1. Origin and Formal Structure of Non-Local Gravitational Self-Energy

Non-local gravitational self-energy terms typically arise in two broad settings: (a) quantum corrections to the gravitational effective action, and (b) geometric or field-theoretic models of continuous, extended mass-energy distributions with intrinsic metric feedback.

In quantum gravity, the one-loop effective action for a massive scalar $\phi$ in a curved background yields after curvature expansion a non-local, zeroth derivative term of the schematic form

$$S_{\text{non-loc}} \sim m^4 \int d^4x\,d^4y\,(\Box^{-1}R)_x\,\langle x|\log(\Box + m^2)|y\rangle\,(\Box^{-1}R)_y,$$

where $\Box$ is the covariant d'Alembertian and $R$ the Ricci scalar. The kernel $\langle x|\log(\Box+m^2)|y\rangle$ couples curvature at $x$ and $y$ with a logarithmic, bi-local dependence dictated by the heat kernel or spectral representations. This term acts as a non-local partner to the cosmological constant and cannot be re-absorbed into local counterterms (Donoghue, 2022).

In analytic non-local gravity models, the action is extended as

$$S_g = \frac{1}{2\kappa^2} \int d^4x\,\sqrt{-g}\left[R + R\,f(\Box^{-1}R)\right],$$

where $f$ is analytic and supplies gravitational "memory" effects. The corresponding field equations and energy-momentum complexes incorporate terms reflecting the collective interaction of curvature over extended spacetime regions (Capozziello et al., 2023).

In monistic and field-mass frameworks, as in the Bulyzhenkov hierarchy, the action includes double integrals over spacetime density distributions $\rho(x)K(x,x')\rho(x')$, with $K(x,x')$ the non-local inertial or self-energy kernel constructed from the local time-potentials or metric elements (Bulyzhenkov, 12 Apr 2025, Bulyzhenkov, 13 Sep 2024).

2. Mathematical Representations and Kernels

Central to the explicit realization of non-local gravitational self-energy is the choice of kernel:

• Quantum-generated kernels: The bi-local kernel in one-loop quantum gravity is represented as $\langle x| \log [(\Box + m^2)/m^2] |y\rangle$, defined via

$$\langle x| \log[(\Box + m^2)/m^2] |y \rangle = - \int_0^{\infty} \frac{ds}{s} \left[ K(s; x, y)e^{-sm^2} - \delta^4(x,y)\,e^{-sm^2} \right],$$

with $K(s; x, y)$ the heat kernel (Donoghue, 2022).

• Infinite-derivative gravity: Non-locality is introduced via form factors, e.g., $F(\Box / \Lambda^2) = e^{-\Box/ \Lambda^2}$, modifying propagators as $G(k^2) \rightarrow e^{-k^2/\Lambda^2}/k^2$ and yielding convergent self-energy integrals (Vinckers et al., 2023).
• T-duality-inspired kernels: Modified potentials have the form $V_G(r) = -GM/{\sqrt{r^2 + l_0^2}}$, with $l_0$ a minimal length. The corresponding bare density is $$\rho_{\text{bare}}(r) = \frac{3l_0^2 M}{4\pi (r^2 + l_0^2)^{5/2}}$$ (Jusufi et al., 17 Dec 2025, Jusufi et al., 10 Sep 2025).
• Monistic matterspace kernels: The inertial memory kernel takes the form $$K(\mathbf{x}, \mathbf{x}') = U(|\mathbf{x}|) U(|\mathbf{x}'|) / |\mathbf{x} - \mathbf{x}'|$$, with $U(r)$ the time-potential, directly encoding self-consistent feedback among all volume elements (Bulyzhenkov, 12 Apr 2025).

3. Self-Energy, Ultraviolet Regularization, and Gravitational Memory

Non-local kernels fundamentally modify the gravitational self-energy:

• Ultraviolet regularization: Standard local self-energy for a point mass diverges, but non-locality (e.g., Gaussian damping in $k$-space or point-mass smearing) yields finite results, such as $E_{\rm self} = - m^2 / (M_p^2 L_0 \sqrt{\pi})$ for infinite-derivative gravity (Vinckers et al., 2023), or more generally removes the $1/r$ singularity at $r \to 0$ in T-duality-based frameworks (Jusufi et al., 10 Sep 2025).
• Cosmological implications: In "upgraded Newtonian" gravity, non-local self-consistency imposes a hard upper bound $E(r) \leq (c^4/G) r$, yielding a finite energy density for vacuum fluctuations and a natural estimate for observable dark energy, with perturbative inhomogeneities behaving as dark-matter-like gravitational clusters (Kauffmann, 2012).
• Quantum-to-classical transition: Non-local gravitational self-energy, when imported into the Schrödinger–Newton equation, produces phase shifts and nonlinear energy corrections leading to model-independent gravitationally-induced wavefunction collapse. The collapse time is $\tau \sim \hbar / \Delta E^{\rm GSE}$ and scales inversely with mass squared, matching Diòsi–Penrose heuristics in the local limit (Jusufi et al., 17 Dec 2025).

4. Nonlocality in Self-Energy: Monistic, Geometrodynamic, and Pseudotensor Approaches

Non-local gravitational self-energy is not strictly a quantum phenomenon. Classical and geometric schemes exhibit intrinsic nonlocality:

• Monistic field-mass theories: The divergence of the four-acceleration vector field $a^\mu$ generates both mass density and Ricci curvature, independent of pairwise forces. The self-energy is spread throughout the entire configuration; local densities are functionals of global field arrangements. The resulting field equations guarantee equality of active and reactive masses, stabilize against gravitational collapse without requiring pressure, and produce positive, finite self-energy densities (Bulyzhenkov, 12 Apr 2025, Bulyzhenkov, 13 Sep 2024).
• Pseudotensor complexes: In non-local gravity, energy–momentum pseudotensors are constructed to accommodate terms like $R f(\Box^{-1}R)$, leading to modifications in gravitational self-energy and stress-energy conservation. These corrections are explicitly non-local and reflect memory effects; in practical applications, they alter the binding energy and radiative properties of compact bodies (Capozziello et al., 2023).
• Flat-space reinterpretation: In "spooky black holes," proper accounting of non-local gravitational self-energy in flat space removes the necessity for intrinsic spacetime curvature, recasting all standard relativistic phenomena as arising from local metric rescalings with non-local mass defects. At the endpoint of gravitational contraction, the negative self-energy cancels rest mass, yielding a zero-energy, non-interacting object (Christillin, 2011).

5. Black Holes, Wormholes, and Regularization

Incorporating non-local gravitational self-energy yields profound changes for black hole and wormhole physics:

• Regular (non-singular) black holes: Non-local self-interaction terms regularize the usual curvature singularities. With mass profiles smeared over a minimal length (e.g., via T-duality), the corresponding energy densities and field strengths remain finite everywhere, and the spacetime admits extremal configurations—thermodynamically stable, neutral, Planck-mass remnants—that serve as natural dark matter candidates (Jusufi et al., 10 Sep 2025).
• Wormholes and ER=EPR: Non-local gravitational self-energy supports horizon structures and wormhole throats without requiring exotic, null energy condition-violating matter. Smeared energy-momentum distributes the requisite negative pressure and density conditions intrinsically. Only zero-throat, non-traversable wormhole solutions fully realize ER=EPR duality between quantum entanglement and Einstein–Rosen bridges within a regular spacetime (Jusufi et al., 4 Dec 2025).
• Energy condition violations and collapse prevention: The effective stress–energy from non-local self-energy can violate the strong energy condition and saturate or mildly violate the null energy condition within throat regions or Planck-scale cores, preventing the formation of singularities and providing natural resistance against total collapse (Jusufi et al., 4 Dec 2025, Bulyzhenkov, 13 Sep 2024).

6. Physical Interpretation, Scale Dependence, and Observational Relevance

The impact and observability of non-local gravitational self-energy are highly scale-dependent:

• Non-local terms become active at curvatures or Hubble scales $H \gtrsim m$ (with $m$ the relevant mass scale), ensuring suppression of their effects in low-energy, late-universe regimes while providing significant corrections in early cosmology, inflation, or compact objects at Planckian densities (Donoghue, 2022, Kauffmann, 2012).
• The coefficients of non-local self-energy terms (e.g., $m^4/(40\pi^2)$ from a massive loop) are set by fundamental scales of the particle content and are not adjustable, implying constraints and potential signatures for quantum-gravitational tests (Donoghue, 2022).
• Gravitational self-energy regularization prevents ultraviolet divergences, sets natural cutoffs for quantum fluctuations (imposing $\Delta m \lesssim m_{\rm Pl}$ for virtual particles), and yields cosmologically plausible dark energy densities given by $\rho_{\text{dark}} \sim c^4 / (G r^2)$ for a universe of radius $r$ (Kauffmann, 2012).
• Large-scale entanglement structures and vacuum fluctuations are conjectured to source networks of Planck-scale, zero-throat wormholes, potentially accounting for the entropy and dynamics associated with dark energy and observable black hole evaporation (Jusufi et al., 4 Dec 2025).

7. Comparative Table: Key Non-Local Gravitational Self-Energy Models

Framework / Paper	Key Non-local Form / Kernel	Physical Consequence
Quantum scalar loop (Donoghue, 2022)	$$m^4(\Box^{-1}R)_x \langle x|\log(\Box+m^2)|y\rangle (\Box^{-1}R)_y$$	Fixed large coefficient, early-universe activation, quantum correction to $\Lambda$
Infinite-derivative gravity (Vinckers et al., 2023)	$G(k^2) \to e^{-k^2/\Lambda^2}/k^2$	Finite point-mass self-energy, UV regularization
T-duality inspired (Jusufi et al., 10 Sep 2025, Jusufi et al., 17 Dec 2025)	$V_G(r) = -GM/\sqrt{r^2 + l_0^2}$, $\rho_{\text{bare}}(r)$	Regular black holes, collapse time, dark matter candidates
Monistic geometrodynamics (Bulyzhenkov, 12 Apr 2025, Bulyzhenkov, 13 Sep 2024)	Double-integral over mass density with $K(x,x')$	Positive, collapse-preventing self-energy, equivalence of active/passive mass
Non-local gravity & memory (Capozziello et al., 2023)	$R f(\Box^{-1}R)$ in action, pseudotensor corrections	Non-local weighting, conservation via Bianchi identities

These approaches converge on the principle that gravitational self-interaction is inherently non-local, with implications that span quantum decoherence, regularization of classical singularities, the structure of spacetime at Planck scales, and the cosmic energy budget. Each mechanism modifies both the energy-momentum localization and the dynamical evolution of gravitating systems, with direct consequences for fundamental theoretical consistency and observational cosmology.