Gravitational Self-Completion in Quantum Gravity

• Gravitational self-completion is a mechanism where attempts to probe trans-Planckian energies lead to black hole formation, thereby establishing a minimal length scale.
• It replaces traditional Wilsonian UV completion by leveraging classical gravitational dynamics to suppress new high-energy quantum fields and maintain unitarity.
• The concept influences quantum gravity, swampland criteria, and phenomenology, impacting black hole remnant behavior and the UV structure of spacetime.

Gravitational self-completion is the principle that gravity dynamically prohibits access to arbitrarily short distances or trans-Planckian energies by ensuring that any attempt to probe such regimes results in the formation of horizons, classically manifested as black holes. This mechanism implies the existence of a minimal observable length scale and fundamentally alters the ultraviolet (UV) structure of quantum gravity and quantum field theory when gravity is included. The concept is sharply distinguished from conventional Wilsonian UV completion: instead of new elementary fields or degrees of freedom entering beyond a cutoff, the underlying gravitational dynamics—via black hole formation, classicalization, or horizon production—provide a self-completing boundary, enforcing unitarity and consistency through IR physics already present in the theory (Dvali et al., 2010, Mureika et al., 2012, Gomez, 2019, Isi et al., 2013, Isi et al., 2014).

1. Self-Completeness: Definitions and Minimal Length

The core of gravitational self-completeness is the interplay between the quantum Compton wavelength $\lambda_C = \hbar c / E$ and the gravitational (Schwarzschild) radius $R_S$ associated to energy $E$ (or mass $M$). For a given energy injection, localization below $\lambda_C$ is forbidden by quantum uncertainty, while for energies above the Planck scale, $R_S$ exceeds the would-be region being probed, and a horizon forms, hiding the region from any further scrutiny:

$\lambda_C(E) = R_S(E)$

Imposing this intersection yields a critical or minimal length $\ell_{\min}$, with a corresponding maximal resolvable energy $E_*$ (often set by the Planck scale). In $D$ dimensions, the condition reads:

$\lambda_C(E^*) = \mu(D)[G_D\,E^*]^{1/(D-3)}$

where $G_D$ is the $D$-dimensional Newton constant and $\mu(D)$ is a dimension-dependent numerical factor. Solving yields (Mureika et al., 2012):

$$E_* = \mu(D)^{-\frac{D-3}{D-2}} G_D^{-1/(D-2)}, \qquad \ell_{\min} \sim G_D^{1/(D-2)}$$

This operationalizes the notion that the Planck length (or its analog in higher dimensions) is an absolute limit to locality; attempts to probe distances below $\ell_{\min}$ inevitably result in macroscopic horizon formation (Dvali et al., 2010, Isi et al., 2014).

2. Non-Wilsonian UV Completion and Black Hole Dominance

Gravitational self-completeness disrupts the standard Wilsonian paradigm of UV completion, in which higher-energy phenomena are resolved by integrating in new weakly-coupled fields. Instead, when scaling to energies above $M_P$, heavy would-be quantum excitations cease to propagate as particle degrees of freedom and manifest as classical black holes. In this regime, the spectrum is reorganized: trans-Planckian scattering is dominated by black hole production, with geometric cross section $\sigma(s)\sim G^2 s$ for $\sqrt{s}\gg M_P$, and non-horizon processes become exponentially suppressed by the entropy of the corresponding black hole (Dvali et al., 2010).

The physical UV/IR mapping is encapsulated by the exponential suppression factor for virtual processes involving trans-Planckian masses $m$:

$$\Gamma \sim \exp(-m^2/M_P^2) = \exp(-S_{\mathrm{BH}})$$

where $S_{\mathrm{BH}}$ is the Bekenstein-Hawking entropy. As a result, propagator poles for $m\gg M_P$ are dynamically decoupled; the UV dynamics is universally controlled by gravitational collapse and not by new local quantum fields (Dvali et al., 2010, Gomez, 2019).

3. Generalization: Dimensionality and Spontaneous Dimensional Reduction

The existence of gravitational self-completeness critically depends on spacetime dimension. In $D\geq 4$, the horizon grows rapidly enough with $E$ to ensure an intersection with the Compton wavelength and a minimal length $\ell_{\min}$ arises. For $D<4$, notably in $D=2,3$, no such intersection occurs—RS grows only logarithmically or not at all, and horizon formation does not prohibit arbitrarily short distances (Mureika et al., 2012). This leads to the following dichotomy:

• For $D\geq 4$: Self-completeness is realized, with a finite $\ell_{\min}$.
• For $D<4$: No self-completing gravitational barrier arises.

At the Planck scale, several approaches to quantum gravity (e.g., causal dynamical triangulations, asymptotic safety) suggest that the effective spectral dimension flows to 2. However, in such two-dimensional regimes, gravity is perturbatively renormalizable but lacks self-completeness: arbitrarily small masses can create finite horizons without a lowest allowed scale. This results in an "exclusive-or" (XOR) between spontaneous dimensional reduction and self-completeness: either spacetime dimensionally reduces, sacrificing self-completeness but gaining renormalizability, or self-completeness enforces $D\geq 4$ down to $\ell_{\min}$ (Mureika et al., 2012).

4. Extensions: GUP, Extra Dimensions, and Alternative Gravity

Gravitational self-completeness is robust but its manifestation is sensitive to modifications of gravitational dynamics. For instance, incorporating a generalized uncertainty principle (GUP) with

$$\Delta x\,\Delta p \geq \frac{\hbar}{2}(1 + \beta (\Delta p)^2)$$

introduces a fundamental minimal length $\sim \sqrt{\beta}$, which is reflected in horizon geometry for black holes sourced by regularized matter profiles. The extremal configuration yields a minimal horizon radius and a corresponding zero-temperature remnant, so that no curvature singularity is ever exposed. However, if $\beta$ exceeds a critical value ($\beta \geq \ell_P^2/\pi$), the self-completing intersection is lost, and the theory ceases to self-complete at the Planck scale (Isi et al., 2013, Isi et al., 2014).

In extra-dimensional scenarios such as Randall-Sundrum models, the minimal black-hole mass and associated minimal length are deformed by bulk geometry (via tidal charges and the AdS length $\ell$). The self-completion mechanism persists but the quantitative value of the cutoff depends on $\ell$, softening or sharpening the Planck barrier depending on the parameter regime (Isi et al., 2014).

5. Phenomenological and Quantum Gravity Implications

The self-completeness framework has extensive consequences for phenomenology and the conceptual development of quantum gravity:

• Black hole remnants: The existence of a minimal black hole mass stabilizes the endpoint of evaporation, yielding possible Planck-scale remnants that can alter astrophysical signatures and collider phenomenology (Mureika et al., 2012).
• Absence of new UV degrees of freedom: All trans-Planckian excitations are absorbed into the black hole sector; low-energy physics is unaffected by any virtual trans-Planckian modes (Dvali et al., 2010).
• Swampland principles: Universal self-completion provides a rationale for multiple quantum gravity swampland conjectures, including the Weak Gravity Conjecture, the Infinite Distance Conjecture, and the de Sitter Conjecture. In this view, gravitational dynamics universally control the UV limit, and the structure of moduli spaces and the absence of genuine de Sitter vacua follow as consequences of entropy and RG flow considerations (Gomez, 2019).
• Completion of non-gravitational divergences: In certain extensions (e.g., the Einstein–Cartan–Sciama–Kibble theory with torsion), inclusion of gravitational effects can self-complete other pathologies, such as the divergent electron electromagnetic self-energy, through algebraic energy balance, obviating the need for bare-mass renormalization (III et al., 2022).

6. Self-Completion in Metric Perturbation and Gravitational Self-Force

In gravitational self-force computations, particularly in Kerr backgrounds, the "completion" problem arises due to the absence of $\ell=0,1$ (mass and angular momentum shift) modes in reconstructed metric perturbations. The full metric perturbation is completed by adding non-radiative modes induced by the source and fixing the residual gauge freedom via physical continuity and Ricci identity conditions. This procedural completion is conceptually distinct from UV self-completeness but reflects the broader role of gravitational dynamics in ensuring physical consistency and completeness in effective field descriptions (Bini et al., 2019).

7. Open Problems, Limitations, and Outlook

Several aspects remain unsettled. It is not possible to realize both self-completeness and spontaneous dimensional reduction in a single model; reconciling these aspects is an open direction in quantum gravity (Mureika et al., 2012). In GUP-based and torsion-modified theories, the physical interpretation of nonlocality, residual curvature at $r=0$, or possible recovery of general relativity at large scales require further analysis (Isi et al., 2013, III et al., 2022). Phenomenologically, the implications for black-hole production, remnant stability, and the observable signatures of UV "bounce-back" physics are under active investigation.

Gravitational self-completion thus constrains theoretical possibilities for quantum gravity, strongly shaping the allowed UV landscape, linking the existence of a minimal length to the structure of black holes, the renormalizability of gravity, and the swampland boundary (Dvali et al., 2010, Mureika et al., 2012, Isi et al., 2014, Gomez, 2019, Isi et al., 2013).