Running Newton Coupling in Gravity

• Running Newton coupling is defined as the scale-dependence of Newton’s constant (G) emerging from quantum corrections, non-minimal couplings, and RG flow in gravitational theories.
• It drives novel cosmological models and black hole thermodynamics by modulating G with energy density, curvature, and horizon area, leading to empirical predictions.
• Advanced numerical simulations and partitioned multi-physics methods leverage running G to bridge quantum field theory with observable gravitational phenomenology across various scales.

The concept of “Running Newton Coupling” addresses the possible scale dependence of Newton’s gravitational constant, $G$, analogous to the “running” of gauge couplings in quantum field theory. This running of $G$ emerges when quantum corrections or non-minimal couplings are taken into account, particularly in curved spacetime, high-energy gravitational processes, scalar-tensor modifications, and models of quantum gravity. The topic spans effective quantum field theory, cosmology, black hole thermodynamics, gravitational phenomenology on galactic scales, and multi-physics simulation methodology.

1. Theoretical Foundations for Running Couplings in Gravity

The canonical framework for running couplings in quantum field theory is the renormalization group (RG), where coupling constants evolve with the change in energy scale or external momenta. In gravity, running of $G$ is usually treated in two distinct contexts:

• Quantum Corrections in Curved Spacetime: Gravitational couplings such as the vacuum energy (cosmological constant) and Newton’s constant $G$ acquire scale dependence from matter loops in curved spacetime, since curvature introduces an intrinsic length scale ($L$), making it possible to define RG flow for $G$ (Kawai et al., 22 May 2025). The one-loop corrected effective action yields RG equations:

$$\rho = \rho_0 + d_1 m^4 \ln\left(\frac{\mu^2}{m^2}\right),\quad \frac{1}{G} = \frac{1}{G_0} + d_2 m^2 \ln\left(\frac{\mu^2}{m^2}\right)$$

where $\mu^2 = f(m^2, 1/L^2)$ matches the physically relevant scale.

• Asymptotically Safe Gravity: In renormalizable, higher-derivative gravity theories, the running of $G$ is determined by beta functions involving higher-derivative couplings. A universal result is the “short-distance freedom” of gravity, with $G \sim 1/\omega$ as the RG scale increases, and $G \to 0$ in the ultraviolet (Narain et al., 2013). The beta function structure is fundamentally distinct from gauge theory:

$$\frac{d}{dt} \left[\frac{1}{G}\right] = \frac{M^2}{\pi}\left( \frac{5}{3} \omega - \frac{7}{24\omega} \right)$$

Here, $t = \ln(\mu/\mu_0)$, and $\omega$ is a dimensionless coupling, manifesting antiscreening behavior.

2. Cosmological Implementation and Observational Consequences

Running of Newton’s constant has prominent implications in cosmology, affecting the expansion rate, structure formation, and the signatures in precision datasets:

• Emergent Cosmological Models: Utilizing the antiscreening property of $G$ at Planck scales, a non-minimal coupling modifies the matter Lagrangian, yielding an effective Newton’s constant having scale dependence with the energy density $\varepsilon$ (Zholdasbek et al., 4 May 2024):

$$G(\varepsilon) = \frac{G_N}{1 + \varepsilon/\varepsilon_c}$$

At high energies ($\varepsilon \gg \varepsilon_c$), $G$ is strongly suppressed, driving a quasi-de Sitter (inflationary) phase without an additional inflaton field. As the universe cools, $G(\varepsilon)$ approaches $G_N$, smoothly transitioning to standard cosmology. This running can be empirically constrained using CMB and large-scale structure, via computed power spectra and slow-roll observables ($n_s$, $r$).

• Scalar-Tensor Gravity with Running $G$: In scalar-tensor models, Newton’s constant is promoted to a function of a scalar field $\phi$ (or redshift $z$) through a coupling function $F(\phi)$, with the direct scalar-matter coupling encoded in $C(\phi)$ (Kim et al., 2015):

$G(z) = G_0 (1 + b z + a z^2)$

Observational fits to SNIa and BAO data yield time variation rates of order $\dot{G}/G |_{z=0} \sim 10^{-12}\,\text{yr}^{-1}$, indicating a potentially observable running. Interaction terms in the matter conservation law signal a vivid, empirically accessible scalar-matter coupling.

3. Quantum Gravity, Black Hole Thermodynamics, and Scale Identification

Quantum gravity programs frequently predict a running $G$, with direct consequences for black hole physics:

• Scale Identification in Asymptotically Safe Gravity: The energy or cutoff scale $k$ introduced by RG flows must be mapped to a physical (geometric) scale in curved backgrounds. The principle advocated is thermodynamic consistency—enforcing the first law of black hole mechanics (Chen et al., 2022). The consistent identification is:

$G = G(k(x_+)),\qquad k_+ \propto 1/\sqrt{A_+}$

leading to a universal entropy formula for black holes,

$S = \int \frac{dx_+}{4 G(x_+)}$

which reproduces the Bekenstein-Hawking entropy in the classical limit and encapsulates quantum corrections in regions of strong gravity.

4. Galactic Spacetime and Phenomenology of Spatially Varying $G$

A spatially running Newton’s constant can mimic dark matter effects and modify gravitational lensing:

• Vacuum Metric Solutions with Mildly Varying $G$: By permitting $G$ to vary as a function $\xi(r)$ in the action, new galactic metrics arise that account for flat rotation curves. The effective mass profile acquires a negative, non-baryonic geometric contribution (Chakraborty et al., 7 May 2025):

$$m(r) = -\frac{\alpha(2-\alpha)}{2(1-\alpha)^2} r + \frac{CR}{2} \left(\frac{r}{R}\right)^\alpha$$

Asymptotically, circular velocity approaches a constant set by the running parameter $\alpha$. The light deflection angle is suppressed compared to Einstein gravity,

$$\delta \approx \frac{4 m_B}{r_0} - \frac{8\alpha m_B}{r_0}$$

This behavior is discriminable from cold dark matter predictions, where lensing is generally enhanced.

5. Methodological Refinements and Coupling Strategies

Running couplings also emerge as critical aspects in numerical and multi-physics simulation methodologies:

• Non-Intrusive and Hierarchical Coupling: Advanced algorithms allow different subdomains or physical theories (e.g., Newtonian, post-Newtonian, or quantum-corrected gravity) to be “glued” together—each with their own time-step and optimization—in multi-scale gravitational simulations (Zwart et al., 2020). Non-intrusive bridging maintains modularity and conservation, supporting higher-order operator splitting and augmentation with dissipative or relativistic physics.
• Partitioned Multi-Physics and Quasi-Newton Coupling: In partitioned fluid-structure interaction simulations, implicit and robust iterative coupling is achieved via waveform relaxation and quasi-Newton acceleration, allowing the interface variables to run over the entire time window and facilitating “strong” coupling per physical time interval (Rüth et al., 2020). This structure is competitive in iteration count and supports black-box solver integration.

6. Analogies and Conceptual Unity with Running Gauge Couplings

The phenomena of running couplings in gravity show analogies with QCD and gauge theories:

• RG Beta Functions and Universality: RG equations in gauge theory and gravity both allow couplings to run with scale. In non-abelian gauge theory, asymptotic freedom is reflected by negative, monotonically decreasing beta functions, e.g.,

$\beta(\alpha) = -b_0 \alpha^2 - b_1 \alpha^3$

Holographic QCD models maintain a similar negative, linear running with coupling, but introduce discontinuities at first-order phase transitions, which are absent in perturbative treatments (Aref'eva et al., 19 Jul 2024).

• Field Redefinitions and Physical Observables: In both gravity and gauge theory, only quantities invariant under field renormalization (wave function rescaling) encode genuine physical running (Kawai et al., 22 May 2025). For gravity, invariants such as $\eta = 16\pi G \sqrt{\rho}$ represent true running under RG flow.

7. Controversies, Scale Setting, and “Physical Running”

Disputes in the literature arise regarding the physical significance and experimental accessibility of “running”:

• Renormalization Scale Choice: The “standard running” is valid when the renormalization point $\mu$ is not artificially fixed to the external momentum. Suitable choice of $\mu$—as a function of relevant scales—ensures correct RG flow. Misidentification may lead to spurious “physical running,” which does not reflect genuine scale dependence in observables (Kawai et al., 22 May 2025).
• Observability and Metric Renormalization: The running of $G$ and the vacuum energy in effective gravitational theories is gauge- and field-redefinition dependent. True physical effects must be extracted only from RG-invariant combinations and preferably tied to empirical observables.

Table: Running Newton Coupling in Major Contexts

Context	Running Law / Mechanism	Principal Phenomenology/Consequence
Quantum Gravity (AS)	$G(\varepsilon) = G_N/(1+\varepsilon/\varepsilon_c)$	Antiscreening, de Sitter inflation, smooth transition to GR (Zholdasbek et al., 4 May 2024)
Black Hole Thermodynamics	$G$ runs with horizon area $A_+$, $S = \int dx_+/(4G(x_+))$	Universal quantum-corrected entropy, consistency with 1st law (Chen et al., 2022)
Galactic Dynamics	$\xi(r)\propto (R/r)^{2\alpha}$ in action	Flat rotation curves, reduced lensing vs. CDM (Chakraborty et al., 7 May 2025)
Scalar-Tensor Cosmology	$G(z) = G_0(1 + bz + az^2)$	Direct constraints from SNIa/BAO, time-variable gravity (Kim et al., 2015)

Summary

Running Newton coupling is a multifaceted phenomenon arising in quantum gravity, cosmology, galactic dynamics, black hole thermodynamics, and computational simulations. Generically, $G$ acquires scale dependence via quantum corrections, non-minimal couplings, or geometric effects; its running can be parametrized by RG beta functions, energy density, curvature invariants, or background scalar fields. Empirical consequences range from improved models of cosmic inflation, structure formation, and galactic dynamics to testable departures in gravitational lensing and the fine structure of black hole entropy. The theoretical basis for running Newton coupling is closely connected to the principles underlying running in gauge theories, but demands careful scale identification, observable invariance, and thermodynamic consistency to yield physically meaningful predictions.