Singularity Resolution to Galactic Rotation: Log-Corrected Quantum Gravity

1. Introduction

Modern astrophysics and gravitational theory have long faced two major cross-scale challenges: At the microscopic scale of black holes, the singularity predicted by classical general relativity exhibits infinite curvature, violating the finiteness requirement of physical quantities in quantum mechanics, and the “information paradox” triggered by Hawking radiation remains unresolved. At the macroscopic scale of galaxies, the observed rotational velocities of peripheral stars and gas are much higher than the limit sustainable by the gravity of visible matter. The mainstream ΛCDM model relies on the unproven hypothesis of dark matter halos, and there is tension between small-scale predictions and observations.

Traditional explanations for these two problems are fragmented: Black hole physics relies on the Kerr metric (requiring post-hoc fitting of spin and inclination), while galactic dynamics depends on dark matter hypotheses, both lacking a unified physical core. More critically, these theories either suffer from inherent incompleteness (e.g., singularities) or lack direct physical carriers (e.g., dark matter particles).

This paper aims to explore a possibility: If quantum gravity, under the effect of nonlocal entanglement, generates a simple logarithmic correction term ($\mathsf{\Phi }\sim \left(\mathrm{l}\mathrm{n}r\right)/r$ ) in the gravitational potential, can it simultaneously solve the problems of black hole singularities and galaxy rotation curves? Subsequently, we will show that quantum vortex topology and nested AdS/CFT duality can provide a possible physical picture for this: at distances close to the black hole singularity ($r<r_{*}\approx 8.792\times {10}^{-11}\mathrm{m}$ ), the negative contribution of $\mathrm{l}\mathrm{n}r$ renders the quantum gravitational potential repulsive, preventing matter from collapsing into a singularity; at large distances in black hole gravitational fields and galaxies, the positive contribution of $\mathrm{l}\mathrm{n}r$ provides additional gravity, replacing dark matter to maintain the high-speed revolution of stars and the flatness of rotation curves. This mechanism requires no renormalization and is based on a clear physical carrier (quantum vortex) and mathematical duality (nested AdS/CFT), placing black hole physics and galaxy dynamics within a single theoretical framework and providing a potential unified solution to cross-scale gravitational challenges.

2. Unified Quantum Gravity Theoretical Framework

2.1. Core Physical Assumptions

The two core pillars of this framework have clear physical images and observational support:

Quantum vortex topological structure: Defined as the statistical average topological carrier of fermion fields, boson fields, and gauge fields. Its operator form (an effective composite operator, characterized by the amplitude + phase of its expectation value on the strong coupling/CFT boundary) is:

$${\mathcal{O}}_{vortex}=⟨\overline{\psi }\psi \varphi {\mathcal{A}}_{\mu \nu }{\mathcal{A}}^{\mu \nu }{⟩}^{1/2}{e}^{iC\theta (x,y)}$$

(1)

Quantum vortex field:

$$\begin{array}{c}{\mathsf{\Phi }}_{vortex}(x,y)={\mathcal{O}}_{vortex}\int d^{4}y\sqrt{-g(y)}K(x,y)\\ =⟨\overline{\psi }\psi \varphi {\mathcal{A}}_{\mu \nu }{\mathcal{A}}^{\mu \nu }{⟩}^{1/2}{e}^{iC\theta (x,y)}\int d^{4}y\sqrt{-g(y)}{\displaystyle \frac{e^{iC\theta (x,y)}}{(|x-y{|}^2+{\mathcal{l}}^2{)}^2}}\end{array}$$

(2)

$\overline{\psi }\psi$: Fermion field, with dimension $\left[\overline{\psi }\psi \right]=L^{-3}$: Boson field, with dimension $\left[\varphi \right]=L^{-1}$${\mathcal{A}}_{\mu \nu }\equiv (B_{\mu \nu },W_{\mu \nu }^a,G_{\mu \nu }^a)$: Unified field strength tensor (macro-photon field), with dimension $\left[{\mathcal{A}}_{\mu \nu }\right]=\left[{\mathcal{A}}^{\mu \nu }\right]=L^{-2}$$e^{iC\theta (x,y)}$: Vortex phase, connecting non-local entanglement (quantum entanglement)

$C$: Central charge (topological charge number)

$\theta (x,y)\sim \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{y_2-x_2}{y_1-x_1}}\right)$: Topological phase

$\mathcal{l}$: Minimum characteristic length (Planck length)

The vortex winding number $W$ is derived from the central charge $C$ and topological phase $\theta (x,y)$ as $W={\oint }_C\nabla \theta \cdot dl$.

It should be noted that the quantum vortex (field) operator does not violate the “Pauli exclusion principle”. Firstly, the vortex phase $e^{iC\theta (x,y)}$ in the operator indicates non-local (entangled) statistical averaging; secondly, the apparent structure of this microscopic topology is mainly located in the “region near black holes” with extremely large spacetime curvature. According to the Heisenberg uncertainty principle: $\mathsf{\Delta }p\approx \hslash /\mathsf{\Delta }x\to \infty$ (since the enormous spacetime curvature makes $\mathsf{\Delta }x\to 0$), thus the Pauli exclusion principle is weakened by the enormous spacetime curvature. This assumption is also indirectly supported by the “quantum tornado” in superfluid helium simulating the vicinity of black holes [1].

Nested AdS/CFT duality: Adopting the hierarchical structure $Ad{S}_4/CF{T}_3\supseteq Ad{S}_3/CF{T}_2\supseteq Ad{S}_2/CF{T}_1$ [2,3] to correlate the quantum spacetime inside black holes with the classical spacetime outside through conformal boundaries, realizing the quantitative description of non-local entanglement.

2.2. Key Formula Derivations

2.2.1. Modified Poisson Equation

Based on the quantum vortex as the carrier of the microscopic topological structure, we regard its statistical average field ${\varphi }_{vortex}$ as a dynamic subsystem satisfying the effective field theory under the high-energy background inside the black hole. Considering the nonlocal entanglement characteristics and scale relativity of this system, its dynamics can be described by a modified d’Alembert operator under the CFT boundary approximation: $\square {\varphi }_{vortex}\approx (k\hslash /c^2){\partial }_t^2{\varphi }_{vortex}-{\nabla }^2{\varphi }_{vortex}=0$, where $k$ is a dimensionless factor characterizing the strength of nonlocal entanglement. Further analysis shows that in the critical region near the boundary, the time evolution derivative term ${\partial }_t^2{\varphi }_{vortex}$ dominates the spatial behavior of the field due to its self-similarity (${\partial }_t^2{\varphi }_{vortex}\approx a^2{\left({\partial }_t{\varphi }_{vortex}\right)}^2\sim a^2{\left(M/t\right)}^2$). Its square term contribution ($a^2{\left(M/t\right)}^2$) is equivalent to a quantum gravity source term inversely proportional to the cube of the distance ($\propto r^{-3}$) (derived from the boundary behavior of the Riemann tensor component $R_{trt}^r\propto r^{-3}$). Introducing this equivalent source term into the classical Poisson equation (${\nabla }^2\mathsf{\Phi }=4\pi G\rho$) yields the modified boundary Poisson equation:

$${\nabla }^2\mathsf{\Phi }=4\pi G\left(M{\delta }^3(r)+{\displaystyle \frac{k{G}_h{M}^2}{4\pi G{r}^3}}\right)$$

(3)

where $M{\delta }^3\left(r\right)$ is the classical gravitational point mass source term, and ${\displaystyle \frac{k{G}_h{M}^2}{4\pi G{r}^3}}$ is the quantum gravitational correction source term. $k$ is the non-local entanglement relative strength factor ($k={\displaystyle \frac{M_{BH,ref}}{M_{BH,topo}}}$, where $M_{BH,ref}$ is the reference black hole mass, and $M_{BH,topo}$ is the target black hole mass providing the quantum gravitational background). The Galactic center black hole Sgr A* is usually taken as the reference: $k={\displaystyle \frac{M_{SgrA*}}{M_{BH,topo}}}$. If another galactic center black hole is used as the reference, the benchmark $G_h$ needs to be relatively transformed. For example, with M87* as the reference: $G_{h,M87*}={\displaystyle \frac{M_{SgrA*}}{M_{M87*}}}{G}_h$, then $k_{M87*}={\displaystyle \frac{M_{M87*}}{M_{BH,topo}}}$, so $k_{M87*}{G}_{h,M87*}={\displaystyle \frac{M_{M87*}}{M_{BH,topo}}}\cdot {\displaystyle \frac{M_{SgrA*}}{M_{M87*}}}{G}_h=k{G}_h$, indicating that the value of $k{G}_h$ is independent of the chosen reference black hole.

$G_h$ is the quantum gravitational constant (fixed value $G_h=\hslash c^2{G}^3/8\approx 3.5224\times 10^{-49}{kg}^{-2}{m}^3{s}^{-2}$), and its unconventional dimension naturally arises in our theoretical framework due to the inclusion of nested AdS/CFT duality ($Ad{S}_4/CF{T}_3\supseteq Ad{S}_3/CF{T}_2\supseteq Ad{S}_2/CF{T}_1$). In this picture, the effective Planck constant at the $CF{T}_1$ boundary, derived from the microscopic quantum vortex structure through duality, undergoes dimensional compactification of the coupled spacetime dimensions (including fluctuation and phase dimensions of the gauge group), leading to a change in its dimension from $kg\cdot m^2\cdot s^{-1}$ to $kg\cdot m^{-8}\cdot s^6$. This dimensional transformation is incorporated into the definition of $G_h$, resulting in its final dimension of $k{g}^{-2}\cdot m^3\cdot s^{-2}$ (when quantum vortices in superfluid helium are confined to nanoscale spaces (simulating dimensional compactification), their vortex phase oscillation energy $E\propto {\hslash }_{eff}\omega$ satisfies ${\hslash }_{eff}\propto d^{-8}$ (d: confinement scale), consistent with the dimension $m^{-8}$ (Nature Phys. 12, 478, 2016) [4], indirectly supporting the rationality of coupled dimensional compactification in the theoretical framework).

Since the theory is in its initial stage of development, there may be mathematical inconsistencies (more physical analogies than rigorous mathematical proofs). However, we believe that the merits outweigh the flaws. At the current initial stage, it can be regarded as a cross-scale semi-effective and semi-phenomenological model (both predictive and fitting). Rigorous mathematical proofs may require the development of new mathematical methods through this model in the future (such as non-local operations in logarithmic coordinates).

2.2.2. Modified Gravitational Potential with Logarithmic Term

Solving the modified Poisson equation yields the core modified gravitational potential:

$$\mathsf{\Phi }(r)=-{\displaystyle \frac{GM}{r}}-{\displaystyle \frac{k{G}_h{M}^2(\ln r+1)}{r}}$$

(4)

This equation consists of two terms:

Classical gravitational term $-{\displaystyle \frac{GM}{r}}$: Dominates conventional gravitational effects, consistent with Newtonian gravity and the weak-field approximation of general relativity.

Quantum gravitational logarithmic term $-{\displaystyle \frac{k{G}_h{M}^2\left(\ln r+1\right)}{r}}$: The core cross-scale correction term, whose effect depends on the magnitude of distance $r$—it is repulsive at short distances (black hole “singularity” scale) and exhibits a gravity-enhancing effect at long distances (galactic scale), essentially representing the macroscopic manifestation of non-local entanglement of quantum vortices (the argument $r$ of the logarithmic term is dimensionless; the theoretical minimum characteristic length $\mathcal{l}$ (Planck length) is normalized to 1 m, i.e., $\mathrm{l}\mathrm{n}\left(r/\mathcal{l}\right)=\mathrm{l}\mathrm{n}\left(r/1\right)=\ln r$, naturally eliminating the dimension of the argument. Thus, the argument $r$ in the logarithmic term of this theory is implicitly normalized).

If the quantum gravitational effect under non-local entanglement of quantum vortices is not considered ($k=0$), the gravitational potential automatically degenerates to the classical gravitational potential: $\mathsf{\Phi }\left(r\right)=-{\displaystyle \frac{GM}{r}}$.