--- title: 'Non-Local Gravity: Theory & Applications' url: https://www.emergentmind.com/topics/non-local-gravity-nlg type: topic --- # Non-Local Gravity: Theory & Applications Non-Local Gravity (NLG) denotes a family of gravitational theories in which the gravitational response is not strictly pointwise but depends on a causal spacetime history or on non-local operators. In the best-developed usage surveyed here, NLG is a classical nonlocal generalization of Einstein gravity built in a teleparallel framework and modeled on nonlocal electrodynamics of media; its defining claim is that nonlocality can simulate dark matter through an effective source term rather than through new particles [1101.3752]. In other parts of the literature, the same label also covers theories with inverse d’Alembertian operators such as \(f(R,\Box^{-1}R)\) and theories with analytic form factors built from \(\Box\), where the emphasis shifts toward cosmology, ultraviolet behavior, localization methods, or compact-object solutions [2201.04512, 1803.00561]. ## 1. Conceptual scope and geometric formulations The teleparallel NLG program starts from the teleparallel equivalent of general relativity (TEGR), where the gravitational variables are tetrads and the Weitzenböck connection, so that gravity is encoded by torsion rather than curvature. In that setting, the local constitutive relation between torsion and gravitational excitation is replaced by a causal integral relation, in close analogy with nonlocal electrodynamics of media [1101.3752, 1411.5411]. The preferred tetrad field is central in this construction: the theory has \(16\) tetrad components, \(10\) determining the metric and \(6\) corresponding to local Lorentz gauge freedom, and the reformulated constitutive law is designed so that the nonlocality acts directly on quantities measured by preferred observers [2209.05817]. In the linear approximation around Minkowski spacetime, this teleparallel construction can be rewritten as linearized general relativity sourced by ordinary matter plus an effective dark component. The key structural relation is \[ G_{ij}(x) + \int K(x,y)\,G_{ij}(y)\,d^4y = \kappa T_{ij}(x), \] which, after introducing the reciprocal kernel, becomes \[ G_{ij} = \kappa \bigl(T_{ij} + T^D_{ij}\bigr), \qquad T^D_{ij}(x) = \int R(x,y)\,T_{ij}(y)\,d^4y. \] In this form, nonlocality is mathematically equivalent to an additional source term and is interpreted as effective dark matter rather than as an extra matter species [0812.1059]. The same corpus also contains a local limit of teleparallel NLG. There the nonlocal kernel is collapsed to a local susceptibility \(S(x)\), with general relativity recovered for \(S=0\) and the constitutive law modified by spacetime-dependent response coefficients [2212.05536, 2308.08281]. Separately, other non-local gravity programs use inverse d’Alembertian operators or analytic form factors instead of a constitutive kernel; these are usually motivated by cosmology, effective actions, renormalizability, or the Cauchy problem rather than by dark-matter phenomenology alone [2201.04512, 1803.00561]. | Usage of “NLG” | Nonlocal ingredient | Representative consequence | |---|---|---| | Teleparallel constitutive NLG | Causal kernel in the constitutive relation | Effective dark matter in the Newtonian limit | | Local limit of teleparallel NLG | Susceptibility \(S(x)\) | Modified flat cosmology; de Sitter excluded for dynamic \(S\) | | Inverse-operator or form-factor models | \(\Box^{-1}\) or analytic functions of \(\Box\) | Exact cosmological solutions, localization methods, or wormholes | A recurring misconception is to treat all of these constructions as interchangeable. The literature does not support that identification. The teleparallel constitutive theory, the local-limit susceptibility theory, inverse-\(\Box\) cosmologies, and analytic-form-factor models share the word “nonlocal,” but they are formulated with different dynamical variables, different constitutive or action principles, and different phenomenological targets [2209.05817, 2201.04512, 1803.00561]. ## 2. Newtonian regime and the effective dark-matter mechanism The Newtonian limit is the most explicit and phenomenologically mature sector of teleparallel NLG. The gravitational force on a test mass remains \[ \mathbf{F}(\mathbf{x})=-m\nabla\Phi(\mathbf{x}), \] but the potential obeys a nonlocal Poisson equation, \[ \nabla^2\Phi(\mathbf{x})+\int \chi(\mathbf{x}-\mathbf{y})\,\nabla^2\Phi(\mathbf{y})\,d^3y =4\pi G\,\rho(\mathbf{x}), \] which can be rewritten in reciprocal form as \[ \nabla^2\Phi = 4\pi G\,(\rho+\rho_D),\qquad \rho_D(\mathbf{x})=\int q(\mathbf{x}-\mathbf{y})\rho(\mathbf{y})\,d^3y. \] In Fourier space, \[ \hat{\rho}_D(\mathbf{k})=\hat{q}(\mathbf{k})\,\hat{\rho}(\mathbf{k}), \] so the nonlocal term acts exactly like an additional mass distribution [2107.05841]. A fundamental point emphasized in this formulation is that \(\rho_D=0\) if \(\rho=0\); there is no dark component without ordinary matter to generate it [2107.05841]. The observationally motivated reciprocal kernels are \[ q_1(r)=\frac{1}{4\pi\lambda_0}\,\frac{1+\mu_0(a_0+r)}{r(a_0+r)}\,e^{-\mu_0 r}, \qquad q_2(r)=\frac{1}{4\pi\lambda_0}\,\frac{1+\mu_0(a_0+r)}{(a_0+r)^2}\,e^{-\mu_0 r}, \] with fitted galactic-scale parameters \[ \alpha_0 = 10.94\pm2.56,\qquad \mu_0 = 0.059\pm0.028~\text{kpc}^{-1},\qquad \lambda_0=\frac{2}{\alpha_0\mu_0}\approx 3\pm2~\text{kpc}. \] The inverse scale \(\mu_0^{-1}\approx 17\) kpc marks the distance beyond which the effective coupling saturates [2107.05841]. In the weak-field acceleration law used in galaxy applications, the kernel is often written in the simpler \(a_0=0\) form \[ q(r)=\frac{1}{4 \pi \lambda_0}\frac{(1+\mu r)}{r^2}e^{-\mu r}, \qquad \lambda_0 = \frac{2}{\mu\alpha}, \] which leads to \[ \mathbf{a}(\mathbf x) = - G \int\frac{\rho(\mathbf x')(\mathbf{x}-\mathbf{x'})} {|{\mathbf x}-{\mathbf x'}|^3} \Big[1+\alpha -\alpha e^{-\mu|{\mathbf x}-{\mathbf x'}|}(1+\frac{\mu}{2}|{\mathbf x}-{\mathbf x'}|)\Big]\, d^3x' \] for NLG [1609.07851]. For a point mass \(m\), the nonlocal Poisson equation becomes \[ \nabla^2\Phi(\mathbf{x})=4\pi G\,m\,[\delta(\mathbf{x})+q(\mathbf{x})], \] so each baryonic point source is accompanied by a spherical cloud of effective dark matter of density \(m\,q(r)\). The total effective dark mass is \[ m_D = m\,\alpha_0\,w, \] with \(w\simeq 1\) in the simplest case, so the effective dark mass is roughly \(\alpha_0 m\) [2107.05841]. In finite systems, however, only part of the halo is enclosed. If the baryonic diameter is \(D\), \[ \phi=\frac{M_D}{M_B}<\alpha_0\,H(D), \qquad H(D)=1-\left(1+\frac{1}{2}\mu_0D\right)e^{-\mu_0D}. \] This yields the scale dependence emphasized in the galactic literature: globular clusters should have little effective dark matter, dwarf galaxies may have only modest amounts, and giant spirals or clusters can show strong dark-matter-like behavior [2107.05841]. The effective dark matter generated by this convolution is also qualitatively constrained. It is described as smooth, positive, spherically symmetric, and independent of the matter density itself once the kernel is fixed; as a consequence, the effective density is finite and cusp-free [2201.12852]. That prediction is nontrivial observationally: it aligns with claims of low effective dark matter in some ultra-diffuse galaxies, but it struggles with systems whose inner rotation curves are too steep for a smooth central profile [2201.12852]. ## 3. Galaxy dynamics, rotation curves, and dwarf-galaxy phenomenology Galaxy-scale tests dominate the phenomenology of teleparallel NLG. In the LITTLE THINGS analysis of \(16\) dwarf galaxies, the universal NLG parameters were kept fixed at \[ \alpha = 10.94\pm 2.56,\qquad \mu = 0.059\pm 0.028\ \mathrm{kpc}^{-1}, \] and only the stellar mass-to-light ratio \(\Upsilon_*^{(3.6)}\) was varied. The discretized radial acceleration was computed under cylindrical symmetry and inserted into the rotation-curve model, with the fit based on \[ \chi^2 = \sum_{i=1}^{N}\frac{(V_i^{th}-V_i^{obs})^2}{\sigma_i^2}. \] The reported average goodness-of-fit values were \[ \overline{\chi^2}_{MOG}=1.14,\qquad \overline{\chi^2}_{NLG}=0.84,\qquad \overline{\chi^2}_{MOND}=1.99, \] while the average stellar mass-to-light ratios were \[ \bar{\Upsilon}_*^{(3.6)}(MOG)=11.37,\qquad \bar{\Upsilon}_*^{(3.6)}(NLG)=6.91,\qquad \bar{\Upsilon}_*^{(3.6)}(MOND)=1.91. \] In that sample, NLG fit the dwarf-galaxy rotation curves reasonably well, but did so with stellar mass-to-light ratios larger than standard Milky Way values [1609.07851]. A second line of evidence comes from \(N\)-body simulations of spiral galaxies. In the comparison between an exponential disk in NLG (ENLG) and a Newtonian exponential disk embedded in a live Plummer halo (EPL), both systems were initialized with the same baryonic distribution, particle velocities, random velocities, and Toomre parameter. With \(\alpha=10.9\) and \(\mu=4.5\times10^{-3}R_d^{-1}\), the initial rotation curves were made nearly identical, but the subsequent evolution differed substantially. The bar instability growth rates were reported as \[ \omega \simeq 0.097 \quad (\text{ENLG}),\qquad \omega \simeq 0.067 \quad (\text{EPL}),\qquad \omega \simeq 0.058 \quad (\text{MOG}), \] so the NLG bar grew about \(45\%\) faster than the standard dark-matter case and about \(67\%\) faster than MOG. At late times the NLG bars were weaker and faster, and the ratio \[ \mathcal{R}=\frac{D_L}{a_B} \] approached the fast-bar boundary \(\mathcal{R}=1.4\), unlike earlier EPL simulations that reached \(\mathcal{R}\simeq 4\) [1901.01087]. The ultra-diffuse galaxy analysis sharpened the cusp-free prediction. AGC 114905 was found to be broadly consistent with the low-effective-dark-matter expectation of NLG, whereas AGC 242019 and AGC 219533 were identified as challenging cases because NLG could not explain the sharp slopes of the rotation curves at small radii [2201.12852]. This contrast is significant because it turns the smooth-kernel construction into an observational discriminator rather than a merely formal property. The most recent dwarf-spheroidal study extended the testing program from rotation curves to Jeans modeling of line-of-sight velocity dispersions in Carina, Draco, Fornax, Leo I, Leo II, Sculptor, Sextans, and Ursa Minor. The model used the Plummer profile for the baryonic stellar density, a constant anisotropy parameter \(\beta\), and an `emcee` MCMC fit over \[ \bm{\theta} = \{\lambda_0, \mu_0, \beta, \Upsilon\}. \] The joint analysis of all eight galaxies yielded \[ \log(\mu_0/\mathrm{kpc}^{-1}) \lesssim 3.2 \text{ at 99.7\% confidence}, \qquad \lambda_0 = 9.7 \pm 1.9\ \text{kpc}, \] while Fornax and Sextans showed the main parameter inconsistency, with \(\mu_0\) only marginally compatible with previous literature at about the \(\sim 1.5\sigma\) level [2507.21986]. This suggests that NLG remains phenomenologically viable in dwarf-galaxy kinematics, but not parameter-free in practice. ## 4. Dynamical friction, gravitational scattering, and barred spirals A major technical result of teleparallel NLG is the derivation of a Chandrasekhar-type dynamical friction law in the Newtonian regime. In the approximation used for gravitational encounters, the nonlocal force enhancement is represented by \[ G(1+\Delta)\to G(1+\eta),\qquad 0<\eta<\alpha_0 w, \] so that the change in the velocity of the massive body \(M\) in a single encounter is computed with \[ G_\eta=G(1+\eta). \] After integrating over a background distribution, the NLG analogue of Chandrasekhar’s formula becomes \[ \frac{d\mathbf{v}_M}{dt} = -16\pi^2 G_\eta^2\,m\,(m+M)\ln\Lambda\, \frac{\mathbf{v}_M}{v_M^3} \int_0^{v_M} f(v)\,v^2\,dv, \] and for \(M\gg m\), \[ \mathcal{F}_{\rm NLG} = -\frac{4\pi G_\eta^2 M^2}{v_M^2}\,\rho(