Local Limit of Nonlocal Gravity

• Local limit of nonlocal gravity is a framework where the nonlocal constitutive kernel is replaced by a Dirac-delta distribution multiplied by a scalar susceptibility S(x), localizing gravitational interactions.
• It alters both teleparallel and Newtonian regimes, modifying the Friedmann equations and effectively recovering general relativity for S=0 while introducing dynamical dark energy features.
• The approach integrates linear perturbation theory, Solar System tests, and symmetry-reduced spacetimes, ensuring compatibility with observations and refining dark matter and dark energy models.

The local limit of nonlocal gravity denotes a class of local gravitational theories obtained when the nonlocal constitutive kernel of teleparallel nonlocal gravity is replaced by a Dirac-delta distribution multiplied by a scalar susceptibility $S(x)$. In this formulation, spacetime retains a memory-inspired constitutive structure, but the field equations become local; $S(x)$ measures deviation from the teleparallel equivalent of general relativity (TEGR), and the general-relativistic limit is recovered when $S=0$. In parallel, the same phrase is also used in the Newtonian regime of classical nonlocal gravity to denote the short-distance or Solar System regime in which the theory approaches the inverse-square law, with residual corrections controlled by kernel parameters such as $a_0$ (Tabatabaei et al., 2022, Chicone et al., 2015).

1. Constitutive origin in teleparallel gravity

The teleparallel formulation uses an orthonormal tetrad $e^\mu{}_{\hat\alpha}$ and the Weitzenböck connection

$$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$

which is curvature-free but has nonvanishing torsion

$$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$

In TEGR, the constitutive relation is local and directly analogous to a simple electromagnetic medium. Nonlocal gravity broadens this analogy by replacing the local constitutive law with an integral relation involving a causal kernel $\mathcal K(x,x')$ and a torsion-built tensor $X_{\mu\nu\rho}$ (Tabatabaei et al., 2022).

The local limit is obtained by taking the kernel to be

$$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$

so that

$S(x)$0

The constitutive relation then becomes

$S(x)$1

with $S(x)$2 the torsion pseudovector. The scalar $S(x)$3 is interpreted as a gravitational susceptibility, directly analogous to electric permittivity or magnetic permeability in electrodynamics of media. The condition $S(x)$4 is imposed for physical solutions, and the dynamical equations remain second order if $S(x)$5 (Tabatabaei et al., 2022).

Within this framework, nonlocal gravitational memory is not discarded but compressed into a local constitutive response. The formal reduction to TEGR for $S(x)$6 supplies the precise sense in which the local limit recovers general relativity.

2. Homogeneous and isotropic cosmology

In the cosmological sector, the local limit selects a particularly restrictive family of models. Among homogeneous and isotropic FLRW spacetimes, only the spatially-flat Cartesian model with tetrad aligned with Cartesian axes allows a time-dependent susceptibility $S(x)$7. In conformal time $S(x)$8,

$S(x)$9

and the modified Friedmann equations take the form

$S=0$0

$S=0$1

with $S=0$2 (Tabatabaei et al., 2023).

In cosmic time, the same structure is

$S=0$3

$S=0$4

A central result is that a true cosmological constant cannot be accommodated unless $S=0$5 is constant. Equivalently, de Sitter spacetime is forbidden when $S=0$6; accelerated expansion must therefore be attributed to a dynamical dark energy component rather than to $S=0$7 in the $S=0$8CDM sense (Tabatabaei et al., 2022, Tabatabaei et al., 2023).

The energy densities of cosmic components evolve differently from standard flat cosmology. For a component with equation-of-state parameter $S=0$9,

$a_0$0

For $a_0$1, the dark-energy density is still time-dependent: $a_0$2 This establishes that the local limit does not simply rescale Newton’s constant or the Planck mass; it alters the redshift scaling of matter, radiation, and dark-energy sectors (Tabatabaei et al., 2022).

A commonly used phenomenological parameterization is

$a_0$3

which leads to

$a_0$4

In this sense, the local limit produces a modified expansion history while retaining the standard cosmological parameter set plus the susceptibility sector (Tabatabaei et al., 2022).

3. Linear perturbations and observational viability

Perturbation theory in the local limit is formulated at the tetrad level,

$a_0$5

with induced metric perturbation

$a_0$6

The standard scalar-vector-tensor decomposition can then be applied. The modified linearized field equations are

$a_0$7

so the perturbation dynamics contain explicit teleparallel and constitutive contributions beyond general relativity (Tabatabaei et al., 2023).

In the scalar sector, for the conformal Newtonian gauge variables $a_0$8 and $a_0$9, the generalized Poisson equation becomes

$e^\mu{}_{\hat\alpha}$0

The perturbation $e^\mu{}_{\hat\alpha}$1 couples directly to the gravitational potentials, so the susceptibility affects both background expansion and linearized structure formation. Vector and tensor modes propagate similarly to GR but with damped corrections involving $e^\mu{}_{\hat\alpha}$2 and its derivatives (Tabatabaei et al., 2023).

For confrontation with data, the perturbation equations are rewritten in synchronous gauge and implemented in CAMB. The resulting system contains coupled ODEs for density contrasts and velocity divergences,

$e^\mu{}_{\hat\alpha}$3

with coefficients depending explicitly on $e^\mu{}_{\hat\alpha}$4, $e^\mu{}_{\hat\alpha}$5, $e^\mu{}_{\hat\alpha}$6, and $e^\mu{}_{\hat\alpha}$7. Parameter constraints are obtained through Monte Carlo Markov Chains using Planck CMB data, BAO, and local $e^\mu{}_{\hat\alpha}$8 measurements. The best-fit model offers a somewhat improved fit to the locally measured $e^\mu{}_{\hat\alpha}$9 compared to the value inferred from the CMB in $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$0CDM; there is a clear positive correlation between $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$1 and $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$2, while the data are largely insensitive to $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$3. A reconstruction approach further indicates that the inferred $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$4 from reconstructed Hubble data lies within the $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$5 region of the model’s $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$6 (Tabatabaei et al., 2023).

These results place the local-limit cosmology in a distinct class: it is not a de Sitter completion of $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$7CDM, but a dynamical dark-energy model generated by constitutive modification of teleparallel gravity.

4. Newtonian and Solar System local regime

A different but related use of “local limit” appears in the Newtonian regime of classical nonlocal gravity. There the gravitational potential obeys a nonlocal Poisson equation,

$$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$8

where $$\Gamma^\mu_{\alpha\beta}=e^\mu{}_{\hat\rho}\,\partial_\alpha e_\beta{}^{\hat\rho},$$9 is the reciprocal kernel. The nonlocal contribution simulates dark matter at galactic scales, while the short-distance regime is constructed to revert to Newtonian gravity inside the Solar System (Chicone et al., 2015).

Two representative reciprocal kernels are

$$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$0

with three spatial parameters: the nonlocality scale $$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$1, the exponential decay length $$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$2, and the short-range parameter $$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$3. The values

$$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$4

are inferred from galactic data, whereas $$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$5 must be constrained locally (Chicone et al., 2015).

For $$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$6, the theory is consistent with the inverse-square law, but small radial perturbations produce anomalous perihelion precession. For the two kernels, the secular precession rates are

$$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$7

$$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$8

Using Mercury and Saturn, preliminary lower bounds were obtained: for Mercury,

$$C_{\mu\nu}{}^\alpha=\Gamma^\alpha_{\mu\nu}-\Gamma^\alpha_{\nu\mu}.$$9

and for Saturn,

$\mathcal K(x,x')$0

These bounds imply that nonlocal gravity effectively reduces to Newtonian gravity locally (Chicone et al., 2015).

Later Solar System analysis tightened the short-range bounds by using Saturn’s perihelion precession, giving

$\mathcal K(x,x')$1

and estimated that out to $\mathcal K(x,x')$2 astronomical units the deviation from the inverse-square law is at or below the $\mathcal K(x,x')$3 level for the minimal acceptable values of $\mathcal K(x,x')$4 (Roshan et al., 2022). In this Newtonian sense, the local limit is the regime in which the effective dark-matter contribution is parametrically suppressed by short-range regularization.

5. Symmetry-reduced exact spacetimes

Beyond isotropic cosmology, the local limit has been studied in several symmetry-reduced settings. The recurrent structural theme is that symmetry often forces the susceptibility to be constant.

Spacetime	Condition on $\mathcal K(x,x')$5	Principal consequence
Modified Cartesian flat FLRW	time-dependent $\mathcal K(x,x')$6 allowed	no de Sitter solution; dynamic dark energy
Bianchi type I	nontrivial effects require $\mathcal K(x,x')$7	anisotropic acceleration modified by $\mathcal K(x,x')$8 and $\mathcal K(x,x')$9
Gödel universe	$X_{\mu\nu\rho}$0	modified relation between $X_{\mu\nu\rho}$1, rotation, $X_{\mu\nu\rho}$2, and $X_{\mu\nu\rho}$3
Schwarzschild spacetime	$X_{\mu\nu\rho}$4	exact solution with mass $X_{\mu\nu\rho}$5

For Bianchi type I,

$X_{\mu\nu\rho}$6

and the field equations contain both $X_{\mu\nu\rho}$7 prefactors and explicit $X_{\mu\nu\rho}$8 terms. For example,

$X_{\mu\nu\rho}$9

while the directional acceleration equations acquire source terms proportional to $$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$0. Constant nonzero $$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$1 amounts to a trivial rescaling of time; genuinely new anisotropic dynamics arise only when $$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$2 is time-dependent. The late-time tendency toward isotropy remains, but transient anisotropic acceleration can be generated by the susceptibility sector (Tabatabaei et al., 2023).

In Gödel spacetime, direct evaluation of the modified field equations yields

$$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$3

so the susceptibility must be constant, in accord with spatial homogeneity. The modified matter-rotation relations become

$$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$4

Setting $$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$5 recovers the standard Gödel solution (Mardaninezhad et al., 20 Apr 2025).

For static spherical symmetry, the local-limit field equations admit the Schwarzschild metric as an exact vacuum solution only when $$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$6 is constant, with effective mass

$$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$7

However, the Weitzenböck torsion invariants

$$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$8

diverge at the Schwarzschild horizon. The Schwarzschild solution therefore cannot be interpreted as a black hole in this theory; only the exterior region is physically admissible (Mohammadi et al., 24 Sep 2025).

6. Terminological scope and relation to other nonlocal-gravity programs

The literature uses the phrase “local limit” in more than one sense, and the teleparallel susceptibility model should be distinguished from other nonlocal-gravity constructions.

In infrared-modified models based on quantum effective actions, such as the RR model, the local limit is the regime $$\mathcal K(x,x')=\frac{S(x)}{\sqrt{-g(x)}}\,\delta(x-x'),$$9 or, equivalently, the high-energy or short-distance regime in which the nonlocal correction decouples and the theory reduces smoothly to Einstein gravity: $S(x)$00 In that framework, the local limit is tied to short-distance recovery of GR rather than to a constitutive susceptibility $S(x)$01 (Belgacem et al., 2017).

Local tests sharply separate models in this infrared-effective-action class. Lunar Laser Ranging constrains

$S(x)$02

Under plausible assumptions, the RR model and the Deser-Woodard model are ruled out because their effective Newton’s constant remains time-dependent locally, whereas the RT model has

$S(x)$03

to sub-horizon scales and is compatible with LLR (Belgacem et al., 2018). Here again, “local limit” denotes local phenomenological viability rather than a teleparallel local constitutive limit.

In weakly nonlocal and infinite-derivative gravities, the local limit usually means the infrared regime $S(x)$04, where the nonlocal potential reduces to the Newtonian form. A standard example is

$S(x)$05

which approaches $S(x)$06 at large $S(x)$07 but becomes finite at the origin; the nonlocal core is singularity-free, and in entire-function models all curvature and curvature-derivative invariants can be regular in the Newtonian limit (Buoninfante et al., 2018, Burzillà et al., 2020). This usage is mathematically distinct from the teleparallel local limit with susceptibility $S(x)$08, even though both are organized by the same general demand: agreement with Einstein or Newtonian gravity in the appropriate local regime.

Taken together, these usages identify a common research program rather than a single universal construction. In teleparallel nonlocal gravity, the local limit is a constitutive theory with gravitational susceptibility. In Solar System studies, it is the short-distance suppression of reciprocal-kernel effects. In quantum-effective-action and infinite-derivative models, it is the decoupling regime in which nonlocal operators become phenomenologically negligible.