Causal Nonlocal Extensions of Gravity

• Causal nonlocal gravity is defined by gravitational dynamics that depend on fields over extended regions via retarded kernels or integral operators.
• Models employ inverse differential operators, constitutive kernels, or entire form factors to maintain causality while addressing UV-completeness and dark sector phenomenology.
• These frameworks are applied in effective actions, teleparallel approaches, and exact solutions to probe gravitational memory, effective dark matter, and chronology protection.

Searching arXiv for the cited nonlocal gravity papers to ground the article in the current literature. Causal nonlocal extensions of gravity are gravitational theories in which the field equations or constitutive relations at a spacetime point depend on fields over extended regions, while the support of the relevant kernels or inverse operators is restricted so that the resulting dynamics remains retarded or otherwise compatible with relativistic causal structure. In the literature surveyed here, “causal” is implemented in more than one technically distinct way: by retarded Green’s functions in effective metric theories, by causal constitutive kernels in teleparallel or Poincaré-gauge formulations, and by solution-space restrictions that dynamically exclude closed timelike-curve sectors of otherwise admissible spacetimes. Across these approaches, nonlocality is motivated by infrared effective dynamics, ultraviolet completion, dark-sector phenomenology, and the classical consistency of exact solutions, rather than by a single universal construction (Cusin et al., 2016).

1. Nonlocality, causality, and the main classes of theories

In one major class of models, nonlocality is introduced through inverse differential operators acting on curvature scalars or tensors, such as $R\,\Box^{-2}R$, $R(-\Box+R/6)^{-2}R$, or $R\,\Delta_4^{-1}R$. These theories are interpreted as effective descriptions, and the causal equations are not obtained by naïvely varying a Lorentzian nonlocal action. Instead, one computes or organizes the effective action and then imposes retarded Green’s functions in the in-in, or Schwinger–Keldysh, formulation, so that the resulting evolution equations are causal (Cusin et al., 2016). A related Euclidean-to-Lorentzian construction treats the nonlocal action as a Euclidean quantum effective action and derives causal Lorentzian equations by analytic continuation followed by the retarded prescription; in that setting the physical equations govern expectation values of the metric and preserve the massless spin-2 spectrum on both flat and $(A)$dS backgrounds (Barvinsky, 2011).

A second class implements nonlocality by replacing local constitutive relations with causal integral kernels in gauge formulations of gravity. In teleparallel nonlocal gravity, the basic field strength is torsion rather than curvature, and the excitation is related to the field strength by a constitutive kernel analogous to nonlocal electrodynamics of media. In the linearized regime, this leads to field equations of the form

$$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$

with ${\cal K}(x,y)$ taken retarded, so that $x$ lies in the future of $y$ (Blome et al., 2010). The same constitutive-kernel logic is extended to full Poincaré gauge theory, where torsion and curvature excitations become nonlocal functionals of field strengths with scalar causal kernels and bitensorial transport (Puetzfeld et al., 2020).

A third class consists of weakly nonlocal, ultraviolet-complete theories built from entire functions of the d’Alembertian. Their propagators are modified by form factors such as $e^{H(\Box)}$, but these are chosen to be entire and nonvanishing so that no extra poles appear. The central causality criterion in that literature is the absence of Shapiro time advance in high-energy scattering. For a broad class of such theories, the eikonal phase remains positive and therefore the Shapiro time delay never turns negative; this is taken as evidence that weakly nonlocal gravity can be simultaneously UV-complete, perturbatively unitary, and causal (Giaccari et al., 2018).

These constructions share the aim of preserving relativistic causal ordering despite nonlocal dependence, but they differ in what “causal” means operationally. In effective metric models it means retarded inverse operators; in teleparallel and gauge formulations it means constitutive kernels supported in the causal past; in exact-solution analyses it can mean the dynamical exclusion of geometries with closed timelike curves (Nascimento et al., 2021). A further, conceptually distinct line of work asks whether one can formulate gravitation that is nonlocal in the Bell sense, namely through bitensors coupling spacelike separated regions while preserving the ordinary Lorentzian light-cone structure; this proposal is exploratory and does not yet provide a retarded-kernel formulation comparable to the other classes (Bonder et al., 2022).

2. Retarded effective actions and causal inverse operators

A standard difficulty of nonlocal theories is that direct variation of an action containing inverse operators generically symmetrizes the kernel. In the scalar example discussed in the conformal nonlocal gravity literature,

$$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$

so even if one inserted a retarded propagator into the action, the Euler–Lagrange equations would involve a symmetric advanced-plus-retarded combination and would therefore be acausal (Cusin et al., 2016). This is why these theories are not treated as fundamental nonlocal microscopic actions.

Instead, the nonlocal actions are interpreted as effective actions. In the in-in formulation, the relevant observables are expectation values such as $R(-\Box+R/6)^{-2}R$0, not in-out matrix elements. The Schwinger–Keldysh formalism then implies that the inverse operators appearing in the physical equations, including $R(-\Box+R/6)^{-2}R$1, $R(-\Box+R/6)^{-2}R$2, and $R(-\Box+R/6)^{-2}R$3, must be taken with retarded Green’s functions (Cusin et al., 2016). This same principle underlies the Euclidean effective-action approach in which one first varies the Euclidean nonlocal action—where the inverse operator obeys the usual symmetric variational identity

$R(-\Box+R/6)^{-2}R$4

—and only পরে imposes retarded boundary conditions upon analytic continuation to Lorentzian signature (Barvinsky, 2011).

In conformally motivated cosmological models, the baseline nonlocal action is

$R(-\Box+R/6)^{-2}R$5

with $R(-\Box+R/6)^{-2}R$6 and $R(-\Box+R/6)^{-2}R$7 (Cusin et al., 2016). Two nonlinear extensions replace $R(-\Box+R/6)^{-2}R$8 by conformally natural operators. The first uses the conformal Laplacian,

$R(-\Box+R/6)^{-2}R$9

leading to

$R\,\Delta_4^{-1}R$0

The second replaces $R\,\Delta_4^{-1}R$1 by the Paneitz operator

$R\,\Delta_4^{-1}R$2

giving

$R\,\Delta_4^{-1}R$3

Both are localized with auxiliary scalars whose initial data are fixed by the retarded prescription, so these auxiliary fields do not represent new independent propagating degrees of freedom (Cusin et al., 2016).

Within this effective-action paradigm, causality is therefore a property of the equations and boundary conditions, not of a naïve Lorentzian variational problem. This distinction is central throughout the causal nonlocal gravity literature: the action organizes the structure of the equations, but the retarded kernel defines the physical theory (Cusin et al., 2016).

3. Entire form factors, ghost freedom, and ultraviolet completion

A separate body of work studies weakly nonlocal gravity built from analytic form factors of the d’Alembertian. A representative gravitational action is

$R\,\Delta_4^{-1}R$4

where

$R\,\Delta_4^{-1}R$5

and $R\,\Delta_4^{-1}R$6 is a local potential built from curvature invariants at least cubic in curvature (Giaccari et al., 2018). The nonlocality scale is $R\,\Delta_4^{-1}R$7, and $R\,\Delta_4^{-1}R$8 is chosen entire so that the propagator contains no extra poles beyond the massless graviton pole. Two explicit families are the Kuzmin-like and Tomboulis-like choices,

$R\,\Delta_4^{-1}R$9

with $(A)$0 a polynomial of degree $(A)$1 and $(A)$2 (Giaccari et al., 2018).

The defining property of these models is “weak nonlocality”: the action contains infinitely many derivatives resummed into entire functions, but the analytic structure remains pole-free apart from the graviton pole. The propagator takes the schematic form

$(A)$3

so the exponential factor softens the ultraviolet behavior without introducing new ghostlike states (Giaccari et al., 2018). The same strategy appears in nonlocal supergravity, where the quadratic action is organized by superfield form factors $(A)$4 and $(A)$5, often taken equal: $(A)$6 When these are entire and nonvanishing, the spectrum is exactly that of local $(A)$7 supergravity: a massless graviton and massless gravitino, with no extra poles (Giaccari et al., 2016).

In the supergravity case, the weakly nonlocal superspace action is

$(A)$8

and power counting gives a superficial degree of divergence

$(A)$9

For $$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$0, all divergences beyond one loop disappear, yielding super-renormalizability; with additional local “super-killer” operators, quantum finiteness can be achieved in dimensional regularization (Giaccari et al., 2016).

A related Euclidean effective-action construction uses the nonlocal tensor operator $$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$1 acting on symmetric rank-2 tensors: $$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$2 When the parameters satisfy the conditions for the existence of a stable $$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$3dS vacuum and

$$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$4

the quadratic action on $$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$5dS reduces to the massless graviton form with effective Planck mass

$$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$6

and the propagating degrees of freedom remain the two transverse-traceless graviton polarizations of Einstein gravity (Barvinsky, 2011).

Across these UV-oriented theories, causality is linked to analyticity, the absence of additional poles, and the structure of amplitudes. The models are not causal merely because they are retarded; rather, they are causal because their nonlocal form factors preserve the correct spectrum and avoid the time-advance pathologies that plague certain higher-curvature local theories (Giaccari et al., 2018).

4. Teleparallel and Poincaré-gauge formulations with constitutive kernels

A distinct nonlocal program formulates gravity in teleparallel or full Poincaré-gauge language, where nonlocality is introduced through constitutive laws rather than through direct insertion of inverse differential operators into a metric action. In the teleparallel equivalent of GR, the tetrad is the fundamental field,

$$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$7

and the linearized torsion field strength is

$$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$8

The linearized field equation takes the form

$$G_{\mu\nu}(x)+\int{\cal K}(x,y)G_{\mu\nu}(y)\,d^4y=\kappa T_{\mu\nu}(x),$$9

with ${\cal K}(x,y)$0 retarded (Blome et al., 2010). Introducing iterated kernels ${\cal K}(x,y)$1 and the reciprocal kernel ${\cal K}(x,y)$2,

${\cal K}(x,y)$3

one can rewrite the equation as

${\cal K}(x,y)$4

In this representation the Einstein operator remains local, but the source becomes nonlocal and can be interpreted as effective dark matter (Blome et al., 2010).

In the Newtonian limit, retardation collapses to a temporal delta function and ${\cal K}(x,y)$5 becomes

${\cal K}(x,y)$6

which yields the nonlocal Poisson equation

${\cal K}(x,y)$7

The simplest Tohline–Kuhn kernel,

${\cal K}(x,y)$8

gives the modified potential

${\cal K}(x,y)$9

which produces asymptotically flat rotation curves (Blome et al., 2010). Later observational work generalizes the kernel and fits galaxy rotation curves and cluster data with a two-parameter force law involving a repulsive Yukawa-type interaction (Rahvar et al., 2014).

The Poincaré-gauge generalization preserves this constitutive viewpoint but extends it to both torsion and curvature. Starting from the general quadratic PGT Lagrangian, the local excitations $x$0 and $x$1 are replaced by nonlocal constitutive relations involving a scalar kernel $x$2 and the parallel propagator. For torsion,

$x$3

and analogous formulas hold for the linear and quadratic curvature excitations (Puetzfeld et al., 2020).

This framework unifies teleparallel nonlocal gravity, nonlocal Einstein–Cartan–Holst theory, and generic nonlocal Poincaré-gauge gravity. The kernel is explicitly causal, meaning it vanishes unless the source point lies in the causal past of the field point. The theory therefore preserves covariance and causal ordering while allowing gravitational memory to generate effective dark components (Puetzfeld et al., 2020). In later teleparallel work on “fundamental tetrads,” the constitutive relation is reformulated so that the nonlocal averaging directly connects measurable tetrad components, and with this revised constitutive law de Sitter spacetime is shown not to be a solution of the theory (Mashhoon, 2022).

5. Exact solutions, Gödel-type spacetimes, and chronology

Causal properties can also be probed nonperturbatively through exact solutions. In the ghost-free infinite-derivative model

$x$4

the form factors are transcendental entire functions of $x$5,

$x$6

so that the theory is ghost-free when the modified propagator acquires no additional poles (Nascimento et al., 2021).

The causality problem is analyzed on the family of ST-homogeneous Gödel-type metrics,

$x$7

with

$x$8

In GR, the hyperbolic class is causal only if $x$9; the original Gödel metric corresponds to $y$0 and contains closed timelike curves. The linear class and trigonometric class also admit noncausal sectors, whereas the degenerate class with $y$1 has no CTCs (Nascimento et al., 2021).

For the simplified nonlocal model containing only $y$2,

$y$3

the Gödel-type metrics have constant scalar curvature

$y$4

so $y$5 for $y$6, and the theory reduces effectively to GR or an $y$7 model on these backgrounds. As a result, all GR Gödel-type solutions survive, including those with CTCs (Nascimento et al., 2021).

The full ghost-free model behaves differently. Using the Newman–Penrose formalism and the CSI structure of Gödel-type spacetimes, the higher-derivative invariants reduce to powers of

$y$8

For example,

$y$9

and all nonlocal contributions vanish when either $e^{H(\Box)}$0 or $e^{H(\Box)}$1 (Nascimento et al., 2021). The only ST-homogeneous Gödel-type exact solutions of the full theory are therefore:

Allowed class	Condition	Causal status
Degenerate	$e^{H(\Box)}$2	No CTCs
Hyperbolic special case	$e^{H(\Box)}$3	No CTCs

All classes that admit CTCs in GR, including the Gödel point $e^{H(\Box)}$4, fail to solve the full nonlocal field equations under the assumptions of the analysis (Nascimento et al., 2021). The result does not amount to a general chronology-protection theorem, but it does show that within this ansatz the full ghost-free nonlocal structure acts as a dynamical filter that selects only causal Gödel-type geometries.

This exact-solution perspective is important because it complements retarded-kernel and eikonal-amplitude notions of causality. A nonlocal theory may be causal in the propagation sense yet still admit pathological global spacetimes; conversely, the full tensorial nonlocal structure can sometimes remove such sectors even when scalar-only nonlocality cannot (Nascimento et al., 2021).

6. Cosmology, dark sectors, and observational viability

Nonlocal gravity has also been developed as a framework for cosmic acceleration and, in some formulations, dark-matter mimicry. In the conformal extension of the RR model,

$e^{H(\Box)}$5

the auxiliary fields $e^{H(\Box)}$6 satisfy

$e^{H(\Box)}$7

and on a flat FLRW background the Friedmann equation can be written as

$e^{H(\Box)}$8

with $e^{H(\Box)}$9 (Cusin et al., 2016). For the conformal value $$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$0, the model admits stable solutions in radiation domination, matter domination, and asymptotic de Sitter, with

$$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$1

for $$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$2. The late-time equation of state is

$$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$3

and the fit parameters are

$$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$4

The future attractor is a de Sitter phase with $$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$5 when $$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$6 (Cusin et al., 2016).

The Paneitz model,

$$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$7

has stable background evolution but predicts much stronger phantom behavior,

$$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$8

which places it well outside the 95% confidence region from Planck 2015 combined with BAO, supernovae, and $$\delta S \sim \int d^4x\,d^4x'\,[G(x;x')+G(x';x)]\,\delta\phi(x)\phi(x'),$$9 data (Cusin et al., 2016). Within this conformally motivated family, the causal implementation via retarded Green’s functions is therefore not sufficient for observational viability; the operator choice matters strongly.

A different dark-sector mechanism appears in teleparallel nonlocal gravity. In the Newtonian regime, the field equation becomes

$R(-\Box+R/6)^{-2}R$00

so nonlocality is equivalent to an effective dark matter density generated by the baryonic source (Blome et al., 2010). With the Tohline–Kuhn-inspired reciprocal kernel and its regulated generalizations, galaxy rotation curves, the Tully–Fisher relation, and cluster dynamics can be fit without particle dark matter (Rahvar et al., 2014). In the weak-field stationary regime of nonlocal gravitomagnetism, the same kernel induces effective source currents,

$R(-\Box+R/6)^{-2}R$01

so both gravitoelectric and gravitomagnetic sectors receive dark-source analogues (Mashhoon et al., 2019).

These observational programs illustrate two sharply different uses of causal nonlocality. In retarded metric effective actions, nonlocality is an infrared modification designed to reproduce late-time acceleration with $R(-\Box+R/6)^{-2}R$02CDM-like expansion (Cusin et al., 2016). In teleparallel constitutive theories, nonlocality generates effective dark matter through causal gravitational memory (Rahvar et al., 2014). A plausible implication is that “causal nonlocal gravity” is not a single phenomenological category but a family of frameworks whose observational roles depend on where nonlocality is inserted: curvature action, constitutive law, or exact-solution sector.

7. Limits, controversies, and open directions

Several constraints recur across the literature. First, nonlocality by itself does not guarantee improved causal behavior. Scalar-only modifications such as $R(-\Box+R/6)^{-2}R$03 can leave Gödel-type CTC solutions intact (Nascimento et al., 2021). Likewise, mathematically elegant conformal operators can be observationally excluded even when they are implemented causally (Cusin et al., 2016). Second, weak nonlocality does not automatically resolve singularities: in nonlocal $R(-\Box+R/6)^{-2}R$04 supergravity, all Ricci-flat vacuum solutions of local supergravity remain exact solutions of the nonlocal theory, including Schwarzschild, so spacetime singularities persist despite finiteness and ghost freedom (Giaccari et al., 2016).

There is also an important conceptual controversy over what sort of nonlocality is acceptable. One line of argument insists that only effective nonlocality is viable: nonlocality should arise from integrating out light fields or from infrared vacuum polarization, with retarded inverse operators and initial data fixed on a physical hypersurface, rather than from a fundamentally nonlocal microscopic action (Woodard, 2018). Another line pursues UV-complete weakly nonlocal actions with entire form factors and argues that, when designed appropriately, such theories are both perturbatively unitary and causal in the eikonal sense (Giaccari et al., 2018). These programs are not identical. The former emphasizes in-in effective dynamics and cosmological IR effects; the latter emphasizes analytic S-matrix structure, field redefinition theorems, and UV softness.

A further conceptual extension asks whether gravitation can be made nonlocal in the Bell sense, meaning through explicitly bilocal geometric structures associated with spacelike separated regions. Three constructions have been proposed: a bitensorial conformal factor $R(-\Box+R/6)^{-2}R$05, a world-function-based Einstein bitensor built from $R(-\Box+R/6)^{-2}R$06, and a bitensorial connection in metric–affine gravity. All reduce to ordinary Einstein gravity in the coincidence limit, but none yet provides a complete dynamical theory with generalized Bianchi identities, a well-posed initial-value problem, or an explicit no-signaling proof (Bonder et al., 2022). This suggests that “nonlocality” in gravitational theory remains a heterogeneous notion ranging from retarded memory to Bell-type correlational structure.

A recent proposal, “CETOmega,” presents a strictly four-dimensional, retarded nonlocal kernel with positive Stieltjes spectral density,

$R(-\Box+R/6)^{-2}R$07

and an action

$R(-\Box+R/6)^{-2}R$08

together with claims of analyticity, spectral positivity, and causal propagation (Balfagon, 15 Mar 2026). This suggests a newer synthesis of retarded-kernel causality, spectral representations, and unified dark-sector phenomenology. A plausible implication is that future work may increasingly combine the retarded-effective-action perspective of infrared cosmology with the spectral-analytic discipline of UV-complete weakly nonlocal gravity. At present, however, the established literature already supports a narrower conclusion: causal nonlocal extensions of gravity are technically viable only when the implementation of nonlocality, the analytic structure of propagators or kernels, and the admissible solution space are controlled with great precision (Giaccari et al., 2018).