Causality and Initial Data in Nonlocal Gravity

A nonlocal equation can be retarded without being the Euler–Lagrange equation of an ordinary single-field classical action. It can also have a finite initial-data prescription without being local. These are distinct claims: causal support concerns which spacetime points enter an equation, variational origin concerns how that equation is derived, and well-posedness concerns existence, uniqueness, and continuous dependence of its solutions. This page compares the formulations rather than assigning one verdict to all nonlocal gravity. The comparison axes are kernel support, variational origin, independent initial functions, constraint preservation, background, and validity regime.

Required background. PDEs and Green Functions for Gravitation supplies causal kernels, domains, and initial-value criteria; Gravitational Actions and Boundary Terms supplies functional variation and its boundary assumptions.

Helpful background. Einstein Equations supplies Bianchi consistency and the GR limit; Formulations of General Relativity supplies constrained initial data and hyperbolicity; Gravity Beyond General Relativity supplies the theory-versus-effective-equation distinction.

Why ordinary variation symmetrizes a kernel

Take a real scalar analogue,

$$S[\phi] =\frac12\int d^4x\,d^4y\, \phi(x)G_{\mathrm{ret}}(x,y)\phi(y).$$

Varying the field at $z$ produces two terms:

$$\frac{\delta S}{\delta\phi(z)} =\frac12\int d^4y \left[ G_{\mathrm{ret}}(z,y)+G_{\mathrm{ret}}(y,z) \right]\phi(y).$$

For a formally self-adjoint wave operator,

$$G_{\mathrm{ret}}(y,z)=G_{\mathrm{adv}}(z,y).$$

The Euler–Lagrange equation therefore contains the time-symmetric combination $(G_{\mathrm{ret}}+G_{\mathrm{adv}})/2$, not the purely retarded kernel written in one slot of the action. A second functional derivative of a regular single-field action must be symmetric under interchange of its arguments; a retarded response kernel is not.

The same obstruction appears in metric actions containing $\Box^{-1}$. Ordinary variation moves inverse operators between factors by formal adjunction and generates both temporal orientations. The derivation and its scope are given in Belgacem et al. 2018, § 2.4.1, Eq. (2.28). It rules out a purely retarded equation from this ordinary variational construction; it does not rule out causal effective equations.

Three routes to an evolution equation

In-in effective equation. A Schwinger–Keldysh construction doubles the fields along forward and backward time contours. After variation and taking the physical limit, expectation values are real and response functions are retarded. The underlying fundamental action may remain local even though the quantum effective equation contains history-dependent kernels. The in-out effective action instead computes amplitudes between different vacua and generally yields Feynman, complex, and acausal intermediate equations. This distinction is developed in Belgacem et al. 2018, § 2.1, equations (2.1)–(2.9).

Equation-level prescription. A phenomenological model may vary a formal covariant expression and then declare every inverse in the final equation to be retarded. The equation is causal in the support sense if its kernels actually vanish outside the past light cone. Its conservation law, auxiliary boundary data, nonlinear consistency, and relation to a microscopic theory must still be checked. This route defines a model, but the formal displayed functional is not by itself a conventional fundamental classical action for that retarded equation.

Fundamental infinite-derivative equation. An analytic kinetic operator can be defined by a Laplace transform or complex contour. Its allowed poles and contour determine the solution space and number of initial functions. Zero-free entire multiplication can leave the local massless pole count, but the position-space kernel and nonlinear domain still decide causal support. Barnaby and Kamran construct the analytic initial-value problem in Barnaby and Kamran 2008, §§ 2.1–2.3, 3.1, and 5.3.

These routes are not interchangeable. A result about in-in expectation values does not establish a fundamental nonlocal classical action, while a flat-space pole count does not establish a retarded curved-spacetime kernel.

Initial data after localization

For

$$U=-\Box_{\mathrm{ret}}^{-1}R,$$

localization gives $\Box U=-R$. The local differential equation admits a particular solution plus arbitrary $U_{\mathrm{hom}}$ with $\Box U_{\mathrm{hom}}=0$. The retarded inverse fixes that homogeneous part, for example through

$$U|_{\Sigma_*}=0, \qquad n^a\nabla_aU|_{\Sigma_*}=0.$$

Thus the localized implementation does not acquire two freely specifiable scalar initial functions. Allowing them creates a different local scalar–tensor theory. Belgacem and collaborators demonstrate the spurious homogeneous solution explicitly in Belgacem et al. 2018, §§ 2.4.4–2.4.5, equations (2.30)–(2.42).

This restriction does not by itself prove well-posedness. One must still show that the retarded integral exists on the data class, respects constraints, and depends continuously on the metric and source. Zero modes, caustics, noncompact memory, changing operator domains, or nonlinear feedback can invalidate a formal inverse.

Results and open questions through 2026-08-19

This discussion reflects results available through 2026-08-19. “Causal nonlocal gravity” has no single status until the formulation, background, kernel, and data class are fixed.

Formulation or evidence	Supported result and competing interpretation	Scope, limitation, or negative result
Ordinary single-field classical bilocal action	Functional variation symmetrizes the kernel, so a retarded kernel in the written action produces retarded-plus-advanced response. Belgacem et al. 2018, § 2.4.1, Eq. (2.28)	This is a negative result for the standard variational route, not a theorem that every nonlocal effective equation is acausal. Boundary singularities or doubled-field formalisms lie outside its hypotheses.
Schwinger–Keldysh in-in effective dynamics	Physical expectation values can obey real equations with retarded response even when the effective functional is nonlocal. Belgacem et al. 2018, § 2.1, equations (2.1)–(2.9)	Causality is inherited from a specified underlying quantum theory and state. A guessed infrared kernel is not thereby derived, and its regime of validity remains unresolved.
Retarded equation plus fixed auxiliary data	A past-supported inverse and restricted homogeneous data define causal history evolution; localization can implement it without new auxiliary initial functions. Belgacem et al. 2018, §§ 2.4.4–2.4.5	Conservation and constraint propagation are not automatic for an arbitrary tensor inverse. Imposing retarded kernels after variation supplies a prescription, not a standard fundamental action.
Entire-function or contour-defined evolution	The number of analytic initial data is governed by propagator poles and the contour, and can be finite. Barnaby and Kamran 2008, §§ 2.1–2.3, 3.1, and 5.3	Finite pole count does not prove light-cone support, curved-background well-posedness, or nonlinear stability. Different contours can define different solution spaces.

The strongest conservative interpretation accepts retarded nonlocality as a controlled effective equation only when it is derived by an in-in construction or supplied with an explicit equation-level prescription, restricted data, conservation proof, and validity regime. The strongest permissive interpretation treats a phenomenological retarded equation as a legitimate predictive starting point even before its microscopic origin is known. The competing positions differ over acceptable foundations, not over the elementary kernel-symmetry calculation.

It remains unresolved whether particular infrared kernels proposed phenomenologically arise with the required magnitude and tensor structure from a controlled underlying theory. It is also not settled that a flat-background entire-function prescription extends to a globally well-posed nonlinear gravitational data class.

The conclusions could change after a doubled-field variational theorem enlarges or limits the causal formulation class, a nonlinear well-posedness result controls a previously open data regime, an explicit microscopic calculation derives an infrared kernel, or a counterexample reveals constraint failure for a currently admitted retarded branch.

Adverse checks and the GR limit

A useful adverse example replaces $G_{\mathrm{ret}}$ by $G_{\mathrm{ret}}+H$, where $\Box_xH=0$. The sourced local equation is unchanged, but the solution and metric correction generally change. This shows why “apply $\Box$ and recover the source” is not a sufficient specification.

For a modification $\epsilon\mathcal N_G[g]$, the formal limit $\epsilon\to0$ recovers Einstein’s equation only if the history-dependent $\mathcal N_G[g]$ remains bounded on the same branch. Secular memory can make $\epsilon\mathcal N_G$ finite at late time. The controlled limit must hold the Green function, initial surface, state, and dimensionless observation interval fixed.

Retarded support also does not imply subluminal characteristics or microcausality. Conversely, a tail inside the past light cone is nonlocal in time but causal in the support sense. Those distinctions should be stated before comparing models.

Common pitfalls

Retarded support implies a standard action. Ordinary variation symmetrizes a single-field kernel. An in-in or equation-level construction is needed.

Finite initial data imply locality. A contour-defined entire operator can have a finite pole count while remaining spatially or temporally nonlocal.

Auxiliary wave equations add particles. Their homogeneous data are fixed when they represent prescribed inverses.

Exercises

A paper writes $S=\frac12\int\phi G_{\mathrm{ret}}\phi$ and claims both a conventional classical variational principle and purely retarded evolution. Diagnose the claim, then give two consistent reformulations and state their initial-data content.

Solution

Direct variation gives

$$\frac{\delta S}{\delta\phi(x)} =\frac12\int d^4y\, \left[G_{\mathrm{ret}}(x,y)+G_{\mathrm{adv}}(x,y)\right]\phi(y).$$

The claimed ordinary action therefore does not yield purely retarded evolution.

One consistent reformulation is an in-in effective theory derived from a specified local microscopic action and state. Its doubled-field variation can produce a retarded equation for the physical expectation value; initial data are those of the underlying theory and state.

A second is to define a phenomenological equation directly with $G_{\mathrm{ret}}$. If it is localized by $\Box U=-F$, the initial data of $U$ are fixed by that retarded inverse and are not new free functions. This formulation must independently verify conservation, constraint propagation, existence, uniqueness, and its validity regime. Either route is coherent, but neither supports the original simultaneous claim about an ordinary single-field action.

References

• Barnaby, Neil, and Niky Kamran (2008), “Dynamics with Infinitely Many Derivatives: The Initial Value Problem,” Journal of High Energy Physics 2008(02), 008. DOI; Open PDF.
• Belgacem, Enis, Yves Dirian, Stefano Foffa, and Michele Maggiore (2018), “Nonlocal Gravity: Conceptual Aspects and Cosmological Predictions,” Journal of Cosmology and Astroparticle Physics 2018(03), 002. DOI; Open PDF.