---
title: Non-Local Palatini Models in Modified Gravity
url: https://www.emergentmind.com/topics/non-local-palatini-models
type: topic
---

# Non-Local Palatini Models in Modified Gravity

Non-local Palatini models constitute a class of gravitational theories in which infinite-derivative, non-local modifications to gravity are formulated in the Palatini approach: the metric and affine connection are treated as independent dynamical variables. These models generalize the standard Einstein–Hilbert action by promoting curvature invariants to analytic functions of the covariant d'Alembertian operator, leading to dynamics non-local in spacetime. Key research themes include the explicit derivation of dynamical equations, field-space structure in equivalence frames, ghost stability, and the implications for singularity resolution and cosmological dynamics, especially inflation [1511.03578, 2412.15064].

## 1. Gravitational Action and Non-local Extensions

The cornerstone of non-local Palatini gravity is an action of the form:
\[
S_{\rm grav}[g,\Gamma] = S_{\rm EH} + S_{\rm Scalar} + S_{\rm Ricci} + S_{\rm Riemann}
\]
with independent metric \(g_{\mu\nu}\) and symmetric affine connection \(\Gamma^{\alpha}_{\;\;\mu\nu}\). The terms are:
- Einstein–Hilbert term:
  \[
  S_{\rm EH} = \frac{1}{2\kappa^2}\int d^4x\,\sqrt{-g}(R+\Lambda_c)
  \]
- Analytic non-local curvature extensions:
  \[
  S_{\rm Scalar} = \int d^4x\,\sqrt{-g}\; R\,h_1(-\Box/\Lambda^2)\,R
  \]
  \[
  S_{\rm Ricci} = \int d^4x\,\sqrt{-g}\; R_{\mu\nu}\,h_2(-\Box/\Lambda^2)\,R^{\mu\nu}
  \]
  \[
  S_{\rm Riemann} = \int d^4x\,\sqrt{-g}\; R_{\mu\nu\rho\sigma}\,h_3(-\Box/\Lambda^2)\,R^{\mu\nu\rho\sigma}
  \]
where each \( h_i(z) \) is analytic around \( z=0 \) and expands into an infinite sum of derivatives, making the model genuinely non-local. The d'Alembertian \(\Box\) is constructed using the affine connection, distinct from purely metric non-local theories [1511.03578].

## 2. Palatini Variational Principle and Field Equations

Variation of the total action proceeds independently in \( g_{\mu\nu} \) and \( \Gamma^{\alpha}_{\;\;\mu\nu} \):

- The metric variation gives modified Einstein equations with an effective non-local stress-energy tensor \( T^{\rm nonloc}_{\mu\nu} \), incorporating the analytic form-factors and their derivative structure.
- The connection variation yields a generalized compatibility condition:
  \[
  \nabla_\alpha(\sqrt{-g}g^{\mu\nu}) + \Delta_\alpha^{\;\mu\nu}[g, \Gamma] = 0
  \]
  where \( \Delta_\alpha^{\;\mu\nu} \) encodes the non-trivial non-local dependence from the series expansions of the \( h_i \) [1511.03578].

Unlike the Einstein–Palatini system, the connection equation cannot, in generic non-local cases, be rewritten as compatibility with a conformally rescaled metric, due to the presence of double sums over non-local terms.

## 3. Vacuum Solutions, Singularities, and the Role of Form-factors

In the regime where the Riemann tensor contributions are absent (\( h_3=0 \)), all classical vacuum solutions of Einstein gravity with (A)dS asymptotics remain exact solutions of the full non-local Palatini system. Explicitly, for vacuum with vanishing cosmological constant and Levi-Civita connection, \( R_{\mu\nu}(g)=0 \implies R(g)=0 \), all non-local corrections vanish identically. Thus, the Schwarzschild, Kerr, and their (A)dS generalizations survive unmodified [1511.03578].

The crucial implication is that non-locality in this formulation fails to regularize black-hole singularities or similar curvature divergences: $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ still diverges at $r=0$ in the Schwarzschild metric. Even inclusion of the $h_3$ (Riemann$^2$) term introduces a far more non-trivial connection equation, and no universal argument guarantees singularity regularization. In contrast, certain purely metric non-local models with suitable exponential form-factors can smooth singularities, but these results do not directly transfer to the Palatini formalism [1511.03578].

## 4. Hybrid Metric–Palatini Non-locality and Ghost Structure

Recent developments incorporate non-locality through the inverse d'Alembertian acting on scalars such as $R$, particularly in hybrid metric–Palatini frameworks. An example is the action:
\[
S_J = \frac{1}{2\kappa^2}\int d^4x\sqrt{-g}\,[R + f(\Box^{-1}R)] + S_m
\]
Auxiliary fields ($\chi = \Box^{-1}R$) and Lagrange multipliers localize the action, yielding a two-field scalar–tensor theory in the Jordan frame. Conformal transformation to the Einstein frame exposes a non-trivial kinetic structure:
\[
K_{IJ}(\Phi) = \begin{pmatrix}
1 & e^{-\sqrt{2/3}\Phi} \\
e^{-\sqrt{2/3}\Phi} & e^{-2\sqrt{2/3}\Phi}
\end{pmatrix}
\]
where $\Phi$ and $\chi$ parameterize the fields. In purely metric $f(\Box^{-1}R)$-models, the kinetic matrix is degenerate ($\det K=0$), signaling the presence of a ghost, i.e., no choice of $f(\cdot)$ ensures complete ghost-freedom unless the action is degenerate or additional couplings are engineered. Hybrid metric–Palatini actions allow the possibility of coupling non-local terms to distinct curvature invariants, such that the resulting multi-field Einstein-frame action removes the ghost via non-degeneracy of $K_{IJ}$ [2412.15064]. This construction is essential for physical viability.

## 5. Inflationary Dynamics in Non-local Palatini and Hybrid Models

In the Einstein frame, the dynamics of the multi-field system is governed by:
\[
S_E = \frac{1}{2\kappa^2} \int d^4x\sqrt{-q}\left\{R[q] - K_{IJ}(\Phi)\partial_\mu\phi^I\partial^\mu\phi^J - U(\Phi, \chi)\right\}
\]
Inflationary solutions are studied in a Friedmann–Robertson–Walker (FRW) background. The non-minimal kinetic mixing term $K_{\Phi\chi}$ implies an initial kick to the spectator-like field $\chi$, generating a turn in field-space trajectory. Subsequently, Hubble damping freezes $\chi$, and inflation proceeds via canonical slow-roll of the inflaton $\Phi$ down a plateau potential, formally equivalent to Starobinsky inflation plus an inert spectator field. Calculations show that key cosmological observables, such as scalar tilt $n_s$ and tensor ratio $r$, align with plateau inflation predictions, with only sub-leading isocurvature corrections from the spectator [2412.15064].

## 6. Open Problems: Singularity Resolution and Consistency

A key motivation for non-local gravity is the hope of resolving curvature singularities. In metric non-local models with exponential form-factors (e.g., $e^{-\Box/\Lambda^2}$), the Newtonian potential becomes non-singular, but in the Palatini formulation, these desirable features do not automatically persist, due to independent connection dynamics and the altered structure of the field equations.

If $h_3\neq0$, the full set of Palatini equations becomes highly non-trivial, and it remains an open problem whether singularity resolution is possible for generic non-local analytic forms. The necessary criteria likely involve the detailed analytic structure of the form-factors $h_i$ and the specific way non-local operators are coupled to independent curvature components [1511.03578].

## 7. Summary Table: Key Features of Non-local Palatini Models

| Feature                          | Pure Metric Non-local Model        | Palatini Non-local Model             | Hybrid Metric–Palatini Non-local Model        |
|-----------------------------------|------------------------------------|--------------------------------------|-----------------------------------------------|
| Dynamical variables               | Metric $g_{\mu\nu}$                | $g_{\mu\nu}$, $\Gamma^\alpha_{\mu\nu}$ | $g_{\mu\nu}$, $\Gamma^\alpha_{\mu\nu}$      |
| Singularities in vacuum           | Removable for some $h_i$           | Not removed for $h_3=0$              | Depends on coupling structure                |
| Ghost freedom                     | Typically has a ghost              | Ghost content dictated by $h_i$       | Ghost-free with suitable coupling            |
| Inflationary dynamics             | Starobinsky plus spectator field   | As above if localized, but details model-dependent | Multi-field kinetic, single effective inflaton|
| Connection–metric relation        | Levi–Civita                        | Nontrivial non-local relation        | Model-dependent                              |

The references for these results are [1511.03578], [2412.15064]. The general expectation is that non-local Palatini models extend the landscape of viable modifications to gravity, but pose significant challenges in achieving ghost freedom and resolving curvature singularities without introducing new pathologies.

Source: https://www.emergentmind.com/topics/non-local-palatini-models