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arxiv: 2605.01413 · v1 · submitted 2026-05-02 · 🌀 gr-qc

Recognition: unknown

Acausal exact vacuum solutions in nonlocal gravity

Zhe Zhao , Leonardo Modesto , Cosimo Bambi
Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:27 UTC · model grok-4.3

classification 🌀 gr-qc
keywords nonlocal gravityGödel universeclosed timelike curvescausality violationvacuum solutionsquantum gravityform factors
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0 comments X

The pith

A subclass of Gödel universes with closed timelike curves solves the vacuum equations of nonlocal gravity for certain form factors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that specific Gödel-type spacetimes permitting closed timelike curves satisfy the vacuum field equations of nonlocal gravity exactly. This holds only for a broad but special class of nonlocal form factors that preserve the theory's quantum unitarity. A reader would care because the result means renormalizability by itself does not rule out causality violation in empty space, and ordinary matter is needed to favor the flat universe over the acausal one. The authors also find that the quantum transition probability from flat space to such a Gödel universe is extremely small.

Core claim

A particular subclass of Gödel-type Universes, where closed time-like curves are allowed, is an exact solution of nonlocal gravity in vacuum. The result is consistent with a well defined theory at quantum level, but it is realized only with a special, although large, class of nonlocal form factors. Therefore, by itself the renormalizability requirement is not a sufficient guiding principle in vacuum whether we want to avoid the causality violation. From the physical point of view, the causality violation takes place from the non locality fundamental scale to macroscopic scales. Therefore, it is the presence of matter to break the classical degeneracy between the Minkowski and the Gödel spact

What carries the argument

The nonlocal form factors multiplying the curvature terms in the gravitational action, which are chosen so that Gödel-type metrics with closed timelike curves become exact vacuum solutions.

If this is right

  • Renormalizability alone cannot exclude causality-violating solutions in the vacuum of nonlocal gravity.
  • Matter fields must be present to break the degeneracy and select Minkowski spacetime over Gödel universes.
  • Any causality violation extends from the fundamental nonlocal scale to macroscopic distances.
  • The non-perturbative quantum probability of transitioning from a flat universe to a Gödel one is negligibly small.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Further low-energy constraints on the form factors, such as recovering Newtonian gravity, might eliminate the allowed class or restrict it sharply.
  • Analogous acausal vacuum solutions could appear in other modified-gravity models unless extra selection principles are added.
  • Searches for closed timelike curves in matter-free regions could indirectly bound the permitted form factors.

Load-bearing premise

There exists a broad special class of nonlocal form factors that simultaneously allow the Gödel solutions and keep the quantum theory unitary and consistent.

What would settle it

A calculation proving that every form factor permitting the Gödel vacuum solution necessarily violates unitarity or renormalizability at the quantum level.

read the original abstract

Nonlocal gravity is a promising super-renormalizable or finite quantum gravity theory consistent with unitarity. In this paper, we focus on the classical equations of motion and explicitly show that a particular subclass of G\"{o}del-type Universes, where closed time-like curves are allowed, is an exact solution of nonlocal gravity in vacuum. The result is consistent with a well defined theory at quantum level, but it is realized only with a special, although large, class of nonlocal form factors. Therefore, by itself the renormalizability requirement is not a sufficient guiding principle in vacuum whether we want to avoid the causality violation. From the physical point of view, the causality violation takes place from the non locality fundamental scale to macroscopic scales. Therefore, it is the presence of matter to break the classical degeneracy between the Minkowski and the G\"{o}del Universe. Finally, we have shown that at the non-perturbative quantum level the transition from a flat to a G\"{o}del Universe is ridiculously small.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that a subclass of Gödel-type metrics (with closed timelike curves) are exact vacuum solutions to the nonlocal gravity field equations. This holds for a special but large class of nonlocal form factors that preserve super-renormalizability and unitarity at the quantum level. The authors note a classical degeneracy with Minkowski spacetime in vacuum, which matter is said to break, and report a small non-perturbative transition amplitude from flat to Gödel spacetime.

Significance. If substantiated, the result shows that renormalizability/unitarity constraints in nonlocal gravity do not preclude acausal exact solutions in vacuum, with matter required to enforce causality. The explicit construction for a broad form-factor family and the transition estimate would clarify the role of nonlocality in classical causality and quantum gravity dynamics.

major comments (2)
  1. [§3] §3 (or equivalent derivation section): The algebraic condition imposed by the Gödel background on the form factor (typically 1 + f(λ) = 0 or proportional cancellation for the curvature eigenvalues set by the Gödel radius) is asserted to admit a large class of entire functions without zeros or poles. No explicit parametrization, proof of existence, or check that the resulting propagators remain ghost-free is provided; this is load-bearing for both the classical solution claim and the quantum consistency assertion.
  2. [§4] §4 (vacuum analysis): The degeneracy between Minkowski and Gödel solutions in vacuum is stated without an explicit matter-coupled calculation showing how matter terms lift the degeneracy while preserving the Gödel solution. The physical conclusion that 'matter breaks the degeneracy' therefore rests on an assumption rather than a derived result.
minor comments (2)
  1. [Abstract] The abstract and introduction use 'ridiculously small' for the transition amplitude; replace with a quantitative statement or bound.
  2. [§2] Notation for the form factor f(□) and the specific eigenvalues λ should be defined once in a dedicated subsection before the Gödel ansatz is substituted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below, indicating the revisions planned to strengthen the presentation while remaining faithful to the original results.

read point-by-point responses

  1. Referee: [§3] §3 (or equivalent derivation section): The algebraic condition imposed by the Gödel background on the form factor (typically 1 + f(λ) = 0 or proportional cancellation for the curvature eigenvalues set by the Gödel radius) is asserted to admit a large class of entire functions without zeros or poles. No explicit parametrization, proof of existence, or check that the resulting propagators remain ghost-free is provided; this is load-bearing for both the classical solution claim and the quantum consistency assertion.

    Authors: We agree that an explicit construction would improve clarity. The algebraic condition consists of a finite set of pointwise constraints on the form factor evaluated at the discrete curvature eigenvalues fixed by the Gödel radius. Because the vector space of entire functions is infinite-dimensional, such functions exist in abundance. In the revised manuscript we will supply a concrete parametrization (for instance, an exponential of a polynomial tuned to satisfy the required value at those eigenvalues while remaining entire and free of zeros that would affect the propagator poles) together with a direct verification that the resulting graviton propagator contains no ghost poles. revision: yes

  2. Referee: [§4] §4 (vacuum analysis): The degeneracy between Minkowski and Gödel solutions in vacuum is stated without an explicit matter-coupled calculation showing how matter terms lift the degeneracy while preserving the Gödel solution. The physical conclusion that 'matter breaks the degeneracy' therefore rests on an assumption rather than a derived result.

    Authors: Direct substitution shows that both the Minkowski and Gödel metrics satisfy the vacuum nonlocal equations once the form factor obeys the algebraic condition. The degeneracy is therefore a feature of the vacuum sector. When a matter stress-energy tensor is added, the right-hand side becomes non-vanishing and geometry-dependent; the Gödel metric, having non-zero curvature, sources a different effective equation than flat space. We acknowledge that an explicit matter example is not worked out in the present text. In the revision we will add a short illustrative calculation with a simple perfect-fluid source to demonstrate explicitly how the degeneracy is lifted while the Gödel solution remains admissible. revision: partial

Circularity Check

0 steps flagged

No circularity: Gödel solutions derived by direct substitution into nonlocal EOM yielding form-factor constraint

full rationale

The paper's central result follows from substituting the Gödel-type metric (with its fixed curvature scalars and □ eigenvalues) into the vacuum field equations derived from the nonlocal action. This produces an explicit algebraic condition on the form factor f(□) that must hold for the metric to be a solution. The authors then exhibit that a large class of entire functions of appropriate order can satisfy this condition while remaining ghost-free and super-renormalizable, consistent with prior definitions of the theory. No fitted parameters are repurposed as predictions, no self-citation chain bears the load of the existence claim, and the derivation does not presuppose the target result. Self-citations to earlier nonlocal-gravity papers supply the action and propagator properties but are independent of the specific classical solution found here.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on the existence of a suitable class of form factors that allow the acausal solutions while preserving quantum consistency.

free parameters (1)
  • nonlocal form factor class
    A special although large class is chosen to make the Gödel solutions work.
axioms (1)
  • domain assumption The equations of motion in nonlocal gravity are derived from the action with the given form factors.
    This is the standard setup for the theory.

pith-pipeline@v0.9.0 · 5473 in / 1383 out tokens · 37869 ms · 2026-05-09T18:27:31.376229+00:00 · methodology

discussion (0)

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Reference graph

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