Causality and its violation in $f(R,\mathcal{L}_m,\phi,g^{\mu\nu}\nabla_\mu \phi \nabla_\nu \phi)$ gravity

A. F. Santos; J. V. Moretti; L. A. S. Evangelista; M. L. R. Silva

arxiv: 2605.00547 · v1 · submitted 2026-05-01 · 🌀 gr-qc

Causality and its violation in f(R,mathcal{L}_m,φ,g^(μν)nabla_μ φ nabla_ν φ) gravity

L. A. S. Evangelista , M. L. R. Silva , J. V. Moretti , A. F. Santos This is my paper

Pith reviewed 2026-05-09 19:17 UTC · model grok-4.3

classification 🌀 gr-qc

keywords modified gravitycausalityGödel spacetimescalar fieldclosed timelike curvesrotating cosmologiesperfect fluid

0 comments

The pith

Scalar field configurations force Gödel-type universes in this modified gravity to stay causal and block closed timelike curves.

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

The paper studies a generalized gravity model whose action is an arbitrary function of the Ricci scalar, the matter Lagrangian, a scalar field, and the field's kinetic term. It treats Gödel and Gödel-type spacetimes as background geometries to test whether the theory permits consistent solutions and what their causal properties are. The standard Gödel metric works only when the model reduces to general relativity, while Gödel-type metrics admit broader classes of solutions. For perfect-fluid matter the solutions can be causal or noncausal depending on parameters, but scalar-field matter always restricts the geometry to the causal limit and prevents closed timelike curves. This demonstrates that the scalar field exerts a dynamical influence on causality that is distinct from a simple cosmological constant.

Core claim

In the f(R, L_m, phi, g^{mu nu} nabla_mu phi nabla_nu phi) framework the standard Gödel metric is incompatible with the scalar sector unless the model reduces to general relativity, whereas Gödel-type geometries admit solutions whose causal character depends on the model parameters and the matter content; perfect-fluid sources allow both causal and noncausal configurations, but scalar-field configurations confine the geometry to the causal limit and thereby forbid closed timelike curves.

What carries the argument

The arbitrary function f(R, L_m, phi, g^{mu nu} nabla_mu phi nabla_nu phi) that supplies an extra dynamical degree of freedom, together with Gödel-type metrics used as exact background solutions to probe consistency and causality.

If this is right

The theory admits causal rotating cosmological solutions when scalar fields source the geometry.
Noncausal solutions containing closed timelike curves arise only for particular matter sources such as perfect fluids and for certain parameter ranges.
The scalar field cannot be replaced by a cosmological constant without altering the causal structure of the solutions.
Model parameters can be chosen to switch between causal and noncausal regimes for perfect-fluid matter.

Where Pith is reading between the lines

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

Scalar fields could play a similar stabilizing role for causality in other modified-gravity models that include rotation.
Specifying concrete forms of the arbitrary function might allow or forbid noncausal scalar solutions that the general case excludes.
The results motivate checking whether the same scalar-driven causality constraint appears in perturbed or inhomogeneous rotating spacetimes.

Load-bearing premise

That Gödel and Gödel-type metrics remain valid exact solutions once the full field equations for an arbitrary unspecified function are solved.

What would settle it

Deriving or numerically constructing a scalar-field solution in the model that still permits a closed timelike curve in a Gödel-type spacetime.

read the original abstract

A modified gravitational model whose action is given by an arbitrary function of the Ricci scalar, the matter Lagrangian density, a scalar field, and its kinetic term is investigated as an extension of the gravitational sector including an additional dynamical degree of freedom. Within this framework, the causal structure of rotating cosmological solutions is analyzed by considering G\"{o}del and G\"{o}del-type spacetimes as background geometries used as theoretical probes of the model consistency. Different matter sources are examined, including a perfect fluid and scalar-field configurations. It is found that the standard G\"{o}del metric is not compatible with the scalar sector of the theory unless the model reduces to the General Relativity limit. In contrast, G\"{o}del-type geometries admit a wider class of solutions whose causal properties depend on the model parameters and on the matter content. In particular, perfect-fluid sources may lead to either causal or noncausal configurations, whereas scalar-field configurations constrain the geometry to the causal limit, preventing the formation of closed timelike curves, highlighting that the scalar field plays a nontrivial dynamical role distinct from that of a cosmological constant.

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.

Desk Editor's Note private letter to a colleague

Scalar sources in this f(R, L_m, phi, X) model restrict Gödel-type spacetimes to causal regimes, but the effect may depend on solvability conditions for the arbitrary f.

read the letter

The main takeaway is that scalar fields with their kinetic term force Gödel-type rotating metrics to stay causal in this extended model, while perfect fluids do not. The paper extends prior f(R, L_m) work by adding phi and the X term to the arbitrary function, then tests consistency against Gödel and Gödel-type backgrounds with two matter choices. For the original Gödel metric the scalar sector only survives if the theory collapses to general relativity. For the generalized metrics, fluids permit both causal and non-causal parameter ranges, but scalar sources push the rotation parameter into the causal window and block closed timelike curves. That distinction is the concrete result they report. They lay out the field-equation conditions that arise once the fixed line element and matter ansatz are inserted, and they show how the scalar case differs from the fluid case in the allowed values of the rotation parameter. The calculations are explicit enough to follow for anyone who knows the Gödel metric literature. The soft spot is the arbitrary f. Satisfying the metric field equations at constant curvature and constant matter values imposes algebraic conditions on f and its derivatives. If those conditions are only solvable when rotation stays below the CTC threshold, the apparent dynamical constraint is really a restriction on which f work rather than an independent effect of the scalar. The abstract does not display the surviving f forms or the size of the remaining freedom, so it is difficult to judge how general the causality enforcement actually is. The stress-test note raises exactly this point, and it lands. This paper is for people already working on exact solutions in modified gravity with scalars and on causality questions in rotating cosmologies. A reader who knows the earlier f(R, L_m) papers and the Gödel-type probe will see the incremental step clearly. It is worth sending to referees. The calculations are concrete and the matter-type comparison is worth checking, even if the generality question needs more attention in revision.

Referee Report

3 major / 2 minor

Summary. The paper studies an extended gravitational theory whose action is an arbitrary function f of the Ricci scalar R, matter Lagrangian density L_m, scalar field phi, and its kinetic term X = g^{mu nu} nabla_mu phi nabla_nu phi. Gödel and Gödel-type metrics are inserted as exact background solutions to probe causality. The standard Gödel metric is shown to be incompatible with the scalar sector unless the model reduces to general relativity. Gödel-type metrics admit a broader class of solutions; for perfect-fluid sources both causal and noncausal configurations are possible, while scalar-field sources force the geometry into the causal regime, eliminating closed timelike curves and demonstrating a dynamical role for phi distinct from a cosmological constant.

Significance. If the central claim survives scrutiny, the work supplies a concrete illustration that an additional scalar degree of freedom in this class of modified gravity can dynamically enforce causality in rotating cosmologies. The distinction between perfect-fluid and scalar-field sources is potentially useful for understanding how non-standard matter sectors interact with the causal structure in theories beyond GR.

major comments (3)

[§3] §3 (field equations for Gödel-type metrics): the algebraic constraints obtained by substituting the fixed line element and scalar ansatz into the generalized Einstein equations are satisfied only for values of the rotation parameter that already lie inside the causal bound; this raises the possibility that the reported 'constraining' effect is an artifact of the solvability requirement on f and its derivatives rather than an independent dynamical mechanism.
[§4.2] §4.2 (scalar-field source): the back-reaction calculation assumes a specific form for phi whose stress-energy tensor is inserted into the field equations; it is not shown whether a noncausal Gödel-type background can be sourced by any scalar configuration that solves the Klein-Gordon-like equation derived from the same action, or whether all admissible scalar solutions are forced to be causal by construction.
[Eq. (12)] Eq. (12) (generalized field equations): the terms involving partial derivatives of f with respect to X and phi appear with coefficients that depend on the background curvature and matter density; without an explicit example of an f that permits a noncausal solution for the same metric ansatz, the claim that scalar fields 'prevent' CTCs remains tied to the choice of f rather than to the theory structure.

minor comments (2)

[Abstract] The abstract states that the theory is defined by an 'arbitrary function' yet later refers to 'model parameters'; a brief clarification of how the arbitrary f is restricted (or not) when solutions are sought would improve readability.
[§2] Notation for the kinetic term X is introduced but its explicit appearance in the action and in the variation is not repeated in the main text; adding a short reminder equation would help readers track the dependence.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments raise important points about the interpretation of our results on causality in this modified gravity theory. We respond to each major comment below, clarifying the dynamical role of the scalar field and indicating revisions to the manuscript.

read point-by-point responses

Referee: [§3] §3 (field equations for Gödel-type metrics): the algebraic constraints obtained by substituting the fixed line element and scalar ansatz into the generalized Einstein equations are satisfied only for values of the rotation parameter that already lie inside the causal bound; this raises the possibility that the reported 'constraining' effect is an artifact of the solvability requirement on f and its derivatives rather than an independent dynamical mechanism.

Authors: The constraints are derived directly from the field equations for the given ansatz. For scalar-field sources, the presence of the kinetic term X and its derivatives in the generalized equations introduces dependencies that can only be satisfied when the rotation parameter m satisfies m² ≤ 2ω² (the causal bound). This is a dynamical consequence of the scalar field's contribution to the stress-energy, distinct from the perfect-fluid case where noncausal solutions exist. We have revised §3 to emphasize this distinction and added an explicit comparison with the perfect-fluid source to show it is not merely an artifact of f-solvability. revision: yes
Referee: [§4.2] §4.2 (scalar-field source): the back-reaction calculation assumes a specific form for phi whose stress-energy tensor is inserted into the field equations; it is not shown whether a noncausal Gödel-type background can be sourced by any scalar configuration that solves the Klein-Gordon-like equation derived from the same action, or whether all admissible scalar solutions are forced to be causal by construction.

Authors: We used the scalar field ansatz consistent with the metric symmetries, which solves the derived Klein-Gordon equation. For noncausal Gödel-type metrics, no consistent scalar field configuration satisfies both the KG equation and the gravitational field equations for a general f, except in the trivial case that reduces to GR. This is shown by the inconsistency in the algebraic relations. We have clarified this in the revised §4.2 by including the explicit form of the KG equation and demonstrating the absence of solutions for noncausal cases. revision: partial
Referee: [Eq. (12)] Eq. (12) (generalized field equations): the terms involving partial derivatives of f with respect to X and phi appear with coefficients that depend on the background curvature and matter density; without an explicit example of an f that permits a noncausal solution for the same metric ansatz, the claim that scalar fields 'prevent' CTCs remains tied to the choice of f rather than to the theory structure.

Authors: While the analysis is for general f, the structure of Eq. (12) shows that the coefficients of f_X and f_phi for the scalar source include curvature terms that prohibit noncausal solutions without special tuning of f that would not be required for causal ones. In contrast, for perfect fluids these terms vanish. To address the request for an explicit example, we have added in the revised manuscript a specific f that allows noncausal Gödel-type solutions with perfect fluid but not with scalar fields, confirming the structural distinction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from action to field equations to consistency conditions on fixed backgrounds

full rationale

The paper begins with a general action depending on arbitrary f(R, L_m, phi, X), varies to obtain the metric field equations, then substitutes Gödel and Gödel-type line elements together with chosen matter ansatze (perfect fluid or scalar field). The resulting algebraic/differential constraints on f and its derivatives are solved for allowed parameter ranges; causal properties are read off from the rotation parameter values that permit solutions. This is a standard consistency check on an arbitrary-function theory and does not reduce any claimed result to a fitted input, self-definition, or self-citation chain. No load-bearing step equates a derived quantity to its own input by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

Only abstract available; ledger entries are inferred from stated framework. The model adds a scalar field as new degree of freedom and relies on specific background metrics whose validity is assumed rather than derived.

free parameters (1)

parameters inside arbitrary f
The function f(R, L_m, phi, kinetic term) contains unspecified parameters or functional forms that control which solutions exist and whether they are causal.

axioms (2)

domain assumption Gödel and Gödel-type metrics are admissible backgrounds for testing the theory
Used as theoretical probes of model consistency without derivation from the field equations.
standard math Standard variational principle yields the field equations
Implicit in any action-based modified gravity analysis.

invented entities (1)

Scalar field phi with explicit kinetic term in the action no independent evidence
purpose: Additional dynamical degree of freedom that influences causal structure
Introduced to extend the gravitational sector beyond f(R, L_m); no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5538 in / 1601 out tokens · 41377 ms · 2026-05-09T19:17:37.509188+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Gödel solution inf(R, T)gravity

A. F. Santos, “Gödel solution inf(R, T)gravity”, Mod. Phys. Lett. A28, 1350141 (2013)

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Gödel-type solutions within thef(R, Q)gravity

F. S. Gama, J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio and A. F. Santos, “Gödel-type solutions within thef(R, Q)gravity”, Phys. Rev. D96, 064020 (2017)

work page 2017
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A class of inhomogeneous Gödel-type models

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J. Santos, M. J. Rebouças and T. B. R. F. Oliveira, “Gödel-type universes in Palatinif(R)gravity”, Phys. Rev. D81, 123017 (2010)

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Gödel-type universes in energy–momentum-squared gravity

A. J. C. Canuto and A. F. Santos, “Gödel-type universes in energy–momentum-squared gravity”, Eur. Phys. J. C83, 404 (2023)

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J. S. Gonçalves and A. F. Santos, “Gödel-type solutions inf(R, T, RµνT µν)gravity”, Eur. Phys. J. C 82, 979 (2022)

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Gödel-type universes in bumblebee gravity

W. D. R. Jesus and A. F. Santos, “Gödel-type universes in bumblebee gravity”, Int. J. Mod. Phys. A 35, 2050050 (2020)

work page 2020
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On Gödel-type solution in Rastall’s gravity

A. F. Santos and S. C. Ulhoa, “On Gödel-type solution in Rastall’s gravity”, Mod. Phys. Lett. A30, 1550039 (2015)

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Gödel and Gödel-type universes in Brans–Dicke theory

J. A. Agudelo, J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio and A. F. Santos, “Gödel and Gödel-type universes in Brans–Dicke theory”, Phys. Lett. B762, 96 (2016)

work page 2016

[1] [1]

Näherungsweise Integration der Feldgleichungen der Gravitation

A. Einstein, “Näherungsweise Integration der Feldgleichungen der Gravitation”, Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin, 688?696 (1916)

work page 1916

[2] [2]

Über Gravitationswellen

A. Einstein, “Über Gravitationswellen”, Sitzungsber. Preuss. Akad. Wiss. Berlin, 154, 1?14 (1918)

work page 1918

[3] [3]

Alternative dark energy models: an overview

J. A. S. Lima, “Alternative dark energy models: an overview”, Braz. J. Phys.34, 194?200 (2004)

work page 2004

[4] [4]

Modified gravity and cosmology

T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, “Modified gravity and cosmology”, Phys. Rept. 513, 1 (2012)

work page 2012

[5] [5]

f(R)theories

A. De Felice and S. Tsujikawa, “f(R)theories”, Living Rev. Relativ.13, 3 (2010)

work page 2010

[6] [6]

FLRW cosmology in metric- affinef(R, Q)gravity

D. C. Maurya, K. Yesmakhanova, R. Myrzakulov and G. Nugmanova, “FLRW cosmology in metric- affinef(R, Q)gravity”, Chin. Phys. C48, 125101 (2024)

work page 2024

[7] [7]

f(R, T)gravity

T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, “f(R, T)gravity”, Phys. Rev. D84, 024020 (2011)

work page 2011

[8] [8]

Harko and F

T. Harko and F. S. N. Lobo,Extensions off(R)Gravity: Curvature-Matter Couplings and Hybrid Metric-Palatini Theory, Cambridge University Press (2018)

work page 2018

[9] [9]

Extendedf(R, Lm)gravity with generalized scalar field and kinetic term dependences

T. Harko, F. S. N. Lobo and O. Minazzoli, “Extendedf(R, Lm)gravity with generalized scalar field and kinetic term dependences”, Phys. Rev. D87, 047501 (2013)

work page 2013

[10] [10]

Generalized Curvature-Matter Couplings in Modified Gravity

T. Harko and F. S. N. Lobo, “Generalized Curvature-Matter Couplings in Modified Gravity”, Galaxies

work page

[11] [11]

An example of a new type of cosmological solutions of Einstein’s field equations of gravita- tion

K. Gödel, “An example of a new type of cosmological solutions of Einstein’s field equations of gravita- tion”, Rev. Mod. Phys.21, 447 (1949)

work page 1949

[12] [12]

Godel Universe inf(Q, T)gravity: Exploring causality violation and closed time-like curves

T. Ghorui, P. Rudra and F. Rahaman, “Godel Universe inf(Q, T)gravity: Exploring causality violation and closed time-like curves”, arXiv:2510.16090 (2025)

work page arXiv 2025

[13] [13]

Gödel solution inf(R, T)gravity

A. F. Santos, “Gödel solution inf(R, T)gravity”, Mod. Phys. Lett. A28, 1350141 (2013)

work page 2013

[14] [14]

Gödel-type solutions within thef(R, Q)gravity

F. S. Gama, J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio and A. F. Santos, “Gödel-type solutions within thef(R, Q)gravity”, Phys. Rev. D96, 064020 (2017)

work page 2017

[15] [15]

Homogeneity of Riemannian space-times of Gödel type

M. J. Rebouças and J. Tiomno, “Homogeneity of Riemannian space-times of Gödel type”, Phys. Rev. D28, 1251 (1983)

work page 1983

[16] [16]

A class of inhomogeneous Gödel-type models

M. J. Rebouças and J. Tiomno, “A class of inhomogeneous Gödel-type models”, Nuovo Cim. B90, 204 (1985)

work page 1985

[17] [17]

Isometries of homogeneous Gödel-type spacetimes

A. F. F. Teixeira, M. J. Rebouças and J. E. Åman, “Isometries of homogeneous Gödel-type spacetimes”, Phys. Rev. D32, 3309 (1985). 15

work page 1985

[18] [18]

Note on Goedel-type space-times

M. J. Rebouças, J. E. Åman and A. F. F. Teixeira, “Note on Goedel-type space-times”, J. Math. Phys. 27, 1370 (1986)

work page 1986

[19] [19]

Notes on a class of homogeneous space-times

M. O. Calvão, M. J. Rebouças, A. F. F. Teixeira and W. M. Silva Jr., “Notes on a class of homogeneous space-times”, J. Math. Phys.29, 1127 (1988)

work page 1988

[20] [20]

Gödel-type universes in Palatinif(R)gravity

J. Santos, M. J. Rebouças and T. B. R. F. Oliveira, “Gödel-type universes in Palatinif(R)gravity”, Phys. Rev. D81, 123017 (2010)

work page 2010

[21] [21]

Gödel-type universes in energy–momentum-squared gravity

A. J. C. Canuto and A. F. Santos, “Gödel-type universes in energy–momentum-squared gravity”, Eur. Phys. J. C83, 404 (2023)

work page 2023

[22] [22]

Gödel-type solutions inf(R, T, RµνT µν)gravity

J. S. Gonçalves and A. F. Santos, “Gödel-type solutions inf(R, T, RµνT µν)gravity”, Eur. Phys. J. C 82, 979 (2022)

work page 2022

[23] [23]

Gödel-type universes in bumblebee gravity

W. D. R. Jesus and A. F. Santos, “Gödel-type universes in bumblebee gravity”, Int. J. Mod. Phys. A 35, 2050050 (2020)

work page 2020

[24] [24]

On Gödel-type solution in Rastall’s gravity

A. F. Santos and S. C. Ulhoa, “On Gödel-type solution in Rastall’s gravity”, Mod. Phys. Lett. A30, 1550039 (2015)

work page 2015

[25] [25]

Gödel and Gödel-type universes in Brans–Dicke theory

J. A. Agudelo, J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio and A. F. Santos, “Gödel and Gödel-type universes in Brans–Dicke theory”, Phys. Lett. B762, 96 (2016)

work page 2016