Traversable Wormholes with Non-Exotic Matter: The Role of Higher Curvature Corrections

M Daniel Ranjan; Sanjit Das; Soumya Chakrabarti

arxiv: 2605.18376 · v1 · pith:GHDKDAGNnew · submitted 2026-05-18 · 🌀 gr-qc

Traversable Wormholes with Non-Exotic Matter: The Role of Higher Curvature Corrections

M Daniel Ranjan , Soumya Chakrabarti , Sanjit Das This is my paper

Pith reviewed 2026-05-20 09:09 UTC · model grok-4.3

classification 🌀 gr-qc

keywords wormholesmodified gravityhigher derivativesnull energy conditiontraversable wormholesf(R, Box R)

0 comments

The pith

Higher-derivative corrections in f(R, □R) gravity can support traversable wormholes without exotic matter.

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

The paper explores static spherically symmetric wormhole solutions in a higher-derivative gravity theory whose action depends on both the Ricci scalar and its d'Alembertian. Field equations are derived and solved analytically and numerically to show that the extra curvature terms contribute an effective stress-energy that can reduce or remove the need for matter violating the null energy condition at the throat. In standard general relativity such wormholes always require exotic matter with negative energy density, so this framework offers a route to geometries that stay open using only ordinary matter plus the gravitational corrections. A reader would care because it addresses a long-standing obstacle to wormholes as physical objects rather than mathematical curiosities.

Core claim

In f(R, □R) gravity the higher-order terms modify the effective stress-energy tensor so that, for suitable choices of the function f and the wormhole shape function, the null energy condition holds at the throat and the wormhole can be supported without exotic matter.

What carries the argument

The effective stress-energy tensor generated by varying the higher-derivative f(R, □R) action, which augments the Einstein tensor and can cancel the usual exotic-matter requirements at the wormhole throat.

If this is right

The quantity of exotic matter required at the throat can be reduced or eliminated for appropriate f.
Classical energy conditions can be satisfied while still permitting traversable wormhole geometries.
Both analytical arguments and numerical integration confirm the existence of such solutions.

Where Pith is reading between the lines

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

Quantum corrections to gravity could therefore make wormhole-like structures more plausible without invoking new matter fields.
The same higher-derivative mechanism might relax energy-condition violations in other exotic spacetimes such as warp drives.
Observational searches for gravitational-wave echoes or lensing signatures could test whether such corrections are active.

Load-bearing premise

The chosen functional form of f(R, □R) together with the assumed static spherical metric ansatz produces an effective stress-energy tensor that obeys the null energy condition at the throat.

What would settle it

Demonstrating that no choice of f(R, □R) and no wormhole shape function ever yields an effective null energy condition that holds at the throat would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.18376 by M Daniel Ranjan, Sanjit Das, Soumya Chakrabarti.

**Figure 1.** Figure 1: Model I: Plots of radial (ρ + pr) and tangential (ρ + pt) null energy condition profiles for the parameters b0 = 2 and k1 = 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Model II: Plots of radial (ρ + pr) and tangential (ρ + pt) null energy condition profiles for the parameters b0 = 2, k1 = 3, and k2 = 1. k2 = 1 k2 = 2 k2 = 3 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -0.002 0.000 0.002 0.004 0.006 0.008 0.010 r ρ+pr k2 = 1 k2 = 2 k2 = 3 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0000 0.0005 0.0010 0.0015 r ρ+pt [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Model II: Plots of radial (ρ + pr) and tangential (ρ + pt) null energy condition profiles for the parameters b0 = 2, k1 = 3, and k3 = 1. upper panels correspond to the radial NEC (ρ + pr), while the lower panels represent the tangential NEC (ρ + pt) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Figure 5: Model III: Plots of radial (ρ + pr) and tangential (ρ + pt) null energy condition profiles for the parameters b0 = 2, k1 = 1, and k3 = −1. Figures 4 and 5 illustrate the behavior of the radial and tangential null energy conditions for Model III. The upper panels correspond to the radial NEC (ρ + pr), while the lower panels represent the tangential NEC (ρ + pt) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Plots of Radial Null Energy condition for model I wi [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: Plots of Tangential Null Energy condition for mode [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 8.** Figure 8: Embedding diagram of the wormhole metric [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 9.** Figure 9: Plots of Radial Null Energy condition for model II w [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

**Figure 10.** Figure 10: Plots of Tangential Null Energy condition for mod [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

**Figure 11.** Figure 11: Plots of Radial Null Energy condition for model II [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗

**Figure 12.** Figure 12: Plots of Tangential Null Energy condition for mod [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗

**Figure 13.** Figure 13: Plots of Radial Null Energy condition for [PITH_FULL_IMAGE:figures/full_fig_p010_13.png] view at source ↗

**Figure 16.** Figure 16: Plots of Tangential Null Energy condition [PITH_FULL_IMAGE:figures/full_fig_p011_16.png] view at source ↗

**Figure 17.** Figure 17: Plots of Radial Null Energy condition for [PITH_FULL_IMAGE:figures/full_fig_p012_17.png] view at source ↗

read the original abstract

In this paper, we explore wormhole solutions in a higher-derivative theory of gravity where the action depends not only on the Ricci scalar $R$, but also on its d'Alembertian, $\Box R$. Such $f(R,\Box R)$ models are motivated by quantum corrections to general relativity and naturally extend the space of possible gravitational geometries. Our goal is to examine whether traversable wormholes can exist in this framework and to understand the role of higher-order curvature terms in supporting them. We derive the field equations for a static, spherically symmetric wormhole and study their solutions using both analytical arguments and numerical methods. Particular attention is given to the classical energy conditions, which are usually violated in wormhole physics. We find that the higher-derivative corrections can effectively contribute to the stress-energy tensor, reducing the amount of exotic matter required at the throat, and in some cases eliminating the need for it altogether.

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

The paper shows wormhole solutions in f(R, □R) gravity where higher-derivative terms can satisfy the null energy condition at the throat for ordinary matter, but only for their specific functional form and metric ansatz.

read the letter

The key point is that higher curvature corrections from the □R term can stand in for exotic matter in traversable wormholes, at least in the cases they solve. This extends earlier f(R) constructions by bringing in the d'Alembertian of the Ricci scalar, which the authors motivate from quantum corrections to GR. They write down the field equations for a static spherically symmetric metric, pick shape and redshift functions, and then find both analytical and numerical solutions that meet the energy conditions at the throat in some parameter ranges. That is the concrete advance they deliver. The calculations appear standard for the subfield and they do check the null energy condition explicitly rather than waving at it. Credit for carrying the work through to numerical checks instead of stopping at the equations. The result is not entirely new in spirit, since similar tricks with higher-order terms already exist in the wormhole literature, but the specific inclusion of □R is a modest step beyond pure f(R). The main limitation is that success is tied to the particular f(R, □R) they chose and the fixed metric ansatz. The stress-test note is right on this: nothing in the setup shows the outcome survives a different functional form or a changed throat geometry. If the effective stress-energy tensor only complies for that narrow choice, the claim that exotic matter can be eliminated altogether needs stronger qualification. The paper does not appear to test robustness against variations, which leaves the generality open. This work is aimed at researchers already active in modified-gravity wormholes. Someone building similar models would find the technique and the numerical approach worth seeing. It is not foundational, but the derivations are grounded enough and the question is well-posed. I would send it to peer review. The evidence is sufficient to justify referee time even if the authors need to tighten the scope and add sensitivity checks.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates traversable wormhole solutions in f(R, □R) gravity. The authors derive the field equations for a static spherically symmetric metric ansatz and employ analytical arguments together with numerical methods to obtain solutions. They report that the higher-derivative corrections contribute to an effective stress-energy tensor that can reduce the amount of exotic matter required at the throat and, for certain choices, allow the null energy condition to be satisfied with ordinary matter alone.

Significance. If the explicit solutions and numerical evidence confirm that the effective NEC can be satisfied at the throat without exotic matter, the result would be significant for modified-gravity approaches to wormhole physics. It would demonstrate a concrete mechanism by which quantum-motivated higher-curvature terms can support exotic geometries, extending earlier f(R) studies and providing a falsifiable link between effective field theory corrections and classical energy conditions.

major comments (2)

[Field-equation derivation and throat analysis] The central claim that higher-derivative terms can eliminate the need for exotic matter rests on the specific functional form of f(R, □R) and the chosen static spherically symmetric ansatz (shape and redshift functions). The manuscript should demonstrate whether the effective stress-energy tensor satisfies ρ + p_r ≥ 0 at the throat only for this choice or more generally; without such a check the result remains tied to the particular model and ansatz.
[Section presenting solutions and energy-condition checks] The abstract asserts that analytical and numerical solutions exist with energy conditions satisfied, yet the provided text supplies neither the explicit modified field equations obtained from varying the f(R, □R) action nor the boundary conditions or numerical data (e.g., plots of ρ + p_r). These elements are load-bearing for verifying the claim that the □R contribution renders exotic matter unnecessary.

minor comments (1)

[Throughout] Notation for the effective stress-energy tensor should be introduced once and used consistently when separating the higher-derivative contributions from the ordinary-matter source.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We have considered each major comment in detail and provide point-by-point responses below. Revisions have been made to address the concerns raised regarding generality and the explicit presentation of results.

read point-by-point responses

Referee: [Field-equation derivation and throat analysis] The central claim that higher-derivative terms can eliminate the need for exotic matter rests on the specific functional form of f(R, □R) and the chosen static spherically symmetric ansatz (shape and redshift functions). The manuscript should demonstrate whether the effective stress-energy tensor satisfies ρ + p_r ≥ 0 at the throat only for this choice or more generally; without such a check the result remains tied to the particular model and ansatz.

Authors: We agree that assessing the dependence on the ansatz is important for the robustness of the claim. Our analysis uses the standard static spherically symmetric metric with general shape function b(r) and redshift function Φ(r), which is the conventional choice in wormhole studies. The □R terms contribute to the effective stress-energy tensor in a manner that permits ρ + p_r ≥ 0 at the throat for suitable parameter ranges within this framework. To address the referee's concern, we have added a new subsection in the revised manuscript that examines the sensitivity of the null energy condition satisfaction to variations in b(r) and Φ(r), including power-law and exponential forms. While a completely general proof for arbitrary metrics lies outside the present scope, these checks indicate that the qualitative effect of the higher-derivative corrections persists across a representative class of functions. revision: yes
Referee: [Section presenting solutions and energy-condition checks] The abstract asserts that analytical and numerical solutions exist with energy conditions satisfied, yet the provided text supplies neither the explicit modified field equations obtained from varying the f(R, □R) action nor the boundary conditions or numerical data (e.g., plots of ρ + p_r). These elements are load-bearing for verifying the claim that the □R contribution renders exotic matter unnecessary.

Authors: We acknowledge that the explicit expressions and supporting data should be more readily accessible. The modified field equations are obtained in Section II by varying the f(R, □R) action for the chosen metric ansatz, yielding the effective Einstein equations with additional higher-derivative contributions to the stress-energy tensor. Boundary conditions are imposed at the throat (r = r_0 with b(r_0) = r_0 and b'(r_0) < 1) together with asymptotic flatness. Numerical solutions are generated for specific f(R, □R) models by integrating the resulting system. To improve verifiability, the revised manuscript now includes the full set of explicit field-equation components, a clear statement of the boundary conditions, and supplementary plots displaying ρ + p_r near the throat for the reported solutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from modified field equations to explicit solutions

full rationale

The paper derives the field equations from the f(R, □R) action for a static spherically symmetric metric, then solves them analytically and numerically to obtain the effective stress-energy contributions. The reduction or elimination of exotic matter follows directly from the higher-derivative terms in those equations evaluated at the throat; no parameter is fitted to the target NEC compliance, no self-citation supplies a uniqueness theorem, and the ansatz is stated explicitly rather than smuggled. The central result is therefore an output of the calculation rather than a redefinition of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; full details of the function f, numerical implementation, and any fitted parameters are unavailable. The ledger therefore records only the explicitly mentioned modeling choices.

axioms (1)

domain assumption The wormhole metric is static and spherically symmetric.
Stated in the abstract as the geometry under study.

pith-pipeline@v0.9.0 · 5695 in / 1123 out tokens · 37730 ms · 2026-05-20T09:09:38.538124+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We derive the field equations for a static, spherically symmetric wormhole and study their solutions using both analytical arguments and numerical methods... higher-derivative corrections can effectively contribute to the stress-energy tensor
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

f(R,□R) = R + k1 R^2 + k2 R □R

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 2 internal anchors

[1]

Lorentzian wormholes

Matt Visser. Lorentzian wormholes. from einstein to hawking. Woodbury, 1995

work page 1995
[2]

The particle problem in the general theory of relativity

Albert Einstein and Nathan Rosen. The particle problem in the general theory of relativity. Phys- ical Review, 48(1):73, 1935

work page 1935
[3]

On the nature of quantum ge- ometrodynamics

John A Wheeler. On the nature of quantum ge- ometrodynamics. Annals of Physics , 2(6):604– 614, 1957

work page 1957
[4]

Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity

Michael S Morris and Kip S Thorne. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity. American Journal of Physics , 56(5):395–412, 1988

work page 1988
[5]

Traversable wormholes: Some simple examples

Matt Visser. Traversable wormholes: Some simple examples. Physical Review D , 39(10):3182, 1989

work page 1989
[6]

Casimir wormholes

Remo Garattini. Casimir wormholes. The Euro- pean Physical Journal C , 79(11):951, 2019

work page 2019
[7]

Ro- tating casimir wormholes

Remo Garattini and Athanasios G Tzikas. Ro- tating casimir wormholes. The European Physical Journal C , 85(3):336, 2025

work page 2025
[8]

Wormholes with scalar ﬁelds

David H Coule and Kei-ichi Maeda. Wormholes with scalar ﬁelds. Classical and Quantum Gravity , 7(6):955–963, 1990

work page 1990
[9]

Vacuum polarization of a scalar ﬁeld in wormhole space- times

Arkadii A Popov and Sergey V Sushkov. Vacuum polarization of a scalar ﬁeld in wormhole space- times. Physical Review D , 63(4):044017, 2001

work page 2001
[10]

A class of wormhole solutions to higher-dimensional general relativity

Gerard Clement. A class of wormhole solutions to higher-dimensional general relativity. General relativity and gravitation , 16(2):131–138, 1984

work page 1984
[11]

Energy conditions in modiﬁed gravity

Salvatore Capozziello, Francisco SN Lobo, and José P Mimoso. Energy conditions in modiﬁed gravity. Physics Letters B , 730:280–283, 2014

work page 2014
[12]

Cosmology without singularity and nonlinear gravitational lagrangians

Richard Kerner. Cosmology without singularity and nonlinear gravitational lagrangians. General Relativity and Gravitation , 14(5):453–469, 1982

work page 1982
[13]

Non-linear lagrangians and cos- mological theory

Hans A Buchdahl. Non-linear lagrangians and cos- mological theory. Monthly Notices of the Royal Astronomical Society, 150(1):1–8, 1970

work page 1970
[14]

Curvature quintessence

Salvatore Capozziello. Curvature quintessence. International Journal of Modern Physics D , 11(04):483–491, 2002

work page 2002
[15]

Dark en- ergy in modiﬁed gauss-bonnet gravity: Late-time acceleration and the hierarchy problem

Guido Cognola, Emilio Elizalde, Shin’ichi Nojiri, Sergei D Odintsov, and Sergio Zerbini. Dark en- ergy in modiﬁed gauss-bonnet gravity: Late-time acceleration and the hierarchy problem. Physi- cal Review D—Particles, Fields, Gravitation, and Cosmology, 73(8):084007, 2006

work page 2006
[16]

Worm- hole geometries in f (R) modiﬁed theories of grav- ity

Francisco SN Lobo and Miguel A Oliveira. Worm- hole geometries in f (R) modiﬁed theories of grav- ity. Physical Review D—Particles, Fields, Gravi- tation, and Cosmology , 80(10):104012, 2009

work page 2009
[17]

Wormhole geometries in f (R, T ) gravity

Tahereh Azizi. Wormhole geometries in f (R, T ) gravity. International Journal of Theoretical Physics, 52(10):3486–3493, 2013

work page 2013
[18]

f (T ) grav- ity and static wormhole solutions

Muhammad Sharif and Shamaila Rani. f (T ) grav- ity and static wormhole solutions. Modern Physics Letters A , 29(27):1450137, 2014

work page 2014
[19]

Worm- hole solutions for diﬀerent shape functions in 4D einstein gauss–bonnet gravity

Riaz Ahmed, G Abbas, and Saira Arshad. Worm- hole solutions for diﬀerent shape functions in 4D einstein gauss–bonnet gravity. International Jour- nal of Geometric Methods in Modern Physics , 19(07):2250109, 2022

work page 2022
[20]

Sixth-order gravity and conformal trans- formations

S Gottlober, H-J Schmidt, and Alexei A Starobin- sky. Sixth-order gravity and conformal trans- formations. Classical and Quantum Gravity , 7(5):893, 1990

work page 1990
[21]

Mutated hybrid inflation in $f(R,{\Box}R)$-gravity

Masao Iihoshi. Mutated hybrid inﬂa- tion in f (R, □R)-gravity. arXiv preprint arXiv:1011.3927, 2010

work page internal anchor Pith review Pith/arXiv arXiv 2010
[22]

Energy Conditions in Higher Derivative $f(R,\Box R,T)$ Gravity

Z Yousaf, M Sharif, M Ilyas, and MZ Bhatti. En- ergy conditions in higher derivative f (R, □R, T ) gravity. arXiv preprint arXiv:1806.09275 , 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018
[23]

Cosmology of f (R, □R) gravity

Sante Carloni, João Luís Rosa, and José PS Lemos. Cosmology of f (R, □R) gravity. Physi- cal Review D , 99(10):104001, 2019. 13

work page 2019

[1] [1]

Lorentzian wormholes

Matt Visser. Lorentzian wormholes. from einstein to hawking. Woodbury, 1995

work page 1995

[2] [2]

The particle problem in the general theory of relativity

Albert Einstein and Nathan Rosen. The particle problem in the general theory of relativity. Phys- ical Review, 48(1):73, 1935

work page 1935

[3] [3]

On the nature of quantum ge- ometrodynamics

John A Wheeler. On the nature of quantum ge- ometrodynamics. Annals of Physics , 2(6):604– 614, 1957

work page 1957

[4] [4]

Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity

Michael S Morris and Kip S Thorne. Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity. American Journal of Physics , 56(5):395–412, 1988

work page 1988

[5] [5]

Traversable wormholes: Some simple examples

Matt Visser. Traversable wormholes: Some simple examples. Physical Review D , 39(10):3182, 1989

work page 1989

[6] [6]

Casimir wormholes

Remo Garattini. Casimir wormholes. The Euro- pean Physical Journal C , 79(11):951, 2019

work page 2019

[7] [7]

Ro- tating casimir wormholes

Remo Garattini and Athanasios G Tzikas. Ro- tating casimir wormholes. The European Physical Journal C , 85(3):336, 2025

work page 2025

[8] [8]

Wormholes with scalar ﬁelds

David H Coule and Kei-ichi Maeda. Wormholes with scalar ﬁelds. Classical and Quantum Gravity , 7(6):955–963, 1990

work page 1990

[9] [9]

Vacuum polarization of a scalar ﬁeld in wormhole space- times

Arkadii A Popov and Sergey V Sushkov. Vacuum polarization of a scalar ﬁeld in wormhole space- times. Physical Review D , 63(4):044017, 2001

work page 2001

[10] [10]

A class of wormhole solutions to higher-dimensional general relativity

Gerard Clement. A class of wormhole solutions to higher-dimensional general relativity. General relativity and gravitation , 16(2):131–138, 1984

work page 1984

[11] [11]

Energy conditions in modiﬁed gravity

Salvatore Capozziello, Francisco SN Lobo, and José P Mimoso. Energy conditions in modiﬁed gravity. Physics Letters B , 730:280–283, 2014

work page 2014

[12] [12]

Cosmology without singularity and nonlinear gravitational lagrangians

Richard Kerner. Cosmology without singularity and nonlinear gravitational lagrangians. General Relativity and Gravitation , 14(5):453–469, 1982

work page 1982

[13] [13]

Non-linear lagrangians and cos- mological theory

Hans A Buchdahl. Non-linear lagrangians and cos- mological theory. Monthly Notices of the Royal Astronomical Society, 150(1):1–8, 1970

work page 1970

[14] [14]

Curvature quintessence

Salvatore Capozziello. Curvature quintessence. International Journal of Modern Physics D , 11(04):483–491, 2002

work page 2002

[15] [15]

Dark en- ergy in modiﬁed gauss-bonnet gravity: Late-time acceleration and the hierarchy problem

Guido Cognola, Emilio Elizalde, Shin’ichi Nojiri, Sergei D Odintsov, and Sergio Zerbini. Dark en- ergy in modiﬁed gauss-bonnet gravity: Late-time acceleration and the hierarchy problem. Physi- cal Review D—Particles, Fields, Gravitation, and Cosmology, 73(8):084007, 2006

work page 2006

[16] [16]

Worm- hole geometries in f (R) modiﬁed theories of grav- ity

Francisco SN Lobo and Miguel A Oliveira. Worm- hole geometries in f (R) modiﬁed theories of grav- ity. Physical Review D—Particles, Fields, Gravi- tation, and Cosmology , 80(10):104012, 2009

work page 2009

[17] [17]

Wormhole geometries in f (R, T ) gravity

Tahereh Azizi. Wormhole geometries in f (R, T ) gravity. International Journal of Theoretical Physics, 52(10):3486–3493, 2013

work page 2013

[18] [18]

f (T ) grav- ity and static wormhole solutions

Muhammad Sharif and Shamaila Rani. f (T ) grav- ity and static wormhole solutions. Modern Physics Letters A , 29(27):1450137, 2014

work page 2014

[19] [19]

Worm- hole solutions for diﬀerent shape functions in 4D einstein gauss–bonnet gravity

Riaz Ahmed, G Abbas, and Saira Arshad. Worm- hole solutions for diﬀerent shape functions in 4D einstein gauss–bonnet gravity. International Jour- nal of Geometric Methods in Modern Physics , 19(07):2250109, 2022

work page 2022

[20] [20]

Sixth-order gravity and conformal trans- formations

S Gottlober, H-J Schmidt, and Alexei A Starobin- sky. Sixth-order gravity and conformal trans- formations. Classical and Quantum Gravity , 7(5):893, 1990

work page 1990

[21] [21]

Mutated hybrid inflation in $f(R,{\Box}R)$-gravity

Masao Iihoshi. Mutated hybrid inﬂa- tion in f (R, □R)-gravity. arXiv preprint arXiv:1011.3927, 2010

work page internal anchor Pith review Pith/arXiv arXiv 2010

[22] [22]

Energy Conditions in Higher Derivative $f(R,\Box R,T)$ Gravity

Z Yousaf, M Sharif, M Ilyas, and MZ Bhatti. En- ergy conditions in higher derivative f (R, □R, T ) gravity. arXiv preprint arXiv:1806.09275 , 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

[23] [23]

Cosmology of f (R, □R) gravity

Sante Carloni, João Luís Rosa, and José PS Lemos. Cosmology of f (R, □R) gravity. Physi- cal Review D , 99(10):104001, 2019. 13

work page 2019