In this paper, we investigate the traversable wormhole solutions in the presence of charge within the framework of <italic>f</italic>( <italic>R, &#981;</italic>) gravity, where <italic>R</italic> represents the Ricci scalar and <italic>&#981;</italic> denotes the scalar potential, respectively. We also consider the Karmarkar condition for the discussion of wormhole geometry. The obtained shape function satisfies all necessary Morris-Thorne criteria for traversability. We derive and graphically examine the physical characteristics, including energy density and pressure components, for a specific linear charge distribution. Our analysis confirms the violation of the null energy condition in the radial direction. The violation of these energy conditions indicates the presence of exotic matter, which is the main candidate for the existence of a wormhole. Under some sensible conditions, the analysis may give an assurance of the existence of wormhole geometries in the <italic>f</italic>( <italic>R, &#981;</italic>) theory of gravity. Furthermore, we investigate the stability of the wormhole through the Tolman-Oppenheimer-Volkov equation, namely the equilibrium condition. By comparing four distinct <italic>f</italic>( <italic>R, &#981;</italic>) models, we demonstrate the robustness of traversable wormhole solutions in this modified gravity setting and identify parameter ranges that minimize energy condition violations. The results affirm that <italic>f</italic>( <italic>R, &#981;</italic>) gravity, augmented by an electromagnetic field, can support stable, traversable wormhole geometries.

Azerbaijan

China

Pakistan

Adnan Malik

M. Waqas Aslam

Wenbin Lin

Muhammad Kashif Shafiq

Saudi Arabia

Fatemah Mofarreh

Ramy M. Hafez

Impact of charge on traversable wormhole solutions and modified <italic>f</italic>( <italic>R, &#981;</italic>) gravity

This paper investigates charged traversable wormhole solutions in <italic>f</italic>( <italic>R, &#981;</italic>) modified gravity, where <italic>R</italic> denotes the Ricci scalar and <italic>&#981;</italic> represents a scalar field. Employing the Karmarkar embedding condition within a spherically symmetric spacetime, we derive exact wormhole geometries and visualize their spatial structure through three-dimensional Euclidean embeddings. Our analysis focuses on three representative <italic>f</italic>( <italic>R, &#981;</italic>) models, demonstrating how the synergy between modified gravity and the scalar field facilitates wormhole solutions that significantly reduce the requirement for exotic matter, as evidenced by characteristic energy condition violations. The inclusion of an electromagnetic field proves crucial for stabilizing the wormhole throat against gravitational collapse. Through a detailed examination of the equilibrium conditions, we establish stable configurations where gravitational, hydrostatic, and anisotropic forces attain precise balance according to a generalized Tolman&#8211;Oppenheimer&#8211;Volkoff equation. These findings indicate that <italic>f</italic>( <italic>R, &#981;</italic>) gravity provides a theoretically robust framework for traversable wormholes, potentially mitigating the need for unrealistic matter content. The resulting solutions satisfy all geometric and physical requirements for traversability while yielding testable predictions for scenarios involving modified gravity.

Zoya Asghar

M. Zeeshan Gul

Effect of scalar potential and Karmarkar condition on charged traversable wormholes

<p>We investigate the photon structure functions via the photon-photon and photon-vector meson scattering within the framework of holographic QCD, focusing on the small Bjorken <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> region and assuming that the Pomeron exchange dominates. The quasireal photon structure functions are formulated as the convolution of the known U(1) vector field wave function with the Brower-Polchinski-Strassler-Tan (BPST) Pomeron exchange kernel in the five-dimensional AdS space. Assuming the vector meson dominance, the photon structure functions can be calculated in a different way with the BPST kernel and vector meson gravitational form factor, which can be obtained in a bottom-up AdS/QCD model, for the Pomeron-vector meson coupling. It is shown that the obtained <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub></math> structure functions in the both ways agree with the experimental data, which implies the realization of the vector meson dominance within the present model setup. Calculations for the longitudinal structure function and the longitudinal-to-transverse ratio are also presented.</p>

Wei Gao

Siming Liu

Japan

Akira Watanabe

Vector meson dominance in photon structure functions at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> from holography

In this work, the orbital energy and angular momentum are calculated based on the DMPR brane-world black holes, followed by the derivation of the solution for the motion of a test particle to the second post-Newtonian order. The charge parameter $\textit{Q}$ is found to influence both the periastron advance and orbital period. In particular, its effect on perihelion precession is observed at both the first and second post-Newtonian orders, while the impact on the orbital period occurs only at the second post-Newtonian order. Furthermore, the validity of our model is confirmed by analyzing the pericenter precession of the star S2 orbiting the Galactic Center supermassive black hole Sgr A*. This nearby high-eccentricity orbit ( $$e \approx 0.88$$ ) provides a unique strong-gravity laboratory, yielding a constraint on the tidal charge parameter of $$Q/M^2 = -\,0.29 \pm 0.56$$ .

Xiaoyan Zhu

Effects of tidal charge on orbital motion in DMPR brane-world black holes

The Hamilton–Jacobi approach offers a natural framework for analyzing inflationary dynamics, relying on the specified Hubble parameter rather than the potential, particularly in extended gravity theories. In this study, we apply this method to investigate inflation with non-minimal coupling, comparing the metric and Palatini formulations. Using a power-law Hubble parameter in the Jordan frame, we find that both formalisms satisfy the attractor condition, with a slight suppression in the Palatini case. Our results align closely with the latest observational data, demonstrating the Palatini formalism’s heightened sensitivity to coupling constants and model parameters. Furthermore, we show that within the model’s viable parameter space, the reheating process can achieve sufficiently high temperatures to support successful leptogenesis.

Feng-Yi Zhang

Hamilton–Jacobi method in non-minimal coupling inflation: metric vs. Palatini

We investigate the precessing motion around the the regular black hole with an asymptotically Minkowski core which has a suppression parameter  $$\ell $$  <math> <mi>ℓ</mi> </math>  . We obtain the third-order post-Newtonian quasi-Keplerian motion of a test particle in this spacetime and find its relativistic periastron advance. Then, we test this model with the precessing motion of the star S2 and of OJ 287 and determine the upper limits of the dimensionless suppression parameter as  $$\ell /m \sim 10^{-3}$$  <math> <mrow> <mi>ℓ</mi> <mo>/</mo> <mi>m</mi> <mo>∼</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> </math>  , where m is the mass of the black hole. Compared with the bound given by the shadow of Sgr A*, our bound on  $$\ell /m$$  <math> <mrow> <mi>ℓ</mi> <mo>/</mo> <mi>m</mi> </mrow> </math>   is reduced by 2 orders of magnitude, although our upper limit of  $$\ell $$  <math> <mi>ℓ</mi> </math>   still needs further improvement.

Bo Yang

Guansheng He

Yi Xie

Probing the regular black hole with an asymptotically Minkowski core by precessing motion of S2 star and OJ 287

We explore the inflationary dynamics and reheating predictions of the fourth-order generalized polynomial inflation model featuring adjustable saddle-point position and flatness near the saddle point for both small and large field cases, based on the latest Planck 2018 Cosmic Microwave Background (CMB) and BICEP/Keck data. We observe that the small field case, with saddle points proximate to the origin, exhibits a significantly lower tensor-to-scalar ratio compared to the large field case. Additionally, the spectral index of the curvature perturbation increases as the flatness near the potential saddle point decreases. Through an analysis of the interplay between inflation and reheating parameters derived from the expansion history of the pivot CMB scale, we establish a permissible range for the reheating parameters. We find that both the small and large cases of polynomial inflation can be consistent with the latest observations due to potential flatness reduction, except for the case where the saddle point deviates significantly from the origin, and the reheating process accentuates the disparities in parameter space between small and large field inflationary scenarios.

Inflation and reheating predictions in polynomial inflation with small and large fields

In this paper, the gravitational deflection of a relativistic massive neutral particle in the Schwarzschild-de Sitter spacetime is studied via the Rindler–Ishak method in the weak-field limit. When the initial velocity  $$v_0$$  <math> <msub> <mi>v</mi> <mn>0</mn> </msub> </math>   of the particle tends to the speed of light, the result is consistent with that obtained in the previous work for the light-bending case. Our result is reduced to the Schwarzschild deflection angle of massive particles up to the second order, if the contributions from the cosmological constant  $$\varLambda $$  <math> <mi>Λ</mi> </math>   are dropped. The observable correctional effects due to the deviation of  $$v_0$$  <math> <msub> <mi>v</mi> <mn>0</mn> </msub> </math>   from light speed on the  $$\varLambda $$  <math> <mi>Λ</mi> </math>  -induced contributions to the deflection angle of light are also analyzed.

Xia Zhou

Zhongwen Feng

Xueling Mu

Hui Wang

Weijun Li

Chaohong Pan

Gravitational deflection of massive particles in Schwarzschild-de Sitter spacetime

The significance of the radial velocity effects on the leading-order gravitational frequency shift in some special cases was reported recently. This paper investigates the gravitational shift of frequency of light propagating in the equatorial plane of a radially moving Kerr–Newman black hole up to the second post-Minkowskian order, and discusses the radial velocity effects on the second-order contributions to the frequency shift. It is found that a new radial velocity effect appears in the second-order Schwarzschild contribution to the frequency shift, in contrast to no radial velocity effect in the first-order contribution, when both the emission and reception events are far away from the lens. Velocity effects on the gravitational frequency shifts induced respectively by the lens’s electrical charge and spin are also analyzed.

Velocity effects on second order contributions to gravitational frequency shift by a moving Kerr–Newman black hole