The present work is focused on the investigation of the existence of compact structures describing anisotropic matter distributions within the framework of modified gravity theories, specifically $f(R,\mathcal{T}$) gravity theory. Additionally, we have taken $f(R,\mathcal{T})$ as a linear function of the Ricci scalar $R$ and the trace of the energy-momentum tensor $\mathcal{T}$ as $f(R,\mathcal{T})=R+2\chi \mathcal{T}$, where $\chi$ is a dimensionless coupling parameter, and the Lagrangian matter ${\mathcal{L}}_m=-\frac{1}{3}(2p_t+p_r)$, to describe the complete set of field equations for the anisotropic matter distribution. We follow the embedding class I procedure using the Eisland condition to obtain a full space-time description inside the stellar configuration. Once the space-time geometry is specified, we determine the complete solution of modified Einstein equations by using the MIT bag model equation of state $p_r=\frac{1}{3}(\rho -4B)$ that describes the strange quark matter (SQM) distribution inside the stellar system, where $B$ denotes a bag constant. The physical validity of our anisotropic solution is confirmed by executing several physical tests. It is worth mentioning that with the help of the observed mass values for the various strange star candidates, we have predicted the exact radii by taking different values for $\chi$ and $B$. These predicted radii show a monotonic decreasing nature as the parameter $\chi$ is moved from $-0.8$ to 0.8 progressively. In this case, our anisotropic stellar system becomes more massive and transforms into more dense compact stars. We also perform a detailed graphical analysis of the compact star. As a result, for $\chi <0$, the current modified $f(R,\mathcal{T})$ gravity seems promising to explain the observed massive compact astrophysical objects, similar to magnetars, massive pulsars, and Chandrasekhar super white dwarfs, which are not justified in the framework of general relativity. Finally, we note that when $\chi =0$, general relativity results for anisotropic matter distributions are recovered.

In this work, we present a new class of analytic and well-behaved solution to Einstein’s field equations describing anisotropic matter distribution. It’s achieved in the embedding class one spacetime framework using Karmarkar’s condition. We perform our analysis by proposing a new metric potential $$g_{rr}$$grr which yields us a physically viable performance of all physical variables. The obtained model is representing the physical features of the solution in detail, analytically as well as graphically for strange star candidate SAX J1808.4-3658 ( $$Mass=0.9 ~M_{\odot }$$Mass=0.9M⊙ , $$radius=7.951$$radius=7.951 km), with different values of parameter n ranging from 0.5 to 3.4. Our suggested solution is free from physical and geometric singularities, satisfies causality condition, Abreu’s criterion and relativistic adiabatic index $$\varGamma $$Γ , and exhibits well-behaved nature, as well as, all energy conditions and equilibrium condition are well-defined, which implies that our model is physically acceptable. The physical sensitivity of the moment of inertia (I) obtained from the solutions is confirmed by the Bejger−Haensel concept, which could provide a precise tool to the matching rigidity of the state equation due to different values of n viz., $$n=0.5, 1.08, 1.66, 2.24, 2.82$$n=0.5,1.08,1.66,2.24,2.82 and 3.4.

The main aim of this work is devoted to studying the existence of compact spherical systems representing anisotropic matter distributions within the scenario of alternative theories of gravitation, specificallyf(R, T) gravity theory. Besides, a noteworthy and achievable choice on the formulation off(R, T) gravity is made. To provide the complete set of field equations for the anisotropic matter distribution, it is considered that the functional form off(R, T) as $$f(R, T)=R+2\chi T$$f(R,T)=R+2χT , whereRandTcorrespond to scalar curvature and trace of the stress–energy tensor, respectively. Following the embedding class one approach employing the Eisland condition to get a full space–time portrayal interior the astrophysical structure. When the space–time geometry is identified, we construct a suitable anisotropic model by using a new gravitational potential $$g_{rr}$$grr which often yields physically motivated solutions that describe the anisotropic matter distribution interior the astrophysical system. The physical availability of the obtained model, represents the physical characteristics of the solution is affirmed by performing several physical tests. It merits referencing that with the help of the observed mass values for six compact stars, we have predicted the exact radii for different values of $$\chi $$χ -coupling parameter. From this one can convince that the solution predicted the radii in good agreement with the observed values. Since the radius of MSP J0740+6620, the most massive neutron star observed yet is still unknown, we have predicted its radii for different values of $$\chi $$χ -coupling parameter. These predicted radii exhibit a monotonic diminishing nature as the parameter $$\chi $$χ going from $$-1$$-1 to 1 gradually. TheM–Rcurve generated from our solution can accommodate a variety of compact stars from the less massive (Her X-1) to super massive (MSP J0740+6620). So the present study uncovers that the modifiedf(R, T) gravity is an appropriate theory to clarify massive astrophysical systems, in any case, for $$\chi =0.0$$χ=0.0 the standard consequences of the general relativity are recovered.

This study is devoted to explore the physical aspects of wormhole geometry under embedded class-1 spacetime in $$f(T,\tau )$$f ( T , τ ) gravity, where $$\tau $$τ denotes the trace of the energy-momentum tensor and T is the torsion. We derive the embedded class-1 solutions by considering spherically symmetric static spacetime. The shape function is calculated in the framework of embedded class-1 spacetime. It is necessary to mention here that the calculated shape function can be used in other modified theories of gravity. To complete this study, we take diagonal and off-diagonal tetrad, and try to build a comparison by considering the validity region of energy conditions in embedded class-1 spacetime. The embedded surface diagram is given to understand the connection between the two different regions of spacetime. The validity regions of all the energy conditions are calculated. A detailed graphical analysis is provided for validity regions of all the energy conditions. The presence of exotic matter is confirmed in both the cases as the null energy condition is violated.