# A generalized Finch-Skea class one static solution

## TL;DR

This paper develops a generalized relativistic stellar model using the Einstein field equations with anisotropic matter, applying embedding class one techniques to produce solutions consistent with physical and mathematical criteria for ultra-dense stars.

## Contribution

It introduces a new generalized Finch-Skea class I solution for static anisotropic stars using embedding techniques, expanding the class of exact solutions in relativistic astrophysics.

## Key findings

- Model satisfies physical and mathematical conditions for dense stellar objects.
- Mass-radius diagram indicates stiffer equation of state with increasing parameter n.
- Mass at maximum moment of inertia is slightly less than maximum mass, implying no strong high-density softening.

## Abstract

In the present article, we discuss relativistic anisotropic solutions of the Einstein field equation for the spherically symmetric line element under the class I condition. To do so we apply the embedding class one technique using Eisland condition. Within this approach, one arrives at a particular differential equation that links the two metric components $e^{\nu}$ and $e^{\lambda}$. In order to obtain the full space-time description inside the stellar configuration we ansatz the generalized form of metric component $g_{rr}$ corresponding to the Finch-Skea solution. Once the space-time geometry is specified we obtain the complete thermodynamic description i.e. the matter density $\rho$, the radial, and tangential pressures $p_r$ and $p_t$, respectively. Graphical analysis shows that the obtained model respects the physical and mathematical requirements that all ultra-high dense collapsed structures must obey. The $M-R$ diagram suggests that the solution yields stiffer EoS as parameter $n$ increases. The $M-I$ graph is in agreement with the concepts of Bejgar et al. \cite{bej} that the mass at $I_{max}$ is lesser by few percent (for this solution $\sim 3\%$) from $M_{max}$. This suggests that the EoSs is without any strong high-density softening due to hyperonization or phase transition to an exotic state.