Compactness bound of Buchdahl–Vaidya–Tikekar anisotropic star in $$D\ge 4$$ dimensional spacetime

Abstract

We study the higher dimensional scenario of an anisotropic compact star using the Buchdahl–Vaidya–Tikekar metric ansatz. In our formalism, the anisotropy is assumed in such a way that, in the absence of it, the solution reduces to Schwarzschild’s interior solution in \(D \ge 4\) dimensions. The model is so developed that it correlates anisotropy to the curvature parameter K which characterizes a departure from spherical geometry of the \(t=\) constant hypersurface of the associated spacetime when embedded in a 4 dimensional Euclidean space. Due to the particular choice of anisotropy, the pressure balancing equation for hydrostatic equilibrium continues to have the same form in higher dimensions. Consequently, our approach permits extending a four-dimensional solution to a higher dimensional spacetime without deforming the sphericity of the configuration. Making use of the model, we propose a higher dimensional anisotropic analogue of the Buchdahl bound on compactness. We show that additional dimension as well as anisotropy reduce the compactification limit. Our technique helps to regain the original Buchdahl limit in \(D=4\) dimensions and also, in the absence of anisotropy, the compactification limit in higher dimensions obtained earlier by Leon and Cruz (Gen Relativ Gravit 32:1207–1216, 2000. https://doi.org/10.1023/A:1001982402392). It turns out that the maximum achievable dimension remains model dependent through the causality condition and the compactification limit. We scrutinize the model under all the requisite physical conditions for a relativistic anisotropic fluid sphere which might serve as the internal structure of a compact star in higher dimensions. We analyze the consequences of the departure from homogeneous spherical distribution and dimensionality on the physical behaviour of the star. The EOS becomes stiffer in higher dimensions and comparatively lower anisotropic stress. Our calculation shows that the central density reduces as we move towards higher dimensions and inclusion of anisotropy increases the rate of fall of the density profile. We also note that the two pressures get reduced considerably in higher dimensions. We show that, for a given curvature parameter specifying the sphericity, an extra dimension is analogous to moving towards a homogeneous distribution of an anisotropic star.