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.

我们使用 Buchdahl-Vaidya-Tikekar 度量拟模型研究了各向异性致密星的高维场景。在我们的形式主义中，假设各向异性的方式是，在不存在各向异性的情况下，解简化为\(D \ge 4\)维中的史瓦西内部解。该模型的发展使得各向异性与曲率参数K相关联，曲率参数 K 表征嵌入到 4 维欧几里德空间中时相关时空的\(t=\)常数超曲面的球面几何形状的偏离。由于各向异性的特殊选择，流体静力平衡的压力平衡方程在更高维度中继续具有相同的形式。因此，我们的方法允许将四维解扩展到更高维度的时空，而不会使构型的球形度变形。利用该模型，我们提出了紧致性布赫达尔界的高维各向异性类似物。我们表明，附加尺寸和各向异性降低了致密化极限。我们的技术有助于恢复\(D=4\)维度中的原始布赫达尔极限，并且在没有各向异性的情况下，恢复 Leon 和 Cruz 早期获得的更高维度的致密化极限 (Gen Relativ Gravit 32:1207–1216, 2000 .https://doi.org/10.1023/A:1001982402392）。事实证明，通过因果关系条件和紧致化极限，最大可实现维度仍然依赖于模型。我们在相对论各向异性流体球的所有必要物理条件下仔细检查该模型，该流体球可能充当更高维度致密恒星的内部结构。我们分析了偏离均匀球形分布和维度对恒星物理行为的影响。 EOS 在更高的维度和相对较低的各向异性应力中变得更硬。我们的计算表明，随着我们向更高维度移动，中心密度会降低，并且各向异性的包含会增加密度分布的下降率。我们还注意到，在更高的维度中，这两个压力会大大降低。我们表明，对于指定球度的给定曲率参数，额外的维度类似于朝着各向异性恒星的均匀分布移动。