{"title":"$D\\ge 4$$ 维时空中布达赫尔-韦迪亚-蒂凯卡各向异性星的紧凑性约束","authors":"Samstuti Chanda, Ranjan Sharma","doi":"10.1007/s10714-024-03231-x","DOIUrl":null,"url":null,"abstract":"<p>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 <span>\\(D \\ge 4\\)</span> dimensions. The model is so developed that it correlates anisotropy to the curvature parameter <i>K</i> which characterizes a departure from spherical geometry of the <span>\\(t=\\)</span> 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 <span>\\(D=4\\)</span> 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.</p>","PeriodicalId":578,"journal":{"name":"General Relativity and Gravitation","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Compactness bound of Buchdahl–Vaidya–Tikekar anisotropic star in $$D\\\\ge 4$$ dimensional spacetime\",\"authors\":\"Samstuti Chanda, Ranjan Sharma\",\"doi\":\"10.1007/s10714-024-03231-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>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 <span>\\\\(D \\\\ge 4\\\\)</span> dimensions. The model is so developed that it correlates anisotropy to the curvature parameter <i>K</i> which characterizes a departure from spherical geometry of the <span>\\\\(t=\\\\)</span> 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 <span>\\\\(D=4\\\\)</span> 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. 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引用次数: 0
摘要
我们利用布赫达赫-瓦伊迪亚-蒂凯卡公设解析法研究了各向异性紧凑恒星的高维情况。在我们的形式主义中,各向异性是以这样一种方式假定的:如果没有各向异性,那么解就会还原为施瓦兹柴尔德的内部解(Schwarzschild's interior solution in \(D\ge 4\) dimensions)。这个模型是这样建立的:它把各向异性与曲率参数K联系起来,而曲率参数K是相关时空的(t=\)恒定超表面嵌入4维欧几里得空间时偏离球形几何的特征。由于各向异性的特殊选择,流体静力学平衡的压力平衡方程在更高维度中仍然具有相同的形式。因此,我们的方法允许将四维解法扩展到更高维度的时空,而不会改变构型的球形性。利用该模型,我们提出了关于紧凑性的布赫达尔约束的高维各向异性类似物。我们证明,额外维度和各向异性会降低紧凑性极限。我们的技术有助于在(D=4)维度上重新获得原始的布赫达尔极限,而且,在没有各向异性的情况下,也有助于获得莱昂和克鲁兹(Gen Relativ Gravit 32:1207-1216, 2000. https://doi.org/10.1023/A:1001982402392)早先在更高维度上获得的紧凑性极限。事实证明,通过因果关系条件和压缩极限,可达到的最大维度仍然取决于模型。我们在相对论各向异性流体球体的所有必要物理条件下仔细研究了这个模型,该流体球体可能是高维度紧凑恒星的内部结构。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。各向异性应力相对较低时,各向异性应力在更高维度下会变得更加坚硬。我们的计算表明,随着维数的增加,中心密度会降低,而各向异性会增加密度曲线的下降速度。我们还注意到,在更高的维度上,两个压力会大大减小。我们表明,对于指定球度的给定曲率参数,额外维度类似于各向异性恒星的均匀分布。
Compactness bound of Buchdahl–Vaidya–Tikekar anisotropic star in $$D\ge 4$$ dimensional spacetime
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.
期刊介绍:
General Relativity and Gravitation is a journal devoted to all aspects of modern gravitational science, and published under the auspices of the International Society on General Relativity and Gravitation.
It welcomes in particular original articles on the following topics of current research:
Analytical general relativity, including its interface with geometrical analysis
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Theoretical and observational cosmology
Relativistic astrophysics
Gravitational waves: data analysis, astrophysical sources and detector science
Extensions of general relativity
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Gravitational aspects of string theory and its extensions
Quantum gravity: canonical approaches, in particular loop quantum gravity, and path integral approaches, in particular spin foams, Regge calculus and dynamical triangulations
Quantum field theory in curved spacetime
Non-commutative geometry and gravitation
Experimental gravity, in particular tests of general relativity
The journal publishes articles on all theoretical and experimental aspects of modern general relativity and gravitation, as well as book reviews and historical articles of special interest.