arXiv 2109.028552021-09gr-qcastro-ph.CO

相对论分数维引力

Relativistic Fractional-Dimension Gravity

Gabriele U. Varieschi

备注：: Minor changes and references added. 18 pages, including one figure. Final version published in Universe, MDPI
发表信息：: Universe 7, no. 10: 387 (2021)
DOI：: 10.3390/universe7100387

摘要：

本文提出了一个相对论版本的牛顿分数维引力（NFDG），最近推出的替代引力模型，并基于分数维空间理论。这个扩展版本-相对论分数维引力（RFDG）-是基于其他现有的理论在文献中，可能是有用的天体物理学和宇宙学的应用。特别是，我们回顾了非整数维空间的数学理论及其与非相对论NFDG的联系。标量场的欧拉-拉格朗日方程也可以通过添加适当的权重因子而扩展到分数维空间，然后可以用于将拉普拉斯算子推广到直角坐标、球面坐标和柱面坐标。此外，同样的权重因子可以被添加到标准希尔伯特作用量中，以获得场方程，遵循用于引力的标量-张量模型，多尺度时空和分数引力理论的方法。然后，我们将场方程应用于标准宇宙学和弗里德曼-勒梅特-罗伯逊-沃克度规。使用一个合适的权重$v_{t}\left(t\right)$，取决于同步时间$t$和一个单一的时间维参数$α_{t}$，我们扩展的弗里德曼方程的RFDG的情况下。这允许计算分数时间维度$α_{t}$的不同值的比例因子$a\left(t\right)$，并与标准宇宙学结果进行比较。未来的其他工作，包括宇宙学的后期加速，Ia型超新星的数据和相关的暗能量理论的研究，将需要建立这个模型作为一个相对论性的替代理论的重力。

英文摘要：

This paper presents a relativistic version of Newtonian Fractional-Dimension Gravity (NFDG), an alternative gravitational model recently introduced and based on the theory of fractional-dimension spaces. This extended version - Relativistic Fractional-Dimension Gravity (RFDG) - is based on other existing theories in the literature and might be useful for astrophysical and cosmological applications. In particular, we review the mathematical theory for spaces with non-integer dimensions and its connections with the non-relativistic NFDG. The Euler-Lagrange equations for scalar fields can also be extended to spaces with fractional dimensions, by adding an appropriate weight factor, and then can be used to generalize the Laplacian operator for rectangular, spherical, and cylindrical coordinates. In addition, the same weight factor can be added to the standard Hilbert action in order to obtain the field equations, following methods used for scalar-tensor models of gravity, multi-scale spacetimes, and fractional gravity theories. We then apply the field equations to standard cosmology and to the Friedmann-Lemaître-Robertson-Walker metric. Using a suitable weight $v_{t}\left(t\right)$, depending on the synchronous time $t$ and on a single time-dimension parameter $α_{t}$, we extend the Friedmann equations to the RFDG case. This allows for the computation of the scale factor $a\left(t\right)$ for different values of the fractional time-dimension $α_{t}$ and the comparison with standard cosmology results. Future additional work on the subject, including studies of the cosmological late-time acceleration, type Ia supernovae data, and related dark energy theory will be needed to establish this model as a relativistic alternative theory of gravity.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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