In this study, Einstein’s field equations are derived based on two dissimilar frameworks: the first is based on the concepts of “fractional velocity” and “fractal action” motivated by Calcagni’s approach to fractional spacetime while the second is derived based on fractal calculus which is a generalization of ordinary calculus that include fractal sets and curves. The fractional theory displays a breakdown of Lorentz invariance. It was observed that a spatially dependent cosmological constant emerges in the fractional theory. A connection between the fractional order parameter and the dimensionless parameter [Formula: see text] arising in the parameterized post-Newtonian (PPN) formalism is observed. A confrontation with very long-baseline radio interferometry targeting quasars 3C273 and 3C279 is done which proves that the fractional order parameter is within the range [Formula: see text]. Moreover, emergence of quantum Hawking radiation is realized in the theory supporting Hawking’s best calculations that black holes are not black. Nevertheless, based on the fractal calculus approach, there is a conservation of the Lorentz invariance and absence of spatially-dependent cosmological constant. The theory depends on the fractal order [Formula: see text] and gives rise to a fractal Schwarzschild radius of the massive body greater than the conventional radius besides a fractal Hawking’s temperature less than the standard one. However, the confrontation with radio interferometry targeting quasars 3C273 and 3C279 gives [Formula: see text].

中文翻译：

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关于分数和分形爱因斯坦场方程

在这项研究中，爱因斯坦的场方程是基于两个不同的框架推导出来的：第一个是基于 Calcagni 的分数时空方法所激发的“分数速度”和“分形作用”的概念，而第二个是基于分形微积分的推导，即包括分形集和曲线的普通微积分的推广。分数理论展示了洛伦兹不变性的分解。据观察，分数理论中出现了一个空间依赖的宇宙学常数。观察到在参数化后牛顿 (PPN) 形式中出现的分数阶参数和无量纲参数 [公式：见正文] 之间的联系。与针对类星体 3C273 和 3C279 的超长基线无线电干涉测量进行了对抗，证明分数阶参数在范围内 [公式：见正文]。此外，量子霍金辐射的出现是在支持霍金最佳计算的理论中实现的，即黑洞不是黑色的。然而，基于分形微积分方法，存在洛伦兹不变性的守恒并且不存在空间相关的宇宙学常数。该理论依赖于分形顺序[公式：见正文]，除了分形霍金温度低于标准半径外，大质量天体的分形史瓦西半径大于常规半径。然而，与针对类星体 3C273 和 3C279 的无线电干涉测量的对抗给出了 [公式：见正文]。