We present in this paper a 4-dimensional formulation of the Newton equations for gravitation on a Lorentzian manifold, inspired from the 1+3 and 3+1 formalisms of general relativity. We first show that the freedom on the coordinate velocity of a general time-parametrised coordinate system with respect to a Galilean reference system is similar to the shift freedom in the 3+1-formalism of general relativity. This allows us to write Newton's theory as living in a 4-dimensional Lorentzian manifold $M^N$. This manifold can be chosen to be curved depending on the dynamics of the Newtonian fluid. In this paper, we focus on a specific choice for $M^N$ leading to what we call the \textit{1+3-Newton equations}. We show that these equations can be recovered from general relativity with a Newtonian limit performed in the rest frames of the relativistic fluid. The 1+3 formulation of the Newton equations along with the Newtonian limit we introduce also allow us to define a dictionary between Newton's theory and general relativity. This dictionary is defined in the rest frames of the dust fluid, i.e. a non-accelerating observer. A consequence of this is that it is only defined for irrotational fluids. As an example supporting the 1+3-Newton equations and our dictionary, we show that the parabolic free-fall solution in 1+3-Newton exactly translates into the Schwarzschild spacetime, and this without any approximations. The dictionary might then be an additional tool to test the validity of Newtonian solutions with respect to general relativity. It however needs to be further tested for non-vacuum, non-stationary and non-isolated Newtonian solutions, as well as to be adapted for rotational fluids. One of the main applications we consider for the 1+3 formulation of Newton's equations is to define new models suited for the study of backreaction and global topology in cosmology.

中文翻译：

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牛顿方程的 1+3 公式

我们在本文中提出了洛伦兹流形上的引力牛顿方程的 4 维公式，其灵感来自广义相对论的 1+3 和 3+1 形式。我们首先证明广义时间参数化坐标系相对于伽利略参考系的坐标速度的自由度类似于广义相对论的 3+1 形式主义中的位移自由度。这使我们可以将牛顿的理论写成生活在 4 维洛伦兹流形 $M^N$ 中。可以根据牛顿流体的动力学选择该流形为弯曲的。在本文中，我们专注于 $M^N$ 的特定选择，导致我们称之为 \textit{1+3-Newton equations}。我们表明，这些方程可以从广义相对论中恢复，并在相对论流体的其余坐标系中执行牛顿限制。牛顿方程的 1+3 公式以及我们引入的牛顿极限也使我们能够定义牛顿理论和广义相对论之间的字典。这本字典是在尘埃流体的其余帧中定义的，即非加速观察者。这样做的结果是它仅适用于无旋流体。作为支持 1+3-Newton 方程和我们的字典的示例，我们表明 1+3-Newton 中的抛物线自由落体解完全转换为 Schwarzschild 时空，并且没有任何近似值。然后，字典可能是测试牛顿解相对于广义相对论的有效性的附加工具。然而，它需要针对非真空、非静止和非隔离牛顿解决方案进行进一步测试，以及适用于旋转流体。我们为牛顿方程的 1+3 公式考虑的主要应用之一是定义适用于宇宙学中逆反应和全局拓扑研究的新模型。