论文标题
1+3牛顿方程式的配方
1+3 formulation of Newton's equations
论文作者
论文摘要
我们在本文中介绍了牛顿方程的4维公式，该方程是在洛伦兹歧管上引力的，这是从1+3和3+1的形式上的启发。我们首先表明，相对于伽利略参考系统的一般时间覆盖坐标系的坐标速度的自由度类似于一般相对性的3+1型正常主义的移位自由。这使我们能够写下牛顿的理论生活在4维洛伦兹歧管$ m^n $中。可以根据牛顿流体的动力学选择该歧管以弯曲。在本文中，我们专注于$ m^n $的特定选择，从而导致我们称为\ textIt {1+3-Newton方程}。我们表明，可以从相对论流体的其余框架中执行的牛顿极限从一般相对论中恢复这些方程。牛顿方程的1+3公式以及我们介绍的牛顿限制也使我们能够在牛顿的理论和一般相对论之间定义字典。该词典在灰尘流体的其余框架中定义，即非加速观察者。结果是它仅针对无旋转流体定义。作为支持1+3-Newton方程和我们的词典的示例，我们表明1+3-Newton中的抛物线自由落体解决方案精确地转化为Schwarzschild时空，并且没有任何近似值。然后，词典可能是测试牛顿解决方案在一般相对性方面的有效性的附加工具。但是，需要进一步测试非效率，非平稳和非分离的牛顿溶液，并适用于旋转流体。我们考虑的牛顿方程式1+3公式的主要应用之一是定义适合研究宇宙学中反应和全球拓扑的新模型。
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.
