### 物理代写|宇宙学代写cosmology代考|Gravitational field action and dynamic equations

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|宇宙学代写cosmology代考|The correspondence principle

Since a Riemannian space-time coincides with Minkowski space (its tangent space) in a small vicinity of each world point, the laws of SR are approximately valid at any point. It is possible to pass on to SR in the whole space in the limit of weak gravity; the weak gravity condition is formulated as the condition that the metric is only weakly deflecting from the flat metric (in any coordinates), or that the Riemann tensor is small in terms of a certain length scale.

It should be noted, however, that the formal transition $x \rightarrow 0$ in a solution to the Einstein equations does not generally lead to a flat metric: instead, it leads to a certain nonflat vacuum solution of the Einstein equations, that is, a solution with $T_{\mu \nu} \equiv 0$.

Newton’s law of gravity follows from GR under the conditions of small velocities $(v \ll c$ ) and weak gravity (that is, the Newtonian gravitational potential, introduced in a proper way, should be small, $V \ll c^{2}$ ). In what follows, we shall verify the existence of such a transition using as an example the Schwarzschild metric, and also the validity of the relation $x=8 \pi G / c^{4}$.

A transition to Newtonian gravity can also be carried out under some additional conditions by the formal transition $c \rightarrow \infty$, and an expansion of the metric in powers of $c^{-1}$ (more precisely, in powers of $v / c$ and $V / c^{2}$ ). Such an expansion of the metric tensor is convenient for describing the observable effects of relativistic gravity in weak gravitational fields (for example, in the Solar system) by using a few first terms. This tool is applicable both in GR (the postNewtonian approximation) and in other metric theories of gravity (the parametrized post-Newtonian (PPN) approximation) [563]. For example, in the first PPN approximation the component $g_{00}$ should be calculated up to $c^{-4}$ (since it is multiplied by $c^{2} d t^{2}$ ), the components $g_{0 i}$ up to $c^{-3}$, and the components $g_{i j}$ up to $c^{-2}$. Higher-order PPN coefficients are also in demand owing to the increasing accuracy of measurements [565].

## 物理代写|宇宙学代写cosmology代考|Macroscopic matter and nongravitational

All kinds of matter other than the gravitational field admit a description in the framework of SR. For their description in GR (and, in general, in any theory formulated in a Riemannian space), most frequently (but not always), the so-called minimal coupling principle is used, according to which all equations known in SR are extended to curved space time by replacing all partial derivatives with covariant derivatives. We note that this trick can even increase the freedom of calculations in the framework of SR without restriction to Minkowski

coordinates, introducing curvilinear coordinates and invoking any accelerations, both translational and rotational ones.

We will present some relations valid for nongravitational matter in curved space-time according to the minimal coupling principle.

## 物理代写|宇宙学代写cosmology代考|Perfect fluids

We have previously presented the expression (2.59) for the SET of a perfect fluid in Riemannian space-time in a derivation of the geodesic equations. Let us now consider a more general conservation law for a perfect fluid and derive the corresponding equations of motion, the general-relativistic analogues of the continuity equation and the Euler equation.
Rewrite the tensor (2.59) in mixed components,
$$T_{\mu}^{\nu}=(\varepsilon+p) u_{\mu} u^{\nu}-p \delta_{\mu}^{\nu},$$
apply to it the operator $\nabla_{\nu}$ and equate the result to zero
$$\nabla_{\nu} T_{\mu}^{\nu}=\left(\partial_{\nu} w\right) u_{\mu} u^{\nu}+w \nabla_{\nu}\left(u_{\mu} u^{\nu}\right)-\partial_{\mu} p=0$$
where $w:=\varepsilon+p$ is the thermal function of the fluid. Contracting (2.64) with $u^{\mu}$ (i.e., finding its projection to the direction of $u^{\mu}$ ) and recalling that $\partial_{\mu}\left(u_{\nu} u^{\nu}\right)=0$, we obtain
$$\nabla_{\nu}\left(w u^{\nu}\right)-u^{\mu} \partial_{\mu} p=0$$
Let us now make a projection of (2.64) onto a direction perpendicular to $u^{\mu}$. Such a projection has the form $\nabla_{\nu} T_{\mu}^{\nu}-u^{\nu} u_{\mu} \nabla_{\lambda} T_{\nu}^{\lambda}=0$. As a result, we arrive at the perfect fluid equation of motion, the generalrelativistic analogue of the Euler equation
$$w u^{\nu} \nabla_{\nu} u_{\mu}=\partial_{\mu} p-u_{\mu} u^{\nu} \partial_{\nu} p$$
In nonrelativistic hydrodynamics, the continuity equation is known to represent the mass conservation law. In SR and, even more in GR, the mass is not conserved, and analogues of the continuity equation are only obtained for conserved quantities such as the number of particles if one can neglect their possible production and absorption. Then one can introduce the particle number current $n^{\mu}=n u^{\mu}$, where $n$ is the particle number density in the RF where the fluid formed by these particles is at rest, and $u^{\mu}$ is the 4-velocity of this fluid. The particle number conservation law (valid in the absence of their creation, annihilation and conversion) is expressed in the equality
$$\nabla_{\mu}\left(n u^{\mu}\right)=0,$$
quite similar to the electric charge conservation law (2.79) (see below).

## 物理代写|宇宙学代写cosmology代考|Perfect fluids

∇ν吨μν=(∂ν在)在μ在ν+在∇ν(在μ在ν)−∂μp=0

∇ν(在在ν)−在μ∂μp=0

∇μ(n在μ)=0,

## 有限元方法代写

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## MATLAB代写

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