### 物理代写|宇宙学代写cosmology代考|Black Holes, Cosmology and Extra Dimensions

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Spherically symmetric gravitational fields

Spherical symmetry is a natural assumption in describing the simplest isolated bodies and island-like configurations. Spherically symmetric space-times are invariant under spatial rotations forming the isometry group G3.

In the general case, a spherically symmetric metric can be written in the form (see, e.g., [366])
$$d s^{2}=\mathrm{e}^{2 \gamma} d t^{2}-\mathrm{e}^{2 \alpha} d u^{2}-\mathrm{e}^{2 \beta} d \Omega^{2}, \quad d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2},$$
where $\alpha, \beta, \gamma$ are, in general, functions of the radial coordinate $u$ and the time coordinate $t$. We will also use the notation $r \equiv \mathrm{e}^{\beta}$; thus $r$ is the radius of a coordinate sphere $u=$ const, $t=$ const, and is also called the Schwarzschild radial coordinate, or the spherical radius, or, sometimes, the areal radius (since the area of such a coordinate sphere is equal to $4 \pi r^{2}$ ). In the expression (3.1), there is a freedom of choosing a reference frame (RF): different RFs correspond to spherically symmetric reference bodies with different radial velocity distributions with respect to each other.
The following two remarks are here in order.
First, let us note from the very beginning that in curved space the spherical radius $r$ has nothing to do with a distance to the center (as is the case in flat space), and in many spherically symmetric space-times there is no center at all.

Second, the “exponential” notations in (3.1) assume positive values of the corresponding metric coefficients. However, the quantities $g_{t t}$ and $g_{u u}$ can change their sign, and we will then accordingly change the notations.

## 物理代写|宇宙学代写cosmology代考|A regular center and asymptotic flatness

A center in a static, spherically symmetric space-time is, by definition, a point, line or surface in its spatial section where $r \equiv \mathrm{e}^{\beta}=0$,

that is, a place where coordinate spheres shrink to points. A center can be regular or singular; and regularity, as at any space-time point, is determined by finiteness of all $K_{i}$ in the expression (3.7). It is necessary to note that there can be no center at all in a spherically symmetric space-time; this happens if the quantity $r$ is nonzero in the whole space-time, or at least in its static region. We will encounter such behavior very soon, while discussing the properties of the Schwarzschild geometry.

With an arbitrary $u$ coordinate, the necessary and sufficient conditions for regularity of the metric at the center $(r=0)$ are obtained in the form
$$\gamma=\gamma_{0}+O\left(r^{2}\right), \quad\left|\beta^{\prime}\right| \mathrm{e}^{-\alpha+\beta}=1+O\left(r^{2}\right)$$
where $\gamma_{0}$ is a constant. The second condition is obtained from the finiteness requirement of the quantity $K_{4}$ in (3.7). Its meaning is that the circumference to radius ratio should take the correct value $(2 \pi)$ for small circles circumscribed around the center. This guarantees local flatness of space at the center and the existence of a tangent space. These are properties of any regular point – but for a center one has to introduce special regularity conditions because a center is a singular point of the class of spherical coordinate systems used.

## 物理代写|宇宙学代写cosmology代考|Solution of the Einstein equations

Let us find an important class of exact static, spherically symmetric solutions to the Einstein equations, characterizing the gravitational fields in vacuum or in the presence of an electromagnetic field (without charges) and a cosmological constant. This class contains the metrics that have the greatest number of astrophysical applications among all spherically symmetric metrics; it will also provide us with explicit examples in our future discussion of general properties of spherically symmetric space-times, including those with BHs.

It proves to be convenient to solve the problem in the curvature coordinates, in which two independent Einstein equations can be written in the form (3.5):
\begin{aligned} &G_{0}^{0}+\Lambda=\mathrm{e}^{-2 \alpha}\left(\frac{1}{r^{2}}-\frac{2 \alpha^{\prime}}{r}\right)-\frac{1}{r^{2}}+\Lambda=-\varkappa T_{0}^{0}, \ &G_{1}^{1}+\Lambda=\mathrm{e}^{-2 \alpha}\left(\frac{1}{r^{2}}+\frac{2 \gamma^{\prime}}{r}\right)-\frac{1}{r^{2}}+\Lambda=-\varkappa T_{1}^{1}, \end{aligned}
where the prime denotes $d / d r$ while the SET in the present case corresponds to the electromagnetic field.
The Lagrangian and SET of the electromagnetic field are
$$L_{\mathrm{e}-\mathrm{m}}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}, \quad T_{\mu}^{\nu}=\frac{1}{4}\left[-4 F_{\mu \alpha} F^{\nu \alpha}+\delta_{\mu}^{\nu} F_{\alpha \beta} F^{\alpha \beta}\right] .$$
The Maxwell equations $\nabla_{\alpha} F^{\alpha \beta}=0$ must now be written for the spherically symmetric case, so that among the components of $F_{\mu \nu}$ only the ones describing a radial electric field $\left(F_{01}=-F_{10}\right)$ and a radial magnetic field $\left(F_{23}=-F_{32}\right)$ can be nonzero. ${ }^{2}$ Let us restrict ourselves to an electric field. Then the only nontrivial Maxwell equation yields
$$\left(\sqrt{-g} F^{01}\right)^{\prime}=0 \Rightarrow F^{01}=\frac{Q \mathrm{e}^{-\alpha-\gamma}}{\sqrt{4 \pi} r^{2}}, \quad F_{10}=\frac{Q \mathrm{e}^{\alpha+\gamma}}{\sqrt{4 \pi} r^{2}},$$

## 物理代写|宇宙学代写cosmology代考|Spherically symmetric gravitational fields

ds2=和2Cd吨2−和2一个d在2−和2bdΩ2,dΩ2=dθ2+罪2⁡θdφ2,

## 物理代写|宇宙学代写cosmology代考|A regular center and asymptotic flatness

C=C0+○(r2),|b′|和−一个+b=1+○(r2)

## 物理代写|宇宙学代写cosmology代考|Solution of the Einstein equations

G00+Λ=和−2一个(1r2−2一个′r)−1r2+Λ=−ε吨00, G11+Λ=和−2一个(1r2+2C′r)−1r2+Λ=−ε吨11,

(−GF01)′=0⇒F01=问和−一个−C4圆周率r2,F10=问和一个+C4圆周率r2,

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## MATLAB代写

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