# 物理代写|广义相对论代写General relativity代考|PHYS760

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## 物理代写|广义相对论代写General relativity代考|Minkowski Spacetime

Exact solutions of Einstein equations mean that the spacetime metric, which satisfies the Einstein field equations with stress-energy tensor $T_{a b}$
$$R_{a b}-\frac{1}{2} g_{a b} R+\Lambda g_{a b}=\frac{8 \pi G}{c^4} T_{a b} .$$
Now, we will study the causal structures of some exact solutions of Einstein field equations. The most simple empty spacetime in general theory of relativity is Minkowski spacetime. This is actually the spacetime in special theory of relativity. Using the natural coordinates $\left(x^1, x^2, x^3, x^4\right)$ on $R^4=M(M=$ manifold $)$, one can express the metric in the form
$$d s^2=\left(d x^4\right)^2-\left(d x^1\right)^2-\left(d x^2\right)^2-\left(d x^3\right)^2$$
with the range of coordinates as $-\infty<x^1, x^2, x^3, x^4<\infty$. In this spacetime, all the components of Riemann tensor $R_{j k l}^i=0$, therefore, it is a flat spacetime. The vector $\frac{\partial}{\partial x^4}$ offers a time orientation of this spacetime.
For the choice of spherical polar coordinates $(t, r, \theta, \phi)$, where
$$x^4=t, x^3=r \cos \theta, x^2=r \sin \theta \cos \phi, x^1=r \sin \theta \sin \phi$$
the metric assumes the following form,
$$d s^2=d t^2-d r^2-r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)$$
In these new coordinate system the ranges are
$$-\infty<t<\infty, 0<r<\infty, 0<\theta<\pi \text { and } 0<\phi<2 \pi$$
Here all the Christoffel symbols $\Gamma_{j k}^i$ will not all vanish. However, due to flat spacetime, all the Riemann curvature components will vanish.

To know the structure of infinity in Minkowski spacetime is our next target. For this, we use the interesting representation of this spacetime proposed by Roger Penrose.

## 物理代写|广义相对论代写General relativity代考|de Sitter Spacetime

Similar to Minkowski space, a de Sitter space is the spacetime of a sphere in ordinary Euclidean space. It is maximally symmetric and simply connected and has constant positive curvature. Willem de Sitter (1872-1934) discovered this spacetime and, therefore, it is named after him. de Sitter space is demarcated as a submanifold of a Minkowski space of one extra dimension and described by the hyperboloid of one sheet,
$$v^2-u^2-x^2-y^2-z^2=\alpha^2$$
Here, the nonzero constant $\alpha$ has the same dimension of length.
The isometry group of four-dimensional de Sitter space is the Lorentz group $O(1,3)$ and the metric has 10 independent Killing vector fields. As de Sitter space is maximally symmetric, therefore, it has constant curvature. The de Sitter metric of constant curvature is locally described as
$$R_{a b c d}=\frac{1}{\alpha^2}\left[g_{a c} g_{b d}-g_{a d} g_{b c}\right]$$
In de Sitter space, the Ricci tensor and the given metric are proportional to each other, i.e.,
$$R_{a b}=\frac{3}{\alpha^2} g_{a b}$$
This indicates that the de Sitter space is nothing but a vacuum solution of Einstein’s field equation in presence of cosmological constant where
$$\Lambda=\frac{3}{\alpha^2}$$

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|Minkowski Spacetime

$$R_{a b}-\frac{1}{2} g_{a b} R+\Lambda g_{a b}=\frac{8 \pi G}{c^4} T_{a b}$$

$$x^4=t, x^3=r \cos \theta, x^2=r \sin \theta \cos \phi, x^1=r \sin \theta$$

$$d s^2=d t^2-d r^2-r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)$$

$-\infty<t<\infty, 0<r<\infty, 0<\theta<\pi$ and $0<\phi<2$

## 物理代写|广义相对论代写General relativity代考|de Sitter Spacetime

$$v^2-u^2-x^2-y^2-z^2=\alpha^2$$

10 个独立的 Killing 向量场。由于德西特空间是最大对称

$$R_{a b c d}=\frac{1}{\alpha^2}\left[g_{a c} g_{b d}-g_{a d} g_{b c}\right]$$

$$R_{a b}=\frac{3}{\alpha^2} g_{a b}$$

$$\Lambda=\frac{3}{\alpha^2}$$

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