## 物理代写|广义相对论代写General relativity代考|PHYS501

2023年4月4日

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## 物理代写|广义相对论代写General relativity代考|Anti-de Sitter Space

Anti-de Sitter space is suitably determined as a quadric in a five-dimensional flat spacetime with signature $(2,3)$, i.e., the coordinate points $(x, y, z, v, w)$ follow the relation
$$-x^2-y^2-z^2+v^2+w^2=1$$
It has the topology $S^1 \times R^3$ and the Lorentzian metric induced from the metric on the five-dimensional flat spacetime is given by
$$d s^2=d v^2+d w^2-d x^2-d y^2-d z^2$$
It is evident that anti-de Sitter spacetime is conformally flat and in this spacetime, the Ricci scalar is a negative constant throughout the spacetime. Let us take the following transformation as
$$v=R \cos t, w=R \sin t$$
then Eq. (9.11) takes the form as
$$-x^2-y^2-z^2+R^2=1$$

The metric on the five-dimensional spacetime becomes
$$d s^2=-d x^2-d y^2-d z^2+d R^2+R^2 d t^2$$
Now, we consider another transformation by
$$R=\sqrt{1+\rho^2} .$$
Again substitute the coordinates $(x, y, z)$ by
$$x=\rho \cos \theta, y=\rho \sin \theta \cos \phi, z=\rho \sin \theta \sin \phi .$$
The induced metric assumes the following form as
\begin{aligned} d s^2 & =-d \rho^2-\rho^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)+\frac{\rho^2 d \rho^2}{1+\rho^2}+\left(1+\rho^2\right) d t^2 \ & =\left(1+\rho^2\right) d t^2-\frac{d \rho^2}{1+\rho^2}-\rho^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right) \ & =\left(1+\rho^2\right) d t^2-\frac{d \rho^2}{1+\rho^2}-\rho^2 d \Omega^2 \end{aligned}
The transformation $\rho=\sinh r$ yields the line element as
$$d s^2=\cosh ^2 r d t^2-d r^2-\sinh ^2 r\left(d \theta^2+\sin ^2 \theta d \phi^2\right) .$$
The whole space can be covered by the surfaces $t=$ constant, which have nongeodesic normals.

## 物理代写|广义相对论代写General relativity代考|Robertson–Walker Spaces

The universe appears to be homogeneous and isotropic (i.e., matter content is uniformly distributed and looks qualitatively the same in all direction) around us at sufficiently large scales (more than a 100 million light years or so). That is on this large scale the density of galaxies is roughly the same and all directions from us seem to be alike. Walker has shown that if all points and all directions of the universe are same (i.e., for exact spherically symmetry about all point), then the universe is spatially homogeneous and admits six isometries whose surfaces of transitivity are three-spaces of constant curvature, which are space-like. This space is known as Robertson-Walker space.
(A displacement of the type for which the displaced space is indistinguishable from its original states known as isometry.)

Under certain conditions Robertson-Walker space can be transformed into Minkowski space, de Sitter, anti-de Sitter spaces.

Robertson-Walker spacetimes are foliated by the three-dimensional hypersurfaces $\Sigma$ of constant curvature. The one-parameter family of constant curvature three-spaces $S$ are characterized by the time coordinate $t=$ constant. For this structure, the metric of a general Robertson-Walker spacetime can be written in the form
$$d s^2=d t^2-S^2(t) d \sigma^2$$
where $d \sigma^2$ is the metric of a three space of constant curvature is independent of time. In Chapter Eleven, the deduction of the Robertson-Walker metric is given.

The geometry of these three spaces may have only three types and characterized by a parameter $k$, which is the sign of their curvature and actually they are three spaces of constant positive, zero, or negative curvature. The three-dimensional space is flat spacetime when $k=0$, a three-sphere $S^3$ when $k=+1$, and a hyperbolic three-space $H^3$ when $k=-1$.

Also, general Robertson-Walker spacetime can be written (by alternative coordinate parametrizations of the three-spaces of constant curvature) as
$$d s^2=d t^2-S^2(t)\left[\frac{d r^2}{1-k r^2}+r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)\right]$$
Now, after rescaling the function $S$, we can normalize this curvature $k$ to be 1,0 , or -1 . According to $k$, we can categorize three possibilities:
$k=0:$ This case corresponds to a flat space and replacing $r$ by $\chi$ in $(9.17)$, metric $\left(d \sigma^2\right)$ of the three-spaces of constant curvature takes the form
$$d \sigma^2=d \chi^2+\chi^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)$$

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|Anti-de Sitter Space

$$-x^2-y^2-z^2+v^2+w^2=1$$

$$d s^2=d v^2+d w^2-d x^2-d y^2-d z^2$$

$$v=R \cos t, w=R \sin t$$

$$-x^2-y^2-z^2+R^2=1$$

$$d s^2=-d x^2-d y^2-d z^2+d R^2+R^2 d t^2$$
，我们考虑另一种变换
$$R=\sqrt{1+\rho^2}$$

$$x=\rho \cos \theta, y=\rho \sin \theta \cos \phi, z=\rho \sin \theta \sin \phi .$$

$$d s^2=-d \rho^2-\rho^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)+\frac{\rho^2 d \rho^2}{1+\rho^2}+(1$$

$$d s^2=\cosh ^2 r d t^2-d r^2-\sinh ^2 r\left(d \theta^2+\sin ^2 \theta d \phi^2\right) \text {. }$$

## 物理代写|广义相对论代写General relativity代考|Robertson–Walker Spaces

(位移空间与其原始状态无法区分的位移类型称为等 距。)

Robertson-Walker 时空由常曲率的三维超曲面组成。的 单参数族的特征在于时间坐标常数。对于这种结构，一 般 Robertson-Walker 时空的度量可以写成形式，其中 是三个空间的度量常曲率空间与时间无关。第十一章给 出了罗伯逊沃克度量的推导。 $\Sigma S t=$
$$d s^2=d t^2-S^2(t) d \sigma^2$$
$d \sigma^2$这三个空间的几何可能只有三种类型，由一个参数来表 征， $k$ 是它们曲率的符号，实际上它们是三个常正、零 或负曲率的空间。三维空间在时是平坦时空，在时是三 球面，在时是双曲三空间。 $k k=0 S^3 k=+1 H^3$ $k=-1$

。根据，我们可以归类三种可能性：这种情况对应于 个平面空间并且在中用，度量的三个空间恒定曲率采取 的形式
$$d s^2=d t^2-S^2(t)\left[\frac{d r^2}{1-k r^2}+r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right)\right]$$
$S k k$
\begin{aligned} k=0 & : r \chi(9.17)\left(d \sigma^2\right) \ & d \sigma^2=d \chi^2+\chi^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right) \end{aligned}

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