## 物理代写|广义相对论代写General relativity代考|KYA424

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|Schwarzschild Solutions

The most interesting general spherically symmetric static vacuum solution of the Einstein field equations is the Schwarzschild solution. The Schwarzschild solution, i.e., the Schwarzschild metric describes the Schwarzschild black hole. It defines the gravitational field in the outer region of a spherical mass. The Schwarzschild line element is written as
$$d s^2=\left(1-\frac{2 m}{r}\right) d t^2-\frac{d r^2}{1-\frac{2 m}{r}}-r^2\left(d \theta^2+\sin ^2 \theta d \phi^2\right),$$
where $m=\frac{G M}{c^2}$
Karl Schwarzschild found this exact static spherically symmetric vacuum solutions of the Einstein field equations in 1915 while fighting during World War I in favor of Germany. To show honor, this solution is dubbed as The Schwarzschild solution. The coordinate $r$ is a radial parameter, which has the property that the surface area of the two-sphere ( $t=$ constant, $r=$ constant) is $4 \pi r^2$.
There are two values of coordinates for which the solution has singularities. A singularity at $r=0$ is an essential singularity, whereas singularity at
$$r=2 m=\frac{2 G M}{c^2}(\text { known as Schwarzschild radius })$$
is dubbed as coordinate singularity. Here, the Kretschmann scalar
$$R_{a b c d} R^{a b c d}=\frac{48 m^2}{r^6}$$
which is finite at $r=2 \mathrm{~m}$ but at $r=0$ it blows up. Thus, singularity at $r=0$ is irremovable, and thus, it is an essential singularity.
[Coordinate singularity is a place where geometry cannot be described properly and it is not essential, i.e., it can be uninvolved by a suitable choice of coordinate system.]

Note that coefficient of $d t^2$, i.e., $g_n=0$ yields infinite redshift. Here $g_n$ vanishes at $r=2 m$, therefore, the surface $r=2 m$ is the surface of infinite redshift. Also note that $r=2 m$ is a null hypersurface, which splits the spacetime into two disconnected regions:
I. $2 m2 m$ represents the external field, whereas usual $t$ is time-like and $r$ is space-like, however, in the region $0<r<2 m$, the role of $r$ and $t$ will be reversed, i.e., here, $r$ is time-like and $t$ is space-like. Thus, topological behavior of Schwarzschild solution is not Euclidean.

## 物理代写|广义相对论代写General relativity代考|Null Curves in Schwarzschild Spacetime

In Schwarzschild geometry, we see that $r=2 m$ is a problematic radius. Here, the metric becomes singular at $r=2 m$. Therefore, it is expected that Schwarzschild solution is not appropriate for investigating the physics in the region $r \leq 2 m$. However, this singular behavior has been occurred due to choice of bad coordinates. To know better the characteristic of the Schwarzschild geometry, it is essential to look after its casual structure, i.e., the light cones.

Now we consider radial null curves $\left(d s^2=0\right.$ ) in the planes $\theta=$ constant and $\phi=$ constant. Hence, we have
$$d s^2=0=\left(1-\frac{2 m}{r}\right) d t^2-\left(1-\frac{2 m}{r}\right)^{-1} d r^2$$
This implies,
$$\frac{d t}{d r}= \pm\left(1-\frac{2 m}{r}\right)^{-1}$$
Integrating the above integral, we get (taking positive sign)
$$t=r+2 m \ln |r-2 m|+\text { constant }$$
We note that for $r>2 m, \frac{d t}{d r}>0$. This indicates $r$ is increasing with $t$. This radial null geodesic is outgoing (see Fig. 81).
For negative sign, the above integral yields
$$t=-(r+2 m \ln |r-2 m|+c o n s t a n t)$$
This radial null geodesic is ingoing (see Fig. 81).
We consider the light cones in $(r, t)$ plane. Note that $\frac{d t}{d r}$ signifies the slope of the light cones at a given value of $r$. For $r \rightarrow \infty, \frac{d t}{d r}= \pm 1$, i.e., slope is \pm 1 as in Minkowski space or flat space. When one approaches to $r=2 m$, one will get
$$\frac{d t}{d r} \rightarrow \pm \infty$$

# 广义相对论代考

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

$$d s^2=\left(1-\frac{2 m}{r}\right) d t^2-\frac{d r^2}{1-\frac{2 m}{r}}-r^2\left(d \theta^2+\sin ^2 \theta c\right.$$

1915 年，卡尔.史瓦西 (Karl Schwarzschild) 在第一次世 界大战期间为德国而战时，发现了爱因斯坦场方程的这 个精确的静态球对称真空解。为了表示敬意，这个解被 称为史瓦西解。坐标 $r$ 是一个径向参数，它具有两个球体 的表面积 ( $t=$ 持续的， $r=$ 常量) 是 $4 \pi r^2$.

$r=2 m=\frac{2 G M}{c^2}($ known as Schwarzschild radius $)$

$$R_{a b c d} R^{a b c d}=\frac{48 m^2}{r^6}$$

I。 $2 m 2 m$ 代表外场，而通常 $t$ 是类似时间的，并且 $r$ 是空 间般的，然而，在该地区 $0<r<2 m$, 的作用 $r$ 和 $t$ 将被 逆转，即，在这里， $r$ 是类似时间的，并且 $t$ 是空间般 的。因此，史瓦西解的拓扑行为不是欧几里得。

## 物理代写|广义相对论代写General relativity代考|Null Curves in Schwarzschild Spacetime

$$d s^2=0=\left(1-\frac{2 m}{r}\right) d t^2-\left(1-\frac{2 m}{r}\right)^{-1} d r^2$$

$$\frac{d t}{d r}= \pm\left(1-\frac{2 m}{r}\right)^{-1}$$

$$t=r+2 m \ln |r-2 m|+\text { constant }$$我们注意到对于 $r>2 m, \frac{d t}{d r}>0$. 这表明 $r$ 随着 $t$. 这个 径向零测地线是传出的 (见图 81)。 对于负号，上述积分产生
$$t=-(r+2 m \ln |r-2 m|+\text { constant })$$

$$\frac{d t}{d r} \rightarrow \pm \infty$$

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