Riemann surface
matlab
无需担心！我们的广义相对论代写专家团队将专业地解决您在广义相对论学习中遇到的各种挑战。我们拥有广泛的专业知识和丰富的经验，可以协助您完成高水平的作业和论文，确保您在学习道路上顺利前行！
以下是一些我们可以帮助您解决的问题：
广义相对论基础概念：涵盖时空、引力、度规等各种常用广义相对论概念的定义、性质和分类。
引力场方程：研究和应用于广义相对论中的引力场方程及其解析解、数值解等。
时空几何与曲率：常见的时空几何性质和曲率计算方法，如测地线、黎曼张量等。
黑洞与宇宙学：广义相对论在黑洞和宇宙学领域的应用，包括黑洞形成、膨胀宇宙模型等。
引力波：介绍引力波的产生、探测与分析方法，如LIGO、VIRGO等引力波探测器。
时空奇异性与时空结构：研究广义相对论中的时空奇异性和时空结构，如黑洞奇点、宇宙奇点等。
广义相对论与量子力学：介绍广义相对论与量子力学的关系和统一理论的研究进展。
无论您面临的广义相对论问题是什么，我们都会竭尽全力提供专业的帮助，确保您的学习之旅顺利无阻！
Exercise 1
Consider a binary system of gravitating objects of masses $M$ and $m$.
First consider the case in which $m \ll M$ and where the small-mass object is in quasi-circular orbit around the more massive object. Draw the trajectory in two-space and the worldline in a $1+1$ – and in a $2+1$-dimensional spacetime [Hint: use a co-ordinate system centred in $M$ ].
Now let $m=M$ and the binary be in circular orbit around the Newtonian centre of mass of the system. Draw the trajectory in two-space and the worldline in a $1+1$ – and in a $2+1$-dimensional spacetime [Hint: use a co-ordinate system centred in the Newtonian centre of mass].
Figure 1: Trajectories in two-space for the cases $m \ll M$ (left) and $m=M$ (right).
Exercise 2
Consider a two-dimensional space and cover it with two co-ordinate maps: a Cartesian map where $\left{x^\mu\right}=(x, y)$ and a polar map where $\left{x^{\mu^{\prime}}\right}=(r, \theta)$.
Find the co-ordinate transformation $f: x^\mu \rightarrow x^{\mu^{\prime}}$
Find the inverse co-ordinate transformation $f^{-1}: x^{\mu^{\prime}} \rightarrow x^\mu$
Find the components of the transformation matrix $\Lambda_\mu^{\mu^{\prime}}$ and its determinant $J^{\prime}:=\left|\partial x^{\mu^{\prime}} / \partial x^\mu\right|$
Find the components of the inverse transformation matrix $\Lambda_{\mu^{\prime}}^\mu$ and its determinant $J:=\left|\partial x^\mu / \partial x^{\mu^{\prime}}\right|$
Show that $\Lambda_\mu^{\mu^{\prime}} \Lambda_{\nu^{\prime}}^\mu=\delta_{\nu^{\prime}}^{\mu^{\prime}}$ and that $J J^{\prime}=1$
Solution 2
The co-ordinate transformation is given by:
$$
f:\left{\begin{array}{l}
r=\left(x^2+y^2\right)^{1 / 2} \
\theta=\arctan (y / x)
\end{array}\right.
$$
The inverse co-ordinate transformation is given by:
$$
f^{-1}:\left{\begin{array}{l}
x=r \cos \theta \
y=r \sin \theta
\end{array}\right.
$$
The transformation matrix is given by:
$$
\begin{aligned}
\Lambda_\mu^{\mu^{\prime}} & =\frac{\partial x^{\mu^{\prime}}}{\partial x^\mu} \
& =\left(\begin{array}{ll}
\partial r / \partial x & \partial r / \partial y \
\partial \theta / \partial x & \partial \theta / \partial y
\end{array}\right) \
& =\left(\begin{array}{cc}
x\left(x^2+y^2\right)^{-1 / 2} & y\left(x^2+y^2\right)^{-1 / 2} \
-y\left(x^2+y^2\right)^{-1} & x\left(x^2+y^2\right)^{-1}
\end{array}\right) \
& \equiv\left(\begin{array}{cc}
\cos \theta & \sin \theta \
-\frac{1}{r} \sin \theta & \frac{1}{r} \cos \theta
\end{array}\right),
\end{aligned}
$$
and its determinant is given by:
$$
\begin{aligned}
J^{\prime} & =\left|\partial x^{\mu^{\prime}} / \partial x^\mu\right| \
& =\frac{x^2}{\left(x^2+y^2\right)^{3 / 2}}+\frac{y^2}{\left(x^2+y^2\right)^{3 / 2}} \
& =\left(x^2+y^2\right)^{-1 / 2},
\end{aligned}
$$
3
or alternatively, using equation (4), is given by:
$$
\begin{aligned}
J^{\prime} & =\frac{\cos ^2 \theta}{r}+\frac{\sin ^2 \theta}{r} \
& =\frac{1}{r} .
\end{aligned}
$$
It is trivial to confirm that both expressions for $J^{\prime}$ are equivalent.
The inverse transformation matrix is given by:
$$
\begin{aligned}
\Lambda_{\mu^{\prime}}^\mu & =\frac{\partial x^\mu}{\partial x^{\mu^{\prime}}} \
& =\left(\begin{array}{ll}
\partial x / \partial r & \partial x / \partial \theta \
\partial y / \partial r & \partial y / \partial \theta
\end{array}\right) \
& =\left(\begin{array}{ll}
\cos \theta & -r \sin \theta \
\sin \theta & r \cos \theta
\end{array}\right) \
& \equiv\left(\begin{array}{cc}
x\left(x^2+y^2\right)^{-1 / 2} & -y \
y\left(x^2+y^2\right)^{-1} & x
\end{array}\right)
\end{aligned}
$$
and its determinant is given by:
$$
\begin{aligned}
J & =\left|\partial x^\mu / \partial x^{\mu^{\prime}}\right| \
& =r \cos ^2 \theta+r \sin ^2 \theta \
& =r
\end{aligned}
$$
or alternatively, using equation (8), is given by
$$
\begin{aligned}
J & =\frac{x^2}{\left(x^2+y^2\right)^{1 / 2}}+\frac{y^2}{\left(x^2+y^2\right)^{1 / 2}} \
& =\left(x^2+y^2\right)^{1 / 2}
\end{aligned}
$$
It is again trivial to confirm that both expressions for $J$ are equivalent.
Matrix multiplication of equations (3) and (8) or equations (4) and (7) yields the identity matrix, confirming the result $\Lambda_\mu^{\mu^{\prime}} \Lambda_{\nu^{\prime}}^\mu=\delta_{\nu^{\prime}}^{\mu^{\prime}}$. It is also straightforward to confirm that $J J^{\prime}=1$ in both co-ordinate systems.
E-mail: help-assignment@gmail.com 微信:shuxuejun
help-assignment™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注于数学，统计，金融，经济，计算机科学，物理的作业代写服务