广义相对论是在四大力学之后物理系的一门重要课程，这门课程容易上手的一个点是，他不需要太多的预备知识，实际上一个人只需要有一点点线性代数的知识和一点点微积分的知识，就可以上手学习广义相对论了，有微分几何或者微分流形的知识是有帮助的，但是并不是必要的，因为在局部坐标系下计算中用到的全部都是$R^n$中的微积分和每个点的切空间上的线性代数，可能主要是因为物理学家通常同时学习物理理论和相关的数学。我要说的学习广义相对论的一个更重要的前提并不是足够的物理知识而是一定程度的数学知识和一定的“数学成熟度”和数学直觉。

对微分几何和抽象数学的任何了解都会对广义相对论的学习有所帮助。而理论力学的知识会让你明白你在学什么。

Transformation of Christoffel symbols:

(a) Show that, under a coordinate transformation, the components of the Christoffel symbol transform as follows:

$$

\Gamma_{\beta^{\prime} \gamma^{\prime}}^{\alpha^{\prime}}=\frac{\partial x^{\alpha^{\prime}}}{\partial x^{\alpha}} \frac{\partial x^{\beta}}{\partial x^{\beta^{\prime}}} \frac{\partial x^{\gamma}}{\partial x^{\gamma^{\prime}}} \Gamma_{\beta \gamma}^{\alpha}-\frac{\partial^{2} x^{\alpha^{\prime}}}{\partial x^{\beta} \partial x^{\gamma}} \frac{\partial x^{\beta}}{\partial x^{\beta^{\prime}}} \frac{\partial x^{\gamma}}{\partial x^{\gamma^{\prime}}}

$$

Do this by considering the form of the Christoffel symbol in terms of derivatives of the metric.

(b) Show that, using this rule, the components of the covariant derivative of a vector transform as tensors should:

$$

\nabla_{\alpha^{\prime}} A^{\beta^{\prime}}=\frac{\partial x^{\alpha}}{\partial x^{\alpha^{\prime}}} \frac{\partial x^{\beta^{\prime}}}{\partial x^{\beta}} \nabla_{\alpha} A^{\beta}

$$

We ask you to show that

$$

\begin{aligned}

\Gamma_{\beta^{\prime} \gamma^{\prime}}^{\alpha^{\prime}} &=\frac{\partial x^{\alpha^{\prime}}}{\partial x^{\alpha}} \frac{\partial x^{\beta}}{\partial x^{\beta^{\prime}}} \frac{\partial x^{\gamma}}{\partial x^{\gamma^{\prime}}} \Gamma_{\beta \gamma}^{\alpha}-\frac{\partial^{2} x^{\alpha^{\prime}}}{\partial x^{\beta} \partial x^{\gamma}} \frac{\partial x^{\beta}}{\partial x^{\beta^{\prime}}} \frac{\partial x^{\gamma}}{\partial x^{\gamma^{\prime}}} \\

&=L^{\alpha^{\prime}}{ }_{\alpha} L^{\beta}{ }_{\beta^{\prime}} L^{\gamma}{ }_{\gamma^{\prime}}^{\gamma} \Gamma_{\beta \gamma}^{\alpha}-L^{\beta}{ }_{\beta^{\prime}} L^{\gamma}{ }_{\gamma^{\prime}} \partial_{\beta} L^{\alpha^{\prime}}{ }_{\gamma}

\end{aligned}

$$

You may find at the end of your calculation that you have instead derived a rule that looks like

$$

\Gamma_{\beta^{\prime} \gamma^{\prime}}^{\alpha^{\prime}}=L^{\alpha^{\prime}}{ }_{\alpha} L^{\beta}{ }_{\beta^{\prime}} L^{\gamma}{ }_{\gamma^{\prime}} \Gamma_{\beta \gamma}^{\alpha}+L_{\beta^{\prime}}^{\beta} L^{\alpha^{\prime}}{ }_{\gamma} \partial_{\beta} L^{\gamma}{ }_{\gamma^{\prime}}^{\gamma}

$$

This may look totally wrong $-$ the sign on the final term is incorrect. However, by inspecting this closely, you’ll see that the matrix being differentiated in the second term is not the same in the two versions $-$ the primed and unprimed indices are in opposite locations. By noting that $L^{\gamma}{ }_{\gamma^{\prime}} L^{\alpha^{\prime}} \gamma=\delta^{\alpha^{\prime}} \gamma^{\prime},$ you should be able to show that these two formulas are equivalent.

Relativistic Euler equation

(a) Starting from the stress-energy tensor for a perfect fluid, $\mathbf{T}=\rho \vec{U} \otimes \vec{U}+P \mathbf{h}$, where $\mathbf{h}=\mathbf{g}^{-1}+\vec{U} \otimes \vec{U},$ using local energy momentum conservation, $\nabla \cdot \mathbf{T}=0$, derive the relativistic Euler equation,

$$

(\rho+P) \nabla_{\vec{U}} \vec{U}=-\mathbf{h} \cdot \nabla P

$$

(Note: Because both $\mathbf{T}$ and $\mathbf{h}$ are symmetric tensors, there is no ambiguity in the dot products that appear in this problem.)

(b) For a nonrelativistic fluid $\left(\rho \gg P, v^{t} \gg v^{i}\right)$ and a cartesian basis, show that this equation reduces to the Euler equation,

$$

\frac{\partial v_{i}}{\partial t}+v_{k} \partial_{k} v_{i}=-\frac{1}{\rho} \partial_{i} P

$$

$(i, k$ are spatial indices running from 1 to 3.) What extra terms are present if the connection is non-zero (e.g., spherical coordinates)?

(c) Apply the relativistic Euler equation to Rindler spacetime for hydrostatic equilibrium. Hydrostatic equilibrium means that the fluid is at rest in the $\bar{x}$ coordinates, i.e. $U^{\bar{x}}=0 .$ Suppose that the equation of state (relation between pressure and density) is $P=w \rho$ where $w$ is a positive constant. Find the general solution $\rho(\bar{x})$ with $\rho(0)=\rho_{0}$.

(d) Suppose now instead that $w=w_{0} /(1+g \bar{x})$ where $w_{0}$ is a constant. Show that the solution is $\rho(\bar{x})=\rho_{0} \exp (-\bar{x} / L$. Find $L,$ the density scale height, in terms of $g$ and $w_{0}$. Convert to “normal” units by inserting appropriate factors of $c-L$ should be a length.

(e) Compare your solution to the density profile of a nonrelativistic, plane-parallel, isothermal atmosphere (for which $P=\rho k T / \mu,$ where $T$ is temperature and $\mu$ is the mean molecular weight) in a constant gravitational field. [Use the nonrelativistic Euler equation with gravity: add a term $-\partial_{i} \Phi=g_{i}$, where $\Phi$ is Newtonian gravitational potential and $g_{i}$ is Newtonian gravitational acceleration, to the right hand side of Eq.

(3).] Why does hydrostatic equilibrium in Rindler spacetime – where there is no gravity – give such similar results to hydrostatic equilibrium in a gravitational field?

relativistic Euler equation

(a) Starting from the stress-energy tensor for a perfect fluid, $\mathbf{T}=\rho \vec{U} \otimes \vec{U}+P \mathbf{h}$, where $\mathbf{h}=\mathbf{g}^{-1}+\vec{U} \otimes \vec{U},$ using local energy momentum conservation, $\nabla \cdot \mathbf{T}=0,$ derive the relativistic Euler equation,

$$

(\rho+P) \nabla_{\vec{U}} \vec{U}=-\mathbf{h} \cdot \nabla P

$$

(Note: Because both $\mathbf{T}$ and $\mathbf{h}$ are symmetric tensors, there is no ambiguity in the dot products that appear in this problem.)

It turns out that if you just evaluate $\nabla \cdot \mathbf{T}=0,$ you are unlikely to get a very useful answer. The result you get is, technically, an Euler equation, but it does not naturally reduce to the form we are hoping to show. In particular, you won’t recover the nonrelativistic limit in a natural way.

A more useful form is obtain by separately equating to zero the components of $\nabla \cdot \mathbf{T}$ parallel to and orthogonal to the fluid’s 4 -velocity, $\vec{u}$. In particular, if we define $j^{\nu} \equiv$ $\nabla_{\mu} T^{\mu \nu},$ then the equations

$$

\begin{aligned}

j^{\nu} u_{\nu} &=0 & &(\text { Component parallel to } \vec{u}) \\

j^{\nu} h^{\lambda}{ }_{\nu} &=0 & &(\text { Component orthogonal to }

\end{aligned}

$$

give us useful information. The second equation in particular can be considered the relativistic analog of the usual Euler equation. (It’s worth examining the first equation and seeing what it corresponds to.)

从上面两个典型的例题可以看出，要学好广义相对论，强大的计算能力和扎实的基本功是很重要的。

##### 在这里推荐一本学习广义相对论的很好的教材：

Spacetime and Geometry: An Introduction to General Relativity

**Spacetime and Geometry: An Introduction to General Relativity** provides a lucid and thoroughly modern **introduction to general relativity** for advanced undergraduates and graduate students. It introduces modern techniques and an accessible and lively writing style to what can often be a formal and intimidating subject.

这本书写的非常具体，例子也非常多，比较适合入门。

##### FYS4160 – The General Theory of Relativity University of Oslo代写请认准UpriviateTA

The course provides a comprehensive introduction to the general theory of relativity where all forms of gravity can be described as a purely geometric effect where the curvature of space and time follows the distribution of energy and the amount momentum the matter has. An overview is given of the classical tests of theory, and how the theory is used to describe black holes, gravitational waves, and the cosmological evolution of the universe. The course also provides an introduction to differential geometry, which is necessary to be able to both formulate and apply the theory.

##### Physics 514: General Relativity, Winter 2021 – McGill Physics代写请认准UpriviateTA

This outline is subject to revisions.

• Introduction (.5 lecture). Special Relativity (1 lecture). The Equivalence Principle (.5).

• Manifolds (1). Tensors (2). Differential Forms (1).

• Metrics (.5). Geodesics (.5). Covariant Derivatives (1). Curvature (2).

• General Relativity (1). The Stress Tensor (1).

• The Lagrangian Formulation (1.5). Symmetries (1.5).

• The Schwarzschild Metric (1.5).

• Tests of GR: Precession of Orbits, Gravitational Lensing and Redshift (1.5).

• Black Holes (2).

• Cosmology (2).

• Linearized Gravity and Gravity Waves (2).

• Time Permitting: Hawking Radiation (1).

##### PHYS 515: General Relativity I – Introduction to Gravity

Spring 2020代写请认准UpriviateTA

What is this class about? General Relativity is an advanced graduate course that teaches the foundations of

Einstein’s theory of General Relativity. This class is a very mathematically intensive, laying the foundations for black

hole theory, post-Newtonian theory and numerical relativity. In fact, General Relativity was initially taught in the

mathematics department of universities! It is impossible to teach this subject without doing a deep-dive into the

mathematics that are important in General Relativity, so the first half of this class is quite mathematically intensive.

The second half of the course presents the physical consequences of Einstein’s theory, with a (very brief) tour of its

greatest hits: non-spinning (Schwarzschild) black holes and neutron stars, Solar System tests of gravitation, gravitational waves and an introduction to cosmology. Students interested in these physical applications are encouraged to

take subsequent courses on General Relativity, physical cosmology and astrophysics.

Who should take this class? This course is intended for advanced graduate students. As such, it is assumed

students have prior knowledge of Einstein’s theory of special relativity, Newtonian gravitation and classical mechanics,

Maxwell’s theory of electrodynamics and advanced mathematics, including differential equations, advanced Calculus

and advanced linear algebra. The purpose of the class is to prepare students who wish to specialize in General

Relativity (analytical or numerical), relativistic astrophysics and cosmology for research. Other students with broader

interests are welcomed to take this class, but they should be advised that there are other (perhaps less intensive)

courses they can take to fulfill their elective requirements.

What is expected of students who take this class? Students are expected to attend class, complete all

homework assignments and complete a midterm exam and a final exam (see breakdown of topics below). In addition,

students are expected to be mature enough to independently do some amount of self-learning outside of class, including

reading the assigned book (see below), reading papers mentioned in class, and watching video lectures recorded by

Prof. Yunes. Since this is a graduate course, readings will not be assigned weekly, but rather, students are expected

to find the topics in the course’s textbook that are being covered in class and read about them in the textbook; in

addition to the required class textbook, there are also other additional (recommended) textbooks that students can

and should refer to if and when needed. Questions are always welcomed, either in class, or outside of class during

office hours

#### Relativity and Gravitation | Part III (MMath/MASt)

*“Wissen Sie, wenn man zu rechnen anfängt, b’scheisst man unwillkürlich.”*

— Albert Einstein, describing the beauty of general relativty to Otto Stern. It roughly translates as:

*“You know, when you start to calculate, you inevitably end up cheating.”*

广义相对论代写，Relativity and Gravitation代写General Relativity代写请认准UpriviateTA. UpriviateTA为您的留学生涯保驾护航。

更多内容请参阅另外一份经济学代写案例

统计代写可以参考此份案例