Posted on Categories:General Relativity, 广义相对论, 物理代写, 相对论

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 物理代写|相对论代写Theory of relativity代考|THE SCHWARZSCHILD METRIC

Introduction. This chapter begins a long set of chapters that explore the physical meaning of the Schwarzschild metric. This metric tensor is a spherically symmetric, time-independent solution to the Einstein equation in a vacuum, and is thus suitable for describing the spacetime in the empty space surrounding a spherical, static object. This metric therefore plays much the same role in general relativity that the formula for the gravitational field of a point mass does in Newtonian gravity and the formula for the electric field of a point charge does in electrostatics.

Spherical Coordinates for Flat Spacetime. According to box 5.6, the metric equation for latitude-longitude coordinates $\theta, \phi$ on a $2 \mathrm{D}$ spherical surface is given by
$$d s^2=R^2 d \theta^2+R^2 \sin ^2 \theta d \phi^2$$
where $R$ is the radius of the spherical surface (in three dimensions). We can label any event in an ordinary flat spacetime with spherical coordinates $r, \theta$, and $\phi$, where $r$ (now a variable) specifies the radius of the sphere around the origin on which the event lies, and $\theta$ and $\phi$ specify the event’s latitude and longitude coordinates on that surface. Therefore, the metric for spherical coordinates in flat spacetime should be
$$d s^2=-d t^2+d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$

## 物理代写|相对论代写Theory of relativity代考|The :Sc.11war:lsc:n1Jlct Metric

The Schwarzschild Metric. In 1916, Karl Schwarzschild (who at the time was fighting in the trenches of World War I and who died shortly thereafter) discovered that the following metric satisfied the Einstein field equations in empty space:
$$d s^2=-\left(1-\frac{r_s}{r}\right) d t^2+\left(1-\frac{r_s}{r}\right)^{-1} d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$
where $r_s$ is a constant with units of length called the Schwarzschild radius. (We will verify that this metric is indeed a solution to the Einstein equation later in the course.) This metric is spherically symmetric (note that surfaces of constant $r$ have the same metric as that for latitude-longitude coordinates on the surface of a sphere and so have the geometry of a sphere, and the other components of the metric depend only on $r$ ), time-independent, and becomes the flat space metric in the limit as $r \rightarrow \infty$. This metric is therefore a suitable candidate for that describing the spacetime in the empty space surrounding a spherical, static object.

## 物理代写|相对论代写Theory of relativity代考|THE SCHWARZSCHILD METRIC

$$d s^2=R^2 d \theta^2+R^2 \sin ^2 \theta d \phi^2$$

$$d s^2=-d t^2+d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$

## 物理代写相对论代写Theory of relativity代考|The :Sc.11war:Isc:n1Jlct Metric

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

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:General Relativity, 广义相对论, 物理代写, 相对论

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 物理代写|相对论代写Theory of relativity代考|The Scalar Product of a Vector and Covector

The Scalar Product of a Vector and Covector. The scalar product of a vector with components $A^\mu$ and a covector with components $B_\mu$ is defined to be $A^\mu B_\mu$ (with the implied sum over $\mu$ ). This scalar product is interesting because its value is independent of coordinate system. The proof is direct:
$$A^{\prime \mu} B_\mu^{\prime}=\left[\frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha\right]\left[\frac{\partial x^\beta}{\partial x^{\prime \mu}} B_\beta\right]=\frac{\partial x^\beta}{\partial x^{\prime \mu}} \frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha B_\beta=\delta^\beta{ }\alpha A^\alpha B\beta=A^\alpha B_\alpha$$
In fact, equation $6.5$ means that this is really the same as $A^\mu g_{\mu v} B^v$, which we have already defined as the scalar product of two vectors, and which we have already seen is frame independent for Lorentz transformations. We now see that $A^\mu B_\mu=A^\mu g_{\mu \nu} B^\nu$ has a coordinate-independent value not only in special relativity, but in any arbitrary coordinate system we use (as long as we use a coordinate basis)!

The Inverse Metric. We have already seen that the components of the metric tensor transform as follows (see equation 5.11):
$$g_{\mu \nu}^{\prime}=\frac{\partial x^\alpha}{\partial x^{\prime \mu}} \frac{\partial x^\beta}{\partial x^{\nu v}} g_{\alpha \beta}$$
Compare this to the transformation equation $6.2$ for a covector. Note that $g_{\alpha \beta}$, which has two subscript indices, has a transformation law that is essentially a doubled version of equation 6.2: there is a partial derivative factor associated with each lower index that has the same form as the single factor associated with the single lower index in equation $6.2$.

## 物理代写|相对论代写Theory of relativity代考|The Gradient of a Tensor Is Not a Tensor

The Gradient of a Tensor Is Not a Tensor. Note that while the gradient of a scalar invariant yields a covector, the gradient of any other tensor does not yield another tensor. For example, imagine that $A^\mu$ is a four-vector. The components of the gradient of $A^\mu$ transform as follows when we change coordinate systems:
\begin{aligned} \partial_v^{\prime} A^{\prime \mu} & \equiv \frac{\partial A^{\prime \mu}}{\partial x^{\prime \nu}}=\frac{\partial}{\partial x^{\prime \nu}}\left(\frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha\right)=\frac{\partial x^\beta}{\partial x^{\prime \nu}} \frac{\partial}{\partial x^\beta}\left(\frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha\right) \ & =\frac{\partial x^\beta}{\partial x^{\prime \nu}} \frac{\partial^2 x^{\prime \mu}}{\partial x^\beta \partial x^\alpha} A^\alpha+\frac{\partial x^\beta}{\partial x^{\prime \nu}} \frac{\partial x^{\prime \mu}}{\partial x^\alpha}\left(\partial_\beta A^\alpha\right) \end{aligned}
by the product rule. The second term appearing in the bottom line, if it were alone, would be the transformation rule for a tensor. However, if the coordinate transformation factors $\partial x^{\prime \mu} / \partial x^\nu$ are not constants, then the second derivative appearing in the first term will not be zero, meaning that the gradient of a vector does not transform as a tensor, and therefore is not a tensor. The same issue arises with derivatives with respect to a particle’s proper time $\tau$ (see problem P6.7). This is an issue we must address in the future, because most physical equations involve such derivatives.
(However, the components of the Lorentz transformation are constants, so if we limit ourselves to transformations between IRFs in cartesian coordinates, the first term in equation $6.14$ is zero and the gradient of a vector does transform like a tensor. This also applies to the gradients of higher-rank tensors.)

## 物理代写|相对论代写Theory of relativity代考|The Scalar Product of a Vector and Covector

$$A^{\prime \mu} B_\mu^{\prime}=\left[\frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha\right]\left[\frac{\partial x^\beta}{\partial x^{\mu \mu}} B_\beta\right]=\frac{\partial x^\beta}{\partial x^{\prime \mu}} \frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha B_\beta=\delta^\beta \alpha A^\alpha B \beta=A^\alpha B_\alpha$$

$$g_{\mu \nu}^{\prime}=\frac{\partial x^\alpha}{\partial x^{\prime \mu}} \frac{\partial x^\beta}{\partial x^{\nu v}} g_{\alpha \beta}$$

## 物理代写|相对论代写Theory of relativity代考|The Gradient of a Tensor Is Not a Tensor

$$\partial_v^{\prime} A^{\prime \mu} \equiv \frac{\partial A^{\prime \mu}}{\partial x^{\prime \nu}}=\frac{\partial}{\partial x^{\prime \nu}}\left(\frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha\right)=\frac{\partial x^\beta}{\partial x^\mu} \frac{\partial}{\partial x^\beta}\left(\frac{\partial x^{\prime \mu}}{\partial x^\alpha} A^\alpha\right) \quad=\frac{\partial x^\beta}{\partial x^{\prime \nu}} \frac{\partial^2 x^{\prime \mu}}{\partial x^\beta \partial x^\alpha} A^\alpha+\frac{\partial x^\beta}{\partial x^{\prime}} \frac{\partial x^{\prime \mu}}{\partial x^\alpha}\left(\partial_\beta A^\alpha\right)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:General Relativity, 广义相对论, 物理代写, 相对论

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 物理代写|相对论代写Theory of relativity代考|Electromagnetic Field Tensor

The Electromagnetic Field Tensor. Another interesting quantity in special relativity is the electromagnetic field tensor, whose components are
$$\left[\begin{array}{llll} F^{t t} & F^{t x} & F^{y y} & F^{t z} \ F^{x t} & F^{x x} & F^{x y} & F^{x z} \ F^{y t} & F^{y x} & F^{y y} & F^{y z} \ F^{y t} & F^{z x} & F^{z y} & F^{z z} \end{array}\right]=\left[\begin{array}{cccc} 0 & E_x & E_y & E_z \ -E_x & 0 & B_z & -B_y \ -E_y & -B_z & 0 & B_x \ -E_z & B_y & -B_x & 0 \end{array}\right]$$
where $\left[E_x, E_y, E_z\right]$ are the components of the electric field vector $\vec{E}$ and $\left[B_x, B_y, B_z\right]$ are the same for the magnetic field vector $\vec{B}$ (in GR units, both vectors have the same units of $\mathrm{kg} \cdot \mathrm{C}^{-1} \mathrm{~m}^{-1}$ : see box $4.2$ ). Using this tensor, we can write a relativistically valid version of the Lorentz force law (which describes the total electromagnetic force acting on particle with charge $q$ moving through an electromagnetic field),
$$\frac{d p^\mu}{d \tau}=q F^{\mu v} \eta_{v \alpha} u^\alpha$$
and Gauss’s law and the Ampere-Maxwell relation become the single equation
$$\frac{\partial F^{\mu \nu}}{\partial x^\nu}=4 \pi k J^\mu$$
where $k$ is the Coulomb constant (in GR units), and $J^t=\rho, J^x=\rho v_x, f^y=\rho v_y$, and $J^2=\rho v_z$ are the components of the four-current $J$ of charge flowing at the event $(\rho$ is the density of charge at the event in question and $v_x, v_y$, and $v_z$ are the usual velocity components of the flowing charge). In equation 4.16, the superscript $\nu$ in the denominator of the derivative is considered to be equivalent to a subscript in the numerator, so there is an implicit sum over $\nu$. See box $4.3$ for a discussion of these equations. However, you can see how the abstract component notation here yields very compact versions of these important equations.

## 物理代写|相对论代写Theory of relativity代考|Free and Bound Indices

Free and Bound Indices. Consider the following example equation regarding the total four-momentum of a two-particle system:
$$p_{\text {tot }}^{\prime \mu}=p_1^{\prime \mu}+p_t^{\prime \mu}=\Lambda_\nu^\mu p_1^\nu+\Lambda_\alpha^\mu p_2^\alpha$$
If $\mu=t$, this equation tells us that the primed-frame $t$ component of the system’s fourmomentum $\boldsymbol{p}_{\text {tot }}$ is the sum of the primed-frame $t$ components of particle 1’s four-momentum $\boldsymbol{p}_1$ and particle 2’s four-momentum $\boldsymbol{p}_2$, which in turn is the sum of the $t$ rows of the matrix equations for the Lorentz transformation of $\boldsymbol{p}_1$ ‘s and $\boldsymbol{p}_2$ ‘s unprimed-frame components. If $\mu=x$, the equation makes the corresponding statement about the $x$ components and so on. This abstract equation therefore actually represents four equations about the four components of the total momentum. Note that there are implicit sums over the indices $\nu$ and $\alpha$.

What I want you to focus on at the moment is the structure of this equation. The index $\mu$ appears in every term of the equation, and it is not summed in any term. We are free to assign any value to $\mu$ that we like in order to specify which component of the equation we are talking about. We therefore call $\mu$ a free index.

In contrast, the sole purpose of the pair of indices labeled $\nu$ in the term $\Lambda^\mu{ }_\nu p_1^v$ is to indicate a sum. We are not free to specify a value for these indices: they must together take on all four possible values and the four resulting terms summed if the equation is to make any sense. We call $\nu$ a bound index (some texts call this a dummy index). The index $\alpha$ in the next term is also a bound index.

## 物理代写|相对论代写Theory of relativity代考|Electromagnetic Field Tensor

$\left[\begin{array}{llllllllllllllllllll}F^{t t} & F^{t x} & F^{y y} & F^{t z} & F^{x t} & F^{x x} & F^{x y} & F^{x z} F^{y t} & F^{y x} & F^{y y} & F^{y z} & F^{y t} & F^{z x} & F^{y y} & F^{z z}\end{array}\right]=\left[\begin{array}{lllllllll}0 & E_x & E_y & E_z-E_x & 0 & B_z & -B_y-E_y & -B_z\end{array}\right.$

$\mathrm{kg} \cdot \mathrm{C}^{-1} \mathrm{~m}^{-1}$ :见框 $4.2$ ). 使用这个张量，我们可以写出洛伦兹力定律的相对论有效版本（它描述了作用在带电粒子上的总电磁力 $q$ 通过电碰场移动)，
$$\frac{d p^\mu}{d \tau}=q F^{\mu v} \eta_{v \alpha} u^\alpha$$

$$\frac{\partial F^{\mu \nu}}{\partial x^\nu}=4 \pi k J^\mu$$

## 物理代写|相对论代写Theory of relativity代考|Free and Bound Indices

$$p_{\text {tot }}^{\prime \mu}=p_1^{\prime \mu}+p_t^{\prime \mu}=\Lambda_\nu^\mu p_1^\nu+\Lambda_\alpha^\mu p_2^\alpha$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。