物理代写|广义相对论代写General relativity代考|MATH7105

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

物理代写|广义相对论代写General relativity代考|Review of Special Relativity

The theory of Special Relativity (SR) was introduced by A. Einstein in 1905. It deals with the observations of inertial observers in the absence of gravity. The theory of General Relativity (GR) that includes gravitation, and thus acceleration, was published in 1915. For English translations, see Einstein (1905). The latter theory predicted the deflection of light near a massive body, like the sun. Shortly after the end of the first world war, a British team, led by A. S. Eddington, confirmed this startling prediction. This made Einstein world famous, even among people who had no particular interest in science.

In relativity, an observation is the assignment of coordinates $x^{\mu}, \mu=$ $0,1,2,3$, for the time and space location of an event. Space is continuous, and functions of the coordinates can be differentiated. Upon partial differentiation with respect to one of the coordinates, the others are held constant. This insures that the coordinates are independent,
$$x^{\mu},{ }{\nu} \equiv \frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta^{\mu}{ }{\nu}=\delta_{\nu}{ }^{\mu}=1, \quad \mu=\nu, \quad \delta^{\mu}{ }{\nu}=0, \quad \mu \neq \nu .$$ As will be seen $\delta^{\mu}{ }{\nu}$ is the Kronecker delta tensor. The superscript, subscript indexes are termed contravariant, covariant. Note the shorthand notation for the partial derivative, by use of a comma. Such a shorthand will keep some of the formulas of GR, with many partial derivatives, to a reasonable length.

物理代写|广义相对论代写General relativity代考|Lorentz Transform

Two observers $\mathrm{O}$ and $\mathrm{O}^{\prime}$ are considered. They use parallel axes and rectangular coordinates. Rotations, like those in Fig. 1.1, allow them to align their $z$-axes along the relative velocity. $\mathrm{O}$ uses $x^{\mu}$, and says $\mathrm{O}^{\prime}$ is moving in the $z$-direction with speed $V(<1)$, while $\mathrm{O}^{\prime}$ uses $x^{\mu^{\prime}}$, and says $\mathrm{O}$ is moving in the $-z$-direction with speed $V$.

When their origins overlapped, the clocks were synchronized $t=x^{0}=$ $t^{\prime}=x^{0^{\prime}}=0$. In this geometry, $(x, y)=\left(x^{\prime}, y^{\prime}\right)$ or $x^{1,2}=x^{1^{\prime}, 2^{\prime}}$, as there is no relative motion in these directions. However, $c=1$ for both observers, so space and time are interconnected, and now termed spacetime. If $\mathrm{O}^{\prime}$ says that events led to changes in coordinates $d z^{\prime}=d x^{3^{\prime}}$ and $d t^{\prime}=d x^{0^{\prime}}$, the components of the displacement vector $d r^{\mu^{\prime}}$, then $\mathrm{O}$ would calculate from the chain rule of differential calculus,
\begin{aligned} d x^{3} &=d z=\frac{\partial z}{\partial z^{\prime}} d z^{\prime}+\frac{\partial z}{\partial t^{\prime}} d t^{\prime}+\frac{\partial z}{\partial x^{\prime}} d x^{\prime}+\frac{\partial z}{\partial y^{\prime}} d y^{\prime} \equiv x^{3}, \mu^{\prime} d x^{\mu^{\prime}} \ &=x^{3}, 3^{\prime} d x^{3^{\prime}}+x^{3}, 0^{\prime} d x^{0^{\prime}} \ d x^{0} &=d t=\frac{\partial t}{\partial z^{\prime}} d z^{\prime}+\frac{\partial t}{\partial t^{\prime}} d t^{\prime}+\frac{\partial t}{\partial x^{\prime}} d x^{\prime}+\frac{\partial t}{\partial y^{\prime}} d y^{\prime} \equiv x^{0}, \mu^{\prime} d x^{\mu^{\prime}} \ &=x^{0}, 3^{\prime} d x^{3^{\prime}}+x^{0}, 0^{\prime} d x^{0^{\prime}} \end{aligned}
One notes that, similar to rotations, this transform can be represented by matrix multiplication,
$$\left(\begin{array}{l} d x^{0} \ d x^{1} \ d x^{2} \ d x^{3} \end{array}\right)=\left(\begin{array}{cccc} x^{0}, 0^{\prime} & 0 & 0 & x^{0}, 3^{\prime} \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ x^{3}, 0^{\prime} & 0 & 0 & x^{3}, 3^{\prime} \end{array}\right)\left(\begin{array}{l} d x^{0^{\prime}} \ d x^{1^{\prime}} \ d x^{2^{\prime}} \ d x^{3^{\prime}} \end{array}\right) .$$

物理代写|广义相对论代写General relativity代考|Review of Special Relativity

$$x^{\mu}, \nu \equiv \frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta^{\mu} \nu=\delta_{\nu}^{\mu}=1, \quad \mu=\nu, \quad \delta^{\mu} \nu=0, \quad \mu \neq \nu .$$

物理代写|广义相对论代写General relativity代考|Lorentz Transform

$$d x^{3}=d z=\frac{\partial z}{\partial z^{\prime}} d z^{\prime}+\frac{\partial z}{\partial t^{\prime}} d t^{\prime}+\frac{\partial z}{\partial x^{\prime}} d x^{\prime}+\frac{\partial z}{\partial y^{\prime}} d y^{\prime} \equiv x^{3}, \mu^{\prime} d x^{\mu^{\prime}} \quad=x^{3}, 3^{\prime} d x^{3^{\prime}}+x^{3}, 0^{\prime} d x^{0^{\prime}} d x^{0}$$

$$\left(d x^{0} d x^{1} d x^{2} d x^{3}\right)=\left(x^{0}, 0^{\prime} \quad 0 \quad 0 \quad x^{0}, 3^{\prime} 0 \quad 1 \quad 0 \quad 0 \quad 0 \quad 0 \quad 0 \quad 1 \quad 0 x^{3}, 0^{\prime} \quad 0 \quad 0 \quad x^{3}, 3^{\prime}\right)\left(d x^{0^{\prime}}\right.$$

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