### 物理代写|宇宙学代写cosmology代考| Special relativity — Minkowski geometry

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|宇宙学代写cosmology代考|Geometry

The simplest example of a Riemannian geometry is Minkowski flat space-time. The basic geometric invariant in this space is the interval between two arbitrary points – events. There is a class of preferable coordinates (the Minkowski coordinates) $x^{\mu}$, and their each possible choice corresponds to a certain inertial reference frame (IRF). The summed force applied to a body at rest in a certain IRF is equal to zero, or, in other words, such a body moves by inertia, uniformly and straightly with respect to any other IRF. In any IRF, the squared four-dimensional “distance” (interval) between the events $1\left[x_{1}^{\mu}=\right.$ $\left.\left(c t_{1}, \vec{x}{1}\right)\right]$ and $2\left[x{2}^{\mu}=\left(c t_{2}, \vec{x}{2}\right)\right]$ is written in the form $$s^{2}(1,2)=c^{2}\left(t{2}-t_{1}\right)^{2}-\left(\vec{x}{2}-\vec{x}{1}\right)^{2},$$
where $c$ is a universal constant coinciding with the propagation velocity of electromagnetic waves (light, in particular) in vacuum and called the speed of light. For close events 1 and 2, the interval (2.1) can be written as
$$d s^{2}=\eta_{\mu \nu} d x^{\mu} d x^{\nu}, \quad \mu=0,1,2,3,$$
where the tensor with the covariant components
$$\eta_{\mu \nu}=\operatorname{diag}(1,-1,-1,-1)$$
is called the Minkowski metric tensor (the Minkowski metric). As usual, summing is assumed over repeated indices if one of them is covariant and the other contravariant. The matrix (2.3), together with its inverse matrix
$$\eta^{\mu \nu}=\operatorname{diag}(1,-1,-1,-1)$$
of the contravariant components of the Minkowski tensor are used for raising and lowering vector and tensor indices, so that, e.g., for an arbitrary vector $a=\left(a^{\mu}\right)$ we have $a^{\mu}=\eta^{\mu \nu} a_{\nu}, a_{\mu}=\eta_{\mu \nu} a^{\nu}$. The Minkowski tensor defines a scalar product $(a b)$ of two arbitrary 4 -vectors $a^{\mu}$ and $b^{\mu}$ as follows:
$$(a b)=\eta_{\mu \nu} a^{\mu} b^{\nu}=\eta^{\mu \nu} a_{\mu} b_{\nu}=a_{\mu} b^{\mu}=a^{\mu} b_{\mu} .$$

## 物理代写|宇宙学代写cosmology代考|Coordinate transformations

A transition from one IRF to another is described in the simplest way if the velocity $\vec{v}$ of the system $S^{\prime}$ with respect to the system $S$ is directed along one of the coordinate axes of the latter, for instance, along the axis $O x$, i.e., in the system $S$ the origin of the system $S^{\prime}$ moves according to the law $x=v t$. In this case, the coordinate transformation (the special Lorentz transformation) that leaves the interval (2.1) invariant, has the form
$$x^{\prime}=\frac{x-v t}{\sqrt{1-v^{2} / c^{2}}}, \quad y^{\prime}=y, \quad z^{\prime}=z, \quad t^{\prime}=\frac{t-v x / c^{2}}{\sqrt{1-v^{2} / c^{2}}}$$
where the primed coordinates belong to the IRF $S^{\prime}$.
A general Lorentz transformation, connecting any two IRFs, includes a boost (a transition of the type (2.7) with a vector $\vec{v}$ of arbitrary direction) and an arbitrary rotation of the spatial coordinate axes. All Lorentz transformations form a six-parameter group called the Lorentz group. In addition to general Lorentz transformations, the interval (2.1) is invariant under space and time translations
$$x^{\prime \mu}=x^{\mu}+a^{\mu}, \quad a^{\mu}=\text { const }$$
forming the four-parameter group of translations. Thus the complete group of isometries (coordinate transformations leaving invariant the metric tensor contains ten parameters. It is called the Poincaré group.
The matrix of an arbitrary Lorentz transformation $A=\left(A_{\mu}^{\nu}\right)$ has the definitive property of pseudo-orthogonality. It is this property that expresses the invariance of the Minkowski metric under such transformations. Namely, let the coordinates $x^{\mu}$ of the system $S$ and the coordinates $y^{\mu}$ of the system $S^{\prime}$ be connected by the linear transformation
$$x^{\mu}=A_{\alpha}^{\mu} y^{\alpha}+a^{\mu}, \quad A_{\alpha}^{\mu}, a^{\mu}=\text { const. }$$
According to $(2.9), A_{\alpha}^{\mu}=\partial x^{\mu} / \partial y^{\alpha}$. Substituting $(2.9)$ to the expression for the interval $(2.2)$, we obtain
$$d s^{2}=\eta_{\mu \nu} A_{\alpha}^{\mu} A_{\beta}^{\nu} d y^{\alpha} d y^{\beta}$$

## 物理代写|宇宙学代写cosmology代考|Kinematic effects

An analysis of the Lorentz transformations leads to the most important kinematic effects of SR. Thus, any motion of a point particle at any fixed time instant can be considered to be approximately inertial, so one can introduce an IRF $S^{\prime}$ in which the particle is at rest at this time instant. Assuming, without loss of generality, that the motion occurs along the axis $O x$, it is easy to find that the time increment $d t^{\prime}$ by a clock connected with the particle (and equal to $d s / c$ ) is related to the time increment $d t$ by the clock of the “laboratory” IRF $S$ according to
$$d t^{\prime}=d t \sqrt{1-v^{2} / c^{2}}$$
Due to arbitrariness of choosing the axes, this formula is valid for a velocity of any direction, $\vec{v}=d \vec{x} / d t$. Consequently, for an arbitrary trajectory of motion $\vec{x}(t)$, the proper time interval $\tau$ of the particle (that is, the time elapsed according to a clock connected with the particle or any object whose size is insignificant) is determined by the relation
$$\tau\left(t_{1}, t_{2}\right)=\frac{1}{c} \int_{t_{1}}^{t_{2}} d s=\int_{t_{1}}^{t_{2}} d t \sqrt{1-v^{2}(t) / c^{2}},$$
if the time $t_{1}$ to $t_{2}$ has elapsed at the clock of an observer at rest.
From (2.12) and (2.13) it follows that the proper time interval of a moving object is always smaller than the time between the same events from the viewpoint of an observer at rest. It is the so-called Lorentzian time slowing-down; it leads, in particular, to the famous twin paradox. If one of the twins is at rest (or moves slowly) in a certain IRF while the other travels with relativistic velocities and, having completed his closed trajectory, meets his brother, then, at their meeting, their ages will be different: the traveler will be younger than the home-sitter. It could seem that if one considers the situation in the RF where the traveler is at rest, the result should be the opposite. However, a careful analysis, taking into account the fact that the traveler must have changed his IRF at least three times (at acceleration, at turning back and at final deceleration) shows that his calculated age will be smaller than that of his brother. The paradox is explained by the asymmetry of the situation: the integral (2.13) turns out to be smaller for an object (or subject in the present case) that has carried out noninertial motion.

## 物理代写|宇宙学代写cosmology代考|Geometry

s2(1,2)=C2(吨2−吨1)2−(X→2−X→1)2,

ds2=这μνdXμdXν,μ=0,1,2,3,

Minkowski 张量的逆变分量的一部分用于提高和降低向量和张量索引，因此，例如，对于任意向量一个=(一个μ)我们有一个μ=这μν一个ν,一个μ=这μν一个ν. Minkowski 张量定义了一个标量积(一个b)两个任意 4 向量的一个μ和bμ如下：

(一个b)=这μν一个μbν=这μν一个μbν=一个μbμ=一个μbμ.

## 物理代写|宇宙学代写cosmology代考|Coordinate transformations

X′=X−在吨1−在2/C2,是′=是,和′=和,吨′=吨−在X/C21−在2/C2

X′μ=Xμ+一个μ,一个μ= 常量

Xμ=一个一个μ是一个+一个μ,一个一个μ,一个μ= 常量。

ds2=这μν一个一个μ一个bνd是一个d是b

## 物理代写|宇宙学代写cosmology代考|Kinematic effects

d吨′=d吨1−在2/C2

τ(吨1,吨2)=1C∫吨1吨2ds=∫吨1吨2d吨1−在2(吨)/C2,

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