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# 物理代写|狭义相对论代写Special Relativity代考|PHYS458 Length Contraction

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## 物理代写|狭义相对论代写Special Relativity代考|Length Contraction

(a) Statement: According to this phenomenon, a moving body appears to be contracted along the direction of motion. But the length of body perpendicular to direction of motion remains unchanged. Mathematically

$$l=l_0\left(1-\frac{v^2}{c^2}\right)^{1 / 2}$$
where $1_0$ is proper length of object (measured in proper frame of reference in which there is no relative motion between object and observer or in which object is actually placed).
$l$ is improper length of object (measured in improper frame of reference in which there is relative motion between object and observer or in which object is not actually placed.

(b) Derivation of Expression:
Step I:
Let inertial frame $S^{\prime}$ is moving with velocity v relative to inertial frame $S$.
In derivation of expression for length contraction the events should be simultaneous in improper frame. (In which body is not actually placed)
Step II:
When body is placed in frame $\mathrm{S}$, inverse Lorentz transformations equations are used.

$$\begin{gathered} x_2-x_1=\frac{\left(x_2-x_1\right)+v\left(t_2-t_1\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}} \ t_2=t_1 \ x_2-x_1=l_0 \ x_2-x_1=l \ l_0=\frac{l+0}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}} \ l=l_0\left(1-\frac{v^2}{c^2}\right)^{1 / 2} \end{gathered}$$
This is required expression which represent that 1 is less than $1 \quad 0$ (moving object appears to be contracted along the direction of motion)

## 物理代写|狭义相对论代写Special Relativity代考|Time Dilation

(a) Statement:
According to this phenomenon a moving clock appears to go slow. Mathematically:
$$\tau=\frac{\tau_0}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}}$$
where $\tau_0$ is proper time interval between events measured in proper frame of reference (in which events actually occurs). It is time interval measured by a single clock at one place because both events occur at same position.
$\tau$ is improper time interval between events measured in improper frame of reference. (in which events actually do not occur). It is time interval measured by two different clocks at two different places because both events occur at different positions.

(b) Derivation of Expression:
Let frame $S^{\prime}$ is moving with velocity $v$ relative to frame $S$. In the derivation of expression for time dilation, the event should be co-local relative to frame $\mathrm{S}$. (In proper frame in which there is no relative motion between event and observer or where events actually occurs.)
Step I: When events occurs in frame $S$ direct LT equations are used
$$t_1=\frac{\left(t_1-\frac{v}{c^2} x_1\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}} \text { and } t_2=\frac{\left(t_2-\frac{v}{c^2} x_2\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}}$$
$$\therefore\left(t_2-t_1\right)=\frac{\left(t_2-t_1\right)-\frac{v}{c^2}\left(x_2-x_1\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}}$$

## 物理代写|狭义相对论代写狭义相对论代考|长度收缩

$$l=l_0\left(1-\frac{v^2}{c^2}\right)^{1 / 2}$$

$l$是不适当的对象长度(在不适当的参照系中测量，在该参照系中，对象和观察者之间存在相对运动或对象没有实际放置

(b)表达式的推导:

$$\begin{gathered} x_2-x_1=\frac{\left(x_2-x_1\right)+v\left(t_2-t_1\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}} \ t_2=t_1 \ x_2-x_1=l_0 \ x_2-x_1=l \ l_0=\frac{l+0}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}} \ l=l_0\left(1-\frac{v^2}{c^2}\right)^{1 / 2} \end{gathered}$$

## 物理代写|狭义相对论代写狭义相对论代考|时间膨胀

(a)语句:

$$\tau=\frac{\tau_0}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}}$$
，其中$\tau_0$是在适当参照系中度量的事件之间的固有时间间隔(事件实际发生在该参照系中)。它是在一个地方用一个时钟测量的时间间隔，因为两个事件发生在同一位置。
$\tau$是在不适当的参照系中测量的事件之间的不适当时间间隔。(在这种情况下，事件实际上不会发生)。它是由两个不同地点的两个不同时钟测量的时间间隔，因为两个事件发生在不同的位置

(b)表达式的推导:

$$t_1=\frac{\left(t_1-\frac{v}{c^2} x_1\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}} \text { and } t_2=\frac{\left(t_2-\frac{v}{c^2} x_2\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}}$$
$$\therefore\left(t_2-t_1\right)=\frac{\left(t_2-t_1\right)-\frac{v}{c^2}\left(x_2-x_1\right)}{\left(1-\frac{v^2}{c^2}\right)^{1 / 2}}$$

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