---
# System prepended metadata

title: Relativity
tags: [modern, mechanics]

---

# Relativity
## Mass–energy equivalence
$c \simeq 3\times 10^8m/s$
$E$ : energy(J)
$M$ : mass(kg)
$$E = mc^2$$
## Relative length
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<img src="https://hackmd.io/_uploads/HySvtZlEeg.png" alt="image"width="300">
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$$l' = l\sqrt{1-\frac{v^2}{c^2}}$$
## Relative mass
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/SkgdKZe4ll.png" alt="image"width="300">
</div>

$$m' = \frac{m_0}{\sqrt{1-\frac{v2}{c2}}}$$
## Relative time
$$\Delta t' = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$
## length in relativity
The position is not equal to the original position plus time multiplied by time
<div style="text-align: center;">

$$\vec u' = \vec u-\vec v$$

$$\vec c' = \vec c-\vec v$$


~~u2 = u1 - v~~

c2 = c1 - ~~v~~

$$x' = x-vt$$

$$t' = t$$

~~x2 = x1 - vt~~

~~t2 = t1~~
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<img src="https://hackmd.io/_uploads/HyNNc-eEge.png" alt="image"width="300">
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$$i = \sqrt{-1}$$

$$x = ct \qquad x' = ct'$$

$$x^2+(cti)^2 = 0$$

$$x'^2+(ct'i)^2 = 0$$
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/HyNRtZgVgx.png" alt="image"width="300">
</div>

$$l = \sqrt{x^2+y^2}$$

$$tan\theta = \frac{x}{y}$$

$$l\cdot sin\theta = y$$

$$l\cdot cos\theta = x$$

$$p(x,y) = (l\cdot cos\theta,l\cdot sin\theta)$$
![image](https://hackmd.io/_uploads/HyyZc-gEee.png)

$$x^2+y^2 = x'^2+y'^2$$

$$x' = l\cdot cos(\theta-\phi)$$

$$= l\cdot sin\theta \cdot cos\phi-l\cdot cos\theta\cdot sin\phi$$

$$= x\cdot cos\phi+y\cdot sin\phi$$

$$y' = l\cdot sin(\theta-\phi)$$

$$= l\cdot cos\theta\cdot cos\phi-l\cdot sin\theta\cdot sin\phi$$

$$= x\cdot sin\phi+y\cdot sin\phi$$
***
$$y = ict$$

$$y' = ict'$$

$$x' =  x\cdot cos\phi+ict\cdot sin\phi$$

$$ict' = x\cdot sin\phi+ict\cdot sin\phi$$

$$t' = x\frac{sin\phi}{ic}+t\cdot cos\phi$$

$$x\frac{sin\phi}{i}+ct\cdot cos\phi \not= x\cdot cos\phi+ict\cdot sin\phi$$

$$ct' \not= x'$$
<div style="text-align: center;">
<img src="https://hackmd.io/_uploads/SJk7qWgVxx.png" alt="image"width="300">
</div>

if $x = Vt$ , $0 = V0\cdot cos\phi+ic0\cdot sin\phi$

$$-v\cdot cos\phi = ic\cdot sin\phi$$

$$-v = ic\cdot tan\phi$$

$$tan\phi = \frac{-v}{ic}$$

$$1-tan^2\phi = sec^2\phi$$

$$sec\phi = \sqrt{1-tan^2\phi}$$

$$x'^2+(vt)^2 = (x-vt)^2$$

$$x' = \frac{x-vt}{\sqrt{1-tan^2\phi}}$$

:::success
$$x-vt = x'\sqrt{1-(\frac{v}{c})^2}$$
:::

## Light cone and gravity
<div style="text-align: center;">

![Light cone](https://upload.wikimedia.org/wikipedia/commons/thumb/1/16/World_line.svg/450px-World_line.svg.png)

![Light cone and gravity](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Light_cones_near_black_hole.svg/300px-Light_cones_near_black_hole.svg.png)
</div>