## General Relativity

### Chapter 1

#### 1.1Two fundamental postulates in SR

- Postulates
- (Galileo)
- (Einstein)

- Galileo velocity transformation $\vec v\to \vec v'=\vec v-\vec V$, it's embodied in Newton's second law, $\vec a$ is invariant, so $\vec F$ is invariant so Newton's third law still hold.

- There's no absolute velocity but absolute acceleration in SR, example: when the train accelerates the water surface will tilt.

#### 1.2 Definition of an inertial observer in SR

- inertial
- The distance between two points are independent of time
- the clocks at everypoint are synchronized and run at the same speed
- The geometry of space at any constant time is Euclidean

#### 1.3 New units

- one meter of time is the time it takes light to travel one year

$\begin{aligned} c &=\frac{\text { distance light travels in any given time interval }}{\text { the given time interval }} \\ &=\frac{1 \mathrm{m}}{\text { the time it takes light to travel one meter }} \\ &=\frac{1 \mathrm{m}}{1 \mathrm{m}}=1 \end{aligned}$

- From SI to New units

$\begin{aligned} 3 \times 10^{8} \mathrm{ms}^{-1} &=1 \\ 1 \mathrm{s} &=3 \times 10^{8} \mathrm{m} \\ 1 \mathrm{m} &=\frac{1}{3 \times 10^{8}} \mathrm{s} \end{aligned}$

#### 1.4&1.5Space time diagrams

- step 1 find t axis

- step 2 find x axis

- Step 3 calibrate with hyperbolae function

#### 1.6 Invariance of the interval

$\Delta s^{2}=-(\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}$

<font color="red">**proof : Not understood yet**</font>

- two events which are simultaneous in one frame are simultaneous in any frame
moving in a direction perpendicular to their separation relative to the first frame

#### 1.7 Invariant hyperbolae

- $-t^{2}+x^{2}=a^{2}$ used to calibrate x-axis

- $-t^{2}+x^{2}=-b^{2}$ used to calibrate t-axis

- tangent of hyperbolae:

tangent line corresponds to $t=const$ in another reference system that under proper Lorentz transformation the origin coincides with the point of tangency

#### 1.8 Particularly important results

- Time dilation

$(\Delta t)_{\text { measured in } \mathcal{O}}=\frac{(\Delta \overline{t})_{\text { measured in } \overline{\mathcal{O}}}}{\sqrt{( 1-v^{2} )}}$

proof **proper time** between R and the origin is defined as the time ticked off by a clock from O to R in O, Choose $\bar O$ so that the clock is at rest in $\bar O$, so in $\bar O$, $\bar x=\bar y=\bar z=0$, $\Delta \tau=\Delta \bar t$, $\Delta s^{2}=-\Delta \overline{t}^{2}=-\Delta \tau^{2}$, according to the invariance of interval, $\begin{aligned} \Delta \tau &=\left[(\Delta t)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}\right]^{1 / 2} \\ &=\Delta t \sqrt{( } 1-v^{2} ) \end{aligned}$

- Lorentz contraction , not understood yet

1.9 The Lorentz transformation

$\overline{t}=\frac{t}{\sqrt{( } 1-v^{2} )}-\frac{v x}{\sqrt{\left(1-v^{2}\right)}}$
$\overline{x}=\frac{-v t}{\sqrt{\left(1-v^{2}\right)}}+\frac{x}{\sqrt{\left(1-v^{2}\right)}}$
$\overline{y}=y$
$\overline{z}=z$

<font color="red">**deduction?**</font>

#### 1.10 The velocity-composition law

$\begin{aligned} W^{\prime} &=\frac{\Delta x}{\Delta t}=\frac{(\Delta \overline{x}+v \Delta \overline{t}) /\left(1-v^{2}\right)^{1 / 2}}{(\Delta \overline{t}+v \Delta \overline{x}) /\left(1-v^{2}\right)^{1 / 2}} \\ &=\frac{\Delta \overline{x} / \Delta \overline{t}+v}{1+v \Delta \overline{x} / \Delta \overline{t}}=\frac{W+v}{1+W v} \end{aligned}$