Relativity - ch. 37

ucla | PHYSICS 1C | 2023-03-18T20:58

---

Table of Contents

• Definitions
• Big Ideas
  • Einstein’s Postulates and Galilean Transforms
    • 1st Postulate (Principle of Relativity)
    • 2nd Postulate (Speed of Light in Vacuum)
    • Galilean Transformation
  • Relativity of Time Intervals
  • Relativity of Length
  • Lorentz Transformations
    • Lorentz Velocity Transformation
  • Doppler Effect on EM Waves
  • Newtonian Mechanics in Relativity
    • Relativistic Momentum
    • 2nd Law
    • Work and Energy
• Resources

Definitions

---

Big Ideas

---

Einstein’s Postulates and Galilean Transforms

1st Postulate (Principle of Relativity)

• Laws of physics are the same in every inertial reference frame

2nd Postulate (Speed of Light in Vacuum)

• Speed of light in a vacuum is the sam in all inertial frames of reference and independent of the motion of the source

Galilean Transformation

• a coordinate transformation between 2 inertial frames

  • Galilean transformations are a set of equations that relate the coordinates of an event as measured by two observers in relative motion. They assume that time and space are absolute and do not depend on the observer. These transformations were the basis of classical mechanics until the discovery of special relativity.

• e.g. two frames $S,S^{\prime }$ where $S^{\prime }$ is moving at velocity $u\hat{i}$, then coordinate for any object in the frame transform to:

$x=x^{\prime }+uty,z=y^{\prime },z^{\prime }⟹v_x=v_x^{\prime }+u$

• Implies $c=c^{\prime }+u$ but this violates Einstein’s 2nd postulate: $c=c^{\prime }$
• Thus the Galilean transformation $v_x=v_x^{\prime }+u$ cannot be correct

Relativity of Time Intervals

$\mathrm{\Delta }{t}_0=\frac{2d}{c}$

• time intervals in different frames are different (time dilation)

$\mathrm{\Delta }t=\frac{\mathrm{\Delta }{t}_0}{\sqrt{1-u^2/c^2}}⟹\mathrm{\Delta }t>\mathrm{\Delta }{t}_0$

• The proper time $\mathrm{\Delta }{t}_0$ describes time interval of 2 events that occur at the same point, thus transformations are given by the Lorentz factor

$\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0⟹\gamma =\frac{1}{\sqrt{1-u^2/c^2}}$

• This is known as time dilation - clocks moving relative to an observer run slow

Relativity of Length

$\mathrm{\Delta }{t}_0=\frac{2l_0}{c}$

$\mathrm{\Delta }t=\frac{2l_0}{c\sqrt{1-u^2/c^2}}{l}_0:\text{proper length}$

• Length Contraction

$l=l_0\sqrt{1-u^2/c^2}=\frac{l_0}{\gamma }$

Lorentz Transformations

• Galilean transforms work as speeds approach 0, while Lorentz transforms are general

$$\begin{gathered} x=ut+\frac{x^{\prime }}{\gamma }=ut+x^{\prime }\sqrt{1-u^2/c^2} \\ implies{x}^{\prime }=\frac{x-ut}{\sqrt{1-u^2/c^2}}=-u{t}^{\prime }+x\sqrt{1-u^2/c^2} \end{gathered}$$

$t^{\prime }=\gamma (t-ux/c^2)=\frac{t-xu/c^2}{\sqrt{1-u^2/c^2}}$

Lorentz Velocity Transformation

$v_x^{\prime }=\frac{v_x-u}{1-u{v}_x/c^2}{v}_x=\frac{v_x^{\prime }+u}{1+u{v}_x^{\prime }/c^2}$

Doppler Effect on EM Waves

• observed frequency of a moving object is greater than emitted: $f>f_0$

$f=f_0\sqrt{\frac{c+u}{c-u}}\quad \text{(approaching)}$

$T=\frac{c{T}_0}{\sqrt{c^2-u^2}}⟹f=\frac{c}{(c-u)T}\mathrm{\&}\lambda =(c-u)T$

Newtonian Mechanics in Relativity

Relativistic Momentum

$\overrightarrow{p}=\frac{m\overrightarrow{v}}{\sqrt{1-v^2/c^2}}=\gamma m\overrightarrow{v}$

2nd Law

• $\overrightarrow{v}\parallel \overrightarrow{F}$

$\overrightarrow{F}=\frac{d}{dt}(\gamma m\overrightarrow{v})⟹F={\gamma }^{3}ma$

• $\overrightarrow{v}\perp \overrightarrow{F}$

$F=\gamma ma$

• General

$\overrightarrow{F}=\gamma m\frac{d\overrightarrow{v}}{dt}+\frac{d\gamma }{dt}m\overrightarrow{v}=\gamma m[\overrightarrow{a}+\frac{{\gamma }^2}{c^2}\overrightarrow{v}(\overrightarrow{v}\cdot \overrightarrow{a})]$

Work and Energy

$W={\int }_{x_1}^{x_2}\frac{ma}{(1-v_x^2/c^2{)}^{3/2}}dx$

$K=(\gamma -1)m{c}^2$

$E=K+mc^2=\gamma mc^2\space$

Resources

---

📌

**SUMMARY
**