Intro to Special Relativity

Reference Frames §

• for event at $(x_0,y_0,z_0,t_0)$
• let $y^{\prime }=y,z^{\prime }=z,t^{\prime }=t,x^{\prime }=x-vt$ for moving frame denoted by ’
• then, we can relate stuff like distances ($r,r^{\prime }$) or forces $F,F^{\prime }$ using $x^{\prime },y^{\prime },t^{\prime }$ aka relative motion
• reality is same for both observers, just relatively transformed

Special Relativity Postulates §

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:0}] Axiom 1. All inertial frames are equally valid - laws of physics are the same, no preferred frame.

[!math|{“type”:“axiom”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:"",“label”:"",“_index”:1}] Axiom 2. Speed of light $c$ is the same in all frames regardless of relative motion of source and observer.

[!math|{“type”:“remark”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:2}] Remark 3. $c$ is the speed at which massless things move - essentially a speed limit of nature. Since we measure events in terms of space and time, $c$ is simply a conversion factor between space and time.

Significance §

• reference frame from Reference Frames is broken!
• light moves at same speed for both observers; thus, relative motion is broken, realities are different
• $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ indicates how impactful relativity is

Implications §

• time dilation
• length contraction
• simultaneity is relative
• intervals: $\Delta r$ and $\Delta t$ are “mixed”
• velocity transformation
• kinematics implications ($E,p,m$)

Time Dilation §

• to the astronaut, the time taken for light to pulse back and forth from the source to the mirror to the receiver is $\frac{2D}{c}$
• to the observer on earth, the light moves at an angle but still at speed $c$; thus, the time taken for light to pulse is $\frac{2s}{c}$
• time is dilated

$$\begin{array}{rl}\Delta t_{\text{between ship’s light clock ticks to earth}}& =\frac{2D}{\sqrt{c^2-V^2}}\end{array}$$ $$t_{\text{moving clock}}=t_{\text{stationary clock}}\sqrt{1-V^2/c^2}$$

Length Contraction §

• to stationary observers earth and alpha centauri, time runs slower on the spaceship according to time dilation
• to the observer on the spaceship, length contraction occurs because $L=VT=V\cdot \frac{T}{\gamma }<L_0$

Horizontal Light Clock §

Imagine a horizontal light clock moving at $v$.

When the light moves forward, clock moves forward:

$$\Delta T_1=\frac{V\Delta T_1+\frac{D}{\gamma }}{c}.$$

When light moves backward,

$$\Delta T_2=\frac{-V\Delta T_2+\frac{D}{\gamma }}{c}.$$

Full trip:

$$T=\frac{2D}{c}\gamma$$

Simultaneity §

If you have two clocks moving together at $v$ and signal is emitted at center, leading clock lags by

$$\Delta t=\frac{vD}{c^2}$$