Special Relativity Fundamentals: Master Notes

Chapter Overview

Special Relativity (SR) revises space, time, and momentum kinematics at speeds approaching the speed of light $c$. In Physics Olympiads (INPhO / IPhO) and advanced competitive examinations, top questions test Time Dilation & Length Contraction, Relativistic Velocity Addition, Energy-Momentum Invariant ($E^2=p^2{c}^2+m_0^2{c}^4$), and Cosmic-ray atmospheric muon decay.

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1. Postulates of Special Relativity & Lorentz Factor ($\gamma$)

1. The Principle of Relativity: The laws of physics take the identical form in all inertial reference frames.
2. Constancy of the Speed of Light: The speed of light in vacuum ($c\approx 3\times {10}^8\text{ m/s}$) is strictly constant for all observers regardless of the motion of the source or the observer.

$$\boxed{{\displaystyle \beta =\frac{v}{c},\gamma =\frac{1}{\sqrt{1-{\beta }^2}}=\frac{1}{\sqrt{1-v^2/c^2}}\ge 1}}$$

Multi-Mode DiagramSpecial Relativity: Spacetime Lightcone & Lorentz Transformations

Option 1: Publication-Grade Scientific Vector SVG

Minkowski spacetime diagram with lightcone x = +-ct, worldlines, and relativistic gamma factor.

Lorentz Factor: $\gamma = \frac{1}{\sqrt{1 - \beta^2}} \quad (\beta = v/c)$

Time Dilation: $\Delta t = \gamma \Delta t_0$

Option 2: Interactive 60 FPS Canvas Physics Engine

Simulate relativistic gamma factor divergence as v -> c.

Speed ratio β (v/c): 0.8

Gamma Factor ($\gamma$):1.67

Option 3: Interactive 3D Spatial Viewport

3D Minkowski future and past lightcone.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Spacetime Invariant:$s^2 = (c\Delta t)^2 - (\Delta x)^2$

Option 5: Reactive Parametric Physics with Live Calculus

Lorentz contraction and time dilation calculator.

Speed β (v/c): 0.8

: 45

Gamma ($\gamma$):$1.67$

Active Mode:Option 1 (Print & Derivation Vector SVG)Switch modes using the tabs above to explore different visualizations.

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2. Space-Time Kinematics

2.1 Time Dilation (Moving Clocks Run Slow)

$$\boxed{{\displaystyle \mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0=\frac{\mathrm{\Delta }{t}_0}{\sqrt{1-v^2/c^2}}}}$$

(where $\mathrm{\Delta }{t}_0$ is the proper time measured in the frame where the clock is at rest).

2.2 Length Contraction (Moving Rods Measure Shorter)

$$\boxed{{\displaystyle L=\frac{L_0}{\gamma }=L_0\sqrt{1-v^2/c^2}}}$$

(where $L_0$ is the proper length measured in the rod's rest frame. Contraction occurs only along the direction of motion!)

2.3 Relativistic Velocity Addition in 1D

If object $A$ moves with velocity $u^{\prime }$ inside a vehicle moving with velocity $v$ relative to the ground:

$$\boxed{{\displaystyle u=\frac{u^{\prime }+v}{1+\frac{u^{\prime }v}{c^2}}}}$$

(Notice: If $u^{\prime }=c$, then $u=\frac{c+v}{1+cv/c^2}=\frac{c(1+v/c)}{1+v/c}=c$. The speed of light cannot be exceeded!)

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3. Relativistic Dynamics & Energy-Momentum Invariant

$$\overrightarrow{p}=\gamma m_0\overrightarrow{v}=\frac{m_0\overrightarrow{v}}{\sqrt{1-v^2/c^2}}$$$$\text{Total Relativistic Energy: }E=\gamma m_0{c}^2=K+m_0{c}^2$$$$\text{Rest Mass Energy: }{E}_0=m_0{c}^2$$$$\text{Relativistic Kinetic Energy: }K=(\gamma -1)m_0{c}^2$$

::: theorem 🌟 The Energy-Momentum Invariant Triangle

$$\boxed{{\displaystyle E^2=(pc{)}^2+(m_0{c}^2{)}^2}}$$

• For massless particles ($m_0=0$, such as photons): $\boxed{{\displaystyle E=pc⟹p=\frac{E}{c}}}$. :::

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4. Authentic Olympiad & Advanced PYQs

PYQ 1: INPhO Olympiad — Atmospheric Muon Decay Paradox

Question:
Cosmic ray muons are created at an altitude of $h=10\text{ km}$ above Earth's surface and travel downward at speed $v=0.995c$ ($\gamma =10$). The proper lifetime of a muon is ${\tau }_0=2.2\mu \text{s}$.

1. In the Earth frame, find the lifetime ${\tau }_{\text{Earth}}$ of the muon and the distance it travels before decaying.
2. In the muon's rest frame, explain how the muon reaches the ground despite living only $2.2\mu \text{s}$.

Step-by-Step Solution:

• 1. In the Earth Frame (Time Dilation):

  $${\tau }_{\text{Earth}}=\gamma {\tau }_0=10\times 2.2\mu \text{s}=22.0\mu \text{s}$$

  Distance traveled before decay:

  $$d=v{\tau }_{\text{Earth}}\approx c{\tau }_{\text{Earth}}=(3\times {10}^8\text{ m/s})\times (22\times {10}^{-6}\text{ s})=\boxed{{\displaystyle 6.6\text{ km}}}$$

  (Because of time dilation, a large fraction survive the 10 km journey to ground level!)

• 2. In the Muon Frame (Length Contraction): In the muon's own frame, the clock ticks normally (${\tau }_0=2.2\mu \text{s}$). However, the 10 km atmosphere is rushing past at $0.995c$, so the atmosphere's height contracts:

  $$h_{\text{contracted}}=\frac{h_0}{\gamma }=\frac{10\text{ km}}{10}=\boxed{{\displaystyle 1.0\text{ km}}}$$

  Distance atmosphere moves during muon lifetime:

  $$d^{\prime }=v{\tau }_0\approx (3\times {10}^8)\times (2.2\times {10}^{-6})=0.66\text{ km}$$

  (Both observers agree on the physical reality that muons reach the ground!)

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PYQ 2: Advanced Olympiad Drill — Speed at Double Rest Mass Energy

Question:
Find the speed $v$ of a particle whose kinetic energy $K$ is equal to twice its rest energy ($K=2m_0{c}^2$).

Step-by-Step Solution:

1. Total Energy:$$E=K+m_0{c}^2=2m_0{c}^2+m_0{c}^2=3m_0{c}^2$$
2. Equating with $E=\gamma m_0{c}^2$:$$\gamma m_0{c}^2=3m_0{c}^2⟹\gamma =3$$
3. Solving for $v$:$$\frac{1}{\sqrt{1-v^2/c^2}}=3⟹1-\frac{v^2}{c^2}=\frac{1}{9}⟹\frac{v^2}{c^2}=1-\frac{1}{9}=\frac{8}{9}$$$$\boxed{{\displaystyle v=\frac{\sqrt{8}}{3}c=\frac{2\sqrt{2}}{3}c\approx 0.943c}}$$

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5. High-Yield Formula Sheet

Entity	Formula	Notes
Lorentz Factor	$\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$	$\gamma \ge 1$
Time Dilation	$\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0$	Moving clocks run slow
Length Contraction	$L=L_0/\gamma$	Along direction of motion
Relativistic Velocity Addition	$u=\frac{u^{\prime }+v}{1+u^{\prime }v/c^2}$	Speed of light $c$ invariant
Energy Invariant	$E^2=p^2{c}^2+m_0^2{c}^4$	Total vs rest mass energy