Relativity Basics

Einstein’s special relativity, introduced in 1905, revolutionized our understanding of space and time by showing they are not absolute but depend on the relative motion of observers. This theory, based on two postulates— the constancy of the speed of light and the principle of relativity—leads to phenomena like time dilation, length contraction, and mass-energy equivalence. This guide explores the fundamentals of special relativity, covering all basic formulas (time dilation, length contraction, relativistic mass, energy-mass equivalence), detailed examples, and practical applications to illustrate its profound implications for physics and technology.

Time Dilation (with Length Contraction, Relativistic Mass, Energy-Mass Equivalence)

Time dilation describes how time slows for an object moving relative to an observer, derived from the Lorentz factor:

\[ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Where:

  • \( t \): Dilated time (s)
  • \( t_0 \): Proper time (s)
  • \( v \): Velocity (m/s)
  • \( c = 3 \times 10^8 \, \text{m/s} \): Speed of light

Related Formulas:

  • Length Contraction: \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \)
  • Relativistic Mass: \( m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \)
  • Energy-Mass Equivalence: \( E = mc^2 \)

Examples

Example 1: Time Dilation

A spaceship travels at \( v = 0.8c \) for \( t_0 = 10 \, \text{s} \):

\[ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] \[ = \frac{10}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}} \] \[ = \frac{10}{\sqrt{1 - 0.64}} \] \[ = \frac{10}{\sqrt{0.36}} \] \[ = \frac{10}{0.6} \] \[ \approx 16.67 \, \text{s} \]

Example 2: Length Contraction

A spaceship of proper length \( L_0 = 100 \, \text{m} \) moves at \( v = 0.6c \):

\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] \[ = 100 \sqrt{1 - \frac{(0.6c)^2}{c^2}} \] \[ = 100 \sqrt{1 - 0.36} \] \[ = 100 \sqrt{0.64} \] \[ = 100 \cdot 0.8 \] \[ = 80 \, \text{m} \]

Example 3: Relativistic Mass

Mass of an object with rest mass \( m_0 = 2 \, \text{kg} \) at \( v = 0.9c \):

\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] \[ = \frac{2}{\sqrt{1 - \frac{(0.9c)^2}{c^2}}} \] \[ = \frac{2}{\sqrt{1 - 0.81}} \] \[ = \frac{2}{\sqrt{0.19}} \] \[ \approx \frac{2}{0.436} \] \[ \approx 4.59 \, \text{kg} \]

Example 4: Energy-Mass Equivalence

Energy equivalent of a mass \( m = 1 \, \text{kg} \) (at rest):

\[ E = mc^2 \] \[ = 1 \cdot (3 \times 10^8)^2 \] \[ = 1 \cdot 9 \times 10^{16} \] \[ = 9 \times 10^{16} \, \text{J} \]

Applications

Relativity has practical and theoretical significance in modern science and technology. Below are examples with calculations:

Example 1: GPS Time Correction

A satellite moves at \( v = 3.87 \times 10^3 \, \text{m/s} \) (approx. 0.0129c). Time dilation for \( t_0 = 1 \, \text{s} \):

\[ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] \[ = \frac{1}{\sqrt{1 - \frac{(3.87 \times 10^3)^2}{(3 \times 10^8)^2}}} \] \[ = \frac{1}{\sqrt{1 - \frac{1.497 \times 10^7}{9 \times 10^{16}}}} \] \[ \approx \frac{1}{\sqrt{1 - 1.663 \times 10^{-10}}} \] \[ \approx \frac{1}{1 - 8.315 \times 10^{-11}} \] \[ \approx 1 + 8.315 \times 10^{-11} \, \text{s} \] \[ \approx 1.00000000008315 \, \text{s} \]

Example 2: Particle Accelerator (Relativistic Mass)

A proton (\( m_0 = 1.67 \times 10^{-27} \, \text{kg} \)) at \( v = 0.99c \):

\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] \[ = \frac{1.67 \times 10^{-27}}{\sqrt{1 - \frac{(0.99c)^2}{c^2}}} \] \[ = \frac{1.67 \times 10^{-27}}{\sqrt{1 - 0.9801}} \] \[ = \frac{1.67 \times 10^{-27}}{\sqrt{0.0199}} \] \[ \approx \frac{1.67 \times 10^{-27}}{0.141} \] \[ \approx 1.18 \times 10^{-26} \, \text{kg} \]

Example 3: Length Contraction in Space Travel

A spacecraft \( L_0 = 50 \, \text{m} \) moves at \( v = 0.7c \):

\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] \[ = 50 \sqrt{1 - \frac{(0.7c)^2}{c^2}} \] \[ = 50 \sqrt{1 - 0.49} \] \[ = 50 \sqrt{0.51} \] \[ \approx 50 \cdot 0.714 \] \[ \approx 35.7 \, \text{m} \]

Example 4: Nuclear Energy

Energy from mass loss of \( \Delta m = 1 \, \text{g} = 0.001 \, \text{kg} \):

\[ E = \Delta m c^2 \] \[ = 0.001 \cdot (3 \times 10^8)^2 \] \[ = 0.001 \cdot 9 \times 10^{16} \] \[ = 9 \times 10^{13} \, \text{J} \]

Example 5: Muon Decay (Time Dilation)

A muon with proper lifetime \( t_0 = 2.2 \times 10^{-6} \, \text{s} \) moves at \( v = 0.98c \):

\[ t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}} \] \[ = \frac{2.2 \times 10^{-6}}{\sqrt{1 - \frac{(0.98c)^2}{c^2}}} \] \[ = \frac{2.2 \times 10^{-6}}{\sqrt{1 - 0.9604}} \] \[ = \frac{2.2 \times 10^{-6}}{\sqrt{0.0396}} \] \[ \approx \frac{2.2 \times 10^{-6}}{0.199} \] \[ \approx 1.11 \times 10^{-5} \, \text{s} \]

Example 6: Relativistic Energy

Energy of a particle with \( m_0 = 1 \times 10^{-27} \, \text{kg} \) at \( v = 0.95c \):

\[ E = \frac{m_0 c^2}{\sqrt{1 - \frac{v^2}{c^2}}} \] \[ = \frac{(1 \times 10^{-27}) (3 \times 10^8)^2}{\sqrt{1 - \frac{(0.95c)^2}{c^2}}} \] \[ = \frac{9 \times 10^{-11}}{\sqrt{1 - 0.9025}} \] \[ = \frac{9 \times 10^{-11}}{\sqrt{0.0975}} \] \[ \approx \frac{9 \times 10^{-11}}{0.312} \] \[ \approx 2.88 \times 10^{-10} \, \text{J} \]