Relativity

Einstein’s special relativity, introduced in 1905, reshapes our understanding of space and time. At MathMultiverse, we explore its core concepts—time dilation, length contraction, relativistic mass, and energy-mass equivalence—through clear formulas, examples, and visualizations.

Key Concepts

Time Dilation

Time slows for a moving object:

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

Length Contraction

Objects contract along the direction of motion:

\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]

Relativistic Mass

Mass increases with speed:

\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Energy-Mass Equivalence

Mass converts to energy:

\[ E = m c^2 \]

Where \( c = 3 \times 10^8 \, \text{m/s} \).

Examples

Time Dilation

Spaceship at \( v = 0.8c \), \( t_0 = 10 \, \text{s} \):

\[ t = \frac{10}{\sqrt{1 - 0.64}} \approx 16.67 \, \text{s} \]

Length Contraction

Spaceship \( L_0 = 100 \, \text{m} \), \(ვ

Relativistic Mass

Object with \( m_0 = 2 \, \text{kg} \), \( v = 0.9c \):

\[ m \approx \frac{2}{\sqrt{0.19}} \approx 4.59 \, \text{kg} \]

Energy-Mass Equivalence

Mass \( m = 1 \, \text{kg} \):

\[ E = 1 \cdot 9 \times 10^{16} = 9 \times 10^{16} \, \text{J} \]

Visualizations

Time Dilation vs. Velocity

Applications

GPS: Time dilation correction, \( \Delta t \approx 8.315 \times 10^{-11} \, \text{s} \) for \( v = 3.87 \times 10^3 \, \text{m/s} \).
Particle Accelerators: Proton mass, \( m \approx 1.18 \times 10^{-26} \, \text{kg} \) at \( v = 0.99c \).
Space Travel: Length contraction, \( L \approx 35.7 \, \text{m} \) for \( L_0 = 50 \, \text{m} \), \( v = 0.7c \).
Nuclear Energy: Energy from \( \Delta m = 0.001 \, \text{kg} \), \( E = 9 \times 10^{13} \, \text{J} \).
Muon Decay: Lifetime, \( t \approx 1.11 \times 10^{-5} \, \text{s} \) at \( v = 0.98c \).