Theory of Relativity

Overview

The Theory of Relativity, developed by Albert Einstein, revolutionized the understanding of space, time, and gravity. It includes two interrelated theories: Special Relativity and General Relativity.

Key Concepts

Special Relativity

Deals with objects moving at constant speed, especially close to the speed of light. Introduces the concepts of time dilation and length contraction.

General Relativity

Explains gravity as the curvature of spacetime caused by mass and energy.

Time Dilation

Time runs slower for objects moving close to the speed of light relative to an observer.

Mass-Energy Equivalence

Energy and mass are interchangeable, as expressed in Einstein’s famous equation.

Important Equations

E = mc²

Mass-energy equivalence. Energy (E) equals mass (m) times the speed of light (c) squared.

t′ = t / √(1 - v²/c²)

Time Dilation: Time measured by a moving observer appears longer.

L′ = L√(1 - v²/c²)

Length Contraction: Length appears shorter in the direction of motion.

F = G(m₁m₂)/r² (Modified in GR)

Newton’s Law is modified in General Relativity to include spacetime curvature.

Applications

GPS Systems: Require time correction due to relativistic effects.
Particle Accelerators: Use relativistic mass and energy calculations.
Astrophysics: Explains black holes, gravitational lensing, and the expansion of the universe.
Nuclear Energy: Based on mass-energy equivalence (E=mc²).

Example Problems

1. A spaceship travels at 0.8c. How much time passes for the astronauts if 10 years pass on Earth?

Solution: t′ = t / √(1 - v²/c²) = 10 / √(1 - 0.64) = 10 / √0.36 = 10 / 0.6 = 16.67 years

2. Find the energy equivalent of 2 kg of mass.

Solution: E = mc² = 2 × (3×10⁸)² = 1.8 × 10¹⁷ J

3. A rod moving at 0.6c has a rest length of 1 m. What is its length to an observer?

Solution: L′ = L√(1 - v²/c²) = 1 × √(1 - 0.36) = √0.64 = 0.8 m