Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Relativity → Special relativity ↓


Time expansion and length contraction


Time dilation and length contraction are two fascinating and interconnected concepts from Einstein's special theory of relativity, a theory that revolutionized our understanding of space, time, and motion. Let's take a deeper look at these concepts using simple language, examples, and some basic mathematics.

Basic concepts of special relativity

Einstein's theory of special relativity is based on two principles:

  1. The laws of physics are the same in all inertial reference frames.
  2. The speed of light in a vacuum is constant and the same for all observers, regardless of the speed of the light source or the observer.

Now, let's explore time dilation and length contraction, which arise directly from these principles.

Time extension

Time dilation refers to the effect of time passing at different rates for observers in different inertial frames. It occurs when we compare time intervals measured by observers in relative motion.

Consider a simple thought experiment with a light clock: Imagine a clock with two mirrors facing each other, and a beam of light colliding between them.

Suppose this light clock is stopped in the reference frame of an observer (let's call this observer Alice), then the distance traveled by the light is simply equal to the distance between the mirrors multiplied by 2.

t0 = 2L/c

In this formula:

  • t0 is the proper time, the time measured by Alice.
  • L is the distance between the mirrors.
  • c is the speed of light.

Now, consider another observer (Bob) moving relative to the light clock. From Bob's perspective, the light does not simply travel up and down, but follows a long, diagonal path due to the motion of the clock. This forms a right-angled triangle from Bob's perspective, and the light travels along the hypotenuse.

For Bob, the time taken by the light to travel back and forth is greater and can be obtained using the Pythagorean theorem:

t = 2L'/c

To add the time intervals of Alice and Bob, we use:

t = t0 / sqrt(1 - v^2/c^2)

Here, v is the relative velocity between Alice and Bob. This shows that Bob, by seeing the clock in motion, sees the time interval as longer, so time dilation occurs.

Example

Imagine a spaceship traveling at 90% of the speed of light, v = 0.9c . An observer inside the spaceship measures a time interval of 10 seconds. For a stationary observer on Earth, the time interval becomes:

t = 10 / sqrt(1 - (0.9)^2) = 10 / sqrt(1 - 0.81) = 10 / sqrt(0.19) ≈ 10 / 0.435 ≈ 22.99 seconds

This means that to an observer on Earth, the time interval appears to increase by about 22.99 seconds.

Visual representation

Light clock

The circles represent the mirrors, and the blue and red lines represent the path of light bouncing between them. The diagonal path represents the longer distance seen by the moving observer.

Length contraction

Length contraction is the phenomenon in which a moving object, relative to the observer, appears shorter in the direction of its motion than it would be at rest. This contraction occurs only in the direction of motion.

To measure length contraction, we consider an object with length L0 that is stationary in its own frame. For an observer moving at velocity v relative to this object, the length L becomes:

L = L0 * sqrt(1 - v^2/c^2)

Example

Imagine that the length of a stationary rod is 5 m. If it moves at a speed of 80% of the speed of light, then its length for a stationary observer becomes:

L = 5 * sqrt(1 - (0.8)^2) = 5 * sqrt(1 - 0.64) = 5 * sqrt(0.36) = 5 * 0.6 = 3 meters

From the observer's point of view this rod appears to be only 3 m long.

Visual depictions

length of rest Length in motion

The top line represents the original length of the rod at rest, while the bottom line represents its compressed length during motion.

Closing thoughts

Time dilation and length contraction reveal the complex and non-intuitive nature of space and time when dealing with high speeds close to the speed of light. These effects, though negligible at everyday speeds, become significant at velocities approaching that of light. They not only reshape our understanding of time and space but are crucial for explaining various high-speed phenomena observed in our universe.


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Time expansion and length contraction

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