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


Relativity is a fundamental concept in physics that was introduced by Albert Einstein. There are two main theories of relativity: special relativity and general relativity. Together, these theories revolutionized our understanding of space, time, and gravity.

Introduction to special relativity

Special relativity, proposed by Einstein in 1905, addresses the physics of objects moving at constant speeds, particularly those close to the speed of light. This theory is based on two main principles:

  • The laws of physics are the same for all observers in uniform motion.
  • The speed of light remains constant for all observers, regardless of their relative speed.

The concept of synchronicity

An important result of special relativity is the relativity of simultaneity. This refers to the idea that two events that are simultaneous for one observer may not be simultaneous for a second observer moving relative to the first.

A B C D

In this diagram, two observers, one at point A and the other at point B, are at rest relative to each other. They both see the light from events C and D at the same time and declare them to be simultaneous. However, an observer moving along the line may receive the light from C and D at different times due to his motion.

Time extension

Time dilation is another important result of special relativity. It shows that a clock moving relative to an observer moves slower than a clock stationary relative to that observer.

This relationship is expressed by the formula:

t' = t / √(1 - v²/c²)

Here, t' is the time interval measured by a moving observer, t is the proper time interval measured by a stationary observer, v is the relative velocity, and c is the speed of light.

For example, consider astronauts traveling at 99% of the speed of light on a trip lasting 5 years. For those in the Earth's reference frame, many more years will have passed due to this time dilation effect.

Length contraction

Length contraction refers to the phenomenon of an object appearing shorter in the direction of motion relative to the observer's point of view.

The contraction formula is given as:

L' = L * √(1 - v²/c²)

where L' is the compressed length, and L is the proper length. Essentially, to a stationary observer, objects moving at high speeds appear smaller than their actual dimensions.

Introduction to general relativity

General relativity, published by Einstein in 1915, extends special relativity to include acceleration and gravity. It is a comprehensive description of gravity, not as a force but as a curvature of space-time due to mass.

The fabric of space-time

Space-time is represented as a four-dimensional entity, consisting of three dimensions of space and one dimension of time. According to general relativity, massive objects such as the Earth and the Sun distort this space-time structure, creating what we observe as gravity. Small objects move along these curves.

Earth

In this view, space-time is represented by a grid. A massive body such as Earth curves space-time, represented here as a dimple in the grid. Objects such as satellites orbiting the planet follow paths in this curved space, which explains orbital motion without using the force of gravity.

Equivalence principle

The equivalence principle is an important concept in general relativity. It holds that the effects of gravity are no different from the effects of acceleration. For example, if you are in an elevator in space and it accelerates upward, you will experience the same force of gravity.

This principle implies that objects freely falling in a gravitational field experience no force and hence follow the straightest possible path in curved space-time, which is here considered as the absence of gravity.

Gravitational time dilation

Gravitational time dilation highlights that time passes more slowly in strong gravitational fields. A clock located close to a massive object will run slower than a clock located farther away. This effect is negligible for everyday experiences, but becomes significant around massive objects such as stars or black holes.

t = t 0 / √(1 - 2GM/(rc²))

This formula calculates the extended time t experienced at a distance r from the center of a massive object, where G is the gravitational constant, M is the mass of the object, and t 0 is the proper time.

Applications and implications

The theory of relativity has significant implications in a variety of fields. For example, GPS systems must take into account the effects of time dilation due to both the motion of satellites (special relativity) and the Earth's gravitational field (general relativity) in order to maintain accurate position.

Another spectacular implication arises in the study of black holes. General relativity predicts these astronomical phenomena, from within which no matter or light can escape. Their study advances the understanding of extreme conditions in the universe.

Experimental evidence

The validity of relativity has been established through numerous experiments. The famous GPS clock synchronization is one such verification of the time dilation effect. Additionally, observations of the bending of light around the Sun during eclipses support the predictions of general relativity.

The discovery of gravitational waves permeating the framework of space-time, first detected in 2015, adds credibility to the theory of general relativity, as they accurately follow Einstein's predictions about such cosmic phenomena.

Thought experiment

Thought experiments popularized by Einstein are key to understanding relativity. Consider the twin paradox, where a twin traveling close to the speed of light ages at a slower rate than his or her Earth-based sibling, illustrating time dilation.

Similarly, if you shoot a beam of light into a car moving at speed v, both observers inside and outside measure the speed of light as c. However, they disagree on the sequence of events because of their relative velocities.

Concluding remarks

Relativity reshapes our understanding of fundamental concepts such as time, space and gravity. It forces us to understand time as flexible and gravity as a geometric property of space. Though complex, its implications lead to a deeper understanding of the beauty of the universe and the complexity of celestial mechanics.

This beautiful theory remains the pinnacle of scientific achievement, guiding modern physics and inspiring new conversations in fields such as quantum mechanics, further enriching the scientific landscape.


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