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 ↓


General relativity


General relativity is a fundamental theory in physics formulated by Albert Einstein. It reshaped our understanding of gravity, space, and time. Before Einstein introduced this theory, Isaac Newton's law of universal gravitation was the dominant explanation of gravitational forces. While Newton's theory was useful and accurate for many practical purposes, it did not adequately explain some phenomena, such as the precise orbit of Mercury. However, general relativity provides an elegant and profound way to understand the universe.

Space, time, and gravity

One of the most important concepts introduced by general relativity is the idea that space and time are interconnected in a four-dimensional continuum known as spacetime. According to Einstein, gravity is not simply a force acting at a distance, as Newton proposed. Instead, it is the effect of massive objects such as stars and planets that distort the fabric of spacetime.

Imagine spacetime as a flexible rubber sheet. When you place a heavy ball on this sheet, it creates a dent. Small balls placed near the heavy ball will roll toward the ball, simulating the effects of gravity. This visual analogy helps explain how general relativity describes gravity as the curvature of spacetime due to mass and energy.

Sun Earth Mars planet

In this view, the large gray circle represents a massive object, such as the Sun, and it distorts the grid or "sheet" of spacetime around it. The smaller circles, representing Earth and Mars, move along the curves of spacetime, which makes it seem to us that these objects are being attracted to the Sun.

Einstein field equations

The essence of general relativity is contained in the Einstein field equations (EFE). These are a set of ten interrelated partial differential equations that describe how matter and energy in the universe affect the curvature of spacetime. Here is a simplified version of the primary equation:

Gμν = 8πTμν

In this equation, Gμν denotes the Einstein tensor, which contains the curvature of spacetime, while Tμν is the stress–energy tensor that contains the distribution of mass–energy in spacetime. The constant 8π comes from the proportionality factor and is related to Newton's gravitational constant.

Solving these equations for different scenarios helps physicists predict how objects will move within a gravitational field. One of the most famous solutions to the EFE is the Schwarzschild solution, which describes spacetime around a spherical, non-rotating massive object such as a star or planet.

Example: orbit of Mercury

Newtonian physics could not accurately predict Mercury's orbit. Observations showed that Mercury's orbit is much further away than Newton's laws would allow. General relativity explains this by showing how space around the Sun is curved more strongly due to its mass, which affects Mercury's path.

This is not just a refinement of Newton's laws, but it shows how mass can affect spacetime more strongly in regions of high curvature, such as near massive stars or black holes.

Black holes and singularities

One of the extraordinary predictions of general relativity is the existence of black holes. A black hole is a region in space where the gravitational pull is so strong that nothing, not even light, can escape its grasp. It occurs when a massive star collapses due to its own gravity at the end of its lifecycle.

At the center of a black hole is a singularity, a point where the curvature of spacetime becomes infinite, and the laws of physics, as we know them, break down. The boundary around a black hole is called the event horizon. Once an object crosses this boundary, it cannot escape.

Example: Exploring mass and radius

To understand how dense an object must be to form a black hole, consider a simple example. If the mass of the Earth could be compressed into a sphere with a radius of just 9 millimeters, it would form a black hole. Such is the strength of the gravitational force interacting with space-time curvature.

Gravitational waves

Another important and verified prediction of general relativity is the existence of gravitational waves. These are ripples in spacetime caused by the acceleration of massive bodies, such as when two black holes orbit each other and eventually merge. Imagine throwing a stone into a pond: the ripples propagating from the point of impact correspond to gravitational waves propagating through spacetime.

Scientists have recently confirmed the existence of gravitational waves with instruments like LIGO and Virgo. This important discovery opens up new possibilities to understand cosmic phenomena like black hole mergers and collisions of neutron stars.

Gravitational wave

The blue wave represents gravitational waves generated by a cosmic event, such as the collision of two stars, which propagate throughout the universe.

Conclusion

General relativity revolutionized the world of physics by providing a deeper understanding of the interrelationship of space, time, and gravity. Through its principles, we can explain complex phenomena such as the bending of light around massive objects, the inexorable pull of black holes, and the oscillations of gravitational waves rippling through spacetime.

As you explore the universe, general relativity serves as a fundamental pillar in the toolkit of modern physics, opening doors to uncovering the mysteries hidden in the cosmos. By conceptualizing gravity not just as a force but as a property of the geometry of spacetime, it provides a coherent framework for exploring the universe at the largest and smallest scales.


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General relativity

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