Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate ↓


General relativity and cosmology


General relativity and cosmology are fascinating areas of physics that explore the nature of gravity and the structure of the universe. This field of study provides profound insight into how massive objects, such as stars and galaxies, affect the fabric of space and time.

Introduction to general relativity

General relativity (GR) was developed by Albert Einstein in 1915. It is a theory of gravity that has reshaped our understanding of gravity, moving away from the Newtonian idea of gravity as a force between masses. Instead, GR describes gravity as a curvature of spacetime caused by mass and energy.

Representation of the bending of spacetime around a mass Mass object

In the visualization above, imagine that a heavy object is placed on a stretched rubber sheet. The object will bend the sheet by stretching it. If you roll a marble on the sheet, it will travel along the curve or path created by the heavy object. Similarly, massive bodies such as planets and stars curve spacetime. Objects moving through this curved spacetime experience this curvature as gravity.

Einstein field equations

At the core of general relativity are the Einstein field equations. These equations relate the geometry of spacetime to the distribution of matter within it. The equations are represented as:

G μν = 8πGT μν

Here, G μν is the Einstein tensor, which describes the curvature of spacetime, and T μν is the stress–energy tensor, which represents the distribution of matter and energy. G is the gravitational constant.

Understanding cosmology

Cosmology is the scientific study of large-scale aspects of the universe, including its origin, its structure, its dynamical behavior, and its ultimate fate. It is the astronomical study of the universe taken as a whole.

The Big Bang theory

The most widely accepted theory about the origin of the universe is the Big Bang Theory. According to this theory, the universe began as an incredibly small, hot and dense point about 13.8 billion years ago. Since then, it has expanded into the vast universe we see today.

Timeline of the universe from the Big Bang Big Bang Creation of elements The first galaxies nature of the solar system Present Day

The visual illustration shows a simplified timeline of the universe from the Big Bang to the present universe. Immediately after the Big Bang, the universe expanded rapidly in a process known as inflation. Then it cooled enough for protons and neutrons to form, and eventually lighter elements such as hydrogen and helium were formed.

Cosmic microwave background radiation

The cosmic microwave background (CMB) radiation is a key piece of evidence supporting the Big Bang theory. It is the thermal remnant from the early hot stages of the universe, which has now dispersed into microwaves due to the expansion of the universe.

Geometry of the universe

The universe can be open, closed or flat, depending on its density and the nature of dark energy. In a closed universe, the geometry resembles that of a sphere, so if you travel in a straight line, you'll eventually come back to your starting point. An open universe resembles a saddle shape, where parallel lines eventually diverge. A flat universe is exactly balanced, with parallel lines staying parallel.

Different geometries of the universe open Closed even

Implications of general relativity in cosmology

General relativity has profound implications in cosmology. It allows us to predict the behavior of the universe over large intervals of time and space. For example:

  • Black holes: Areas whose gravitational force is so strong that nothing, not even light, can escape them.
  • Expansion of the universe: Observations of distant galaxies show that the universe is expanding, a prediction that Albert Einstein initially found surprising.
  • Gravitational waves: Ripples in spacetime caused by sudden changes in massive objects, confirmed by direct detection in 2015.

Black holes

A black hole is a point in space where gravity pulls so strong that even light cannot escape. Gravity is so strong because matter is packed into a tiny space. These objects were one of the earliest predictions of general relativity. The boundary around a black hole is called the event horizon, beyond which events cannot affect an outside observer.

Illustration of a black hole event horizon

Gravitational waves

Gravitational waves are ripples in the framework of space-time that are caused by some of the most violent and energetic processes in the universe, such as colliding black holes. They were directly detected by the LIGO observatory in 2015, further confirming Einstein's theory of general relativity.

Conclusion

The study of general relativity and cosmology has revolutionised our understanding of the universe. What once seemed to be a simple force of attraction, gravity, is now understood as a complex interaction between mass and the geometry of spacetime. This understanding not only enriches our understanding of the physical universe, but also challenges us to push the boundaries of knowledge even further.

As we continue to explore these theories, we move closer to resolving fundamental questions about the origin and ultimate fate of our universe.


Graduate → 5


U
username
0%
completed in Graduate

← Prev (4.4.5)
Spontaneous symmetry breaking and the Higgs mechanism
Next (5.1) →
Tensor calculus and differential geometry

Comments 



General relativity and cosmology

https://www.physicsflow.com/g/5?pfal=en