PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → General relativity and gravity ↓


Cosmology


Cosmology is the branch of astrophysics that studies the universe as a whole, including its birth, evolution, and ultimate fate. It is a central focus in the fields of general relativity and gravitation, a theory formulated by Albert Einstein in 1915. General relativity revolutionized our understanding of gravity, describing it not as a force between masses, as Newton proposed, but as a curvature of spacetime due to the presence of mass and energy.

Background of general relativity

Before diving into cosmology, it is essential to understand the fundamentals of general relativity. This theory expanded the concept of space and time into a unified framework known as spacetime. In Einstein's view, massive objects such as stars and planets bend the framework of spacetime, and this curvature affects the paths of objects as they move through the universe, leading to what we experience as gravity.

The central equation of general relativity is Einstein's field equation, which relates the geometry of spacetime to the distribution of mass and energy:

R μν – 1/2 g μν R + g μν Λ = (8πg/c⁴) T μν

In this equation:

  • Rμν is the Ricci curvature tensor, which represents the gravitational effects due to mass–energy.
  • gμν is the metric tensor that describes the geometry of spacetime.
  • R is the Ricci scalar, the trace of the Ricci tensor.
  • Λ is the cosmological constant, which accounts for the energy density of empty space, or "dark energy."
  • G is the gravitational constant.
  • c is the speed of light.
  • T μν is the stress–energy tensor, which represents the distribution of matter and energy.

Understanding spacetime

The concept of spacetime can be quite challenging to understand because it combines three dimensions of space with one dimension of time into a four-dimensional continuum. Imagine a two-dimensional analogy: a rubber sheet. When a heavy ball is placed on this sheet, it causes the sheet to curve. Similarly, massive objects such as the Earth create curvature in spacetime. Objects moving around will naturally follow the outline of this curve.

Illustration of how mass distorts spacetime (view from above, without physical representation of the time dimension).

Cosmological principle

In cosmology, two fundamental principles are often considered to apply to the universe:

  • Homogeneity: On a large scale, the universe is uniform, with no special locations. This means that when averaged over a large enough volume, the density of matter and energy is the same throughout the universe.
  • Isotropy: The universe looks the same in all directions. There is no preferred direction in the universe when viewed from any point.

Together, these principles form the cosmological principle, which implies that the universe is both homogeneous and isotropic on large scales.

The Big Bang theory

One of the most important theories in cosmology is the Big Bang theory. It proposes that the universe originated from an extremely hot and dense singularity about 13.8 billion years ago and has been expanding ever since. As the universe expanded, it cooled, leading to the formation of subatomic particles and simple atoms. Eventually, these particles combine to form stars and galaxies.

Expansion of the universe

An important observation that supports the Big Bang theory is the expansion of the universe. Edwin Hubble, an American astronomer, discovered in the 1920s that distant galaxies are moving away from us, and that their speed is proportional to their distance. This is summed up in Hubble's law:

V = H₀D

Where:

  • v is the velocity of a galaxy.
  • d is the distance to the galaxy.
  • H₀ is the Hubble constant, which represents the rate of expansion of the universe.

Cosmic microwave background radiation

Another pillar supporting the Big Bang theory is the existence of the cosmic microwave background (CMB) radiation. In 1965, Arno Penzias and Robert Wilson discovered this faint glow of microwave light, which fills the universe and is a relic of the early hot stages after the Big Bang.

Nucleosynthesis

Big Bang nucleosynthesis refers to the production of lighter elements such as helium, deuterium, and lithium during the first few minutes after the Big Bang. The observed abundances of these elements provide important evidence in favor of the Big Bang model.

Dark matter and dark energy

Remarkably, the universe is primarily composed of two forms of energy and matter that do not emit or absorb light, making them nearly invisible to direct observation. These forms are dark matter and dark energy.

Dark matter

Dark matter is a form of matter that does not emit, absorb or reflect light, making it invisible. However, its presence is inferred from gravitational effects on visible matter, radiation and the large-scale structure of the universe. Studies of galaxy rotation curves suggest that dark matter makes up about 27% of the mass-energy content of the universe.

Dark energy

Dark energy is a mysterious form of energy that accounts for about 68% of the mass-energy content of the universe. It is thought to cause the observed acceleration of the expansion of the universe. The cosmological constant (Λ) in Einstein's field equations is often associated with dark energy.

Conclusion

Cosmology, under the framework of general relativity, provides profound insights into the structure, origin, evolution, and fate of the universe. From the revolutionary Big Bang theory to the mysterious nature of dark matter and dark energy, the study of cosmology challenges our understanding and compels further exploration into the mysteries of the universe.

As we continue to explore the universe, new discoveries keep emerging that promise to reshape our knowledge of the cosmos, as well as uncover new aspects of fundamental physics.


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Cosmology

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