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 ↓


Cosmology and the Universe


Cosmology is the large-scale study of the universe, considering its origin, structure, evolution, and ultimate fate. It explores questions such as how the universe began, what it is made of, how it evolves, and what will happen to it in the future.

Role of general relativity

General relativity (GR), a theory of gravity developed by Albert Einstein, is a fundamental framework for understanding cosmology. GR describes gravity not as a force, as in Newtonian physics, but as a result of the curvature of spacetime due to mass and energy.

The Einstein field equations are the core of general relativity and are expressed as:

G μν + Λg μν = (8πG/c 4 )T μν

Here:

  • G μν is the Einstein tensor, which describes the curvature of spacetime.
  • Λ is the cosmological constant introduced by Einstein, which characterizes the energy density that fills space homogeneously.
  • T μν is the energy–momentum tensor, which describes the distribution of matter and energy in spacetime.

A simple example of curved space

To understand spacetime curvature, imagine a stretched rubber sheet. If you place a heavy ball on it, the sheet bends. Similarly, a heavy object like a planet or a star distorts the framework of space around it.

Mass

Main components of the universe

The universe is composed primarily of three elements:

  • Dark Energy: It is believed to make up about 70% of the universe and is causing it to expand at an accelerated rate.
  • Dark matter: Comprising about 25% of the universe, it exerts gravitational effects but does not emit light, making it invisible to telescopes.
  • Ordinary matter: Everything we see, which contributes about 5% to the composition of the entire universe.

The presence of dark matter can be inferred from gravitational effects, such as how quickly galaxies move around within clusters, which cannot be accounted for by visible matter alone. Similarly, data from distant supernovae and the cosmic microwave background radiation suggest the existence of dark energy.

Visualizing the structure of the universe

Dark Energy dark matter Common Substances

The Big Bang Theory

The prevailing cosmological model that explains the early evolution of the universe is known as the Big Bang theory. According to this model, the universe began as a very hot, dense point about 13.8 billion years ago and has been expanding ever since.

Shortly after the Big Bang, the universe underwent an exponential expansion called inflation. This period was important because it smoothed out initial irregularities and explained the large-scale uniformity of the observable universe.

Expanded Universe

In 1929 Edwin Hubble discovered that galaxies are moving away from us, which means the universe is expanding. This was a pivotal moment for cosmology because it provided strong evidence in support of the Big Bang theory.

Hubble's law quantifies this expansion:

v = H 0 d

Here:

  • v is the recession velocity of the galaxy.
  • H 0 is the Hubble constant, which characterizes the rate of expansion.
  • d is the distance of the galaxy from Earth.

Portrayal of the expanding universe

Think of the universe as a balloon with dots printed on its surface, representing galaxies. As the balloon inflates, the dots move farther apart, reflecting the concept of an expanding universe.

initial state Extended state

Observable universe and the cosmic microwave background

The observable universe is limited to the regions from which light has had time to reach us since the Big Bang. Beyond that, the universe may continue indefinitely, but this is not observable with current technologies.

The cosmic microwave background (CMB) is a key discovery that solidified the Big Bang theory. It is the thermal radiation left over from the time of recombination, which occurred about 380,000 years after the Big Bang, when electrons and protons first combined to form neutral atoms, allowing photons to travel freely.

Understanding the CMB

The CMB is remarkably uniform, but the minute fluctuations in temperature provide clues to the structure of the early universe. These fluctuations eventually led to the large-scale structures we see today.

CMB Cold Spot CMB Hot Spot

The fate of the universe

The fate of the universe is one of the biggest questions in cosmology, deeply connected to the properties of dark energy. Various scenarios exist, some of which are as follows:

  • Big Stagnation: Continued expansion leads to a colder, thinner universe as galaxies move away from each other.
  • Big Trouble: If the expansion is reversed due to gravitational attraction, it can result in a recollapse into a hotter, denser state.
  • Big Bang: Rapid expansion may eventually destroy galaxies, stars, planets, and even atomic structures.

Visualizing evolutionary paths

Imagine these scenarios as timelines. An expanding line represents the expansion of the universe, while the divergence or collapse indicates the corresponding fate.

Now Expansion Future

Conclusion

Cosmology is an exciting field at the intersection of physics and astronomy, exploring profound questions about the origin, structure, and fate of the universe. General relativity provides an important foundation for formulating our understanding of cosmic phenomena.

Although many questions remain, advances in observation and theory are improving our understanding of the complexities of the universe, from its tiniest particles to the vast expanses of space and time.


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Cosmology and the Universe

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