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 ↓


Dark matter and dark energy


The mysteries of the universe fascinate the minds of both young and old. Two of the biggest riddles facing cosmology are "dark matter" and "dark energy", the invisible components that make up much of the universe. Despite extensive research, they remain some of the least understood elements, especially as wrought by Einstein's theory of general relativity.

What is dark matter?

In the vast universe, about 27% is composed of dark matter. Unlike normal matter, which is composed of the atoms that make up stars, planets and living beings, dark matter does not emit, absorb or reflect light. We cannot see it directly, but infer its existence through gravitational effects on visible matter.

For example, galaxies spin so fast that the visible matter we see is not enough to prevent them from falling apart. Additional mass invisible to electromagnetic detection must exert the necessary gravitational force to hold them together. This hidden mass is what we call dark matter.

In a simplified visual example, if we imagine a merry-go-round spinning rapidly, the riders stay seated due to gravitational pull and structural integrity. In galaxies, visible stars are like these riders, and dark matter plays the role of an invisible but strong force that keeps everything intact.

Gravitational Effects

Dark matter in the framework of general relativity

General relativity has revolutionised our understanding of gravity, showing that it is a curvature in spacetime caused by mass. Dark matter interacts through the gravitational force, bending light and affecting the trajectories of visible matter. One example of such interactions is gravitational lensing, where we see light bending around objects with mass in ways that cannot be explained visually.

E = mc^2

In this famous equation, energy and mass are equivalent, with "c" being the speed of light. Although dark matter does not emit energy as light, its mass still plays an important role in the cosmic architecture according to Einstein's theories.

What is dark energy?

Dark energy, which is even more elusive, is responsible for about 68% of the universe's mass, causing its accelerating expansion. While dark matter pulls things together with gravity, dark energy acts as a kind of anti-gravity, pushing the universe apart at ever-increasing rates.

Einstein initially introduced a "cosmological constant", symbolized Λ, into his equations of general relativity in order to maintain a static universe model. However, the realization that the universe is expanding led to a reconsideration of Einstein's "biggest mistake" as a possible explanation for dark energy.

Expanded Universe

Inferred from supernova studies and cosmic microwave background measurements, dark energy appears to counteract the gravitational attraction of matter provinces, suggesting a pervasive "cosmic repulsion" that guides galaxy clusters.

The role of Einstein's general theory of relativity

To shape the large-scale coordinates of the universe, the field equations of general relativity become important. Operating in cosmological terms, these equations determine how the mass-energy content shapes the spacetime curvature. Solutions to these equations in the mass distribution allow for models that incorporate both dark matter and dark energy.

R_{μν} - frac{1}{2}g_{μν}R + g_{μν}Λ = frac{8πG}{c^4}T_{μν}

Here, R_{μν} denotes the Ricci curvature tensor, g_{μν} is the metric tensor, T_{μν} denotes the stress–energy tensor, and G is the gravitational constant. The term Λ denotes the cosmological constant, which is a placeholder for the effects of dark energy.

Challenges and secrets

Despite numerous speculative models and experiments, neither dark matter nor dark energy can be directly observed. Dark matter may someday be revealed as a new type of particle or wave beyond the standard model of particle physics. Dark energy may involve quantum fluctuations of vacuum space, as suggested in several theories.

Moreover, fitting dark matter and dark energy into the elegant elegance of general relativity is a major challenge. Some try alternative theories such as Modified Newtonian Dynamics (MOND) or expand the scope through methods such as loop quantum gravity, string theory or hypothetical quintessence interacting fields.

Conclusion

The vast and boundless universe continues to present puzzles that inspire humanity to explore beyond the realm of knowledge. Dark matter and dark energy remain integral to understanding cosmic evolution and structure, linked by the bridge of the effects of general relativity. Pioneering technological advancements, theories, and passionate exploration may one day unravel these unknowns of the universe, revealing the mysteries hidden in the grand and mysterious book of the universe.


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Dark matter and dark energy

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