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

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General relativity and gravity


Albert Einstein's theory of general relativity is one of the pillars of modern physics. It has revolutionized our understanding of gravity and the fundamental nature of the universe. Before Einstein, gravity was best explained by Isaac Newton, whose laws of motion and universal gravitation provided a framework for understanding how objects interact through the force of gravity. However, Newton's theory viewed gravity as a force acting at a distance, which could not satisfactorily explain phenomena such as the precession of Mercury's orbit or how gravity can affect the path of light.

The basic idea of general relativity

Einstein proposed a different approach: rather than being a force in the traditional sense, gravity is the result of warped spacetime. In general relativity, massive objects create curvature in the spacetime fabric, and this curvature guides the objects' motion, which is what we observe as gravity. An easy way to visualize this is to imagine a trampoline on which a heavy ball is placed. The ball creates a depression in the surface of the trampoline, and any smaller balls placed nearby will roll toward the larger ball due to the curvature.

Spacetime and the equivalence principle

General relativity is fundamentally about the geometry of spacetime. Minkowski spacetime, used in special relativity, is extended to curved spacetime in general relativity. The equivalence principle is important, which states that locally (in small regions of spacetime), the effects of gravity are indistinguishable from acceleration.

Consider an experiment in an elevator: if the elevator is accelerating upward, an observer will feel a force pushing them into the floor. In a stationary elevator within a gravitational field, the same force is felt. This observation leads to the formulation that gravity and acceleration are locally indistinguishable.

Einstein field equations

The mathematical formulation of general relativity is contained in the Einstein field equations (EFE), which relate the geometry of spacetime to the distribution of matter within it. The equations can be expressed as:

R μν - 1/2 g μν R = 8πGT μν

Here, R μν is the Ricci curvature tensor, g μν is the metric tensor, R is the Ricci scalar, G is the gravitational constant, and T μν is the stress–energy tensor, which describes the matter and energy content.

Curvature of spacetime: geodesics

An important aspect of understanding general relativity is knowing how objects move in curved spacetime. The path that an object follows is called a geodesic. The geodesic is essentially the "straight line" path for an object moving only under gravity. In the absence of any forces, an object will follow the geodesic determined by the curvature of spacetime.

Imagine a globe. The shortest path between two points is a great circle. This concept of great circles is similar to geodesics in spacetime. Objects in orbit are not moving randomly; they follow geodesic paths through curved spacetime caused by a massive body such as a planet or star.

Leading evidence and predictions

Orbit of Mercury

One of the earliest confirmations of general relativity came from its ability to explain the precession of Mercury's orbit. Newtonian physics could not fully explain the observed precession, but curved spacetime around the Sun provided a natural explanation.

Gravitational lensing

General relativity theory predicts that light will bend around massive objects, a phenomenon called gravitational lensing. Light from a distant star or galaxy bends as it passes through massive cosmic structures such as galaxies or dark matter, which can create multiple images of the same celestial object or magnify it.

This effect was observed by Arthur Eddington during a solar eclipse in 1919, providing the first empirical confirmation of Einstein's theory.

Time extension and GPS

Time dilation is another prediction of general relativity. Clocks in strong gravitational fields run slower than those in weaker fields. This effect is also particularly important for the Global Positioning System (GPS), where satellites operate in a weaker Earth gravitational field than receivers on the ground. Corrections based on general relativity ensure accurate positioning.

Black holes and singularities

One of the strangest predictions of general relativity is black holes, regions in space where gravity is so intense that nothing can escape, not even light. Inside a black hole, all matter collapses to a singularity, a point of infinite density.

Black holes dramatically distort the spacetime around them. The boundary around a black hole beyond which nothing can return is called the event horizon. Anything that crosses this horizon is doomed to fall into the singularity.

Gravitational waves

In 2015, a century after Einstein's prediction, gravitational waves - ripples in spacetime caused by the acceleration of massive objects, such as merging black holes - were directly detected by LIGO. This unprecedented discovery ushered in a new era in astronomy, allowing scientists to "hear" the vibrations of great cosmic events.

Gravitational waves can be visualized as rings expanding outward across a pond after a stone is thrown into it, with the stone representing massive astronomical events and the surface of the pond symbolizing the nature of spacetime.

Understanding on a scale: the universe

General relativity also plays an important role in cosmology, which is the study of the large-scale structure and evolution of the universe. It helps explain phenomena such as the expansion of the universe, the Big Bang, and the cosmic microwave background. It provides a framework to explore ideas such as dark energy and dark matter, which affect cosmic evolution and structure.

Mathematical challenges

The mathematics involved in general relativity is extremely complex, often requiring differential geometry and tensor calculus. Despite the complexity, the profound impact of the theory cannot be underestimated. It has influenced countless fields of study and is a fundamental component of modern theoretical physics.

Educational and philosophical influences

Einstein's theory doesn't just teach us about the universe; it challenges our understanding of concepts like time and space. By recognizing that time and space are interconnected and malleable, general relativity fundamentally changes our perception of reality.

Conclusion

In conclusion, general relativity is a profound and revolutionary framework that redefines our understanding of gravity, space, and time. From its mathematical beauty to its cosmic-scale implications, it remains a central and enduring theory in the quest to understand our universe. Whether predicting the bending of light or the existence of mysterious black holes, it expands our horizons and challenges the mind to accept a universe that is far more interconnected and dynamic than ever imagined.


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