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 ↓


Einstein's field equations


In the early 20th century, Albert Einstein revolutionized the way we understand gravity. Before Einstein, the dominant theory was Newton's law of universal gravitation, which described gravity as a force acting over a distance between two masses. However, this explanation could not account for all observed phenomena, especially those involving large gravitational fields or very high velocities.

Einstein introduced the general theory of relativity in 1915, and at its core are Einstein's field equations (EFE). These are a set of ten interrelated partial differential equations in general relativity that describe the fundamental interactions of gravity as a result of spacetime being curved by matter and energy.

Basics of General Relativity

General relativity is a theory of gravity that Einstein published in 1915. According to general relativity, what we understand as the force of gravity arises from the curvature of spacetime due to the presence of mass. A massive object makes a "dent" in the spacetime structure, and then other objects move along paths determined by the curvature.

In this simplified visualization, imagine that the black circle represents a massive object such as a planet. The gray curve represents the bending of spacetime around this object. Smaller objects will travel along curved paths created by this "dent" in spacetime.

Einstein's formulation of the field equations

Einstein's field equation can be briefly written as follows:

    R μν - ½g μν R + g μν Λ = (8πG/c⁴)T μν

Here is a description of each component:

  • R μν is the Ricci curvature tensor, which represents the gravitational effects due to the local energy density.
  • g μν is the metric tensor, which defines the geometry of spacetime.
  • R is the scalar curvature, indicating the degree of curvature of spacetime.
  • Λ is the cosmological constant, which is associated with the energy density of the vacuum of space.
  • T μν is the stress–energy tensor, which accounts for the matter and energy content in spacetime.
  • G is the gravitational constant.
  • c is the speed of light.

Einstein's field equations relate the geometry of spacetime to the distribution of matter within it. Simplifying this complex relationship, these equations essentially say, "Matter tells spacetime how to curve, and curved spacetime tells matter how to move."

Illustration of Einstein's field equations

Let's consider a more concrete analogy using a rubber sheet. Imagine that a stretched rubber sheet represents a two-dimensional slice of our 4-dimensional spacetime. Placing a heavy ball on this sheet will create a crater.

The heavy ball represents a massive object like a star or planet, and the crater around it represents Einstein's idea of spacetime curvature. When another smaller ball rolls on the sheet, it follows the curve created by the giant ball. This simple experiment mimics how planets orbit stars due to the curvature of spacetime.

Understanding the components

Metric tensor (g μν)

The metric tensor is a fundamental part of Einstein's equations. It describes the geometry of spacetime. In flat spacetime, like special relativity, the metric tensor can be represented with the Minkowski metric:

    g μν = diag(-1, 1, 1, 1)

In curved spacetime, this tensor changes to describe how each point in spacetime is stretched or compressed.

The Ricci curvature tensor (R μν) and the scalar curvature (R)

The Ricci curvature tensor and scalar curvature provide a measure of how much the volume of a small geodesic ball in spacetime deviates from that in flat space. Scalar curvature is a single number at each point in spacetime that characterizes the overall curvature of spacetime.

Stress–energy tensor (T μν)

The stress–energy tensor includes not only mass but also energy, momentum, and pressure. In relativistic physics these quantities are interconnected, and the stress–energy tensor is important for describing how different forms of matter and energy distribute and interact in spacetime.

Cosmological constant (Λ)

The cosmological constant, Λ, was initially introduced by Einstein as a means of achieving a static universe. However, after the discovery of the expanding universe, it was largely ignored, only to be revived later to explain dark energy and the accelerating cosmic expansion.

Solving the field equations

Solving Einstein's field equations can be extremely complex, and often requires assumptions or approximations to make the problems tractable. Some examples of solutions include:

  • Schwarzschild solution: describes spacetime around a stationary, non-rotating, spherically symmetric mass.
  • Kerr solution: generalizes the Schwarzschild solution to include rotating masses.
  • Friedmann-Lemaître-Robertson-Walker (FLRW) metric: describes a homogeneous and isotropic expanding or contracting universe, fundamental in cosmology.

Schwarzschild solution example

An important milestone was the Schwarzschild solution, an exact solution of Einstein's equations representing the external gravitational field of a spherical mass. It is given as:

    dS² = -(1 - 2GM/RC²)c² dt² + (1 - 2GM/RC²)-1dR² + R²(dθ² + sin²θ dφ²)

This solution helped predict the existence of black holes, which are regions of spacetime where gravity is so strong that nothing, not even light, can escape.

Implications of EFE

Einstein's field equations have profound implications in many areas of physics, from predicting the bending of light around massive objects (gravitational lensing) to the expansion of the universe and the behavior of black holes. Here are some of the implications:

  • Gravitational lensing: The bending of light as it passes near a massive object, which confirms that light follows the curvature of spacetime.
  • GPS technology: Relativity corrections are essential to the accuracy of satellite-based global positioning systems.
  • Cosmic expansion: Provides a framework for understanding phenomena such as the expansion of the universe and dark energy.

Conclusion

Einstein's field equations are a cornerstone of modern physics. Although they are mathematically intensive and often challenging to solve, their implications span many areas of physics, opening new avenues of research and deepening our understanding of the universe.

Through this beautiful formulation, Einstein gave us a new lens for examining the dance between matter and space, revealing a profound interconnection where the shape of space is molded by the matter in it and where time itself is part of this grand tapestry we call the universe.

Understanding and using Einstein's field equations remains a central task in theoretical physics, linking our understanding of large phenomena within cosmology, black hole physics, and even attempts to unify gravity with the fundamental forces that govern the universe at the quantum scale.


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Einstein's field equations

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