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 → Tensor calculus and differential geometry ↓


Einstein field equations


The Einstein field equations (EFE) are at the heart of Einstein's theory of general relativity, a framework that revolutionized our understanding of gravity. These equations describe how matter and energy in the universe distort spacetime, causing the phenomenon we observe as gravity. In this exposition, we will take a deep look at each fundamental aspect of the Einstein field equations using tensor calculus and differential geometry, tools essential in graduate-level physics.

Background: What are the Einstein field equations?

Einstein's insight was to view gravity not as a conventional force but as a geometric property of spacetime. The presence of mass and energy makes spacetime curve, and this curvature affects the motion of objects. The Einstein field equations quantify this relationship.

G μν = 8πG T μν

The EFE is a system of ten interrelated differential equations whose solutions describe the gravitational field given the distribution of mass-energy. Here, G μν is the Einstein tensor that comprises the curvature of spacetime, and T μν is the energy-momentum tensor that describes the distribution and flow of energy and momentum in spacetime.

Basic concepts in differential geometry

To understand EFE, it is necessary to be familiar with differential geometry and tensor calculus. These concepts provide the language for describing the curvature of spacetime.

Tensors

Tensors are multidimensional arrays of numerical values that extend the concept of scalars and vectors. They have properties that make them particularly suitable for describing physical laws in any coordinate system.

Metric tensor

The metric tensor g μν is central to describing spacetime in general relativity. It allows us to calculate distances and angles in curved spacetime. The line element in general relativity is expressed as:

ds² = g μν dx μ dx ν

This expression shows how to measure an infinitesimal distance ds using the coordinates dx μ and the metric tensor g μν.

Curvature

Spacetime curvature is characterized by the Riemann curvature tensor R ρ σμν, which has 20 independent components in four dimensions. This tensor reduces to simpler forms, such as the Ricci tensor R μν and the scalar curvature R, which are important in formulating the Einstein tensor:

R = g μν R μν

The Einstein tensor itself is defined as follows:

G μν = R μν - 1/2 g μν R

Interpretation of the Einstein field equations

EFE can be understood as "spacetime curvature equals matter-energy content." This concise idea is equivalent in mathematics to the Einstein tensor, the energy-momentum tensor, as related by a constant factor that includes Newton's gravitational constant G and the speed of light c.

Energy–momentum tensor

The energy–momentum tensor T μν is an important concept because it not only incorporates the energy and momentum densities but also describes how they flow and interact within spacetime using its elements. Its components are as follows:

- T 00: denotes the energy density. - T 0i and T i0: components describing the energy flow or momentum density. - T ij: stress components including pressure and shear forces.

In essence, it describes what occupies and acts within spacetime, affecting its curvature.

Cosmological constant

An often-discussed extension of the EFE involves the cosmological constant Λ, originally introduced by Einstein for a static universe, which was later interpreted to account for the accelerating expansion of the universe:

G μν + Λ g μν = 8πG T μν

Applications and implications of the Einstein field equations

EFEs are used to describe a wide variety of physical phenomena, from the behavior around black holes to the large-scale structure and dynamics of the universe.

Schwarzschild solution

One of the simplest and most notable solutions is the Schwarzschild solution, which describes spacetime around a non-rotating, spherically symmetric mass. The Schwarzschild metric is given as:

ds² = -(1 - 2GM/c²r) c²dt² + (1 - 2GM/c²r) -1 dr² + r²(dθ² + sin²θ dφ²)

Here, it describes the structure around massive bodies such as planets, stars, and especially black holes.

Friedmann–Lemaître–Robertson–Walker (FLRW) metric

The FLRW metric is important in cosmology, describing a homogeneous, isotropic expanding or contracting universe:

ds² = -c²dt² + a(t)² [dr²/(1 - kr²) + r²(dθ² + sin²θ dφ²)]

This metric includes the scale factor a(t) which describes how the size of the universe changes over time and is helpful in modeling the Big Bang and inflation theories.

Intuitive visual representation

Spacetime curvature

Mass

This simplified visualization shows how mass (in blue) curves the path of an object (black line) moving through spacetime, which appears as an effect of gravity.

Gravitational field around a body

Mass

The convergence of the green lines toward the central mass represents a distortion of spacetime, explaining gravitational attraction as a geometric property rather than a conventional force.

Conclusion

The Einstein field equations are a cornerstone of our understanding of modern physics, shedding light on how the universe behaves on large scales. They embody Einstein's revolutionary notion that we live in a universe where geometry and gravity are inextricably linked through the fabric of spacetime. Understanding the mathematical intricacies and profound physical insights of the EFE continues to fascinate and challenge physicists around the world, promising new understandings of reality.


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Einstein field equations

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