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 → Black holes and wormholes ↓


Event horizon in black holes and wormholes


In the fascinating field of general relativity and gravity, the concepts of black holes and wormholes are very interesting. At the center of these phenomena is the concept of the "event horizon". The purpose of this lesson is to unravel the mysteries surrounding event horizons, describe their properties in black holes and wormholes, as well as discuss their importance in physics. With simple language and visuals, the intention is to make these complex topics accessible.

Understanding the event horizon

The event horizon is essentially a boundary in spacetime. It marks the point of no return - beyond which, events cannot affect an outside observer. Imagine a sphere of influence around a black hole, where the escape velocity is equal to the speed of light. Once crossed, nothing, not even light, can escape. This boundary is the event horizon.

Visual representation of the event horizon

event horizon Black holes A simplified illustration showing the event horizon as the boundary around a black hole.

Mathematics of the event horizon

To delve deeper, we can explore the equation governing the event horizon of a non-rotating black hole, also known as a Schwarzschild black hole. The Schwarzschild radius ( rs ) is given by:

r_s = 2GM/c^2

Here:

  • G is the gravitational constant,
  • M is the mass of the black hole,
  • c is the speed of light.
Any mass with a radius smaller than this will become a black hole.

Example calculation

For a black hole with the mass of our Sun (about 2 x 10 30 kg), the Schwarzschild radius is calculated as:

r_s = 2 * (6.67430 x 10^-11 m^3 kg^-1 s^-2) * (2 x 10^30 kg) / (299792458 m/s)^2 ≈ 2.95 kilometers

This means that if the Sun became a black hole, its event horizon would be about 3 kilometers wide.

Event horizon in a rotating black hole

When we consider spinning black holes, known as Kerr black holes, the situation becomes more complicated. The event horizon is no longer a perfect sphere, but is flattened due to the rotation. In addition, an additional surface appears, called the "ergosphere", beyond which objects cannot remain stable.

The Kerr metric describes the geometry of spacetime around a spinning black hole. It introduces parameters for both mass and angular momentum, making the event horizon equation more complicated. The inner (event horizon) and outer (ergosphere) radii are given as:

r_± = GM/c^2 ± sqrt((GM/c^2)^2 - (J/Mc)^2)

where J represents the angular momentum of the black hole. These calculations provide more information about the behavior of light and matter around such cosmic bodies.

Wormholes: Theoretical pathways

Wormholes are theoretical passages through spacetime, predicted by solutions of the Einstein field equations. Unlike black holes, wormholes connect two different points in spacetime, which may be different places or even different times.

Event horizons and wormholes

In the traditional model of wormholes, there may or may not be an event horizon. For example, the traversable wormholes described by Kip Thorne would not have event horizons like black holes. The math involved suggests that exotic matter may be needed to manipulate these cosmic tunnels and keep them open for travel.

Visualizing wormholes

Entry exit A simplified representation showing a wormhole as a tunnel connecting two points in spacetime.

Event horizons: Key features

Event horizons have several fascinating properties. They act like one-way membranes because information that crosses them from the outside cannot return. This has intriguing implications for the laws of information conservation in physics.

The information paradox

The concept of the information paradox arises because according to classical mechanics, it is basically impossible to destroy information. However, if something falls into a black hole beyond its event horizon, it vanishes from the universe, raising questions about the ultimate fate of that information.

Hawking radiation

Hawking radiation offers a possible solution to the information paradox. This theory, discovered by Stephen Hawking in the 1970s, suggests that black holes may emit thermal radiation due to quantum effects near the event horizon. Over vast timescales, this radiation could cause the black hole to evaporate, potentially leaking information out.

Event horizon and observer

An interesting aspect is how different observers see the event horizon. To a distant observer an object falling into a black hole will appear to slow down as it approaches the event horizon, never actually crossing it. This results in a feeling of time "standing still", even though the object itself may experience normal acceleration due to the intense gravitational field.

Visualizing time dilation

event horizon Black holes supervisor An example of time dilation, where an object appears to stop at the event horizon to an observer.

Event horizon in quantum theory

The nature of event horizons also challenges the coherence between quantum mechanics and general relativity. Quantum fields in curved spacetime present a variety of phenomena that are difficult to explain with current physics, and event horizons are at the heart of these puzzles.

Understanding event horizons, especially in black holes and wormholes, pushes the boundaries of human knowledge, and presents both exciting opportunities and profound challenges for theoretical physics.

Closing thoughts

The study of event horizons remains one of the central questions in modern physics. They probe the theoretical limits of space, time, and reality. With advances in both observational technology and theoretical mathematics, our understanding of these cosmic phenomena will continue to evolve, revealing the deepest mysteries of our universe.


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Event horizon in black holes and wormholes

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