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If a closed surface does not have any charge inside where an electric field line can terminate, then the electric field line entering the surface at one point must necessarily exit at some other point of the surface. Therefore, if a closed surface does not have any charges inside the enclosed volume, then the electric flux through the surface is zero. What happens to the electric flux if there are some charges inside the enclosed volume? Gauss's law gives a quantitative answer to this question.
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Gauss's Law: Cylindrical Symmetry01:20

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Although black holes were theoretically postulated in the 1920s, they remained outside the domain of observational astronomy until the 1970s.
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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half...
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Gauss's Law: Planar Symmetry01:27

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...
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Random Pure Gaussian States and Hawking Radiation.

Erik Aurell1, Lucas Hackl2,3, Paweł Horodecki4,5

  • 1<a href="https://ror.org/026vcq606">KTH-Royal Institute of Technology</a>, Alba Nova University Center, SE-106 91 Stockholm, Sweden.

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Black holes evaporate via Hawking radiation, requiring entanglement for a pure state. This study finds minimal entanglement is needed, suggesting unitarity restoration doesn't rely on significant quantum entanglement between Hawking modes.

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Area of Science:

  • Quantum Gravity
  • Black Hole Thermodynamics
  • Quantum Information Theory

Background:

  • Black holes are theorized to evaporate through Hawking radiation.
  • Maintaining a pure quantum state during evaporation necessitates entanglement between radiation modes.
  • Quantifying this entanglement has been a significant challenge.

Purpose of the Study:

  • To develop a new theoretical framework for estimating the minimum entanglement in Hawking radiation.
  • To analyze the role of entanglement in restoring black hole unitarity.
  • To provide general expressions for mode-mode correlations in pure, Gaussian states.

Main Methods:

  • Developed a novel theory of constrained random symplectic transformations.
  • Assumed a pure and Gaussian total state with given marginals.
  • Computed mode-mode correlations to bound mode-mode entanglement.

Main Results:

  • Entanglement is strongly suppressed in thinly populated modes (high-frequency or late-time).
  • Highly populated modes (early-time, low-frequency) show suppressed entanglement despite strong correlations.
  • Unitarity restoration does not require significant entanglement between Hawking radiation modes.

Conclusions:

  • The study establishes that significant quantum entanglement is not essential for restoring unitarity in black hole evaporation.
  • The developed methods offer exact expressions for correlation distributions in pure, Gaussian states, with potential applications beyond black hole physics.