Brian Cox breaks down the most mysterious scale in the cosmos

Key Concepts

Planck length, Planck constant, speed of light, gravitational constant, quantum mechanics, uncertainty principle, black holes, event horizon, entropy, extra dimensions, white dwarf stars, Chandrasekhar limit, Planck mass, Pauli exclusion principle, nuclear fusion, quantum mechanical pressure.

Planck Length: A Fundamental Property

The Planck length, approximately 10^-35 meters, is proposed as a fundamental property of the universe, derived from the speed of light (c), the gravitational constant (G), and Planck's constant (h). It's calculated as the square root of (hG/c^3). While our current measurement places it at this value, hypothetical configurations of the universe, such as those with extra dimensions, could potentially result in a larger Planck length.

Planck Units and Alien Civilizations

The discussion highlights the human-centric nature of common units of measurement like meters and feet, which are based on biological properties. Max Planck sought to define fundamental units based on universal constants, aiming for a common language with hypothetical alien civilizations that might not share our biological characteristics. These constants include:

• Speed of Light (c): A universal speed limit, a property of the universe related to spacetime geometry.
• Gravitational Constant (G): Measures the strength of the gravitational force or how matter/energy distorts spacetime.
• Planck's Constant (h): Associated with quantum theory, defining a fundamental limit on the accuracy with which we can simultaneously know a particle's position and momentum. It also relates to the energy of a photon (E = hf, where f is frequency).

Planck Length and Black Holes

Jacob Bekenstein's work in the 1970s revealed a connection between the Planck length and black holes. The entropy (information stored) of a black hole is equal to the surface area of its event horizon measured in square Planck lengths. This suggests that the Planck length is fundamentally important in determining the information capacity of space.

Planck Length and Observational Limits

Attempting to observe structures at the Planck length scale presents a fundamental limit. Due to quantum mechanics, probing smaller distances requires light with smaller wavelengths and thus higher energy photons. At the Planck length, the energy becomes so high that it forms a black hole, preventing further resolution of the structure. Adding more energy only increases the size of the black hole.

Planck Length and the Structure of Space

The discussion touches upon the idea that the Planck length might represent the "pixels" of spacetime, potentially acting as building blocks of space. However, it's emphasized that this is at the edge of our current understanding.

Caveats: Extra Dimensions

The speaker introduces a caveat: the measured value of the Planck length (10^-35 meters) is based on our current understanding of the universe. Theories involving extra dimensions, potentially detectable at facilities like the Large Hadron Collider, could alter the strength of gravity at high energies. This could lead to a larger effective Planck length at those energy scales.

Chandrasekhar Limit: A Real-World Application

The Chandrasekhar limit, calculated by Subrahmanyan Chandrasekhar in the 1930s, provides a concrete example of how the Planck scale affects observable phenomena. It defines the maximum mass of a white dwarf star (approximately 1.4 times the mass of the Sun) that can be supported against gravitational collapse by electron degeneracy pressure.

White Dwarf Star Formation

Stars are formed by the gravitational collapse of hydrogen and helium. As the star contracts, the core heats up, initiating nuclear fusion reactions (e.g., hydrogen fusing into helium in our Sun). These reactions release energy, creating pressure that balances the inward force of gravity.

Electron Degeneracy Pressure

When a star exhausts its nuclear fuel, it begins to collapse. In a white dwarf, the collapse is halted by the quantum mechanical effect of electron degeneracy pressure. As electrons are confined to smaller volumes, the uncertainty principle dictates that their momentum (and thus their speed) must increase. This rapid "jiggling" creates pressure that counteracts gravity.

Relativistic Limit

There's a limit to how much the electrons can jiggle. As the star collapses further, the electrons' speeds approach the speed of light. At this point, relativistic effects become significant, and the electron degeneracy pressure can no longer increase sufficiently to support the star.

Calculation and Fundamental Constants

The Chandrasekhar limit can be calculated using fundamental constants: Planck's constant, the speed of light, and the gravitational constant. The limit is roughly equal to the Planck mass cubed divided by the proton mass squared. The Planck mass itself is derived from the same fundamental constants.

Significance

The Chandrasekhar limit demonstrates a direct link between abstract quantum mechanical principles and observable astronomical phenomena. It shows how the fundamental properties of the universe dictate the maximum mass of a white dwarf star.

Synthesis/Conclusion

The discussion centers on the Planck length as a fundamental property of the universe, derived from the speed of light, gravitational constant, and Planck's constant. While its exact nature and implications are still under investigation, it appears to play a crucial role in phenomena ranging from black hole entropy to the stability of white dwarf stars. The Chandrasekhar limit serves as a compelling example of how these seemingly abstract concepts manifest in observable astrophysical phenomena, highlighting the interconnectedness of quantum mechanics, gravity, and the large-scale structure of the cosmos. The possibility of extra dimensions adds a layer of complexity, suggesting that our current understanding of the Planck length might evolve as we probe the universe at ever-higher energies.