Deriving Hawking's most famous equation: What is the temperature of a black hole?

This paragraph introduces the concept of black holes according to Einstein's general theory of relativity, describing them as regions of space-time with such strong gravity that not even light can escape. It discusses the event horizon and the classical view that nothing can escape a black hole once it passes this boundary. The paragraph then delves into quantum mechanics, mentioning Stephen Hawking's demonstration in 1974 that black holes can radiate and eventually evaporate. The speaker expresses a personal interest in black holes and introduces the method of dimensional analysis as a way to understand their properties using high school mathematics.

The paragraph explains the technique of dimensional analysis, a valuable tool in theoretical physics for constructing equations without extensive calculations. It introduces the concept of dimensions and fundamental constants in physics, such as the speed of light, gravitational constant, reduced Planck's constant, and Boltzmann's constant. The speaker uses examples to illustrate how these constants can be used to understand the dimensions of various physical quantities. The paragraph sets the foundation for using these concepts to explore black hole properties.

This section focuses on the no hair theorem, which states that black holes are characterized by three external parameters: mass, electric charge, and angular momentum. The speaker narrows the discussion to Schwarzschild black holes, which lack charge and angular momentum, and are thus characterized solely by mass. Using dimensional analysis, the speaker derives a relationship between the area of a black hole's event horizon and its mass. The area is found to be proportional to the square of the mass divided by the speed of light to the fourth power, highlighting that the event horizon area increases with mass.

The paragraph delves into the concept of thermodynamic entropy, introduced by Rudolf Clausius and further explained by Ludwig Boltzmann. It discusses the second law of thermodynamics, which states that entropy of an isolated system must increase over time. Boltzmann's entropy law relates the entropy of a system to the number of ways microscopic components can be rearranged while maintaining macroscopic properties. The speaker suggests that black holes, as systems with many possible internal states for a given external appearance, likely have a high entropy value.

The speaker continues the discussion on black hole entropy, proposing a relationship between the entropy of a black hole and the area of its event horizon. Using dimensional analysis and fundamental constants, the speaker derives a relationship for the proportionality constant, known as eta. The paragraph then connects this to Hawking radiation, a phenomenon discovered by Stephen Hawking where black holes emit faint radiation, leading to their eventual evaporation. The speaker explains the mechanism behind Hawking radiation and its implications for black hole temperature and mass loss.

This section focuses on calculating the Hawking temperature of a black hole and its evaporation time. The speaker uses the first law of thermodynamics and the concept of thermodynamic entropy to derive an expression for the black hole temperature, which is found to be inversely proportional to its mass. The evaporation time is then estimated by calculating the power output of the black hole and relating it to the black hole's energy content. The speaker provides an example of a black hole with three solar masses and discusses the incredibly long time it would take for such a black hole to evaporate.

The speaker explores the possibility of primordial black holes, which could have formed from early universe fluctuations, and their potential evaporation. The paragraph discusses how adjusting the black hole mass could lead to an evaporation time comparable to the age of the universe. The speaker estimates the mass of such a black hole and describes the final moments of its life, including the energy released in the last second. The potential for detecting this energy as gamma rays is also discussed, highlighting the theoretical possibility of observing a black hole's final moments.

The paragraph addresses the black hole information paradox, which arises from the apparent loss of information when a black hole evaporates. The speaker introduces the holographic principle as a potential solution, suggesting that the information content of a black hole is encoded on its event horizon. The concept is illustrated using the Planck length and area, and the speaker estimates the immense amount of information that could be stored in a black hole using Planck-sized cells. The holographic principle is presented as a revolutionary idea in theoretical physics, suggesting that all information in a region of space can be described by data on its boundary.

The speaker concludes the discussion on the holographic principle, emphasizing its profound implications for our understanding of the universe. The principle suggests that all the information within a volume of space can be represented by bits of data on the boundary of that space. The speaker highlights the potential for future exploration of the holographic principle within string theory and shares a quote from physicist Leonid Susskind, who describes the universe as a hologram with reality coded on a distant two-dimensional surface. The video ends with a teaser for future content on this topic.