World Science Scholars
3.2 Black Holes Review
summary
One of the fascinating edges of our universe is a black hole.

• If you launch a fuel-less rocket from the surface of the Earth, depending on its initial velocity, it will either fall back to the surface, or leave the planet entirely, never to return. This is the escape velocity for a given planet.
• For a massive object, the escape velocity is given by $$v_e = \sqrt \frac{2 G M}{r}$$where $G$ is the universal gravitational constant, $M$ is the mass of the object, and $r$ is the radius of the object.
• On Earth, the escape velocity happens to be about 11.2 km/s, whereas the moon’s escape velocity is about 2.38 km/s.
• We can imagine an object with so much mass and such a small radius that the escape velocity becomes equal to (or greater than) the speed of light. Then nothing can leave the surface. This is a black hole. Nothing can escape a black hole, not even light.

Black holes are an implication of general relativity.

• In 1916, Schwarzschild discovered that black holes were emergent from the Einstein equations. They were so strange and confusing that people thought they just couldn’t exist.
• Even Einstein himself didn’t believe black holes existed:

The Schwarzschild singularity does not appear for the reason that matter cannot be concentrated arbitrarily … otherwise the constituting particles would reach the velocity of light.

• Of course, we know a lot more about general relativity today. Most graduate students understand general relativity better than Einstein did. But this illustrated how difficult it was at the time to understand what a black hole is. It took 60 years to understand equations that were written down in a completely mathematically precise form.
• By the mid-1970s, we understood the classical properties of black holes (before we included quantum mechanics) and today we can observe their effects. We believe that at the center of the Milky Way is a supermassive black hole called Sagittarius A* (pronounced “Sagittarius A-star”).
• One of the things that bothered Einstein about a black hole is that it means there is a kind of “edge” of space. There’s nothing inside a black hole. Not empty space—space itself comes to an end. In a sense that means they’re the simplest objects in the universe—they have no features whatsoever.

• After 60 years of hard work by physicists and mathematicians, we reached some understanding of what a black hole is, provided you ignore quantum mechanics. In 1974, Stephen Hawking posed the question: what does quantum mechanics have to say about black holes?
• Hawking applied the uncertainty principle to black holes and showed that they are not “black,” exactly—they radiate. He gave a very explicit formula for the temperature at which they radiate:$$T_H=\frac{\hbar c^3}{8 \pi G M k_B}$$where $\hbar$ is the reduced Planck’s constant, $c$ is the speed of light, $G$ is the universal gravitational constant, $M$ is the mass of the black hole, and $k_B$ is the Boltzmann constant.
• This is a thermodynamic law which relates the temperature of a black hole to its mass. It’s puzzling to understand how radiation can come out of a hole of nothing, but Hawking argued that it must be so, mandated by quantum physics.

Black holes can store information.

• This relationship between the temperature and mass of a black hole emerges from a thermodynamic property that has been studied for hundreds of years.
• In one of the most beautiful developments in the history of scientific thought, Boltzmann and others showed that thermodynamic laws follow from statistical properties of molecules.
• A key ingredient is the notion of entropy, which in a sense is a measure of the complexity of a system.$$S\sim \text{log } N$$where $N$ is the number of microstates of a system. You can think of it as a measure of the information storage capacity of a system.
• Applying Boltzmann’s reasoning to the thermodynamic laws of black holes ascribes a certain number of microstates to them. In other words, the mass/temperature relationship that Hawking found infers the black hole has an entropy, known as the Bekenstein-Hawking entropy.$$S_{BH}=\frac{\text{Area}}{4 G \hbar}$$Here, the Area refers to the area of the black hole’s horizon.
• This is a formula that relates concepts from across physics: thermodynamics, gravity, and quantum mechanics. This entropy turns out to be a huge number by any standards. Even a very small black hole can store a very large amount of information. For instance, all of Google’s stored data could fit inside a black hole that is only 10−24 mm across. We believe that the black hole is the upper bound of information storage.

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