Amateur Astronomers Association of Princeton

by S. Prasad Ganti

In the century following Albert Einstein’s General theory of relativity, the term “Black holes” entered our lexicon. In addition to a greater understanding, Black holes have been confirmed via indirect means and also photographed in the last few ears. Greater understanding resulted from relating Black holes to a thermodynamic concept called Entropy, and also relating to information theory or quantum computing. 

Albert Einstein did not come up with the idea of Black holes. It came up as a solution to the equations of Einstein’s General theory of relativity. Solved by a German Physicist Karl Schwarzschild. A Black hole is a very compact and dense, a few miles to a few hundred miles in diameter and weighing millions of solar masses, an object which sucks in all the matter in its vicinity. It warps space and time to a great extent that the time literally stops at the singularity within the Black hole. Super massive Black holes are at the center of all galaxies including our own Milky way. Our Black hole is known as Sagittarius A*. It probably started off as a graveyard of a massive star or a huge lump of coalescing gas, heavier than 1.4 solar masses which is known as Chandrasekhar limit. It probably grew by sucking in matter in its neighborhood. Like a corporation growing using the mergers and acquisitions route.   

It was presumed that the Black holes live forever and accumulate matter as they go along. Steven Hawking found that Black holes do emit radiation. They are not at absolute zero cold temperatures. The temperature of a Black hole is inversely proportional to its mass. Heavier Black holes are colder while lighter ones are hotter. As per laws of Physics, any object which is hotter than absolute zero, known as a black body, produces radiation. This radiation is known as Hawking radiation. The formula for the temperature of a Black hole is engraved on the floor of Westminster Abbey where Hawking is buried. The following link points to details and picture of the formula:

https://www.westminster-abbey.org/abbey-commemorations/commemorations/stephen-hawking

As the Black holes “evaporate”, they become lighter and hotter and more radiation they emit. Becoming hotter and shrinking in size until they disappear. But they accumulate mass as well and become heavy. This dynamics plays out over billions of years, closer to the age of the Universe. Entropy is a pretty abstract concept. I struggled with understanding it for several years. I don’t pretend to understand it fully. Basically, it is a measure of disorder in a system. Things being scattered around a house or molecules being in different energy levels in a closed surrounding etc. It also involves the number of combinations in which things can be arranged, like a deck of cards. Thermodynamics is an area which deals with energy associated with heat. Two of the three laws of Thermodynamics are relevant to our discussion. One is the familiar law that energy is conserved – neither created nor destroyed but just converted from one form to another. 

The second law states that the overall entropy in the Universe keeps increasing over time. The Universe started with zero Entropy and will keep increasing with its age. Rudolph Clausius, the Physicist who came up with the idea of Entropy,  summarized by saying “the energy of the world is constant, the entropy strives for a maximum”. Entropy is what prevents heat from flowing from cold to hot or for time to go back into the past, although neither of them would violate the principle of energy conservation. The famous equation quantifying Entropy is engraved on the tomb of Ludwig Boltzmann, the Austrian Physicist. The picture below, courtesy Wikipedia, has his bust with the formula on the top of his tombstone. 

The Black holes do have entropy by the virtue of having a temperature. Neither the entropy nor the temperature of a Black hole are zero. Information is represented as bits in computers. The number of bits required to store information is closely related to the Boltzman equation for Entropy. Information does share something with Entropy.

The basis for Hawking radiation is the generation of a virtual pair of submicroscopic particles close to the Event horizon. Event horizon is the point of no return for matter falling into a Black hole. Quantum mechanics tells us that these particles are “entangled”, which means that by studying one particle we can predict what the other particle would look like. For example, their spins are interrelated even after they leave their origin and travel vast distances. Even if one of the particles falls into the Black hole and another escapes in the form of Hawking radiation. 

In normal circumstances such virtual pairs are created and destroyed all the time contributing to the vacuum energy of space. Such “entangled” particles near the Event horizon of the Black hole ensure that the information on matter which gets sucked into a Black hole is not lost, as was perceived earlier. It is stored in the Hawking radiation which emanates from the Black hole. In the book “Einstein’s shadow’”, Seth Fletcher writes “The idea that all the information about the interior of a black hole is engraved on a surface just outside the event horizon is called the holographic principle. Entanglement holds no matter how much distance separates the two particles-even if they are separated by the event horizon of  a black hole. Entanglement plays an important factor in black hole evaporation. In quantum theory, a vacuum is never entirely empty. Spontaneous creation and annihilation of pairs of virtual particles. Near the event horizon, gravity rips a pair of virtual particles, one could fall into the black hole and other could escape. The escaping one becomes a real particle of Hawking radiation”. 

The recent book titled  “Black Holes” by Brian Cox and Jeff Forshaw  summarizes thus “Entangled systems have a rich structure that encodes information in the correlations across the system. The information is lost if the entanglements are broken”. The behavior of particles at quantum levels is the basis of modern day Quantum computing. Entanglement is one of the key features which is exploited by the Quantum computers.  

The notion that the Black holes and Quantum computers share commonality is an interesting one. Cross pollination of ideas not just to understand the abstract, but put to use to the benefit of humanity.