Since we seem to have found ourselves in the business of throwing stuff into a black hole, and waiting to see what pops out. A natural question might be to ask, how does the black hole change when something falls in or comes back out? Of course we do our best to respect the No-hair theorem. But we can ask more about black holes than just what happens to their mass, spin, and charge. After all, if all you've been given are those three values for any particular black hole, I can't tell you anything about its history compared to a black hole with the exact same three characteristics. Somehow, part of the black hole's history is hidden from observation. But astrophysicists have yet another trick up their sleeves: entropy. Entropy is a property that all matter has and is usually explained as a measurement of disorder within a system. That's a pretty good explanation but it's quite a bit easier to explain it by example. Your bedroom. Think about the current state of your bedroom and on a scale of one to 10. Think about how messy you'd rate it. One being perfectly clean and 10 representing the aftermath of a bull in a China shop scenario. How many different ways can your bedroom be dirty? Your socks specifically, instead of being bundled up in a drawer, might be strewn on the floor. There are many places that you could find messy socks, on your bed, on the windowsill, under your desk. In fact there are a huge number of ways that your bedroom can be messy, but there's only one way that it could be perfectly clean. So by measuring how messy something is, we have learned something about how much entropy it contains. Entropy is one of the main reasons why we don't experience reversibility in the universe. Just like your bedroom, there's only one way that a mug is unshattered, but an uncountable number of ways that it can be smashed. What this means is that we rarely see mugs spontaneously going from shattered to unshattered or dirty rooms spontaneously becoming clean, because there are many ways for those systems to evolve. Most of them messier than before. For this reason, entropy is not considered to be time reversible. The behavior of all matter as it relates to entropy, temperature, and energy is codified in a field of physics called thermodynamics. The word thermodynamics comes from the Greek words Therm meaning heat and dynamics which means power. Thermodynamics is therefore the study of converting heat into power. Historically, it was motivated through the 1800s by the invention of steam engines and later combustion engines. However, thermodynamics has become an essential tool for astrophysicists, since stars are basically the equivalent of nuclear engines. There are four basic laws in thermodynamics, and you'll find them surprisingly simple. The Zeroth law, is a statement that the temperature of two things are the same when they're in thermal equilibrium. This is a great law because if it weren't true we could never trust thermometers. The first law is a statement that energy is conserved in isolated systems. This is the principle that thermoses operate on. They attempt to keep your food hot or cold by isolating the interior of the thermos from the outside world. Isolation means that ideally no energy is lost to the outside environment. So for a thermos your food stays hot. The second law is a statement that heat flows from hotter objects to colder objects and not the other way around. This might make some intuitive sense. When we warm ourselves by a campfire, we don't think about cold flowing out of our bodies, we think about hot flowing in. The third law is a statement that as we cool things down to absolute zero all processes stop. This is an extension of our experience since humans can't survive for long below about minus 40 Celsius. But using a fridge is exactly the third law in practice. Cold food last longer because decay processes slow down at low temperatures. The second law of thermodynamics seems pretty obvious. Heat flows from hot things to cold things, and this usually progresses until both objects are the same temperature, right? But how would you go about measuring how far along this equalization is? The answer is entropy. Just like the example with your messy bedroom, entropy measures how messy a system is. As it turns out this is directly related to an object's temperature. The formulaic definition of entropy is this: S is equal to K log(W) where S is the entropy of the system, K is Boltzmann constant and W is the number of equiprobable states the system can be in. Ludwig Boltzmann was so proud of this equation that it was engraved on his gravestone. It's easiest to think back to your messy bedroom for this one. There's only one equiprobable state for a clean room, so the entropy is low. There are millions of ways that your room can be dirty so the entropy of a messy room is high. Here is the formula to calculate the entropy of a black hole based on the area of its event horizon. S black hole is equal to K times A divided by four times LP squared. Here s is the total entropy equal to K Boltzmann constant, times the area of the event horizon divided by four times the square of LP, the plank length. The Planck Length is defined by LP equals the square root of h bar times G divided by C cubed is equal to 1.6 times 10 to the power of minus 35 meters. The Planck Length is incredibly small which is commonly thought of as the smallest distance that can be described by Relativity without quantum mechanics. In some sense, the equation for black holes entropy says that each little bit of entropy that falls into a black hole is encoded as a small patch on the event horizon, one Planck length across. Before we try to apply entropy to a black hole, take a moment to think, does a black hole represent a clean state of all the matter that has fallen in? Or is it messy? Black holes are strange objects within the study of thermodynamics, a fact recognized by scientist Jacob Bekenstein. In the early 1970s, Bekenstein asked himself a question very similar to the one I just asked you. At the beginning of the 1970s, it was still believed that nothing could escape from a black hole. So question like, "How hot is a black hole?" Made absolutely no sense since nothing comes out of a black hole, the temperature should be zero. Bekenstein recognize that if black holes didn't have well-defined temperatures, maybe they had well-defined entropies. He went on to prove that a black hole's entropy is proportional to the area of the black hole's event horizon. But since entropy was well characterized, shouldn't temperature follow? We can't just stick a thermometer into a black hole in order to see if it's running a fever. Instead we have to consider what's emitted from the black hole as Hawking radiation causes it to slowly evaporate. Indeed this was the motivating question that led Stephen Hawking to derive the framework for hawking radiation. Since quantum mechanics permits black holes to evaporate, they then certainly don't have zero temperature. Hawking student at the time, Don Page, took the concept of Hawking radiation further deriving what temperature we could expect a black hole to radiate. So what do you think? Are black holes hot or are they cold? If you said cold you're mostly right. The temperatures of black holes are given by the simple equation, t is equal to Kappa divided by two pi, where t is the temperature, two pi are just constants. This Kappa is just the surface gravity of the black hole. The surface gravity is a way of expressing acceleration due to gravity at the event horizon, but in strange units. Let's skip the lengthy derivation for a Schwarzschild black hole and get right to the result. The surface gravity of a black hole is proportional to one over four M. You recognize this right? The surface gravity is inversely proportional to the mass of a black hole. So small black holes have high surface gravities and big black holes have small surface gravities. This will become important shortly. Since we can now calculate the temperature and entropy of a black hole, we can reconsider the meaning of the thermodynamical laws that got us here. In black hole physics, the four laws of thermodynamics can be restated as they apply to black holes like this: The zeroth law states that a black hole's surface gravity is the same across its surface. While gravity and temperature are different things, they are very closely related in black hole physics. The first law states that changes to a black hole depend on the mass and energy consumed by the black hole. This is a statement of energy conservation. The second law states that the area of a black hole always increases as matter falls in. The event horizon can shrink when Hawking radiation is emitted. However, there is a generalized second law that states that the sum of the area of the black hole and the entropy of the emitted Hawking radiation always increases. So now that we have a theory of black holes compatible with the laws of thermodynamics, what can we say about them that we didn't know before? Simply this black holes have well-defined temperatures and well-defined entropies. So a black hole has temperature. So what? Did you notice how the mass of the black hole relates to its temperature? This is where things get strange. The more massive a black hole is the colder it becomes and the opposite is true too, the smaller a black hole is the hotter it becomes. Now that's strange. To calculate the temperature of a Schwarzschild black hole. This is the full equation T equals h bar c cubed divided by eight PI GM kB. Using this equation with a one solar mass black hole yields a chilling temperature of 60 nano Kelvin. That's closer to absolute zero than most scientists have been able to achieve in the lab. The lowest recorded temperature of anything on earth was a molecular gas created here at the University of Alberta by Professor Lindsay LeBlanc, at a mere 40 nano Kelvins, that's 20 nano Kelvins colder than a solar mass sized black hole. So how small do you think a black hole would need to be for it to be room temperature? What would happen as a black hole cools off through Hawking radiation?
