Brian Greene takes you from the beginning of time (the big bang) to several possible scenarios for ending of time. I consider most of this information from the big bang to the present day as common knowledge and after a brief introduction, I will jump to the end of the book where the information is not common knowledge.
Two leading actors
If you think of the accumulated knowledge about the universe as a collection of stories, there are two leading actors connecting these stories. The first actor is the entropy and the second actor is the evolution.
The second law of thermodynamics states that the entropy almost always increases. The second law of thermodynamics in colloquial terms is "the production of waste is unavoidable". Loosely speaking, everything in the universe has an overwhelming tendency to run down, to degrade, to wither. In other words, over time there is an overwhelming tendency of entropy to increase.
What is life? In purely physical terms, the "life is nothing but an electron looking for a place to rest". In my opinion, Greene elaborates what this means successfully. Here is a short passage from the beginning of the explanation.
An Oxygen atom contains 8 electrons, 2 on the ground level, and 6 on the next level. The maximum occupancy of the second level is 8. A Hydrogen atom has one electron in its ground state. The maximum occupancy of the ground state is 2. So, two Hydrogen atoms and an Oxygen atom agree to share a communal pair of electrons making Oxygen atom has 8 electrons in the second level and the two Hydrogen atoms have 2 electrons in their ground levels. The sharing of these electrons forms the water molecule.
Although water molecule has no net electric charge, the Oxygen atom at its vertex hoards the shared electrons, resulting in a lopsided distribution of charge across the molecule. The vertex has a net negative charge and the two tips (Hydrogen atoms) have a net negative charge. Because of this lopsided charge distribution, the water can dissolve nearly everything. Negatively charged vertex grabs anything positively charged and the positively tips grabs anything negatively charged. In tandem, the vertex and the tips pull apart most anything that submerged for a sufficient time.
This lopsided charge distribution of water is essential to the emergence of life. Cell interiors are like a laboratory. It takes nutrients in, commingling with chemicals to synthesize substances required for cellular function and spits out the waste. Water makes this possible. \(70\%\) of cell’s mass is water.
Nobel laureate Albert Szent-Györgyi summarizes: “Water is life’s matter, matrix, mother, and medium. There is no life without water. Life could leave the ocean when it learns to grow a skin to carry water with it. We are still living in water, having the water now inside.”
What he fails to elaborate successfully are the consciousness and how the brain works. The main reason for this is that both disciplines are at an infant stage and these stories are much more incomplete than the others.
Now, I will move to the last chapters of the book or to the end of time.
Everything will fall into Black Holes in about \(10^{30}\) years.
When a star rotates around the central black hole of a galaxy, it radiates away energy via gravitational waves. The distance between the star and the central black hole starts to decrease as a result. This is a very slow process, however, after a very long time, when the distance between the star and the black hole reduces to a critical value, the gravitational pull of the black hole will dominate and the star will fall into the black hole. After about \(10^{30}\) years there will be nothing but black holes in the universe.
Constant Cosmological Constant (Dark Energy) leads to a universe with a cosmological horizon in about \(10^{50 }\) years.
The expansion of the universe is accelerating due to the dark energy that was experimentally discovered at the end of the 20th century. If this acceleration of the expansion continues unabated, then the distance galaxies will disappear creating a cosmological horizon. The boundary recedes at greater than the speed of light and eventually, the observable universe will be just our own galaxy or whatever it becomes by then.
Black Holes will disappear in about \(10^{68}\) years.
John Wheeler in the early 70s thought of the following scenario. Imagine tossing (a hot cup of) tea into a black hole. Since the inside of a black hole is inaccessible to those on the outside of the blackhole, the tea with its entropy seem to have disappeared. This seems to violate the second low of thermodynamics.
The entropy of a system counts the number of rearrangements of the microscopic constituents of a system that leaves the macroscopic state of the system “looking the same”. If the entropy of tea transfers to the black hole, then the entropy of the blackhole should show up in the number of internal rearrangements of the black hole that has no effect on the black hole’s macroscopic features. That is the problem.
Physicists Werner Israel and Brandon Carter In the late 1960s and early 1970s showed that according to general relativity a black hole is fully determined by just three numbers: the black hole’s mass, the black hole’s angular momentum, and the black hole’s electric charge. Once you measure these three macroscopic features, you can fully describe a black hole. Two black holes which have the same mass, same angular momentum, and the same electric charge are identical. With no other configurations to count, no look alike to enumerate, it would seem like black holes have no entropy.
Toss in a cup of tea into a black hole, its entropy seems to vanish violating the second law of thermodynamics.
Jacob Bekenstain, a student of Wheeler, came up with an answer. The tea’s entropy is not gone. It is transferred to the blackhole.
The radius of the event horizon of a black hole is proportional to its mass. So, when a cup of tea is thrown into a black hole, its mass increases and therefore, the radius of the event horizon increases.
Imagine throwing a very low frequency photon whose wavelength is so long—who’s possible locations are so spread out—that when it encounters the black hole, the most precise description we can give of the outcome is expressed by a single unit of information: either the photon fell into the black hole or it did not fall into the black hole. That is, such a photon carry a single unit of entropy. Since the photon has energy by the equation \(E=mc^2\), if the black hole consumes the photon, then its energy increases slightly and its event horizon expands slightly. Bekenstein noted (mathematically) that by tossing in one unit of entropy, the black hole’s even horizon expands by one Planck unit of area (\(~10^{-70}\) square meters.). The event horizon thus seems to keep track of the entropy that the black hole ingested.
Bekenstein proposed that the total entropy of a black hole is given by the total area of the event horizon of the black hole (measured in Plank units).
Bekenstein could not explain this. This is a surprise because the entropy of an ordinary object like a cup of tea is contained in its interior, its volume. Also, he could not explain how his proposal related to the conventional framework in which entropy should enumerate the possible rearrangements of the interior microscopic ingredients.
This issue was not resolved until the mid 1990’s, when string theory would provide insights.
(See “Black Hole Wars”
https://skoswatt.blogspot.com/2021/05/the-black-hole-warmy-battle-with.html.)
Bekenstein’s proposal saves the second law of thermodynamics. By throwing tea into black hole reduces the entropy in your breakfast table, but the entropy decrease in your home is offset by the increase in the surface area of the event horizon of the black hole.
When Stephan Hawking heard Bekenstein’s proposal he thought it was ludicrous. He sat down to do calculations to disprove this proposal. Instead of disproving the proposal, Hawking not only confirmed it, but also revealed a complementary surprise. Black holes have temperature and black holes glow.
According to quantum mechanics, vacuum (empty space) has quantum fluctuations even though the average energy is zero. Through \(E=mc^2\), these energy fluctuations shows up as mass fluctuations—elementary particles and their anti-particles popping up in empty space. Imagine this is happening just outside the event horizon. Some times, particle-anti particle pair annihilate quickly, but sometimes, one member of the pair get stuck in the black hole. Since the outgoing particle carry positive energy (mass), the in-falling particle carry negative energy (mass) to preserve the conservation of energy, overall decreasing the mass of the black hole. The net effect is the black hole decrease its mass by radiating energy. This radiation is now known as the Hawking radiation.
The following is the Hawking formula.
The black body radiation emitted by a Schwarzschild black hole (an uncharged, non-rotating black hole, i.e., a black hole with only one property, it’s mass \(M\)) is given by
\(T=\frac{hc^2}{16\pi^2GMk}\)
Where, \(T\) is the Hawking temperature, \(h\) is the Plank’s constant, \(c\) is the speed of light, \(G\) is the Newton’s constant, and \(k\) is the Boltzmann’s constant.
The temperature of the super massive black hole at the center of our galaxy has temperature of \(10^{-14}\) degrees Kelvin.
Since the temperate of the microwave background radiation is about 2.7 degrees Kelvin, the supermassive black hole is continue to absorb more energy from the environment that the energy emitted by the Hawking radiation. The net effect is, the supermassive black hole continues to grow even if it has only the space surrounding it to consume.
In a future time, when the temperature of the background radiation falls below that of the temperature of the black hole, it will radiates away by the Hawking radiation. It will take \(10^{68}\) years for the black hole at the center of the galaxy to disappear.
The temperature of a black hole is inversely proportional to mass. (See the above equation.) As a black hole radiates away, its mass decreases, and therefore, its temperature increases. When the black hole at the center of the galaxy reduces to the size of an orange, its temperature will be a whopping \(10^{24}\) degrees Kelvin.
Boltzmann Brains form in about \(10^{10^{68}}\) years.
The second law of thermodynamics is a probabilistic law. That is, there is a very low probability that the entropy can decrease.
If a random collection of particles flitting through the void of structureless, high entropy universe, should by chance, spontaneously dip to a lower -entropy configuration that just happens to match the that of the particles currently constituting your brain. Such a coming together of particles into a special, highly ordered configuration is known as a Boltzmann brain.
The entropy of a system is the natural logarithm of the number of accessible quantum states. So, a system has entropy \(S\), then the number of accessible quantum states is \(e^S\). If we assume that a system spends equal time in any of the microstates comparable with its macrostate, then the probability \(P\) of a fluctuation from an initial state of entropy \(S_1\) to a state of final entropy \(S_2\) is given by \(P=\frac{e^{S_2}}{e^{S_1}}=e^{S_2-S_1}\). Id \(D\) stands for the “drop” in entropy, then \(D=S_1-S_2\), so \(P=e^{-D}\).
What is the probability of forming a Boltzmann Brain?
At a given temperature \(T\), the energies of particles in the thermal bath is very equal to \(T\), by taking \(k=1\). To build a brain of mass \(M\) we need to siphon off about \(\frac{M}{T}\) particles, by taking \(c=1\). The entropy of the bath tracks the number of particles, the drop \(D\) is equal to \(\frac{M}{T}\).
So, the probability of forming a Boltzmann brain is about \(e^{-\frac{M}{T}}\). In a very distant future, we can take \(T\) as the blackbody radiation of the cosmological horizon, i.e., \(T=10^{-30} K\), which is about \(10^{-41}\) GeV, which is roughly the mass of a proton. Since a brain has about \(10^{27}\) protons, \(\frac{M}{T}\approx\frac{10^{27}}{10^{-41}}=10^{68}\). Therefore, the probability of spontaneously forming a brain is about \(e^{-10^{68}}\). The time requires to have a reasonable chance of such a rare event is \(\frac{1}{P}\), namely \(e^{10^{68}}\).
If the cosmological constant changes, then what?
Central to the argument of Boltzmann brains is the formation of a distance cosmological horizon continuously radiating particles, the raw materials for building complex structures including brains. If the "cosmological constant" \(\Lambda\) is not a constant but changes with time, then it is possible to imagine that the accelerating expansion of the universe may stop and the cosmological horizon may not form. In that case, Boltzmann brains will not form.
If the Higgs field changes, then what?
The current value of the Higgs field is ~ 246. Why 246? No one knows. As a result, for the most if not all of the past 13.8 billion years, the masses of elementary particles remain constant. If the Higgs filed jumps to a different value by a process called quantum tunneling, then the masses of elementary particles change and the physics, chemistry, biology all change and life as we know it cease to exist. The projected time scale that Higgs field probably tunnel to a different values is somewhere between \(10^{102}\) and \(10^{359}\) years. If this happens, then again Boltzmann brains will not form.
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