Dash27 said:
Does anyone here know of any book, or preferably online video or text that Greene talks specifically about the math that leads him and others to believe there might be a multiverse? He brings it up often but I never heard or read of any specific example of what sort of math would lead him to that conclusion.
There are a few paths to it that I can think of. First, I'll go with
General Relativity:
It is in GR that the concept of a black hole becomes possible. The gravitational collapse of a star after its life is done is sometimes strong enough to create spacetime configurations with singularities in them. The gravitational pull of that singularity is, in the parlance of GR, strong enough that all radial geodesics point toward the singularity past a certain hypersurface in spacetime. This hypersurface is called the event horizon of the black hole. Mathematically, this manifests itself through a variety of different coordinatizations of the Schwarzchild solution to the Einstein equations.
There are a few different types of black holes. Contrary to something like, say, a planet, a black hole is a very simple object. You can look up something called a "no-hair theorem", but essentially it says that black holes have no microscopic structure that we are aware of to distinguish them beyond three parameters: mass, angular momentum and charge. The Schwarzchild black hole is an idealization; there are probably no true Schwarzchild black holes because they are virtually guaranteed to have net angular momentum and charge. Black holes with charge correspond to the Reissner Nordstrom solution, black holes that are spinning (angular momentum) correspond to the Kerr solution.
These charged and spinning black holes have some extraordinary properties regarding their event horizons. They have more than one horizon, and this permits the possibility of going inside the outer horizon where all trajectories are inwards, then inside the inner horizon where they are NOT all inwards. It is possible to escape charged and spinning black holes, but there is a catch. Diagrammatically, you'll encounter something called Penrose or conformal diagrams that compactify infinite spacetimes into little squares through something in GR called conformal equivalence. When you draw the diagrams for these black hole spacetimes, you find that when you leave these charged and spinning black holes, you do not return to the spacetime you started with, but rather to another copy of it. In essence, you enter another universe.
Obviously there are a fuckload of details that I am leaving out. You'd have to read half of a textbook on GR to get to this stuff, but the core point is that there are black holes that can theoretically be escaped and when you escape them you end up in other universes. These feed into a concept called a white hole, which you can read about on Wikipedia if you so desire.
Quantum Mechanics:
The source of a multiverse in quantum mechanics is a question of interpretation of how the probabilistic nature of quantum mechanics actually works. This is sometimes called the EPR or measurement paradox. In traditional QM (sometimes called the Copenhagen interpretation), we are taught to think that a quantum system is somehow separated from our "physical" reality by an intermediate object called a wave function or state vector that describes the state of a quantum mechanical system. Mathematically, it lives in something called a Hilbert space, and we define measurement of the object through operating on that state vector with a Hermitian operator with a spectrum of possible outcomes that correspond to physical states and values for the measurement. This measurement results in something called wavefunction collapse, where the quantum state collapses into an eigenstate of the operator with a corresponding eigenvalue.
This is just one convenient interpretation. This leaves a big question of "what happened to all the other states that it could have collapsed into?" In terms of Schrodinger's cat, if we measured him and found him alive, what happened to the possibility that it was dead? The alternative that leads to a multiverse theory is called the many-worlds interpretation, and it answers this question in a way that is actually somewhat satisfying. In this interpretation, all the other outcomes that are precluded when a measurement occurs and our universe is observed to be in precisely one of the possible states, all of the other possible states still happen in other universes. In terms of the cat example, there is one universe where the cat is dead and one where it is alive. Every time a measurement occurs, there is a branching off of universes where all the outcomes happen, and we happen to exist consciously in one of them.
This is a somehow satisfying solution to the paradoxes in my opinion, but also hugely difficult to test and believe to be true. It basically says that there is an infinity of other universes in which every conceivable quantum outcome in every quantum interaction ever has come true. There are an infinite number of states in even very simple quantum states. There are different scales of infinity in the study of pure mathematics, and on those scales, this infinity of universes is pretty fucking huge. There are many guesses that go into this interpretation, and many mechanisms we obviously don't understand and we may never even be able to test.
. I'll be sure to check this one out.