Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature.- Hermann Weyl
Perspective has been push back in a reductionistic sense and understanding in a cosmological sense. The limit to which this process could incorporate a relativistic explanation would have been a glorious one indeed?
The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances such as liquids and gases. These equations establish that changes in momentum in infinitesimal volumes of fluid are simply the sum of dissipative viscous forces (similar to friction), changes in pressure, gravity, and other forces acting inside the fluid: an application of Newton's second law to fluid.
They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They may be used to model weather, ocean currents, water flow in a pipe, flow around an airfoil (wing), and motion of stars inside a galaxy. As such, these equations in both full and simplified forms, are used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics.
The Navier-Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have yet to prove that in three dimensions solutions always exist (existence), or that if they do exist they do not contain any infinities, singularities or discontinuities (smoothness). These are called the Navier-Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics, and offered a $1,000,000 prize for a solution or a counter-example.
So where is that? Where is the "perfect fluid" and what has this to do with the current state of the universe? The "effect of collisions" which produce a Cerenkov effect. Is this a "faster then the speed of light" from such a process being encapsulating in that early universe condition?
As perplexing as this sounds, it sets up the understanding that Super Cosmologists have to think outside the box. If no information is lost, then where did the information come from? There is a topological unfolding here that speak to mathematical designs all the while it integrates the Navier-Stokes equations in terms of it's relativistic explanation, derived from the very moments of that creation?
"Helium-3 experiment replicates colliding-brane theory of cosmology."
So as silly as some would have you believe that new models do not have any chance from a mathematical perspective of having lost touch with reality, is the need to explain the process in terms of natural occurrences that are going on around us, which we were not previously aware of.
Information Scrambled, Yet Reassembled
My area of research is superstring theory, a theory that purports to give us a quantum theory of gravity as well as a unified theory of all forces and all matter. As such, superstring theory has the potential to realize Einstein's long sought dream of a single, all encompassing, theory of the universe. One of the strangest features of superstring theory is that it requires the universe to have more than three spatial dimensions. Much of my research has focused on the physical implications and mathematical properties of these extra dimensions --- studies that collectively go under the heading "quantum geometry".
Quantum geometry differs in substantial ways from the classical geometry underlying general relativity. For instance, topology change (the "tearing" of space) is a sensible feature of quantum geometry even though, from a classical perspective, it involves singularities. As another example, two different classical spacetime geometries can give rise to identical physical implications, again at odds with conclusions based on classical general relativity.
Superstring theory is most relevant under extreme physical conditions such as those that existed at the time of the big bang. Recently, we have formed a new institute at Columbia called ISCAP (Institute for Strings, Cosmology, and Astroparticle Physics) dedicated to understanding the interface of superstring theory and cosmology. One primary focus of ISCAP is the search for subtle signatures of string theory that may be imprinted in the precision cosmological data that will be collected through a variety of experiments over the next decade.
In levitation post I try to explain how using Susskind's thought experiment we may derive information about the geometrical conditions being developed from "Bob" entering the blackhole on the back of a elephant.
First let me remind you of where you had been taken in terms of your view of the universe. Had you realized that you are now given a micro perspective on the very nature of this universe? That given the circumstance, the elephant takes on a whole new meaning in terms of searching to understand quantum gravity at a level not considered before.
It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind.
The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
"God bless me! but the Elephant
Is very like a WALL!"
The Second, feeling of the tusk,
Cried, "Ho, what have we here,
So very round and smooth and sharp?
To me 'tis mighty clear
This wonder of an Elephant
Is very like a SPEAR!"
The Third approached the animal,
And happening to take
The squirming trunk within his hands,
Thus boldly up and spake:
"I see," quoth he, "the Elephant
Is very like a SNAKE!"
The Fourth reached out an eager hand,
And felt about the knee
"What most this wondrous beast is like
Is mighty plain," quoth he:
"'Tis clear enough the Elephant
Is very like a TREE!"
The Fifth, who chanced to touch the ear,
Said: "E'en the blindest man
Can tell what this resembles most;
Deny the fact who can,
This marvel of an Elephant
Is very like a FAN!"
The Sixth no sooner had begun
About the beast to grope,
Than seizing on the swinging tail
That fell within his scope,
"I see," quoth he, "the Elephant
Is very like a ROPE!"
And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
So what does Susskind do? You see the very question about interpreting events in this way, ask that we push our perceptive toward topological inferences of continuity? There are no current geometrics that can be explained from inside the blackhole. Pushing perspective needed a method to help us orientate what is happening at that geometrical level.
Quantum Gravity: A physical theory describing the gravitational interactions of matter and energy in which matter and energy are described by quantum theory. In most, but not all, theories of quantum gravity, gravity is also quantized. Since the contemporary theory of gravity, general relativity, describes gravitation as the curvature of spacetime by matter and energy, a quantization of gravity implies some sort of quantization of spacetime itself. Insofar as all extant physical theories rely on a classical spacetime background, this presents profound methodological and ontological challenges for the philosopher and the physicist.
Unfortunately I lost the link to a introduction of a book below yet showed this, to help one define the context of the work that has to be done.
Quantum gravity is perhaps the most important open problem in fundamental physics. It is the problem of merging quantum mechanics and general relativity, the two great conceptual revolutions in the physics of the twentieth century. The loop and spinfoam approach, presented in this book, is one of the leading research programs in the field. The first part of the book discusses the reformulation of the basis of classical and quantum Hamiltonian physics required by general relativity. The second part covers the basic technical research directions. Appendices include a detailed history of the subject of quantum gravity, hard-to-find mathematical material, and a discussion of some philosophical issues raised by the subject. This fascinating text is ideal for graduate students entering the field, as well as researchers already working in quantum gravity. It will also appeal to philosophers and other scholars interested in the nature of space and time.