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Solitary Physics

In many different physical systems, complex (i.e. non-linear) and delicately balanced interactions between microscopic constituents can collude to form configurations known as solitons. This phenomenon arises in systems as disparate as fluid dynamics, superconductivity, and string theory. In all cases, solitons share common attributes and play a central role in understanding the physical system as a whole. In addition, solitons are some of the most beautiful mathematical constructs that arise in physics.

Imagine a long chain of pendulae, connected to each other by flexible springs. You may want to watch the first video attached to this post at this point to visualize the setup. If you perturb a pendulum in the chain by a small amount - say by bumping it slightly - you'll generate a small disturbance that travels along the chain. A much more interesting thing happens if you grab one of the pendulae and rotate it a full circle: this creates a twist in the chain. When you let go, the localized twist in the chain propagates along it much like a particle would (see first video). This is a soliton: a configuration arising from the strong interactions between the constituents of a system (i.e. the pendulae) - hence forming a collective entity (i.e. the twist) that propagates and behaves coherently much like a solitary particle. All solitons carry energy (or mass) inversely proportional to the strength of the interactions in the system: the stronger the interactions, the lighter or less energy the soliton would have. All solitons preserve their shape and a finite size (which is again inversely proportional to the strength of the interactions) as they propagate. In realistic scenarios like the one depicted in the videos, frictional effects eventually dissipate and unravel the soliton. But notice how long the pendulum twist lives before dissipating! All solitons have a "topological" mechanism that assure their stability. In the case of the pendulae chain, this topological element is the twist in the chain: to disentangle the soliton, you would need to untwist the entire chain which requires a rather large effort - hence the robustness and stability of the pendulum soliton. 

The second video shows a soliton propagating on the surface of water. In this case, you also can see the interactions of two solitons! The two water wave solitons repel each other, yet preserve their shapes after the collision - much like particles! Tsunamis in the ocean can be solitonic and pack a devastating blow when they hit a shoreline. The last video shows toroidal bubble solitons created by dolphins and whales, and even a smoke toroid generated by a nuclear explosion… Other examples of solitons include vortices in superconductivity (see previous post on type II superconductors), vortices in superfluids, and even D-branes in string theory (more on this later)…


The M(other) Theory, part 1 


This post requires reading a previous post titled "Finding The Exit".

Imagine the world of physical systems as an imaginary landscape of lengths, times, and masses - a vast three dimensional expanse where every point corresponds to a particular physical phenomenon, with an associated characteristic length, time, and mass;  an abstract space that visually organizes all natural phenomena we are capable of ever observing. Three fundamental constants of Nature delineate this magical space. They define boundaries that cannot be crossed - three planes that form a pyramid without a base, in-between which the laws of physics confine all our observations of the world… 

One boundary is determined by the speed of light: we may not observe any phenomenon involving speeds faster than that of light… That's the statement of Special Relativity, discovered in 1905.

Another boundary is set by Newton's gravitational constant. If we land on this boundary, we are looking at physical systems that have collapsed under their own weight to form black holes… Beyond the boundary, we are studying physics inside a black hole - a mysterious place disconnected from the rest of the universe, a region where space itself comes to an end. Our understanding of this physical realm started with Newtonian gravity in 1687 and ended with General Relativity in 1915.

The last boundary is defined by Planck's constant of Quantum Mechanics. Beyond it, we are not allowed to peek since even Nature itself does not know what goes on… This is the crazy realm of uncertainties - the world of Quantum Mechanics -formulated by 1927.

The three walls of this pyramid intersect in three lines - one between each pair of planar boundaries. And the three intersection lines intersect at a single point - the tip of the pyramid. The junction between the Quantum and Special Relativity planes is captured by a set of physical laws collectively known as Quantum Field Theories. The small world of particle physics - of electrons, quarks, and the sort - lies in this region of our magical landscape. The junction between the Special Relativity and the Gravitational walls is described by the framework of General Relativity - also known as Einsteinian gravity. It is the realm of astrophysics and cosmology - the large scale structure of space. Quantum Field Theory and General Relativity together form the basis of all the laws of Physics that we can currently write down to describe our entire world; and they do so with much success, perhaps a little too much...

But this still leaves us with one junction - the one between the Quantum and Gravitational planes. Physical phenomenon in this region require the use of the laws Quantum Mechanics and of General Relativity… While this is possible, any attempt at understanding phenomena in this region - such as the physics near the surface of a black hole - quickly leads to puzzles and inconsistent conclusions. Quantum Mechanics and General Relativity do not fit together! It is safe to say that the theoretical framework for understanding this region - formally known as Quantum Gravity - is still work in progress…

How about the apex or tip of the pyramid? Near this region, we have a confluence of all the laws of physics that we know - a region where gravity, quantum mechanics, and special relativity are all simultaneously important… We know with certainty that this is a realm that we simply do not yet understand - a realm that probes the tear in the fabric of space and time at the center of a black hole! 

The field of String Theory purports to capture all the physics inside our pyramid of physical phenomena - even the junction of Quantum Gravity, and all the way up to the enigmatic tip of the pyramid. To do so, String Theory has to render a judgement with respect to the following grand question: if Quantum Mechanics and General Relativity do not fit together, at least one of the two is to blame; if so, which one? String Theory proposes that it is General Relativity that is at fault. It adopts wholeheartedly the frameworks of Quantum Mechanics and Special Relativity - that is the frameworks underlying Planck's constant and the speed of light. But it rejects the framework of General Relativity, proposing instead that General Relativity - along with its gravitational constant - is an approximate theory; a theory that is not fundamental and that needs to be replaced with a more fundamental framework if it is to make sense in conjunction with Quantum Mechanics.

Hence, String Theory starts with the premise that there are three fundamental constants in Nature: the speed of light of Special Relativity, Planck's constant of Quantum Mechanics, and a new third scale in lieu of Newton's gravitational constant. Newton's constant is then proposed to be a derived concept, built up from the new third fundamental constant. We will call this new third fundamental constant, cryptically, "the tension"... 

We just described the first of two postulates that underly the subject of String Theory. The second postulate is a simple one, yet surprisingly powerful and restrictive. It is proposed that all observables in the physical world should be dynamical and computable. That is, we do not allow for any additional scales in Nature beyond the three fundamental constants: the speed of light, Planck's constant, and the "tension". After two decades of hard work by hundreds of lonely and isolated string theorists, these two postulates lead to the proposition that the building blocks of all energy in the world consist of membranes flopping around in a ten dimensional world… And the "tension" constant introduced earlier is simply the stretching tension of these membranes - a measure of their stiffness. We call this theory "Membrane Theory" or "M-theory" for short. In many computationally practical situations however, these membranes appear collapsed as strings - hence arises the alternative historical name of our new theory, "String Theory" - along with a constant known as the "string tension".

We now need to describe two main aspects of this new framework: (1) How does the three dimensional world that we see around us possibly arise from such a (putting things mildly) crazy suggestion?; (2) How does this framework take us beyond what we already know? In particular how does it resolve the original problems and inconsistencies between Quantum Mechanics and General Relativity? Stay tuned for answers to these questions in future posts… In the meantime, have a look at the remaining four parts of video that present a historical narrative of the subject.


Finding The Exit

The process of discovery in physics comes in two basic flavors. Imagine you are thrown in a large and totally dark room, and you are asked to map it out - or even to find the exit door to leave it. You can adopt one of two basic strategies. You could start from a nearby object you can touch and feel, and go from there around the room discovering one adjacent bit of the room at a time. Alternatively, you could "shoot in the dark" by walking towards a random direction until you hit something; if nothing interesting is found, just head into another random direction; and repeat until perhaps you would luck out and even find the door. 

This large dark room represents the physical world around us. The first method of exploration corresponds to a process of meticulous experimentation, discovering bit by bit new phenomena - often from a basis of already well established physics. The second method starts with a pen and paper, without any urgent need for explaining new and exotic data from some experiment. It is driven by logical and sometimes esthetic considerations, and lots of imagination. The first method is robust and the norm for all the sciences, including physics. The second method is rarer and leads mostly to disappointments; but, when it works, it leads to a revolution, to a sudden leap of knowledge perhaps worth centuries of traditional work - to finding the room's exit door, to great enlightenment. 

In the history of physics, both methods have played crucial roles in bringing us to our current state of knowledge of the world. The first method is well known and appreciated; the second less so since it often leads to nowhere. However, its impact on physics is immense. The most dramatic example of a revolutionary leap driven by theoretical considerations was the development of the theory of Special Relativity, and soon after that, the theory of General Relativity by Einstein. In both cases, experiments only followed the theories to quickly confirm the counter-intuitive predictions that were implied by the theories. Soon after these developments, quantum mechanics was formulated in a tour force of physical ingenuity through the first method of physics exploration - through meticulous experiments and ideas developed to explain the results. The beginning of the 20th century thus consisted of a two punch discovery extravaganza - first with theory leading experiment, then experiment leading theory.

String theory is  an exercise of "shooting in the dark" akin to General Relativity. It is motivated by strong indications that our current best understanding of the gravitational force - given by the theory of General Relativity - is logically (and ironically) inconsistent with quantum mechanics... Experimentally, we do not currently have the technology to demonstrate and understand this paradoxical inconsistency. However, theoretical considerations rather overwhelmingly force us to accept that General Relativity cannot be a fundamental description of gravity and spacetime; rather, it must be an approximate framework that would fail to describe the real world correctly when one looks at gravity at very small distances - distances of the order of ten to the power minus 33 centimeters...

Historically, string theory was developed with very different motivations. In a series of posts - of which this is the first introductory one - I will try to give an overview of string theory with the benefit of historical hindsight - using a chain of logic instead of a historical thread. Hence, let's put history aside and talk about this very intriguing new framework of theoretical physics with an attitude that we are attempting to take a giant leap, find the exit, and reach enlightenment that brings unity to all pillars of modern physics: gravity, quantum mechanics, and special relativity. Until the next post in this sequence, have a look at the accompanying four part videos about the subject - presented from a historical perspective. 


The Spinning Carpet

According to the General Theory of Relativity, the space around a massive planet or star is distorted and curved due to the energy content of the planet or star. The distance between two fixed points in space changes when the two points find themselves in the vicinity of an astrophysical object such as the Earth; and this effect is what is perceived by us as Earth's gravitational pull. Things get much more interesting when an astrophysical object is spinning… The fabric of space is dragged around by the spinning object - pulling with it nearby probes. The effect is known as frame-dragging and is expected to be particularly dramatic near the surface of black holes. Near a spinning black hole - a rather common scenario - the space is spun so violently that a nearby spaceship, planet or star is forced to spin with the black hole, whether it wants to do so or not! This same frame-dragging effect is also present close to our spinning planet. But the Earth's gravitational pull is much weaker and the corresponding frame-dragging effect is rather subtle. Despite this, recently, a group of experimental physicists attempted to measure this phenomenon using a special satellite. 

The first video talks briefly about the general aspects of frame-dragging near a black hole. The video starts with a very interesting and somewhat related recent phenomenon involving a jet from a supermassive black hole blowing through another galaxy… The remaining three videos (they are 3 parts of a 25 minute long documentary) present a detailed discussion of the recent experiment that studied frame-dragging near the Earth's surface - the so-called Gravity Probe B experiment.


On Grave Matters About Gravity

Gravity is the force law that acts universally on all forms of matter and energy. If something exists, it must gravitate. Furthermore, gravity exerts the same acceleration on objects of different masses and energies! A feather is pulled by the Earth with the same acceleration as is an elephant - even a rather large elephant with weight management issues. This is known as the "equivalence principle": the principle that the amount of stuff within an object - i.e. its inertial mass - is directly related to the amount that gravitates - i.e. its gravitational mass.

These universal aspects of gravity lead Einstein to propose that gravity is really not a force, but the result of distortions in distances in space. Since all objects have to sail through time and space in a universal way, if the fabric of space-time gets deformed, all things moving on it feel it and can get deflected through a common trajectory. This is the framework of General Relativity: gravity is the curvature of space-time itself; things appear to change course under a gravitational "force" simply because the fabric over which they move is distorted by nearby massive objects. The first video shows a cartoon of this idea. 

Very recently, the string theorist Erik Verlinde proposed an intriguing new perspective on gravity - one that further capitalizes on the universal character of this force law. Verlinde proposed that gravity may be a statistical effect, an entropic force - not a fundamental one akin to electromagnetism. He illustrates his proposal through an analogy with other known entropic effects such as osmosis or polymer dynamics.

In the second video, we see the well-known processes of diffusion and osmosis. When two different solutions of salt water of different concentrations are separated by a semipermeable membrane, over time the two sides equilibrate in salt density. The details of any fundamental force laws acting between the molecules of the water and salt are not important. The water from the less dense solution flows towards the higher density side for statistical reasons: the flow would maximize the level of disorder or entropy in the system (see previous post on the arrow of time). Simply put, the configuration of equilibrated uniform solutions is a more likely arrangement of water and salt molecules and hence will be attained eventually due to statistical considerations. We could be misled to think of this process as one driven by some force that pushes water from one side to the other. However, we should know that this "force" is not a fundamental one, but a fictitious one: it is statistically driven, a collective effect, entropic in origin. This was a larger dose of biology than I intended… so, let's quickly move onto the physics of gravity.

Verlinde proposes that gravity is also an entropic effect rather than a fundamental force law. This is motivated from recent results in string theory, but it can in fact be phrased in an independent and self-contained manner. The universal attributes of gravity encourage such a general scheme or interpretation. And recent suggestions from string theory about gravity having a "holographic" nature further bolster it (I'll write more about this in a future post). 

Verlinde's proposal may suggest that there is no reason to worry about developing a theory of quantum gravity - a rather elusive project of theoretical physics for decades that has only recently been realized within string theory. If gravity is a statistical effect, perhaps there is no sense to worry about its quantum realm at small distances - as much as we would want to worry about forces between molecules in a gas in an effort to understand osmosis. Hence also lies the problem with Verlinde's proposal. At the end of the day, this proposal amounts to the following curious observation: that gravity at large distances has all the hallmarks of a statistical effect. This is not inconsistent however with any microscopic formulation of gravity through string theory - nor does it preclude its need for understanding the small scale structure of space-time… The proposal is a statement about long distance gravity only. It is still an intriguing exercise since it attempts to identify a minimal set of independent physical principles that can realize the well-known long distance attributes of gravity from a statistical point of view.

So, the entropic proposal for gravity to me is simply an interesting story - with a great deal of elegance and pedagogical value. However, it is not a framework that can complete or replace a description of space-time at the smallest distances - a description of quantum gravity. Quoting the ghost of Niels Bohr from a century ago, the proposal is crazy - but it is not crazy enough to be correct…


Mixing The Small With The Big

In the context of standard physics - basically all the physics we know so far and all that we have tested - there is a clear hierarchy between physics at small distances and physics at large distances. We expect that, for any physical system, laws valid at smaller distances are more fundamental, and they can be used to derive less fundamental or approximate laws that work well at larger distances (see previous post on renormalization). This even makes intuitive sense. And when things make intuitive sense, expect new physics to disappoint you at worse, or shock you at best… 

String theory is a framework that unifies the description of all the natural laws within a single consistent language. It relies on quantum mechanics and relativity - two of the three pillars of modern physics - but replaces the third, gravity, with new concepts. As a result, string theory successfully formulates a theory of quantum gravity - a task that had eluded physicists for many decades. It is interesting to try to dissect the mechanism responsible for successfully marrying quantum mechanics and gravity within string theory: what are the key ingredients in string theory - the minimum set - needed to achieve this goal? This is important for two reasons: first, string theory can get elaborate and mathematically complicated, yet the basic physics underpinning should be straightforward and useful to identify to understand the grand picture. Second, it is possible that string theory will not survive the test of time in its current form and may need to evolve and adapt to future experiments; it is then important to pick out from string theory the key pieces that one needs to hold on to, throughout any future evolution.

One of these new key ingredients in the string theory soup is the concept of mixing the big and small. In string theory, when one tries to probe smaller and smaller distances - say by throwing things at each other at high speeds - one finds that, after a certain minimal distance, one cannot probe any smaller distances! Trying to do so leads you to a physical process that instead probes larger and larger distances… the big and small gets mixed and confused! There is an easy way to understand this very strange behavior. In string theory, everything is made of, well, strings… the building blocks of Nature are tiny strings. If you dump energy into a string, say to accelerate it to high speeds, you will initially find that the size of the string shrinks. This is natural and expected from quantum mechanics. With higher energies, you thus can probe smaller distances since your probe - the string - is smaller. However, when you try to dump too much energy into the string, it instead prefers to stretch! This phenomenon is dictated by the properties of the string and good old thermodynamics. The string finds it more favorable to store the extra energy by extending its length against its tension… Hence, since everything is made of strings, you cannot "see" distances smaller than a certain minimum distance no matter how hard you try: the probe that you would use - which is a string - cannot get smaller than a certain minimum size. This seems to be crucial to combining quantum mechanics and gravity. The trouble with traditional gravity is that, at very small distances, it makes no sense. String theory caps things off by providing a mechanism for forbidding you from reaching arbitrarily small distances - and hence resolves the quantum gravity paradox through a very elegant mechanism.

Nowhere this mixing of big and small is more dramatic within string theory than in the process of understanding a black hole horizon (see post on black holes). A black hole - basically a hole in the fabric of space and time - has typically a spherical surface called the horizon that hides the hole in its center. Traditional physics proposes that the horizon of a large black hole is an imaginary boundary that anyone could cross without any hindrance; of course, once you cross it, you are trapped inside the black hole forever (see post on the information paradox). Recent computations in string theory have been suggesting that this may not be the case. When a string falls towards a black hole and reaches the horizon, its size starts to stretch; it stretches so much that, by the time it crosses the horizon, the string is about the size of the horizon! That means you may never be able to reach the problematic hole or tear in space-time behind the horizon since you cannot get a probe smaller than the horizon to penetrate it… 

The accompanying video is an animation I had prepared for a colloquium some time ago showing one version of this picture that arises from string theory: the ball of stringy goo that is sloshing around is the black hole in this framework (known as the Matrix theory formulation). Notice how the goo tends to fuzz up to the surface, sitting at the would-be black hole horizon. In this picture, the center of the black hole may be hollow, everything that is the black hole sits at its horizon, like on a spherical membrane. If you try to cross the horizon, you will be absorbed, consumed, and thermalized by this membrane of goo… Pre-string theory physics proposes instead that all the black hole mass is concentrated at its center, punching a hole in spacetime.


Romancing The Sun


Our Sun, like most stars, spews copious amounts of light energy to heat up its immediate neighborhood; but it also spits out a flux of subatomic particles that forms a halo enveloping the whole of the solar system. These particles - mostly protons and electrons - are energetic enough to escape from the hot surface of the Sun: think of them as vapor off the boiling pot. This halo is called rather poetically the "solar wind", as it blows away from the Sun against all the planets around it. The solar wind interacts with the Earth's upper atmosphere and has many interesting effects - from generating beautiful aurorae near the Poles, to occasionally disrupting satellite communication. 

However, the solar wind pales - in terms of energetic content - compared to the amount of light energy that the Sun floods the solar system with. And you may think of the Sun's light as a sort of a solar wind as well: a solar wind of photons - particles of light - streaming through space. There is even enough photon wind in this light to propel a spaceship… 

Much like flowing air or water, flowing light exerts pressure and momentum. A spaceship with a large reflecting surface - i.e. a mirror - can literally sail through the solar system using the Sun's wind of photons! The effect is not a large one, but it can add up over time to generate appreciable speeds. To put things in perspective, the pressure from sunlight near the Earth is about 100 billion times smaller that air pressure; but it is a thousand times greater than pressure arising from the proton/electron solar wind. Most importantly, the pressure and hence propulsion provided by the photon wind is free… 

Recently, there has been several romantic as well as practical attempts at building sailing spaceships that can ride the Sun's light across the solar system. The attached videos describe this technology. The first video is a general discussion of solar wind of the weaker first kind - made of electrons and protons. The second video is about a prototype solar sail based on a blade configuration that uses photon wind for propulsion. The last video describes the first solar sailship that was recently launched into space by Japan. All we now need are a few solar sailship pirates and we'll be back to the Middle Ages…