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Sunday, November 28, 2010

Cambridge College Programme General Relativity and Superstring Theory Essays

This past summer, I attended the Cambridge College Programme where I spent 3 weeks at Cambridge University taking various classes of my choice as well as having loads of fun, of course.  The experience was quite amazing!!!! One of my classes, Superstrings: The Theory of Everything (taught by Dr. Yves F. J.-M. Gaspar) was extremely engaging and interesting; at times, it was so exciting that I got goosebumps just talking about some of the complex topics that we discussed.  Below are two essays that I wrote regarding General Relativity and Superstring Theory.  Even though these complex topics are quite different from kinematics and dynamics, they are topics that I love wholeheartedly with a great passion.  I thought I might post them for anyone who would like a general description of these two topics.  I encourage everyone who is interested to delve deep into these topics for they are very interesting.

Superstring Theory – Essay Questions

1.     Describe the fundamental postulates of General Relativity.  Explain the role of the curvature of Space-time in General Relativity.  Give the experimental verification of General Relativity.

When Einstein was creating his special theory of relativity in response to problems with the Maxwell field equations, Galilean coordinate translations, and violations of the Galilean relativity principle, he probably was not aware of the impact that it would have on cosmology far into the future.  In spite of this astonishing breakthrough, Einstein was still not satisfied; there were still many questions left unanswered.  Special relativity did not include gravity and was not compatible in non-inertial reference frames.  When unifying the heavens and the earth, Newton had “discovered” gravity, defined it to the best of his ability, and wrote equations that described how it affected bodies.  However, there were many problems with his theory.  For instance, it did not describe the nature of gravity (where it comes from, what creates it, nor the speed of propagation of gravity waves), and there were still problems with how his theory calculated the orbit of Mercury.  Einstein recognized that there were problems and gaps in Newtonian gravity and special relativity.  For instance, in Newton’s equation of gravitation (F=(Gm1m2)/r2), mass and distance can be relative and thus can conflict with the Galilean relativity principle.  As a consequence of all this, Einstein created his theory of general relativity.
General relativity is based upon three main ideas or postulates.  The first is an idea that has its roots in special relativity: the equivalence principle.  The equivalence principle states that the inertial mass of an object is equivalent to the gravitational mass of the object (mi=mg).  Basically, this means that if no outside information is being leaked into a system, one could not perceive whether he/she is in a gravitational field being pulled downward or in a cabin accelerating upward (g=-A).  A consequence of the equivalence principle is that inertial effects cannot be differentiated from gravitational effects, a concept that eventually led to the ideas of the gravitational redshift and gravitational time dilation.  The second postulate is the general covariance principle.  This expansion on the Galilean relativity principle states that the laws of nature do not change form in any reference frame, whether they are inertial, non-inertial, rotational, etc.  Thus, the general covariance principle implies that the laws of general relativity are viable in all situations or circumstances that arise.  The third and final main postulate states that at low speeds and weak gravitational fields, the laws of general relativity should simplify down to and be approximated by the laws of Newtonian gravity; basically, general relativity is insignificant at low speeds and in weak gravitational fields.  While special relativity describes the relativity of time and space, space-time, how the speed of light is constant, and other extremely important concepts, general relativity does something entirely different.  It describes gravity not as a force but as a result of the geometry of space-time; it describes gravity in a topological view of the universe.
Essentially, general relativity views our universe as actually a four-dimensional “fabric-like” continuum that is composed of the three well-known special dimensions fused with time.  Masses that are in this “space-time” cause it to bend around the mass, thus causing space-time to be curved rather than flat; the larger the mass, the sharper the curve and deeper the “well” in space-time that is created.  Because this 4D phenomenon cannot be viewed/visualized well in our 3D world, it is best to visualize it with a simpler example.  If one were to view space-time as a flexible rubber sheet (a 2D analogue of 4D space-time), then one could see how placing a bowling ball or other heavy object would cause a dip or dent in the rubber sheet, indicating a curve in space-time.  Also, using this example, one could see how a smaller mass such as a marble would cause less of a dent than a larger mass such as a bowling ball. 
According to Einstein, what is interpreted as gravity is not the result of masses that exert some sort of force or “field” that attract other masses to them.  In fact the answer is far from it.  General relativity states that what is seen as gravity is actually a result of the curvature of space-time.  The trajectories of objects thrown near large masses – like a baseball thrown on the earth – are not curved because of a force originating from the mass that is pulling on them and changing their trajectories; rather, the trajectories are curved because space-time itself is curved.  They cannot travel in a straight line – despite their inertial tendencies – because they are traveling through bent non-Euclidean non-Minkowski space-time.  If space-time were to be flat, the geodesics of particles and objects would be straight lines.  If space-time is curved, then the geodesics of particles moving through it have to be curved as well (thus making their paths in “our” three special dimensions be curved as well).  It can easily be seen that for length contraction and time dilation to be correct, space-time cannot be described with Euclidian geometry because it breaks some of the major concepts of Euclidean geometry such as the definition of pi (Let, for example, a person be standing within a giant ring that is spinning extremely quickly.  He begins to measure the circumference of the ring with a meter stick and then proceeds to measure the diameter as well.  For an outside observer, however, while the measurement of the diameter is correct because the meter stick underwent length contraction in the direction of measurement when measuring the circumference, more meter sticks were “fit” in the ring then there should have been based upon the measured diameter.  Thus, the ratio of circumference to diameter is greater than pi, breaking some of the principles of Euclidean geometry.)
Einstein’s theory of general relativity has been proven and verified directly and indirectly many times.  In 1919, Sir Arthur Stanley Eddington proved that massive celestial bodies could substantially bend light (large gravitational fields can change the trajectory of light).  During a solar eclipse, the position of a known star near the edge of the sun was measured.  This location was then compared to the known position of the star; they did not match.  The star that was seen was, in reality, behind the sun and should not have been seen.  However, as described by general relativity, the mass of the sun caused space-time to curve, bending the light rays so they were viewable from Earth.  Sir A. Eddington’s measurements of the angle between the actual location of the star and the viewed location of the star perfectly matched Einstein’s predictions, verifying his theory.  A second experimental verification is the Pound-Rebka experiment in 1959.  The experiment consisted of placing sources of gamma rays in a tower (one at the top and one at the bottom) and receivers to detect the gamma rays opposite of the emitters.  By performing the experiment, they noticed that rays that were emitted on the Earth’s surface and traveled upward toward the top of the tower away from Earth were slightly redshifted when the rays that were emitted at the top of the tower and traveled downward toward the Earth were slightly blueshifted.  This experiment proves that time dilation and therefore gravitational redshifts and blueshifts described by general relativity are true.  For the rays that are moving away from Earth, they are redshifted because as they move from a stronger gravitational field to a weaker gravitational field, their rates of time speed up making their frequencies decrease and the rays shift toward the red end of the spectrum.  For the rays that are moving toward Earth, they are blueshifted for just the opposite reason.  As they move from a weaker gravitational field to a stronger gravitational field their rates of time slow down making their frequencies increase and the rays shift toward the blue end of the spectrum.  A third experimental verification of general relativity has to do with observations of binary pulsar systems.  When two heavy celestial bodies come close enough together (usually a white dwarf or a neutron star and another giant, heavy, and dense celestial body), due to their large gravitational fields and effects on space-time, they usually begin to circle around each other.  As they swing around and get closer together, they lose energy by emitting gravity waves (ripples in space-time).  Often, the larger body will start to draw matter from the smaller body.  During this exchange, the matter will often heat up, create strong electromagnetic fields, and emit strong radio signals and X-rays at perfectly regular steady rates.  By these X-rays and radio waves, scientist are able to accurately measure the movements of these pulsar systems and prove that they must be emitting some type of gravity waves to account for the rapid loss of energy.  These results have corresponded to the predictions of general relativity to 10-14 orders of accuracy, thus proving its validity. 

2.     What is the origin of string theory?  How is string theory trying to quantize gravitation?  What are the eventual advantages to use strong theory in order to quantize gravity rather than other quantum field theory techniques?

When first hearing about string theory, many people would think that it is a figment of some writer’s imagination from some science fiction novel.  Its basic structure and elements seem so strange and unconventional to every day life that they can at times seem impossible.  String theory, however, is a viable physical theory that has great potential to solve many of the problems in cosmology and theoretical physics, especially in the fields of quantum physics, quantum field theory, special relativity, and general relativity.  Ironically, for such a complex and heavy mathematical theory, it has some fairly simple origins.  String theory was first created by nuclear physicists who were looking for a mathematical model that would help them understand nuclear interactions by treating nuclei and other objects not as point particles but as extended objects with at least one dimension (strings), thus eliminating many of the problems associated with point particles such as infinities or divergences.  The so-called “string theory,” however, quickly came to the attention of many physicists and cosmologist the world over due to its implications as a possible “grand unification theory.”
When string theory was still in its stages of infancy in the hands of nuclear physicists, there was one major problem that continuously arose from their calculations: the constant appearance of closed strings (loops with no ends, like rubber bands or rings).  This was a major problem for nuclear physicists because closed strings had no physical meaning in their interpretations of their theory; they were simply mathematical nuisances that made things difficult and non-coherent.  Surprisingly, when quantum physicists, cosmologists, and many other physicists began to examine these closed strings, something extraordinary happened; their problem was transformed into a triumph.  These closed loops resembled particles with a spin of two: gravitons.  This discovery was unlike anything seen before.  Not only were closed strings with spin two modes included in the model, but they must be included or the theory is not coherent and violates unitarity, one of the principle concepts of quantum mechanics that describes how in any system, the total probability has to be conserved, or the total amount of information in the system has to be conserved (particles cannot just suddenly disappear forever; something has to happen to them).  This in itself was a stunning revelation.  For the first time in a small way, elements of quantum mechanics (unitarity) were being related to elements of general relativity (gravity).
In essence, the modern goals of string theory were to unify the four interactions (forces) of nature, include a quantum theory of gravity, while at the same time unifying all particles and fields in nature.  One of the largest problems in cosmology and theoretical physics at this current point in history is that every time scientists tried to combine different theories, they ended up with many divergences and infinite sets of infinities.  For example, when quantum mechanical principles were applied to quantum field theory, everything lead to divergences due to the fact that point particles have no dimensional limitations and a field, by its definition, has infinity degrees of freedom.  Take, for example, two point masses that are nearing each other; they can get infinitely close.  According to Newtonian gravity (F=(Gm1m2)/r2), as the distance between them approaches zero, the force between them tends toward infinity.  By Newton’s second law of motion (F=ma), as F approaches infinity and the mass remains the same, the acceleration would have to approach infinity as well eventually surpassing the speed of light, an occurrence that is blatantly a problem.  The same dilemma also occurs if one tries to combine quantum field theory with general relativity; all of the equations blow up and result in divergences.  At the time, these problems were not being solved, possibly due, at least in part, to the fact that general relativity and quantum physics were both non-renormalizable.  When scientists tried to renormalize these two theories, they became stuck with free parameters, an infinity of infinite parameters. 
Conversely, one of the major differences between string theory and many of its contemporaries is that instead of dealing with point particles, it is working with extended objects, thus limiting the number of interactions and creating smoother transitions.  Essentially, there were three main postulates that string theory was based upon.  First, it had to abide and be compatible with special relativity, meaning that it had to be invariant under the Lorentz-Fitzgerald transformations.  Second, it had to follow causality, meaning that the effect of an interaction could not be before the cause.  Third, it had to abide by unitarity, the quantum mechanical principle that was discussed earlier and is related to the conservation of information.  In the more recent years, the addition of supersymmetry to string theory to form superstring theory brought scientists one-step closer to achieving a grand unification theory.  Supersymmetry eliminated the need for tachyons (imaginary particles that move faster than the speed of light and are not coherent with special relativity) and has brought a sense of symmetry to the model between fermions and bosons.  Also, it solved a problem in quantum field theory relating to infinite energy densities in vacuums.
While there are still many roadblocks, superstring theory, which has now been expanded to a more universal M-theory with 6 different sub-theories (all relatable through duality transformations), has many amazing possibilities.  It is quite amazing that in 11-Dimentional Supergravity (one of the low energy limits of M-theory), all forces and particles can be described as different manifestations of the geometry of extra dimensional hyperspace.  M-theory can also solve problems related to the motion of stars in the outskirts of galaxies.  It has been observed that the stars on the outer edges of certain galaxies are spinning faster than they should be according to Kepler’s laws.  On possible explanation of these strange phenomena is that gravitons (closed strings) from other universes that are located in different “brane-worlds” near our own are permeating through hyperspace and affect measurably affecting matter on the edge of galaxies.  Because gravitons are closed strings and not connected to any one brane, they can propagate through hyperspace freely.  This could also be the explanation why gravity seems so weak when compared to the other three forces known in our universe.  Gravity might not be constrained to our universe or brane, but might be distributed throughout all of hyperspace.  Now that superstring theory is being applied in superspace (normal space-time plus extra Grassmann dimensions) and M-theory has been formed, the power of these theories are beginning to grow, it seems as if given time, scientists may be able to complete this theory and finally have a grand unification theory that describes the physical world.  Maybe, those humble nuclear physicists stumbled upon something truly great.

Thursday, July 15, 2010

A Wonderous Wormhole - Honors Geometry Picture Graphing Project

In the field of cosmology and particle physics, the possibility of wormholes is a very mind-boggling and exciting subject. In essence, a wormhole is a hypothetical "shortcut" through spacetime, a theoretical "tunnel" with its end at two different points in spacetime. Even though there has never been any physical/observable evidence of wormholes in physics, there are some solutions to the equations of general relativity that could support wormholes. One possibility for a real wormhole is the Schwarzschild wormhole. However, this was discovered to be too unstable to last long enough to allow any matter to pass through it. Traversable wormholes, wormholes that someone/something could travel through, would only be possible if some type of exotic matter with negative energy density could be used to make it stable.

When I was assigned a project in Honors Geometry to create an image solely by writing equations and graphing them, a wormhole immediately came to mind. I knew that it was going to be a challenge with the many ellipses parabolas, and curves. However, I knew it was something I could do, and wanted to do. I worked on it by hand and on the computer. For the sake of giving due credit, the final picture was graphed and colored with the application GeoGebra.


Some Wormhole Information:
Wormhole Photo:

Tuesday, May 18, 2010

AC/DC - Not the band!?!?

When you hear the acronym AC/DC, what is the first thing that pops into your head? Is it the Australian rock band formed in 1973 by brothers Malcolm and Angus Young? Possibly. For me, however, it has to be electricity, for AC/DC also stands for Alternating Current and Direct Current.

Ok. So we're talking about electricity here, an obscure subject for some but an important one nonetheless.

For all intensive purposes, electricity (electric current) can be defined as the flow of charges (typically electrons) from one place to another with a capability of doing work. For an electric circuit to be present, there must be a closed path in which an electric current can travel from an energy source (positive terminal), to an element in the circuit, and back to the energy source (negative terminal). Conventionally, the direction of the current is considered to be the movement of positive charges from + to - when in reality, it is the movement of negatively charged electrons from - to +. The common elements of a circuit are a battery (energy source), wires, and loads (lights, resistors, etc.) that are arranged in a closed loop with no breaks or gaps.

A DC (Direct Current) circuit is one in which the current travels in one direction continuously (a constant flow of electrons).  An AC (Alternating Current) circuit is one in which the current is continuously and rapidly changing directions (the electrons are moving back and forth).

A Series Circuit:
A series circuit is one in which the elements in the circuit (lights, in this example) are arranged in line with each other, from end to end. Take, for example, the circuit below:
In a series circuit, the current is the same along all points in the wire.  This is due to the fact that there is only one path for electrons to take (like cars traveling along a one lane road; they cannot pass each other).  The equivalent (total) resistance of the circuit equals the sum of all the individual resistances of all the elements in the circuit.  Even though the current is the same at all points, the voltage (potential electrical difference) across two points in the circuit is not.  The voltage drop across each individual element in the circuit equals the voltage drop across the entire circuit (the voltage drop of the energy source).  Basically, the voltages of the batteries are distributed between the elements in the circuit. In this example, the total voltage of the circuit is 75V. Because the two lights have the same resistance, they both have a voltage of 37.5V. Note: If elements in a circuit have different resistances, the distribution of the voltage will not be equal.

A Parallel Circuit:
A parallel circuit is one in which the elements are connected in parallel, meaning that each have a direct connection to the energy source (the lead from the energy source is split into many paths that lead to the elements).
In a series circuit, the voltage drop across each "branch" equals the voltage drop of the total circuit (voltage drop of the energy source).  Even thought there are multiple paths, the voltage in each path is the same because it has, in essence, a direct path to the energy source (battery). However, the current drawn by each branch varies with the the resistance of each branch. Furthermore, as can be derived from Kirchhoff's current law (Conservation of Charge), the total current of the circuit equals the sum of the currents through each branch.  The equivalent resistance of the circuit can be found by taking the inverse of the sum of the inverses of each individual resistance (each new resistance added decreases the equivalent resistance).

A Complex Circuit:
A complex circuit is a combination of elements in series and elements in parallel into one circuit.
In a complex circuit, things become more complicated. To find the equivalent resistance you combine the resistances of individual components (parallel segments and series segments). For this example, the equivalent resistance would be found by adding the resistances of the two series lights and the resistance of the parallel segment (as described above). The total current would be found by dividing the voltage drop of the energy source by the equivalent resistance of the circuit. To find individual voltage drops and currents at specific points, look at each component as a single entity and incorporate it into the larger arrangement. In this example, the parallel segment can be considered in series with the two other lights. In general, in situations similar to the one pictured above, to find the voltages of series elements multiply the individual resistances by the total current.  To find the voltage of the parallel branch, subtract the series voltages from the total voltage. Now, you can find the different currents in the individual branches of the parallel segment: divide the voltage of the parallel branch by each individual resistance.

When working with circuits, do not memorize a certain method that can be used in every situation. Circuits are like puzzles; they take thought to work through every situation.

Have Fun!!!

AC/DC Photo: Photo by Yannick Croissant -
Lightning Photo: Photo by Fort Photo -
Electric Circuit Pictures: Screen shots while working in PhET simulation: