String Theory and extra dimensions

Theory of Relativity and Spacetime Coordinates

The Scheme of Things

Standard Model and the Higgs Field

The Standard Model unifies the nuclear, electromagnetic, and weak forces and enumerates the fundamental building blocks of the universe:

6 leptons: electron, electron neutrino, muon, muon neutrino, tau, tau neutrino


6 quarks: d (down), u (up), s (strange), c (charm), b (bottom), t (top)

Each of these has half-integral spin (called fermions) and each has an anti-particle equivalent.

4 Bosons(integral spin): gluon (nuclear force), photon (electromagnetic force), W and Z bosons (weak force).

The model also has serious flaws--it does not account for gravity, does not explain or predict the masses of the various particles, and requires a number of parameters to be measured and inserted into the theory.

According to the Standard Model, the vacuum in which all particle interactions take place is not actually empty, but is instead filled with a condensate of Higgs particles. The quarks, leptons, and W and Z bosons continuously collide with these Higgs particles as they travel through the "vacuum". The Higgs condensate acts like molasses and slows down anything that interacts with it. The stronger the interactions between the particles and the Higgs condensate are, the heavier the particles become.

Quantum electrodynamics requires the photon to have zero mass, but early attempts to develop an electroweak theory required the bosons to be massless, which is bad because then they would be as abundant as the photons in the universe, which indeed they are not. Peter Higgs and other researchers (who worked independently of Higgs) came across the same idea for settling the puzzle. If there is an otherwise undetectable field filling the universe (now called the Higgs field), it could have associated with it a previously unknown kind of boson, the Higgs particle, which has mass. This would allow any photon-like particle to become massive by swallowing up a Higgs boson. It is thought that all-massive particles get their mass this way.

Hamiltonian and Lagrangian Formalism

Isaac Newton discovered the relationship between force and acceleration,
which he expressed as

Afterwards Leonhard Euler developed the calculus of variations that was to become the most important tool in the tool kit of the theoretical physicist. The calculus of variations was useful for finding curves that were the maximal or minimal length given some set of conditions. Joseph-Louis Lagrange took Euler's results and applied them to Newtonian mechanics, now called the Principle of Least Action, in which the differential equations of motion of a given physical system are derived by minimizing the action of the system in question. For a finite system of objects, the action S is an integral over time of a function called the Lagrange function or Lagrangian L(q, dq/dt), which depends on the set of generalized coordinates and velocities (q, dq/dt) of the system in question.

The differential equations that describe the motion of the system are found by demanding that the action be at its minimum (or maximum) value, where the functional differential of the action vanishes:

This condition gives rise to the Euler-Lagrange equations

which, when applied to the Lagrangian of the system in question, gives the equations of motion for the system. As an example, take the system of a single massive particle with space coordinate x (in zero gravity). The Lagrangian is just the kinetic energy, and the action is the energy integrated over time:

The Euler-Lagrange equations that minimize the action just reproduce Newton's equation of motion for a free particle with no external forces:

The set of mathematical methods described above are collectively known as the Lagrangian formalism of mechanics.
In 1834, William Rowan Hamilton applied his work on characteristic functions in optics to Newtonian mechanics, and what is now called the Hamiltonian formalism of mechanics. The idea that Hamilton borrowed from optics was the concept of a function whose value remains constant along any path in the configuration space of the system, unless the final and initial points are varied. This function in mechanics is now called the Hamiltonian and represents the total energy of the system. The Hamiltonian formalism is related to the Lagrangian formalism by a transformation, called a Legendre transformation, from coordinates and velocities (q, dq/dt) to coordinates and momenta (q,p):

The equations of motions are derived from the Hamiltonian through the Hamiltonian equivalent of the Euler-Lagrange equations:

For a massive particle in zero gravity moving in one dimension, the Hamiltonian is just the kinetic energy, which in terms of momentum, not velocity, is just:

If the coordinate q is just the position of the particle along the x axis then the equations of motion become:

which is equivalent to the answer derived from the Lagrangian formalism. Classical mechanics would have had a brief history if only the motion of finite objects such as cannonballs and planets could be studied. But the Lagrangian formalism and the method of differential equations proved well adaptable to the study of continuous media, including the flows of fluids and vibrations of continuous n-dimensional objects such as one-dimensional strings and two-dimensional membranes. The Lagrangian formalism is extended to continuous systems by the use of a Lagrangian density integrated over time and the D-dimensional spatial volume of the system, instead of a Lagrange function integrated just over time. The generalized coordinates q are now the fields q(x) distributed over space, and we have made a transition from classical mechanics to classical field theory. The action is now written:

Here the coordinate xa refers to both time and space, and repetition implies a sum over all D+1 dimensions of space and time. For continuous media the Euler-Lagrange equations become

with functional differentiation of the Lagrange density replacing ordinary differentiation of the Lagrange function. What is the meaning of the abstract symbol q(x)? This type of function in physics that depends on space and time is called a field, and the physics of fields is called, of course, field theory. The first important classical field theory was Newton's Law of Gravitation, where the gravitational force between two particles of masses m1 and m2 can be written as:

The gravitation force F can be seen as deriving from a gravitational field G, which if we set x1=0 and x2=x, can be written as:

Newton's Law of Gravitation was the beginning of classical field theory. But the greatest achievement of classical field theory came 200 years later and gave birth to the modern era of telecommunications. Physicists and mathematicians in the 19th century were intensely occupied with understanding electricity and magnetism. In the late 19th century, James Clerk Maxwell found unified equations of motion of the electric and magnetic fields, now known as Maxwell's equations. The Maxwell equations in the absence of any charges or currents are:

Maxwell discovered that there exist electromagnetic traveling wave solutions to these equations, which can be rewritten as

and in 1873 he postulated that these electromagnetic waves solved the ongoing question as to the nature of light. The greatest year in classical field theory came in 1884 when Heinrich Hertz generated and studied the first radio waves in his laboratory. Hertz confirmed Maxwell's prediction and changed the world, and physics, forever. Maxwell's theoretical unification of electricity and magnetism was engineered into the modern human power to communicate across space at the speed of light. This was a stunning and powerful achievement for theoretical physics, one that shaped the face of coming 20th century as the century of global telecommunications.

The Cyclic Theory

A new paradigm

In 2003 Steinhardt from Princeton and Turok from Cambridge published their landmark paper on what will be the next paradigm: the cyclic theory.

Here are its salient features:
- space and time exist forever
- the big bang is not the beginning of time; rather, it is a bridge to a pre-existing contracting era
- the Universe undergoes an endless sequence of cycles in which it contracts in a big crunch and re-emerges in an expanding big bang, with trillions of years of evolution in between
- the temperature and density of the universe do not become infinite at any point in the cycle; indeed, they never exceed a finite bound (about a trillion trillions degrees)
- no inflation has taken place since the big bang; the current homogeneity and flatness were created by events that occurred before the most recent big bang
- the seeds for galaxy formation were created by instabilities arising as the Universe was collapsing towards a big crunch, prior to our big bang

Why a new theory? For different reasons, but one very important one was the fact that the universe is accelerating.

Now in the Big Bang Theory (BBT), one uses the analogy of sending a rocket into space. To do that, one must calculate the escape velocity of the rocket. What we do is to look at its total energy, its kinetic energy – how fast must it go – and its potential energy – due to the attraction of the Earth on the rocket. To escape the Earth’s gravitational attraction, we calculate that the rocket’s energy at infinity will be zero.

½m(v squared) + - GMm/R = 0

Where: v is the escape velocity of the rocket
m is the rocket’s mass
M is the Earth’s mass
R is the Earth’s radius
G is a universal constant in Newton’s law of universal gravitation

All values are known except for the escape velocity, which can be worked out from the above equation.

v = sqrt(2GM/R) or v = sqrt(2gR) , Where g is acceleration of gravity on the earth's surface.

The value of v is approximately 11100 m/s (40200 km/h or 25000 mi/hr).

Now the BBT uses this notion to look at how the galaxies are moving away from each others. Notice in this calculation that after attaining escape velocity, the rocket will go to infinity with zero velocity. Should we launch the rocket with a velocity less than that, it will fall back to earth. If its velocity is greater than this escape velocity, it will reach infinity with some velocity to spare. But in none of these scenarios, the rocket – after attaining its escape velocity – will it be accelerating. Now in the BBT, the big bang, when all the matter in the universe was launched into space – the mechanism is obscured about how this would be done – that is comparable to the launching of the rocket. So the question was: did the galaxies have enough velocity to escape? If yes, the universe would expand forever and die in a wimp. If not, the universe would eventually reverse course and die in the big crunch. But when it was found out that the galaxies were accelerating that brought major headaches to the theory. One way out was to postulate Dark Energy. But this ad hoc hypothesis was like doing some patch work. Science doesn’t like patch working, it wants a comprehensive theory. The cyclic theory is one such theory that seems to have a lot of wind in its sails.

See: http://wwwphy.princeton.edu/~steinh/

Entropy, black holes, string theory

One of the most dramatic recent results in string theory is the derivation of the Bekenstein-Hawking entropy formula for black holes obtained by counting the microscopic string states which form a black hole. Bekenstein noted that black holes obey an "area law", dM = K dA, where 'A' is the area of the event horizon and 'K' is a constant of proportionality. Since the total mass 'M' of a black hole is just its rest energy, Bekenstein realized that this is similar to the thermodynamic law for entropy, dE = T dS. Hawking later performed a semiclassical calculation to show that the temperature of a black hole is given by T = 4 k [where k is a constant called the "surface gravity"]. Therefore the entropy of a black hole should be written as S = A/4. Physicists Andrew Strominger and Cumrin Vafa, showed that this exact entropy formula can be derived microscopically (including the factor of 1/4) by counting the degeneracy of quantum states of configurations of strings and D-branes which correspond to black holes in string theory. This is compelling evidence that D-branes can provide a short distance weak coupling description of certain black holes! For example, the class of black holes studied by Strominger and Vafa are described by 5-branes, 1-branes and open strings traveling down the 1-brane all wrapped on a 5-dimensional torus, which gives an effective one dimensional object -- a black hole.
( from: http://www.sukidog.com/jpierre/strings/bholes.htm)

Some have compared this extraordinary result to the significance of Boltzman’s kinetic theory of gases. By postulating that a gas was made of microscopic molecules, he was able to derive the ideal gas formula that had already been known using macroscopic quantities such as pressure, volume and temperature. It was this landmark idea that promoted the concept of atoms and molecules. One would hope that the Strominger and Vafa calculations will do the same for the concept of strings as the basic block of matter/energy.