User:Kasparov

Welcome to my laboratory and tour of spacetime amid the Millennium Falcon!

My twin brother and I are physics majors at UNLV in our last year and eager Wikipedians. Our interests primarily lie in Einstein's theory of relativity, quantum physics, statistical mechanics, mathematical logic, and number theory. We will be working with NASA at the Gravitational Astrophysics Laboratory (JPL) in a research program next summer (2007) numerically modeling ripples in spacetime (i.e., gravitational waves). Our goal is to each obtain a doctorate in theoretical physics from CalTech with Kip Thorne as our thesis advisor. We also have a presence guided by the Force on Wookieepedia under the nom de guerre Shon Kon Ray.


 * I, at any rate, am convinced that He [God] does not throw dice. --Albert Einstein


 * Stop telling God what He must do! --Niels Bohr

Special Relativity
The special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies." Some three centuries earlier, Galileo's principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest; a person on the deck of a ship may be at rest in his opinion, but someone observing from the shore would say that he was moving. Einstein's theory combines Galilean relativity with the postulate that all observers will always measure the speed of light to be the same no matter what their state of uniform linear motion is.

This theory has a variety of surprising consequences that seem to violate common sense, but which have been verified experimentally. Special relativity overthrows Newtonian notions of absolute space and time by stating that distance and time depend on the observer, and that time and space are perceived differently, depending on the observer. It yields the equivalence of matter and energy, as expressed in the famous equation E=mc2, where c is the speed of light. Special relativity agrees with Newtonian mechanics in their common realm of applicability, in experiments in which all velocities are small compared to the speed of light.

The theory was called "special" because it applies the principle of relativity only to inertial frames. Einstein developed general relativity to apply the principle generally, that is, to any frame, and that theory includes the effects of gravity. Special relativity doesn't account for gravity, but it can deal with accelerations.

Although special relativity makes relative some quantities, such as time, that we would have imagined to be absolute based on everyday experience, it also makes absolute some others that we would have thought were relative. In particular, it states that the speed of light is the same for all observers, even if they are in motion relative to one another. Special relativity reveals that c is not just the velocity of a certain phenomenon -- light -- but rather a fundamental feature of the way space and time are tied together. In particular, special relativity states that it is impossible for any material object to travel as fast as light.



Postulates of the Special Theory

 * 1) First postulate - The Principle of Relativity - The laws of physics are the same in all inertial frames of reference.
 * 2) Second postulate - The Invariance of c -  The speed of light in a vacuum is a universal constant (c) which is independent of the motion of the light source.

Relativistic mass, momentum, and energy
In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.

There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.

Given an object of invariant mass m0 traveling at velocity v the energy and momentum are given (and even defined) by


 * $$E = \gamma m_0 c^2 \,\!$$


 * $$\vec p = \gamma m_0 \vec v \,\!$$

where &gamma; (the Lorentz factor) is given by


 * $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \,\!$$

and c is the speed of light. The term &gamma; occurs frequently in relativity, and comes from the Lorentz transformation equations.

Relativistic energy and momentum can be related through the formula


 * $$ E^2 - (p c)^2 = (m_0 c^2)^2 \,\!$$

which is referred to as the relativistic energy-momentum equation.

For velocities much smaller than those of light, &gamma; can be approximated using a Taylor series expansion and one finds that


 * $$ E \approx m_0 c^2 + \begin{matrix} \frac{1}{2} \end{matrix} m_0 v^2 \,\!$$


 * $$\vec p \approx m_0 \vec v \,\!$$

Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.

Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and &gamma; = 1) there is a non-zero energy remaining:


 * $$E = m_0 c^2 \,\!$$

This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which are meaningful.

Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:

Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.

This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. The implications of this formula on 20th century life have made it one of the most famous equations in all of science.

General Relativity


General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. It unifies special relativity and Isaac Newton's law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time, this curvature being produced by the mass-energy and momentum content of the spacetime. General relativity is distinguished from other metric theories of gravitation by its use of the Einstein field equations to relate spacetime content and spacetime curvature.

In this theory, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of mass, energy, and momentum (or stress-energy) within it. The relationship between stress-energy and the curvature of spacetime is governed by the Einstein field equations. The motion of objects being influenced solely by the geometry of spacetime (inertial motion) occurs along special paths called timelike and null geodesics of spacetime.

One of the defining features of general relativity is the idea that gravitational 'force' is replaced by geometry. In general relativity, phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion in a curved spacetime. So what people standing on the surface of the Earth perceive as the 'force of gravity' is a result of their undergoing a continuous physical acceleration caused by the mechanical resistance of the surface on which they are standing.

The Einstein Field Equation
The Einstein field equation (EFE) is usually written in the form


 * $$R_{ab} - {1 \over 2}R\,g_{ab} = {8 \pi G \over c^4} T_{ab}.$$

Here $$R_{ab}$$ is the Ricci tensor, $$R$$ is the Ricci scalar, $$g_{ab}$$ is the metric tensor, $$T_{ab}$$ is the stress-energy tensor, and the constants are $$\pi$$ (pi), $$G$$ (the gravitational constant) and $$c$$ (the speed of light).

Wikipedia tags
