Saturday, November 1, 2008

Einstein's Theory of Relativity

What is Relativity?


Theory of Relativity Concepts

What is Relativity?

Classical relativity (defined initially by Galileo Galilei and refined by Sir Isaac Newton) involves a simple transformation between a moving object and an observer in another inertial frame of reference. If you are walking in a moving train, and someone stationary on the ground is watching, your speed relative to the observer will be the sum of your speed relative to the train and the train's speed relative to the observer. You're in one inertial frame of reference, the train itself (and anyone sitting still on it) are in another, and the observer is in still another.

The problem with this is that light was believed, in the majority of the 1800s, to propagate as a wave through a universal substance known as the ether, which would have counted as a separate frame of reference (similar to the train in the above example). The famed Michelson-Morley experiment, however, had failed to detect Earth's motion relative to the ether and no one could explain why. Something was wrong with the classical interpretation of relativity as it applied to light ... and so the field was ripe for a new interpretation when Einstein came along.

Einstein's Theory of Relativity - Index


Einstein's Theory of Special Relativity


Introduction of Special Relativity

In 1905, Albert Einstein published (among other things) a paper called "On the Electrodynamics of Moving Bodies" in the journal Annalen der Physik. The paper presented the theory of special relativity, based upon two postulates:

Einstein's Postulates

Principle of Relativity (First Postulate): The laws of physics are the same for all inertial reference frames.

Principle of Constancy of the Speed of Light (Second Postulate): Light always propagates through a vacuum (i.e. empty space or "free space") at a definite velocity, c, which is independent of the state of motion of the emitting body.

Actually, the paper presents a more formal, mathematical formulation of the postulates. The phrasing of the postulates are slightly different from textbook to textbook because of translation issues, from mathematical German to comprehensible English.

The second postulate is often mistakenly written to include that the speed of light in a vacuum is c in all frames of reference. This is actually a derived result of the two postulates, rather than part of the second postulate itself.

The first postulate is pretty much common sense. The second postulate, however, was the revolution. Einstein had already introduced the photon theory of light in his paper on the photoelectric effect (which rendered the ether unnecessary). The second postulate, therefore, was a consequence of massless photons moving at the velocity c in a vacuum. The ether no longer had a special role as an "absolute" inertial frame of reference, so it was not only unnecessary but qualitatively useless under special relativity.

As for the paper itself, the goal was to reconcile Maxwell's equations for electricity and magnetism with the motion of electrons near the speed of light. The result of Einstein's paper was to introduce new coordinate transformations, called Lorentz transformations, between inertial frames of reference. At slow speeds, these transformations were essentially identical to the classical model, but at high speeds, near the speed of light, they produced radically different results.

Effects of Special Relativity

Special relativity yields several consequences from applying Lorentz transformations at high velocities (near the speed of light). Among them are:
  • Time dilation (including the popular "twin paradox")
  • Length contraction
  • Velocity transformation
  • Relativistic velocity addition
  • Relativistic doppler effect
  • Simultaneity & clock synchronization
  • Relativistic momentum
  • Relativistic kinetic energy
  • Relativistic mass
  • Relativistic total energy
In addition, simple algebraic manipulations of the above concepts yield two significant results that deserve individual mention.

Mass-Energy Relationship

Einstein was able to show that mass and energy were related, through the famous formula E=mc2. This relationship was proven most dramatically to the world when nuclear bombs released the energy of mass in Hiroshima and Nagasaki at the end of World War II.

Speed of Light

No object with mass can accelerate to precisely the speed of light. A massless object, like a photon, can move at the speed of light. (A photon doesn't actually accelerate, though, since it always moves exactly at the speed of light.)

But for a physical object, the speed of light is a limit. The kinetic energy at the speed of light goes to infinity, so it can never be reached by acceleration.

Some have pointed out that an object could in theory move at greater than the speed of light, so long as it did not accelerate to reach that speed. So far no physical entities have ever displayed that property, however.

Adopting Special Relativity

In 1908, Max Planck applied the term "theory of relativity" to describe these concepts, because of the key role relativity played in them. At the time, of course, the term applied only to special relativity, because there was not yet any general relativity.

Einstein's relativity was not immediately embraced by physicists as a whole, because it seemed so theoretical and counterintuitive. When he received his 1921 Nobel Prize, it was specifically for his solution to the photoelectric effect and for his "contributions to Theoretical Physics." Relativity was still too controversial to be specifically referenced.

Over time, however, the predictions of special relativity have been shown to be true. For example, clocks flown around the world have been shown to slow down by the duration predicted by the theory.

Lorentz Transformations



Albert Einstein didn't create the coordinate transformations needed for special relativity. He didn't have to, because the Lorentz transformations that he needed already existed. Einstein was a master at taking previous work and adapting it to new situations, and he did so with the Lorentz transformations just as he had used Planck's 1900 solution to the ultraviolet catastrophe in black body radiation to craft his solution to the photoelectric effect, and thus develop the photon theory of light.

Origins of Lorentz Transformations

The transformations were actually first published by Joseph Larmor in 1897. A slightly different version had been published a decade earlier by Woldemar Voigt, but his version had a square in the time dilation equation. Still, both versions of the equation were shown to be invariant under Maxwell's equation.

The mathematician and physicist Hendrik Antoon Lorentz proposed the idea of a "local time" to explain relative simultaneity in 1895, though, and began working independently on similar transformations to explain the null result in the Michelson-Morley experiment. He published his coordinate transformations in 1899, apparently still unaware of Larmor's publication, and added time dilation in 1904.

In 1905, Henri Poincare modified the algebraic formulations and attributed them to Lorentz with the name "Lorentz transformations," thus changing Larmor's chance at immortality in this regard. Poincare's formulation of the transformation was, essentially, identical to that which Einstein would use.

The transformations apply to a four-dimensional coordinate system, with three spatial coordinates (x, y, & z) and one time coordinate (t). The new coordinates are denoted with an apostrophe, pronounced "prime," such that x' is pronounced x-prime. In the example below, the velocity is in the xx' direction, with velocity u:

x' = ( x - ut ) / sqrt ( 1 - u2 / c2 )

y' = y

z' = z

t' = { t - ( u / c2 ) x } / sqrt ( 1 - u2 / c2 )

The transformations are provided primarily for demonstration purposes. Specific applications of them will be dealt with separately. The term 1/sqrt (1 - u2/c2) so frequently appears in relativity that it is denoted with the Greek symbol gamma in some representations.

It should be noted that in the cases when u << c, the denominator collapses to essentially the sqrt(1), which is just 1. Gamma just becomes 1 in these cases. Similarly, the u/c2 term also becomes very small. Therefore, both dilation of space and time are non-existent to any significant level at speeds much slower than the speed of light in a vacuum.

Consequences of the Transformations

Special relativity yields several consequences from applying Lorentz transformations at high velocities (near the speed of light). Among them are:
  • Time dilation (including the popular "twin paradox")
  • Length contraction
  • Velocity transformation
  • Relativistic velocity addition
  • Relativistic doppler effect
  • Simultaneity & clock synchronization
  • Relativistic momentum
  • Relativistic kinetic energy
  • Relativistic mass
  • Relativistic total energy

Lorentz & Einstein Controversy

Some people point out that most of the actual work for the special relativity had already been done by the time Einstein presented it. The concepts of dilation and simultaneity for moving bodies were already in place and the mathematics had already been developed by Lorentz & Poincare. Some go so far as to call Einstein a plagiarist.

There is some validity to these charges. Certainly, the "revolution" of Einstein was built on the shoulders of a lot of other work, and Einstein got far more credit for his role than those who did the grunt work.

At the same time, it must be considered that Einstein took these basic concepts and mounted them on a theoretical framework which made them not merely mathematical tricks to save a dying theory (i.e. the ether), but rather fundamental aspects of nature in their own right. It is unclear that Larmor, Lorentz, or Poincare intended so bold a move, and history has rewarded Einstein for this insight & boldness.


Einstein's Theory of General Relativity

In Albert Einstein's 1905 theory (special relativity), he showed that among inertial frames of reference there was no "preferred" frame. The development of general relativity came about, in part, as an attempt to show that this was true among non-inertial (i.e. accelerating) frames of reference as well.

Evolution of General Relativity

In 1907, Einstein published his first article on gravitational effects on light under special relativity. In this paper, Einstein outlined his "equivalence principle," which stated that observing an experiment on the Earth (with gravitational acceleration g) would be identical to observing an experiment in a rocket ship that moved at a speed of g. The equivalence principle can be formulated as:
we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.

as Einstein said or, alternately, as one Modern Physics book presents it:

There is no local experiment that can be done to distinguish between the effects of a uniform gravitational field in a nonaccelerating inertial frame and the effects of a uniformly accelerating (noninertial) reference frame.

A second article on the subject appeared in 1911, and by 1912 Einstein was actively working to conceive of a general theory of relativity that would explain special relativity, but would also explain gravitation as a geometric phenomenon.

In 1915, Einstein published a set of differential equations known as the Einstein field equations. Einstein's general relativity depicted the universe as a geometric system of three spatial and one time dimensions. The presence of mass, energy, and momentum (collectively quantified as mass-energy density or stress-energy) resulted in a bending of this space-time coordinate system. Gravity, therefore, was movement along the "simplest" or least-energetic route along this curved space-time.

The Math of General Relativity

In the simplest possible terms, and stripping away the complex mathematics, Einstein found the following relationship between the curvature of space-time and mass-energy density:

(curvature of space-time) = (mass-energy density) * 8 pi G / c4
The equation shows a direct, constant proportion. The gravitational constant, G, comes from Newton's law of gravity, while the dependence upon the speed of light, c, is expected from the theory of special relativity. In a case of zero (or near zero) mass-energy density (i.e. empty space), space-time is flat. Classical gravitation is a special case of gravity's manifestation in a relatively weak gravitational field, where the c4 term (a very big denominator) and G (a very small numerator) make the curvature correction small.

Again, Einstein didn't pull this out of a hat. He worked heavily with Riemannian geometry (a non-Euclidean geometry developed by mathematician Bernhard Riemann years earlier), though the resulting space was a 4-dimensional Lorentzian manifold rather than a strictly Riemannian geometry. Still, Riemann's work was essential for Einstein's own field equations to be complete.

What Does General Relativity Mean?

For an analogy to general relativity, consider that you stretched out a bedsheet or piece of elastic flat, attaching the corners firmly to some secured posts. Now you begin placing things of various weights on the sheet. Where you place something very light, the sheet will curve downward under the weight of it a little bit. If you put something heavy, however, the curvature would be even greater.

Assume there's a heavy object sitting on the sheet and you place a second, lighter, object on the sheet. The curvature created by the heavier object will cause the lighter object to "slip" along the curve toward it, trying to reach a point of equilibrium where it no longer moves. (In this case, of course, there are other considerations -- a ball will roll further than a cube would slide, due to frictional effects and such.)

This is similar to how general relativity explains gravity. The curvature of a light object doesn't affect the heavy object much, but the curvature created by the heavy object is what keeps us from floating off into space. The curvature created by the Earth keeps the moon in orbit, but at the same time the curvature created by the moon is enough to affect the tides.

Proving General Relativity

All of the findings of special relativity also support general relativity, since the theories are consistent. General relativity also explains all of the phenomena of classical mechanics, as they too are consistent. In addition, several findings support the unique predictions of general relativity: Findings such as gravitational radiation, dark matter, & dark energy are predicted, but have not yet been identified.

Einstein's Theory of Special Relativity


Introduction of Special Relativity

In 1905, Albert Einstein published (among other things) a paper called "On the Electrodynamics of Moving Bodies" in the journal Annalen der Physik. The paper presented the theory of special relativity, based upon two postulates:

Einstein's Postulates

Principle of Relativity (First Postulate): The laws of physics are the same for all inertial reference frames.

Principle of Constancy of the Speed of Light (Second Postulate): Light always propagates through a vacuum (i.e. empty space or "free space") at a definite velocity, c, which is independent of the state of motion of the emitting body.

Actually, the paper presents a more formal, mathematical formulation of the postulates. The phrasing of the postulates are slightly different from textbook to textbook because of translation issues, from mathematical German to comprehensible English.

The second postulate is often mistakenly written to include that the speed of light in a vacuum is c in all frames of reference. This is actually a derived result of the two postulates, rather than part of the second postulate itself.

The first postulate is pretty much common sense. The second postulate, however, was the revolution. Einstein had already introduced the photon theory of light in his paper on the photoelectric effect (which rendered the ether unnecessary). The second postulate, therefore, was a consequence of massless photons moving at the velocity c in a vacuum. The ether no longer had a special role as an "absolute" inertial frame of reference, so it was not only unnecessary but qualitatively useless under special relativity.

As for the paper itself, the goal was to reconcile Maxwell's equations for electricity and magnetism with the motion of electrons near the speed of light. The result of Einstein's paper was to introduce new coordinate transformations, called Lorentz transformations, between inertial frames of reference. At slow speeds, these transformations were essentially identical to the classical model, but at high speeds, near the speed of light, they produced radically different results.

Effects of Special Relativity

Special relativity yields several consequences from applying Lorentz transformations at high velocities (near the speed of light). Among them are:
  • Time dilation (including the popular "twin paradox")
  • Length contraction
  • Velocity transformation
  • Relativistic velocity addition
  • Relativistic doppler effect
  • Simultaneity & clock synchronization
  • Relativistic momentum
  • Relativistic kinetic energy
  • Relativistic mass
  • Relativistic total energy
In addition, simple algebraic manipulations of the above concepts yield two significant results that deserve individual mention.

Mass-Energy Relationship

Einstein was able to show that mass and energy were related, through the famous formula E=mc2. This relationship was proven most dramatically to the world when nuclear bombs released the energy of mass in Hiroshima and Nagasaki at the end of World War II.

Speed of Light

No object with mass can accelerate to precisely the speed of light. A massless object, like a photon, can move at the speed of light. (A photon doesn't actually accelerate, though, since it always moves exactly at the speed of light.)

But for a physical object, the speed of light is a limit. The kinetic energy at the speed of light goes to infinity, so it can never be reached by acceleration.

Some have pointed out that an object could in theory move at greater than the speed of light, so long as it did not accelerate to reach that speed. So far no physical entities have ever displayed that property, however.

Adopting Special Relativity

In 1908, Max Planck applied the term "theory of relativity" to describe these concepts, because of the key role relativity played in them. At the time, of course, the term applied only to special relativity, because there was not yet any general relativity.

Einstein's relativity was not immediately embraced by physicists as a whole, because it seemed so theoretical and counterintuitive. When he received his 1921 Nobel Prize, it was specifically for his solution to the photoelectric effect and for his "contributions to Theoretical Physics." Relativity was still too controversial to be specifically referenced.

Over time, however, the predictions of special relativity have been shown to be true. For example, clocks flown around the world have been shown to slow down by the duration predicted by the theory.



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