Wednesday, March 14, 2007

Speed of Light


Constant velocity from all inertial reference frames

It is important to realise that the speed of light is not a "speed limit" in the conventional sense. An observer chasing a beam of light will measure it moving away from him at the same speed as will a stationary observer. This leads to some unusual consequences for velocities.

Most individuals are accustomed to the addition rule of velocities: if two cars approach each other from opposite directions, each travelling at a speed of 50 km/h, relative to the road surface, one expects that each car will perceive the other as approaching at a combined speed of 50 + 50 = 100 km/h to a very high degree of accuracy.

At velocities at or approaching the speed of light, however, it becomes clear from experimental results that this rule does not apply. Two spaceships approaching each other, each travelling at 90% the speed of light relative to some third observer between them, do not perceive each other as approaching at 90% + 90% = 180% the speed of light; instead they each perceive the other as approaching at slightly less than 99.5% the speed of light.

This last result is given by the Einstein velocity addition formula:

u = {v + w \over 1 + v w / c^2} \,\!

where v and w are the speeds of the spaceships as observed by the third observer, and u is the speed of either space ship as observed by the other.

Contrary to one's usual intuitions, regardless of the speed at which one observer is moving relative to another observer, both will measure the speed of an incoming light beam as the same constant value, the speed of light.

The above equation was derived by Albert Einstein from his theory of special relativity, which takes the principle of relativity as a main premise. This principle (originally proposed by Galileo Galilei) requires physical laws to act in the same way in all reference frames. Maxwell’s equations predict a speed of light, in much the same way as is the speed of sound in water. The speed of sound in water is a function of physical constants proper to water. The speed of light was believed to be relative to characteristics of the medium of transmission for light that acted as does water for the transmission of sound -- the luminiferous aether. But the Michelson-Morley experiment, arguably the most famous and useful failed experiment in the history of physics, could not find any trace of this luminiferous aether, suggesting, as a result, that it is impossible to detect one's presumed absolute motion, i.e., motion with respect to the hypothesized luminiferous aether. It should be noted that the Michelson-Morley experiment said little about the speed of light relative to the light’s source and observer’s velocity, as both the source and observer in this experiment were travelling at the same velocity together in space.

Technical impossibility of travel faster than the speed of light

To understand why an object cannot travel faster than light, it is useful to understand the concept of spacetime. Spacetime is an extension of the concept of three-dimensional space to a form of four-dimensional space-time. Having the classical concepts of height, width, and depth as the first three dimensions, the new, fourth dimension is that of time. Graphically it can be imagined as a series of static,three-dimensional 'bubbles', positioned along an arbitrarily chosen line, each bubble representing a separate position along one of the four dimensions. That graphical approach is analogous to using a sequence of two-dimensional cross-sections taken at some standard interval along the third dimension to represent a three-dimensional object on a two-dimensional surface. (Imagine a map of a multi-story building that is created by giving the floor plan for each story of the building on a new page.) The mapping of space and time can be rotated so that, e.g., the x dimension is replaced by the t dimension, and each "bubble" represents a cross-section taken along the x dimension. Supposing that travel is occurring along the y and or the z dimension, what one will observe is that change along the t dimension will decrease from "bubble" to "bubble" as change across the y-z plane increases from "bubble" to "bubble."

With this understood, there is a clear implication that an object has a total velocity through space-time at any instant, and for all particles of matter this velocity is equal to the speed of light. While this result may seem contradictory to the idea of speed-of-light travel being impossible, it in fact proves it, taking into account the fact that faster-than-light travel was a spatial, or three-dimensional concept, not a four-dimensional concept. In the case of four-dimensions, all of the total velocity of an object not accounted for in three-dimensional space is in the fourth dimension, or time. To go back to our bubble picture, if an object is remaining at the same x, y, z positions it will make maximum progress in the t dimension. And that is just to say that any clock associated with whatever we are watching at x, y, z is ticking away at its maximum rate according to a static observer in the same frame of reference, e.g., somebody at x+3, y+4, z+5 or any other position that is not changing with respect to x, y, and z. But the greater the changes of x, y, and z according to the clock of the other observer, the smaller will be the changes in t. But using the Phythagorean theorem to calculate the distances between a point at x,y,z,t and some later point x', y', z', t', then those distances will always be the same.

While this may seem confusing, it shows that as displacement through space increases, measured time will decrease to maintain the overall space-time velocity. If this is the case, it makes speed-of-light travel impossible, since when as an object approaches the speed of light spacially, it will have to approach zero velocity temporally. Another implication is that an object might be said to travel through four-dimensional space-time at the speed of light, but only in cases wherein its velocity through space is zero. That statement is just a counter-intuitive way of expressing the idea that when one is motionless (according to another observer) one's clock is ticking away most rapidly, and that as one moves faster and faster (according to the other observer) one's clock is ticking at slower and slower rates that approach zero.

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