Welcome to Slider142 and Alsakit54's Homepage!

v.06.06.2003

"now once, you had a dream of oceans...and sunken cities,
memories of things you've never known.
that you have never known..."
- R.E.M.-"The Lifting"-"Reveal"

The image above I made as a sample trace and tribute to HSW. Click on it for a better look. It's been forever and a day since I last ray-traced anything.
NEWS : I found an excellent general relativity tutorial at http://pancake.uchicago.edu/~carroll/notes/. There is an excellent description of the Riemann tensor in relation to gravity at this page as well.
I apologize for the general disarray of the site at the moment; I'm in the process of converting everything from nasty table formatting and scattered HTML to more elegant centralized CSS and DHTML. =)

QOTW #4
What is so special about the speed of light ? Why can nothing go faster than it ?

After Maxwell predicted the existence of electromagnetic waves, and calculated their velocity, numerous experiments came to show that visible light was an electromagnetic wave, and others discovered the rest of the spectrum of electromagnetic waves (from here on to be referred to as light), which included radio, microwaves, infrared, ultraviolet, x-ray and gamma rays. The spectrum is continuous, the names only arbitrary to refer to a range of energy (the frequency of the wave is directly related to the energy it can transfer). Einstein later discovered that light was made up of discrete units of energy, called photons. While light travels at different speeds in different media, photons themselves always travel at the speed of light in vacuum, called c. It is their absorption and emission by electrons of the substance that causes the group velocity of light to differ in various media.

What's special about light ? Light can propagate in a vacuum. This raised the interesting question of exactly what medium light was being carried by and with respect to what medium light was travelling at c. For example, sound waves travel at the speed of sound with respect to the medium in which they're travelling, but light didn't seem to have one. What exactly was carrying the energy ? Many theoretical physicists proposed the existence of an aether, which light propagated through. This would mean that light would have differing velocities depending on how one travelled with respect to the aether. Many experiments were done in order to observe such relative velocities, but curiously, light seemed to have the same velocity, no matter how one moved with respect to it! Other physicists, Einstein included, theorized without an aether, and came up with very interesting theories. Einstein's theory, what became known as the special theory of relativity, made use of Lorentz geometry based on Maxwell's equations to predict what should be observed if indeed light propagated independent of a medium. It was only later that photons were brought into play (by Einstein as well) as the carriers of light. It was these photons, then, that travelled at the invariant velocity c. Many experiments were done before the physical world could accept the strange conclusions of special relativity.

Einstein's theory also predicted that massive objects (objects with mass) could not accelerate to c with respect to any inertial frame (It is a misconception that this applies to "everything". It only applies to the specific circumstances mentioned above). An inertial frame is a frame in which the laws of physics (inertial) look the same. Ie., F=ma yields the correct equations of motion in such a frame. Previously, frames with uniform velocity were deemed to be equivalent. Einstein extended this concept to accelerating frames, under certain conditions. As you can see from above, the equivalence of all these frames combined with the invariance of the speed of light with respect to them yielded the most interesting conclusions of special relativity, including length contraction and time dilation, both effects of one frame measuring events in another frame.

Many explanations of the limit of c on massive objects use relativistic mass. While it is easier to explain in that fashion, I will not use that concept as it's a bit misleading. Relativistic mass is a measure of energy, not mass, and is generally not used in theoretical physics. The equation E=mc ² is a solution of E ² = m ² c ⁴ + p ² c ² , when momentum (denoted by p) is zero. In other words, the equation is (Energy in rest frame) = (Rest mass)*(speed of light) ² . A rest frame, or proper frame, is a frame in which the object of interest has no velocity (is at rest). The m in this equation stands for invariant mass, or mass, which does not change with velocity.

Both energy and momentum are functions of velocity, which is transformed by the Lorentz relationship between frames. Mass remains constant. The outcome is that energy behaves asymptotically when measured from differing frames, so you can never measure a massive object to move at c, and the energy put into the object is measured accordingly. Specifically, you get the equation v ² = 1 - m ² /E ² . (How does the graph of this equation behave ? Remember we let c=1). Remove the mass in the equation, however, and the object always travels at c, as photons do, with momentum directly proportional to energy.

Every Sunday, I tackle an interesting question, mostly from the HSW forums. Feedback and suggestions are welcomed. (See my e-mail address below) Take a browse through the store for excellent popular science books and other stuff. :-)

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