David M. Raley
If you ask most physicists why nothing can go faster than the speed of light they usually get out of explaining it by saying that they can't do so without math. When they say that they either:
1. don't want to fool with you and assume you will be frightened away.
2. can't explain it with math either.
You can beat him at his own game by asking to see the math, tolerating the rolling of the eyes, studying it closely and then saying: "Hey, you divided by zero?" He will always look at what he considers the weakest part of his proof and try to explain to you why he is not dividing by zero. You may trust David in this matter.
On this occasion I will undertake to explain only why no object can be observed to approach at the speed of light relative to the observer. Without formulas! The formula in the following paragraph is not part of the explanation and may be skipped if such makes you perspire.
Math is actually shorthand. It should not be confused with arithmetic. It should be taught as a second language. The reason many things are difficult to explain without math is that the complexity and length of the statement is such that you can't hold it all in your, pardon the computerfication of the human mind, RAM. Here is an example that your mind can hold all the longhand of and its math equivalent. Suppose I say to you: "The inductive reactance of a certain coil of wire is equal to twice the ratio of the circumference of a circle to its diameter multiplied by the frequency in cycles per second of the signal applied to said coil and thus multiplied by the reactance of said coil measured in Henrys ." To one who knows the shorthand meanings of the symbols I need only say XL=2piFL. This is one of the more simple electronic formulas. I suspect that the reader will not crave a more complex one.
Light is a form of electromagnetic wave that has been observed to have some particle-like tendencies. Matter is made of particles and particles have been observed to have some wave-like tendencies. Whether these tendencies are true or are artifacts of the observation process is a matter of hair-pulling controversy with some. For this discussion it is assumed that electromagnetic waves are purely waves. Properly understood, waves do not travel, they propagate, I will none the less use the word travel and trust you to know what I mean. Furthermore electromagnetic waves are transverse waves. They develop at right angles to the direction of propagation. Drop a weight into a tub of water and waves propagate on the surface. It used to be thought that light travelled through ether, (aether) . Ether was the fifth element of the ancient sciences. The quintessence. (and) No, the ancients did not think that their four physical elements were all of the forms of matter. They thought that varying quantities of the four physical elements made up all forms of matter; much as you and I believe that electrons, protons, neutrons and the minor subatomics make up all forms of matter. The main reason that ether fell out of favor is not that it was proven nonexistent. (Finally, a legitimate way to employ a double negative.) They just couldn't find the math to make it be entirely a surface! Someone may do so yet but that will be after the fight.
I have found that the easiest way to show that something cannot be done is to assume that it has and describe the result. The recurring question that led to this mindless screed is the question: What would happen if a ship travelling at the speed of light turned on its headlight? Let us, (no, I'm not pregnant, I mean you and me), assume that there is such a ship and that it has a headlight, and a tail light and two side lights. To simplify the discussion we will also assume that all of the lights are producing a unidirectional single wave and that there is no gravitational field. The headlight's wave propagates forward and so forth, the side lights swivel to always point to the side observers. There are four observers. The ship is moving toward one observer at the speed of light, away from one observer at the speed of light and is passing two observers at an angular speed that is determined by the distance separating them. The approached observer will see nothing because the light coming in his direction has a zero wavelength, (infinite frequency). The observer behind will see nothing because the light coming in his direction will have an infinite wavelength, (zero frequency). The side observers will see light of a falling wavelength as the ship comes a'beam to 90 degrees and a rising wavelength as it continues past that point. The ship will not be moving at the speed of light relative to the side observers and the farther away they are from it the slower it will move relative to them.
I will analyze the headlight and the sidelights and leave you to amuse yourself with the tail light. Suppose that the peak to peak power of each light wave is 20. Never mind 20 what, I don't want to pick up a pen pal with nothing on his mind but the proper measure of light intensity. Also assume that the generator somehow puts out the wave in 40 equally spaced packets. Let us measure the wave when the output of the generator says its power is 0, when its power is 1, when its power is 2 and so on. Our first measurement will show zero and the second will show 1. No surprize here, but the third will show 3. Why? The headlight has moved forward the same distance that the wave has and the second packet is on top of the first. Additional packets will be added as the wave peaks at 10 and goes down the other side. When the positive part of the wave has been completed we will be measuring the total of all packets and they will dwell right at the aperture of the headlight. The first measurement after the wave goes negative will show less, for the negative packet will have subtracted from the total. By the time the wave has been completed the resultant light power will be zero. The wave has cancelled itself out. This is why no object can be observed to approach at the speed of light.
It may occur to you at this point that if the ship had been traveling at a slower speed the peak power of the wave would have been increased at the expense of the shortening of the wavelength.
The sidelights will not cancel their own radiation in this manner because they do not overtake their own waves. Therefore the side observers will see the sidelights. If the ship had a single omni-directional light and was circled by observers the ones at 90 degrees would see the light in its true color, those toward the front would see a more and more energetic wave until head on they would see nothing. Actually they would "see" nothing long before the angle where the wave canceled, I use "see" in the sense that they could detect the wave by some means. Those toward the back would see a less and less energetic wave until... But wait, I promised that you could analyze the back side.
P.S. If you really want to know what is observed in the vicinity of a spaceship traveling 99.999999999+ percent of the speed of light, by those who are traveling with it, when the headlight is turned on, do this: Paint a sign that says "Spaceship" and attach it to your car. Turn on the headlights. Observe what happens. Everything in the universe is travelling at 99.999999999+ percent of the speed of light relative to something.