Headlights, Light Waves, Sound WavesDate: 01/26/98 at 23:20:16 From: Greg Gowens Subject: Speed of light Dear Dr. Math, I had a Physics class that dealt with Relativity and a question came up that seemed to be manipulated by the mathematics: If you are in a car traveling at the speed of light, what happens when you turn your lights on? Some people say that the lights will not project outward if you are already traveling at the speed of light, since that is as fast as anything can go. Other people say that your lights come on just as usual. When you put the problem through the mathematical formulas involving the speed of light, you can never exceed c where c is the speed of light. I understand that sometimes light acts like a wave and other times it acts like a particle. Is this another case of mathematics getting in the way of common sense? Thanks for your help, Greg Gowens Date: 01/30/98 at 09:07:26 From: Doctor Luis Subject: Re: Speed of light The answer is: the lights turn on as normal. The reason is that all observers measure the same speed of light, so that for the observer B in the hypothetical car moving at the speed of light with respect to another stationary observer A, BOTH A and B measure the same speed of light. Now, all of this may seem contradictory, since supposedly there is no way to exceed the speed of light, so how can both observers measure the same speed for light. However, this is because you cannot just add the velocities for these two different frames. Velocity is the time rate of change of position, but the two observers don't even agree on the measurements of position and time! (simultaneous events in one frame are not simultaneous in the moving frame, and you also have lorentz contraction along the direction of motion) If you denote the moving observer (speed u) by primed coordinates, and a stationary observer with unprimed coords, the velocity transforms as follows: x' = x - u*t or x = x' + u*t t' = t dx d v = -- = -- (x'+u*t') dt dt' dx' dt' = -- + u * -- dt' dt dx' dt' dt' = -- * -- + u * -- dt' dt dt Notice that in the moving frame we are measuring the time rate of change of x' with respect to t', and in the stationary frame with respect to t. Now, for the Galilean transformation, t = t', so dt'/dt = 1, and so dx dx' -- = -- * 1 + u * 1 dt dt' or dx/dt = u+v. The interpretation of this is: to obtain the speed of an object (in the stationary frame) that is moving with a speed v (in the moving frame) you just add the speed of the frame to the speed of the object in the moving frame This argument is fine, except that the galilean transformation is wrong! (It's a pretty good approximation for low speeds though...) Remember that from the lorentz transformation, t does NOT equal t'. In fact: t' + ux'/c^2 x' + ut' t = -------------- and x = ---------------- sqrt(1-(u/c)^2) sqrt(1-(u/c)^2) so our previous expressions become, dt 1 + (u/c^2)*(dx'/dt') 1 + (uv/c^2) -- = --------------------- = ------------- dt' sqrt(1-(u/c)^2) sqrt(1-(u/c)^2) and so, dx (dx'/dt) + u(dt'/dt) -- = ---------------------- dt sqrt(1-(u/c)^2) (dx'/dt) is just (dx'/dt')*(dt'/dt) by the chain rule, so dx (dx'/dt')*(dt'/dt) + u(dt'/dt) -- = --------------------------------- dt sqrt(1-(u/c)^2) if you factor out the dt'/dt which is just 1/(dt/dt') you get (with dx'/dt' = v) dx v + u sqrt(1-(u/c)^2) -- = --------------- * --------------- dt sqrt(1-(u/c)^2) 1 + (uv/c^2) u + v = ------------- 1 + (uv/c^2) As you can see, this is different from our usual addition law, in that there is an extra factor, and you can see that for low velocities the product uv is way less than c^2 so that you get the familiar (u+v)/(1+really small term) = u+v So what happens when v = c? (that is, when the moving frame moves at the speed of light) and u = c (say, a flashlight is on in the moving frame)? what does the stationary observer measure for the speed of the photons coming out of the flashlight? (dx/dt) = (c+c)/(1+(cc/c^2)) = 2c/2 = c He measures the same speed of light! This is consistent with Einstein's postulate of the constancy of the speed of light. Now, say I am in a frame that is moving at half the speed of light with respect to some stationary frame, and I throw a baseball at half the speed of light. What would be the speed of the baseball that the stationary frame measures? Is it 0.5c+0.5c = c ? (dx/dt) = (0.5c + 0.5c)/(1+(0.5c*0.5c/c^2)) = c/(1+(1/4)) = c/(5/4) = (4/5)c = 0.8 c As you can see, it is still less than c ! What is so mysterious or special about light? Why do all observers HAVE to measure the same speed for light? You can think of it in terms of sound. The sound waves do not propagate with the moving source, but rather on a stationary medium (air). That is why you hear the pitch of the sound increase when a sound source approaches you (the source is moving so the sound is compressed (thus higher frequency)). By the way, there are cases when particles travel faster than the speed of light, as in Cerenkov radiation. Of course, that's because light travels slower in a medium (the index of refraction is a measure of how much slower), and so some particles may travel faster than light in a medium. However, no particle may exceed the vacuum speed of light. This is because in order to have conservation of momentum, the mass of an object must increase with the speed. This means that in order to get closer and closer to the the speed of light, you have to accelerate faster and faster since your inertia is also increasing, but since you can an get arbitrarily large mass if you get close enough to the speed of light, you will never have enough energy to reach the speed of light. To do that you would need an infinite amount of energy to accelerate an infinitely large mass! Photons, by the way, have no rest mass to begin with, so they're fine! Now, mathematics does not get in the way of common sense. Common sense just falls over and trips quite frequently :) In fact, the mathematics is merely a deductive process whereby we may formally set down the consequences of a certain set of axioms. You could lay down the entire foundation of mathematics on a mere verbal description of your arguments, but you wouldn't get very far that way. Mathematics gives you the psychological advantage of expressing your ideas concisely in a rigorous and unambiguous form. -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/30/98 at 23:01:37 From: Greg Gowens Subject: Re: Speed of light Thanks for your help. I agree with everything you have described. One other question: If the frames of reference were in an atmosphere instead of the vacuum of space, would there be any difference detectable for the measurements of the speed of light? I know about the red phase shift and the blue phase shift, which are similar to the Doppler effect for sound. Do they have anything to do with the actual measurements taken by two different observers in two frames of reference that are within an atmospheric region (or even if one is within an atmosphere and the other is in space) ? Thanks again for your help, Greg Date: 01/31/98 at 13:54:21 From: Doctor Luis Subject: Re: Speed of light It doesn't matter if the two observers are in the same atmospheric region, or one in space moving at some speed, the light still experiences a Doppler shift relative to the stationary observer. In fact, if you observe distant stars, you will find that the light from the great majority of them is red shifted (red being a lower frequency) moving away from us as if some great explosion occured a very long time ago (hint hint). The reason I chose the analogy of sound is to illustrate the point that wave motion takes place with respect to an "absolute" stationary frame (the atmosphere for the case of air). Now, when the source of the ondulations moves with respect to that stationary frame, the waves still originate according to Huygen's principle as if they were created in the medium (which defines the rest frame). Basically, they still move at the same speed independent of the motion of the source... however, the frequency of the waves does depend on how fast the source travels, and that is the famous Doppler frequency shift. (Higher frequency (short wavelength) as the source approaches the stationary observer, and Lower frequency (longer wavelength) as the source goes away from the stationary observer.) Sound is essentially a compression wave, defined by the local pressure of the air molecules, so matter is the medium of sound. Sound still goes through liquids and solids, and much faster than in air. The reason you don't hear very well across a solid wall, or when you dive under a swimming pool, is because of refraction: the sound waves lose an awful lot of energy when they make a transition to a medium of different density, and so their amplitude (or intensity) decreases. For light, the situation is entirely different. Over a hundred years ago, when Maxwell introduced his famous equations which describe the electromagnetic field, it was found that the electric and magnetic fields satisfied a special partial differential equation called the wave equation. The speed that these equations give corresponds exactly to the speed of light that was found experimentally, so Maxwell concluded that light is a wave! Maxwell's discovery was to revolutionize physics to an extraordinary extent. Only one problem: the speed of light with respect to what? Since light is a wave, it doesn't need matter to propagate. It can even propagate across a vacuum. This puzzled physicists of the XIXth century, and with good reason, for how can you have wave motion with no medium to propagate in? Thus, they postulated a medium across which light could propagate and called this medium the aether (or ether). The ether provided the absolute frame of reference in which the speed of light given in Maxwell's equations would be calculated. This ether would permeate all of space, and so light could propagate through the ether. Naturally, if the ether defines a rest frame, then, since we are moving with respect to the ether, we should be able to measure an "ether wind" as we orbit the sun. A famous experiment was set up in order to measure this "ether wind" by measuring the speed of light in different directions. (cf. Michelson-Morley experiment) No motion was ever detected; the speed of light was the same in all directions. This result was completely unexpected. The failure of the Michelson Morley experiment seemed to suggest that there was no ether. But how could that be? H. A.Lorentz suggested that in order to have the negative results of the Michelson-Morley experiment, lengths should be contracted along the direction of motion by some factor ( sqrt(1-(v/c)^2) ). But this explanation was rather ad hoc, and it would seem rather awkward and artificial to seek or construct an explanation every time you got a negative result. The true explanation should be derived from fundamental principles. And yet another thing: according to the Galilean principle of relativity (which Einstein happily adopted later on), there is no way for an inertial observer to tell that he is moving "without looking outside." That is to say, there is no mechanical experiment (although Einstein generalized it as to include electromagnetic experiments or any other experiment, as well) which an inertial observer may do that will enable him to determine his speed. He very well could be at rest and everything else moving with respect to him! This implies that there is no absolute inertial frame of reference. But how? The ether certainly defines an absolute frame of reference. Maxwell's equations themselves explained that light is a wave.. Something was terribly wrong with the laws of physics! Maxwell's equations are not invariant with respect to the Galilean transformation. That transformation introduced extra terms into the form of the equations, and this meant that different inertial observers would observe different electromagnetic effects and therefore, by performing a suitable experiment you would be able to determine your speed with respect to the ether. Several experiments were done in order to observe the effects that these extra terms introduced, in an attempt to measure the speed of the ether wind. Naturally, they all failed to discover those effects, thus people began to believe that, somehow, Maxwell's equations were wrong. Interestingly enough, Maxwell's equations turn out to be invariant with respect to the Lorentz transformation. This was all very confusing. Apparently, either Maxwell's equations or the Galilean transformation had to be wrong. They couldn't possibly both be correct! Of course, rejecting the Galilean transformation would be like rejecting the entire foundation of Newtonian physics! That same theory that explained everything so well. Most people were ready to reject Maxwell's equations, thinking that the correct form (invariant with respect to the Galilean transformation) would soon be discovered, but Einstein took the opposite course, recognizing the shaky foundation upon which classical newtonian physics was built. He explained all of those effects (lorentz contraction, time dilation) not with some ad hoc thoery designed to fit observation, but rather from two simple postulates. A remarkable achievement indeed! i) The laws of physics are the same for every observer ii) The speed of light is the same for every observer The second postulate is really part of the first one, since all physical constants (including the speed of light) are the same for every observer. The rest is the story of special relativity.. Thus light, the very thing without which life could not have existed, seems to be intricately linked with the structure of the universe. Of course, that isn't the end of the story. Light also turns out to have particle-like properties - which takes us into the realm of Quantum Theory. As you can see, there is still much to be learned from Nature. You might want to check out a few physics links I have found at: http://www.cco.caltech.edu/~luisa/useful.html --Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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