Headlights, Light Waves, Sound Waves

Date: 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/