Absolute Speed

Date: 08/12/97 at 17:57:08
From: Chris MacPhee
Subject: Absolute speed

The absolute speed, within the Einsteinian universe, is c (the speed 
of light). If a ship going 1/100 of c shines a flashlight, what could 
happen to the light emitted? If it goes 101/100 of c, the speed of 
light is not a constant. If it goes at c, what happens to the law 
of Conservation of Momentum as well as the law of Conservation of 
Energy?

Thanks, 
Chris MacPhee

Date: 08/14/97 at 12:01:24
From: Doctor Marko
Subject: Re: Absolute speed

Hi Chris,

Incidentally, strictly speaking, you are asking a physics question and 
not a math question.

You have to be careful when asking questions like this because it is
important to regard speed as a relative concept and not an absolute 
one. What I mean is that someone can answer your question by saying: 
"Well, with respect to the ship, the light travels at c." and think to 
be done.

But since I believe that your question had to do with the light's 
speed relative to the observer at rest (the one that sees the ship 
moving at .01c) I will continue. The answer to your question lies 
within your first sentence, which is also one of the postulates of 
Einstein's theory of relativity - which is to say that the speed of 
light is c, and always c, and c from all the inertial reference frames 
(that is the unaccelerating frames). Sounds weird and we ordinary 
people have no intuition for that, because after all we do not see 
light as having speed since it is so blazing fast, so we think of it 
as instantaneous. Yet this notion seems to be verified by experiments 
(such as the Michelson-Morley experiment).

So, then your next question is the right one - what happens to the
conservation of momentum and energy laws?  Well, Einstein felt that 
the laws of conservation of momentum and energy were so compelling 
that he wondered if it was possible to keep them in some form. In 
order to keep the conservation laws, he discovered that he needed to 
change the definition of momentum and energy, to what is called 
relativistic momentum and energy.

Now the formulas go something like this:

   Newtonian momentum: 
      p = mv
   relativistic momentum:  
      p = gamma mv, where m is the mass, v is the speed, and gamma is 
   the relativistic factor:
      gamma^-1 = Sqrt(1-(v/c)^2).

To sort of conclude (and if you want to, please keep asking more 
questions), what is remarkable about Einstein's theory is that to the 
best of our knowledge, the relativistic momentum and energy are 
conserved, as tested by countless experiments. Also note that when 
speeds are small compared to the speed of light (we write that as 
"v<<c", i.e. "v much, much less than c") then gamma is approximately 
1, and the relativistic theory and the Newtonian theory differ almost 
indistinguishably from each other.

Cool?

-Doctor Marko,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/