1.2 Reasoning for its Existence (Special Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
1.2 Reasoning for its Existence (Special Relativity)
Before Einstein, there was Newton, and Newtonian physics had its own
concept of relativity; however, it was incomplete. Remember that relativity
involves figuring out what an observation would seem like to one observer
once you knew what it looked like to another observer who is moving with
respect to the first. Before Einstein, this transformation from one frame to
another was not completely correct, but it seemed so in the realm of small
speeds.
Here is an example of the Newtonian idea of transforming from one frame
of reference to another. Consider two observers, you and me, for example.
Let's say I am on a train (in some enclosed, see-through car--if you want to
visualize the situation) that passes you at 30 miles per hour. I throw a
ball in the direction the train is moving such that the ball moves at 10 mph
in MY point of view. Now consider a mark on the train tracks which starts
out ahead of the train. As I am holding the ball (before I throw it), you
will see it moving along at the same speed I am moving (the speed of the
train). When I throw the ball, you will see that the ball is able to reach
the mark on the track before I do. So to you, the ball is moving even faster
than I (and the train). Obviously, it seems as if the speed of the ball with
respect to you is just the speed of the ball with respect to me plus the
speed of me with respect to you. So, the speed of the ball with respect to
you = 10 mph + 30 mph = 40 mph.
This was the first, simple idea for transforming velocities from one
frame of reference to another. It tries to explain a bit about observations
of one observer relative to another observer's observations. In other words,
this was part of the first concept of relativity, but it is incomplete.
Now I introduce you to an important postulate that leads to the concept
of relativity that we have today. I believe it will seem quite reasonable. I
state it as it appears in a physics text by Serway: "the laws of physics are
the same in every inertial frame of reference." (Note that by "inertial
frame of reference" we basically mean a frame of reference which is not
accelerating.) What the postulate means is that if two observers are moving
at a constant speed with respect to one another, and one observes any
physical laws for a given situation in their frame of reference, then the
other observer must also agree that those physical laws apply to that
situation.
As an example, consider the conservation of momentum (which I will
briefly explain here). Say that there are two balls coming straight at one
another. They collide and go off in opposite directions. Conservation of
momentum says that if you add up the total momentum (which for small
velocities is given by the mass of the ball times its velocity) of both the
balls before the collision and after the collision, then the two should be
identical. Now, let this experiment be performed on a train where one ball
is moving in the same direction as the train, and the other is moving in the
opposite direction. An outside observer would say that the initial and final
velocities of the balls are one thing, while an observer on the train would
say they were something different. However, BOTH observers must agree that
the total momentum is conserved. One will say that momentum was conserved
because the total momentum before AND after the collision were both some
number, A; while the other will say that momentum was conserved because the
total momentum before AND after were both some other number, B. They will
disagree on what the actual numbers are, but they will agree that the law
holds. We should be able to apply this postulate to any physical law. If
not, (i.e., if physical laws were different for different frames of
reference) then we could change the laws of physics just by traveling in a
particular reference frame.
A very interesting result occurs when you apply this postulate to the
laws of electrodynamics (the area of physics which deals with electricity
and magnetism). What one finds is that in order for the laws of
electrodynamics to be the same in all inertial reference frames, it must be
true that the speed of electro-magnetic waves (such as light) is the same
for all inertial observers. Perhaps the easiest way to explain why this is
so is to discuss two constants used in basic electrodynamics. They are
denoted as epsilon_0 and mu_0. Epsilon_0 is used in the basic equation which
describes the attraction or repulsion between two electrically charged
particles while mu_0 is used in the basic equation which describes the
magnetic force on a charged particle. According to electrodynamics, these
two constants are properties of the universe, and if any observer in any
frame of reference does an electro-magnetic experiment to measure those
constants, he or she must always come up with the same answers. However, it
is also a property of electrodynamics that the speed (c) of an
electro-magnetic wave (such as light) can be expressed in terms of those two
constants: c = 1/sqrt(mu_0*epsilon_0). If epsilon_0 and mu_0 are constants
for all inertial observers, then so is c.
Thus, requiring the laws of electrodynamics to be the same for all
inertial observers suggests that the speed of light should be the same for
all inertial observers. Simply stating that may not make you think that
there is anything that interesting about it, but it has amazing and
far-reaching consequences. Consider letting a beam of light take the place
of the ball in our earlier example (the one where I was on a train throwing
a ball, and you were outside the train). If the train is moving at half the
velocity of light (c), and I say that the light beam is traveling at the
speed c with respect to me, wouldn't you expect the light beam to look as if
it were traveling one and a half that speed with respect to you? Well,
because of the postulate above, this is not the case, and the old ideas of
relativity in Newton's day fail to explain the situation. All observers must
agree that the speed of any light beam is c, regardless of their frame of
reference. Thus, even though I measure the speed of the light beam to be c
with respect to me, and you see me traveling past you and one half that
speed, still, you must also agree that the light is traveling at the speed c
with respect to you. This obviously seems odd at first glance, but time
dilation and length contraction are what account for the peculiarity.
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