Lorentz Contraction for the even dumber

I recently posted for the benefit of newbies to relativity a
proof that SR does actually predict the Lorentz Contraction,
despite recent postings to the contrary from critics.

I received a reply from "Larry," who posts under the nickname
"blackhead." He reminded me of a simpler way that makes the problem
trivial.

-------------------------------

Prerequisite: The Lorentz Transformation in one dimension:

With t in seconds and x in light-seconds,

x' = gamma * (x - vt)
t' = gamma * (t - vx)
x = gamma * (x' + vt')
t = gamma * (t' + vx')

where v is the velocity of a primed system of coordinates moving
towards +x with respect to an unprimed system, and

gamma = 1/sqrt(1 - v^2), which is always >= 1.

Known already in the Nineteenth Century, these equations follow from
the Einstein's SR postulates as well.

------------------------------

Let's pose the question as the pole-and-barn problem: What is the
length of a speeding pole in a barn?

Let the primed coordinates (x',t') be the pole frame, in which
the pole is at rest. (Primed = Pole) Let the ends of the pole
be at x'= 0, and x'= L no matter what the time.

In the barn frame, time is important: We will need two
*simultaneous* events to mark the two ends of the moving pole. we
choose them to be at t=0, when the origins of the two frames coincide.

We consider an event at one end of the pole with coordinates
x' = 0, t = 0, and x = 0
and an event at the other end of the pole where
x' = L, t = 0,
and we want to know x, the length of the pole in the barn
coordinates.

Among the four equations above, eliminate the three that involve t',
the variable we don't care about, and what is left is

x' = gamma * (x - vt)

If x' = L and t = 0, then x = L/gamma.
---------------------------------

This is the Lorentz Contraction.

It can't get any easier than this. Consider me "even dumber" for not
thinking of this for myself.

Thanks, Larry!

Uncle Ben
.

Relevant Pages

  • Re: Pole and Barn for Dummies
    ... -- except for a brief discussion on how to tell which frame has the ... The reader bored with the Pole and Bar "Paradox" may wish to read ... The Pole and Barn question: ... back door, fixed to the earth. ...
    (sci.physics.relativity)
  • Re: Pole and Barn for Dummies
    ... -- except for a brief discussion on how to tell which frame has the ... The reader bored with the Pole and Bar "Paradox" may wish to read ... The Pole and Barn question: ... back door, fixed to the earth. ...
    (sci.physics.relativity)
  • Re: Pole and Barn for Dummies
    ... -- except for a brief discussion on how to tell which frame has the ... The reader bored with the Pole and Bar "Paradox" may wish to read ... The Pole and Barn question: ... back door, fixed to the earth. ...
    (sci.physics.relativity)
  • Re: Pole and Barn for Dummies
    ... -- except for a brief discussion on how to tell which frame has the ... The reader bored with the Pole and Bar "Paradox" may wish to read ... The Pole and Barn question: ... back door, fixed to the earth. ...
    (sci.physics.relativity)
  • Re: Pole and Barn for Dummies
    ... -- except for a brief discussion on how to tell which frame has the ... The reader bored with the Pole and Bar "Paradox" may wish to read ... The Pole and Barn question: ... back door, fixed to the earth. ...
    (sci.physics.relativity)