Re: Lorentz Contraction for the even dumber


"harry" <harald.vanlintelButNotThis@xxxxxxx> wrote in message news:4a3aa074$1_6@xxxxxxxxxxxxxxxxxx

"Uncle Ben" <ben@xxxxxxxxxxx> wrote in message news:e03d135d-3333-4c63-9434-c2b4c7514f5e@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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.

Yes this is roughly the way the Lorentz contraction is derived in usual textbooks. It's very straightforward indeed. Note that it's only very slightly simpler to choose the origin for one end - the main point is to have t=constant for the position coordinates at both ends.
In detail, the very elegant textbook variant that I happen to have at hand (BUT WATCH OUT: here the definition of L is the length as measured in S; while the length in S' is indicated with L'):

L = x_b - x_a

For time t at each end (= choosing simultaneity according to the "stationary" system S):

x'_a = gamma * ( x_a - v t )
x'_b = gamma * ( x_b - v t )
--------------------------- - subtraction:
x'_a - x'-b = gamma * ( x_b - x_a )

=> for t_a = t_b: L' = gamma * L or L = L' / gamma.

In words: a stick that is measured to be L' long in the "moving" frame, is measured to be L'/gamma long in the "rest" frame.

Cheers,
Harald

OOPS I inversed x'_a and x'_b as I subtracted the other way round than the textbook. Sorry for the glitch!

.

Relevant Pages

  • Re: Lorentz Contraction for the even dumber
    ... proof that SR does actually predict the Lorentz Contraction, ... the pole is at rest. ... end of the pole is located at x'=L, t' = T' in the frame of the pole. ...
    (sci.physics.relativity)
  • Re: Lorentz Contraction for the even dumber
    ... proof that SR does actually predict the Lorentz Contraction, ... the pole is at rest. ... end of the pole is located at x'=L, t' = T' in the frame of the pole. ...
    (sci.physics.relativity)
  • Re: Lorentz Contraction for the even dumber
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    (sci.physics.relativity)
  • Re: Lorentz Contraction for the even dumber
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  • Re: Lorentz Contraction for the even dumber
    ... proof that SR does actually predict the Lorentz Contraction, ... length of a speeding pole in a barn? ... In the barn frame, time is important: ...
    (sci.physics.relativity)