# "Symmetrical Length Contraction" - Demystified - Millennium Relativity

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10 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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The “Symmetrical Length Contraction” - Demystified

In a previous paper
1
we have dealt with demystifying the more famous error of
“symmetrical time dilation” or the so-called “Twins or Dingle Paradox”. There is a less
famous error that stems from a misinterpretation of chapter 4 “Physical Meaning of the
Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks”
2
and it has
to do with the incorrect impression that there is a “symmetrical length contraction”. In
the following we will maintain Einstein’s notation in order to allow for an easier
comparison with the original paper. In this chapter Einstein starts from the well known
Lorentz transformation:

ξ=γ(x-vt) (1.1)

He assumes that there is a system (k) that has a clock situated in the origin ξ=0 that
measures the time τ. The system (or inertial frame) (k) is characterized by the coordinates
ξ,τ. The system (k) moves with speed v with respect to another system (K), characterized
by the coordinates x,t.
The inverse transformation to (1.1) is :

x=γ(ξ+vτ) (1.2)

Assuming that there is a rod extending in the direction of the positive x axis from x
1
to x
2
,
its length in (K) will be x
2
-x
1

According to (1.1), the length of the same rod, as viewed from (k) will be:

ξ
2

1
=γ(x
2
-x
1
) (1.3)

Conversely, if there is another rod extending from ξ
1
to ξ
2
in (k) of length ξ
2

1
as
measured in (k), from the perspective of (K) , its length will be:

x
2
-x
1
=γ(ξ
2

1
) (1.4)

Exactly as in the case of Dingle, some anti-relativists hasten to conclude that:

x
2
-x
1
=γ(ξ
2

1
)=γ
2
(x
2
-x
1
) (1.5)

From (1.5) it follows the absurd conclusion that γ
2
=1. What went wrong?

It is easy to figure out what went wrong if we remember that the two rods are completely
independent of each other and if we also write (1.1) and (1.2) correctly, as functions:

ξ(x,t)=γ(x-vt) (1.6)
x(ξ,τ)=γ(ξ+vτ) (1.7)

Since the endpoints of the two different rods (one in (k) and the other in (K)) do not
move we can ignore the time variable and we can write that the rod extending from x
1
to
x
2
in (K) is perceived as extending from ξ(x
1
) to ξ(x
2
) in (K) and having the length:

ξ(x
2
)-ξ(x
1
)=γ(x
2
-x
1
) (1.8)

Conversely, the rod extending from ξ
1
to ξ
2
in (k) is perceived to extend from x(ξ
1
) to
x(ξ
2
) in (K) having the length:

x(ξ
2
)- x(ξ
1
)= γ(ξ
2

1
) (1.9)

Since ( )
i i
x
x
ξ
≠ and ( )
i i
x
ξ
ξ≠ there is no danger of making the improper chain of
substitutions seen in (1.5). The “mutual length contraction paradox” is gone.

1. A. Sfarti “ The Twins Paradox”-Demystified http://www.wbabin.net/sfarti/sfarti14.pdf
2
.
A. Einstein “On the Electrodynamics of Moving Bodies”,Annalen der Physik 17, 1905,

http://www.fourmilab.ch/etexts/einstein/specrel/www/