# A new thought experiment on relativistic length contraction

Biplab Raychaudhuri<EMAIL_ADDRESS>Department of Physics,
Visva-Bharati University, Santiniketan, West Bengal, India, 731235  Souvik
Ghose<EMAIL_ADDRESS>High Energy & Cosmic Ray Research Center, North
Bengal University, Siliguri, Darjeeling, West Bengal, India, 734013  Arunabha
Bhadra<EMAIL_ADDRESS>High Energy & Cosmic Ray Research Center, North
Bengal University, Siliguri, Darjeeling, West Bengal, India, 734013

###### Abstract

Relativistic length contraction is revisited and a simple but new thought
experiment is proposed in which an apparent asymmetric situation is developed
between two different inertial frames regarding detection of light that comes
from a chamber to an adjacent chamber through a movable slit. The proposed
experiment does not involve gravity, rigidity or any other dynamical aspect
apart from the kinematics of relative motion; neither does it involve any kind
of non-uniformity in motion. The resolution of the seemingly paradoxical
situation has finally been discussed.

##  1 Introduction

Special theory of relativity (STR), which assumes Lorentz invariance (LI) as a
fundamental symmetry of nature, has a central place in modern physics.
According to one of the two postulates of STR, the speed of light in vacuum,
unlike that of any other particle, remains invariant to any inertial observer
independent of the motion of the source or the observer [1]. This idea of
constant light speed is so counter intuitive that since the formulation of the
theory several researchers have critically examined its consistency and
explored different surprising consequences of the Lorentz invariance,
particularly in search of a real paradox in the theory. Length contraction and
time dilation as well as their apparently paradoxical consequences are
discussed in introductory textbooks to familiarize new students with the
curious consequences of the postulates of STR [2]. Several such incongruities
were proposed and subsequently resolved in the literature. Resolution of most
of these alleged paradoxes, if not all, involves the presence of an external
force, rigidity, or some other subtle issues. While the most agreed upon view
on length-contraction and time dilation is that the effects are real and can
be demonstrated through experiment [3], some people have remained skeptical
[4]. Nevertheless, the thought experiments on the consequences of STR have
retained their importance, more so from the pedagogical view point. In the
present work we propose a new variant of the length contraction paradox (and
also advance a resolution of it) which can be discussed based on the kinematic
aspects of special relativistic effects only. While, like all the other length
contraction gedanken in the literature (see Appendix), the present one does
not advance any new idea, it would, hopefully, help students understand the
curious aspect of special theory of relativity without exposing them to the
more confounding issues which can be introduced at a much later stage.

##  2 An apparent paradox

Let us consider two adjacent chambers $V_{1}$ and $V_{2}$ as shown in Fig 1. A
slit $AB$ of length $L$ connects $V_{1}$ and $V_{2}$. There are two identical
light sources $\lambda_{1}$ and $\lambda_{2}$ in chamber $V_{2}$, just
infinitesimal distance below the points $A$ and $B$. A $100\%$ efficient
photon detector $D$ sits in $V_{1}$ and infinitesimally above the middle point
of the slit. As long as the slit is open the photons from $\lambda_{1}$ and
$\lambda_{2}$ pass grazing the edge of the slit and are detected at $D$. The
detector, however, is equipped with an explosive device which would be
detonated if at any moment the detector count fell to zero. There is also a
bar $PQ$ of rest length $L$ that is moving freely with relativistic speed
along the boundary of two chambers towards positive $X$-axis as per an
observer $O$ who is at rest with respect to the two chambers, slit, light
sources and detector. Let us associate a frame of reference $S(t,x,y,z$) with
the observer $O$ and a frame of reference
$S^{\prime}(t^{\prime},x^{\prime},y^{\prime},z^{\prime})$ ($c=1$) with an
observer $O^{\prime}$ which is at rest with the bar $PQ$. The axes of both
frames coincide at the point $A$ at a certain time to which we assign the
origin of the two frames $S$ and $S^{\prime}$. Thus the $X$ coordinates of the
points $A$ and $B$ are $0$ and $x_{B}$ respectively in frame $S$ and $0$ and
$x_{B}^{\prime}$ in frame $S^{\prime}$.

Figure 1: From laboratory frame: detector always receives photon.

Let us first consider the observation of the observer $O$. The bar is moving
towards the slit from its edge $A$ in its frame. Initially the front end of
the bar $P$ coincides with $A$. As the bar moves towards $B$, the light from
the source $\lambda_{1}$ will be blocked; but the detector $D$ will continue
receiving light from $\lambda_{2}$. At some instant $t_{1}$ the back end ($Q$)
of the bar will reach the point $A$. The front end $P$ of the bar is then,
say, at point $x_{1}$. The length of the bar in frame $S$ is $x_{1}$. Due to
length contraction, the length of the bar in the frame $S$ is $L/\gamma$,
where $\gamma=1/\sqrt{(1-v^{2}/c^{2})}$. So $x_{1}=L/\gamma<L$, which implies
that a part of the slit is still open and the detector $D$ is receiving light
from $\lambda_{2}$. Next, consider the situation when the front end $P$ of the
bar reaches $B$ (at $x_{B}$) at a time $t_{2}$. At that instant the back end
$Q$ of the bar is at $x_{2}$ and the length of the bar in frame $S$ is
$x_{B}-x_{2}$. The length contraction implies $x_{2}>0$. So light from the
source $\lambda_{2}$ is now blocked to reach the detector $D$, but light from
the source $\lambda_{1}$ still reaches $D$. Thus, the observer in frame $S$
concludes that the detector $D$ is continuously receiving light. Now, let us
consider the experiment in the frame of the observer $O^{\prime}$ (Fig. 2).
She observes that the slit, light sources and the detectors are moving towards
the bar in the $-$ve $X$ direction after $t^{\prime}=0$.

Figure 2: From rod-bound frame: both sources are blocked for some time.

At time ${t^{\prime}}_{1}$ the far end $B$ of the slit will reach point $P$
and at the same instant the point $A$ will reach, say, $x^{\prime}_{1}$. The
length of the slit, according to the observer $S^{\prime}$, will be
$x^{\prime}_{1}$ . Due to length contraction $x^{\prime}_{1}=L/\gamma<L$.
Since the slit is now shorter than the bar, no light reaches the detector from
any of the two sources, $\lambda_{1}$ or $\lambda_{2}$. This situation will
continue till the point $A$ of the slit reaches the point $Q$ of the bar. The
observer $O^{\prime}$ then definitely sees an explosion!

##  3 Resolution of the proposed paradox

So far we have concentrated on passing of light through the slit and not on
detection of light by the detector. The detector is at rest with respect to
the observer $O$ but is in motion according to the observer $O^{\prime}$. The
light will take exactly the same time while propagating from the point $A$ to
$D$ and the point $B$ to $D$ in frame $O$. But the same is not true in frame
$O^{\prime}$. In the view point of $O^{\prime}$ after the point B coincides
with the point P light from the sources $\lambda_{1}$ and $\lambda_{2}$ cannot
reach the upper chamber till the point $A$ reaches $Q$. So we need to
calculate the time difference in frame $O^{\prime}$ between the propagation of
light from the point $B$, when it coincides with $P$, to $D$ and the point
$A$, as it arrives at $Q$, to $D$.

Let $t_{BD}^{\prime}$ is the time taken by a photon to reach the detector $D$
from the point $B$ as $B$ coincides with $P$. The vertical distance between
B/A and D is say $y$ and the horizontal distance (in $x^{\prime}$ coordinates)
between B/A and D is $\frac{L}{2\gamma}$. Therefore, one can write

$\sqrt{\left(\frac{L}{2\gamma}+vt_{BD}^{\prime}\right)^{2}+y^{2}}=ct_{BD}^{\prime}$
(1)

which leads to

$t_{BD}^{\prime}=\frac{\frac{Lv}{\gamma}\pm\sqrt{\frac{L^{2}v^{2}}{\gamma^{2}}+4\left(c^{2}-v^{2}\right)\left(\frac{L^{2}}{4\gamma^{2}}+y^{2}\right)}}{2\left(c^{2}-v^{2}\right)}$
(2)

Since $t_{BD}^{\prime}$ has to be positive, we have to take only positive root
of the above solution. Similarly,

$t_{AD}^{\prime}=\frac{-\frac{Lv}{\gamma}\pm\sqrt{\frac{L^{2}v^{2}}{\gamma^{2}}+4\left(c^{2}-v^{2}\right)\left[\frac{L^{2}}{4\gamma^{2}}+y^{2}\right]}}{2\left(c^{2}-v^{2}\right)}$
(3)

Thus the difference in propagation time of photon from $B$ to $D$ and $A$ to
$D$ $\Delta t_{prop}^{\prime}=\frac{L\gamma v}{c^{2}}$ which is interestingly
independent of y i.e. vertical position of the detector.

In the primed frame (rest frame of the rod) the last photon that escapes the
rod, when $B$ coincides with $P$, reaches $D$ at
$t^{\prime}_{D1}=t^{\prime}_{1}+t_{BD}^{\prime}$. When $A$ reaches $Q$, the
first photon that escapes, reaches $D$ at
$t^{\prime}_{D2}=t^{\prime}_{2}+t_{AD}^{\prime}$. The time difference in
arrival of these two photons at $D$ is $\Delta
t^{\prime}_{D}=t^{\prime}_{D1}-t^{\prime}_{D2}$

$\Delta t^{\prime}_{D}=\frac{L}{v}\left(\gamma-1\right)$ (4)

which is positive. Thus the the first photon that enters the upper chamber
when $A$ coincides $Q$ reaches the detector $D$ before the last photon that
enters the upper chamber when $B$ coincides $P$. This implies that even in the
primed frame the detector $D$ continuously receives photons and manages to
avoid an explosion.

##  4 Conclusion

The above puzzle relies only on the length contraction phenomenon. Its
formation and resolution do not involve any external force; neither does it
involve any non-uniform motion. The seemingly paradoxical situations in STR
arise from the counter intuitive nature of the time dilation and the length
contraction. In the present work an apparent inconsistency is resolved by
taking the leverage of using length contraction in one frame and time dilation
in the other (i.e. essentially using the invariance of four dimensional length
element ($ds^{2}$)) like in the famous muon decay problem. The (apparent)
paradoxes involving pure time dilation or pure length contraction are rather
difficult to resolve in a straight forward way. The resolution of twin paradox
problem orients around acceleration of one of the twin while the resolution of
the pole and garage paradox relies on rigidity/stress of pole material. STR is
a very successful physical theory. Not only the theory has been found to
explain a wide range of experimental results but it has also been well tested
experimentally with an enormous range and diversity, and the agreement between
theory and experiment is excellent. Many of the predictions of the theory are
at times counter-intuitive and over the years researchers have explored many
enigmatic situations. However, subsequent resolutions of these (apparent)
paradoxes only re-establish the great inner consistency of the theory. In this
ambit the present work seeks to serve some pedagogical purpose by inviting a
student to the curious world of the STR.

## Acknowledgement

Authors are grateful to Professor R. H. Price for pointing the direction of
possible resolution of the proposed paradox. BR likes to thank Inter-
University Centre for Astronomy and Astrophysics (IUCAA), Pune for their
hospitality and facilities they extended towards him during his visits under
their Visiting Associateship programme.

## References

  * [1] A. Einstein, The meaning of relativity (Routledge, 2003).
  * [2] D. Halliday, R. Resnick and J. Walker, Fundamentals of physics (John Wiley & Sons, 2013).
  * [3] J. Bailey, K. Borer, F. Combley, H. Drumm, F. Krienen, F. Lange, E. Picasso, W. Von Ruden, F. Farley, J. Field et al., Nature 268, 301 (1977).
  * [4] O. D. Jefimenko, Zeitschrift für Naturforschung A 53, 977 (1998).
  * [5] G. FitzGerald, Lorentz And Poincare Invariance: 100 Years Of Relativity 8, p. 25 (2001).
  * [6] H. A. Lorentz, Zittingsverlag Akad. V. Wet 1, 74 (1892).
  * [7] W. Rindler, American Journal of Physics 29, 365 (1961).
  * [8] W. Rindler, American Journal of Physics 29, 859 (1961).
  * [9] R. Shaw, American Journal of Physics 30, 72 (1962).
  * [10] E. Pierce, American Journal of Physics 75, 610 (2007).
  * [11] E. Marx, American Journal of Physics 35, 1127 (1967).
  * [12] E. M. Dewan, American Journal of Physics 31, 383 (1963).
  * [13] G. Sastry, American Journal of Physics 55, 943 (1987).

## Appendix

Historically the concept of length contraction of moving objects in the
direction of motion was introduced independently in an ad hoc manner by
FitzGerald [5] and Lorentz [6] in order to explain the negative results of the
Michelson-Morley experiment. It was realized later that length contraction is
an inevitable consequence of Lorentz transformation. This effect also leads to
many puzzling situation many of which found their way into standard special
relativity text books. Some of the well-known (apparent) paradoxes are
mentioned below.
Rod and hole problem:
A rod, moving on a smooth table, approaches a hole of the same dimension [7,
8, 9]. For the observer on the rod the hole is Lorentz contacted and the rod
avoids falling through it. For an observer inside the hole the rod is length-
contracted and should fall through it. The paradox is resolved challenging the
validity of the rigid-rod assumption within relativistic framework (for more
on rigidity see [10]) . The force of gravity is also present to fecilitate the
fall in the first place. This paradox was latter modifid to avoid the force of
gravity [11]. In the new scenario both the rod and the hole moves in mutually
perpendicular direction and happens to be at the same place at some instant.
The paradox is resolved showing that the rod would appear to be titled in one
frame.
The pole and barn/garage paradox:
Here [12], a person runs into a garage with a horizontal pole, the pole and
the garage being of the same length. From the perspective of the runner the
garage is length contracted and the pole can not be fitted within. For a
garage observer, on the other hand, the pole is shortened and should easily
fit inside it. Consequently, in one frame both the garage doors can be closed
with the pole trapped inside but in the other frame that is simply not
possible. Once again the paradox is resolved based on an argument that the
relativistic pole is no longer rigid in the non-relativistic sense and the
back end of it can not know instantly when the front end hits the garage door.
Paradox with an electrical circuit:
Sastri [13] proposed an interesting paradox with a simple electrical circuit
which is completed by a moving shuttle. Consequently a bulb is turn on. From
the frame of the circuit the length of the shuttle is shorter than the gap in
the circuit and it fails to connect the two ends. From the frame of the
shuttle however, the gap is shorter and successfully bridged by the shuttle
for an instant, turning on the light. Although successfully resolved this
paradox brings in the propagation of electrical signal into consideration.