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# Deformation-less Acceleration of a Stick in Special Relativity

Authors:
Research Article Open Access
Keilman, J Phys Math 2018, 9:2
DOI: 10.4172/2090-0902.1000271
Commentary Open Access
Journal of Physical Mathematics
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ISSN: 2090-0902
Volume 9 • Issue 2 • 1000271J Phys Math, an open access journal
ISSN: 2090-0902
Introduction
Can we accelerate a stick in SR so that it remains not deformed
during acceleration? e innitely rigid stick will require that. It was
long time that this question was not answered and it was thought
that innitely rigid stick does not exist in SR. It appears that we can
accelerate without deformation in SR (no matter if the stick is so or
hard).
Explanation
A resting clock usually described in SR by the line parallel to time
axis and displaying time T on its dial which coincides with the local
coordinate time t. But if the same clock moves with a constant velocity
V in x direction then the local time diers from the dial time: t=T∕√1-V2.
(I assume c=1). It is so called “time dilation”. ere is no actual change
of the dial time T (the coordinate time t gets dilated). e dial time T is
invariant and called proper time.
A resting stick of the physical length L usually described in SR by
2 lines parallel to the time axis and cutting x-coordinate length l=L on
the x-axis. But if the same stick moves with a constant velocity V in x
direction then the lines (the tracks of the end points of the stick) make
an angle with the time axis and they cut the length l=L√1-V2 on x-axis.
It is so called “length contraction”. ere is no actual deformation
(contraction) of the physical length of the stick (called proper length)
because in both cases we have inertial movement.
In his article Einstein  put stress on coordinate time and
x-coordinate length that change with frame. But for the practical
reasons (like measurements or calculating deformation) we need to
put stress on proper time and proper length which are invariant. All
physical clocks and meter sticks are invariant and can be used in any
coordinate system. We do not need any additional “normalization” like
proposed by Winkler  Now let us turn to acceleration. Physically we
can imagine deformation-less movement with acceleration in a case of
gravitational forces, or in a case of absolutely rigid stick.
Let us prescribe the trajectory of the rst end of the stick in the
parametric form:
( )
11 1
;x xp t p= = (1)
In particular, in the case of acceleration it can be:
( )
2
1
2
ap
x p Vp= +
(2)
We want to nd the trajectory of the 2nd world line (another end of
a stick). Let us take the arbitrary point P1(t1,x1) on the line (1). Let us
build a normal to the 1st world line at the point P1. Suppose the point
P(x,t) is located on this normal. e invariant distance between P1 and
P is:
( )
( )
( )
22
1
S xxp t p= − −−
(3)
is distance has to be maximum with respect to a variation of p
(we keep the point P xed while P1 is moving along the 1st world line
by the variation of p). We have:
( ) ( )
1
0; (p) ;
()
S tp
x x Vp kp
p Vp
∂−
==+=
(4)
where V is just a notation. With p xed the last equation indicates
that we can move the point P only along the line. is line represents
the normal to the 1st world line in the point P1. Substituting this back
to eqn. (3) we get:
( )
1
22
;
11
LV L
t p x xp
VV
=±=±
−−
(5)
Where L is the value of S at maximum. Changing L we will move
the point P(t,x) along the normal. is normal will intersect the 2nd
world line in some point P2(t2,x2). e t2 and x2 will satisfy to the
equation of normal (5):
( )
2 21
22
;
11
LV L
t p x xp
VV
=±=±
−−
(6)
Given s and changing parameter p we can nd the world line of the
second end of a stick in the parametric form with the same parameter
p : t2(p);x2(p). Considering that the invariant length of the stick L is
constant and changing p (moving the point P1 along the 1st world
line) we actually nd the 2nd world line in a parametric form (eqn. (6)).
Namely the time of the reciprocal point (we consider the points P1 and
P2 are reciprocal) on the world line of the rst end of the stick, so it uses
the same parameter as the 1st world line .
By direct dierentiation of eqn. (6) we can nd:
( )
( )
( )
2
2
xp Vp
tp
=
(7)
at means that the reciprocal points have the same velocity.
Let us build a normal to the 2nd world line in the point P2. Suppose
the point P(x,t) is located on this normal. e invariant distance
between P2 and P is:
( )
( )
( )
( )
22
22
S xxp ttp= − −−
(8)
is distance has to be maximum with respect to a small variation
of p:
( ) ( )
( )
( )
( )
2
22
2
0; tp
Sx xp tt p
pxp
= = +−
(9)
where the last equation represents another normal. is normal will
intersect the 1st world line at some point P1'(t1',x1') (t1' and x1' can
be found as a joint solution of both eqns. (1) and (8)). It is easy to see
from eqns. (4) and (8) that as a result of eqn. (7) P1' coincides with P1.
at means that the points P1 and P2 are “reciprocal” points. Given
*Corresponding author: Keilman Y, University of Oslo Oslo, Oslo, United
State, Tel: +47 22 85 50 50; E-mail: altsci1@gmail.com
Received April 22, 2018; Accepted June 11, 2018; Published June 18, 2018
Citation: Keilman Y (2018) Deformation-less Acceleration of a Stick in Special
Relativity. J Phys Math 9: 271. doi: 10.4172/2090-0902.1000271
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Deformation-less Acceleration of a Stick in Special Relativity
Keilman Y*
University of Oslo Oslo, Oslo, USA
Citation: Keilman Y (2018) Deformation-less Acceleration of a Stick in Special Relativity. J Phys Math 9: 271. doi: 10.4172/2090-0902.1000271
Page 2 of 2
Volume 9 • Issue 2 • 1000271J Phys Math, an open access journal
ISSN: 2090-0902
References
1. Einstein A (1905) Zur Electrodynamik bewegter Korper. Annalen der Physik
16: 895-896.
2. Winkler FG (2005) The Normalization Problem in Special Relativity.
3. https://arxiv.org/pdf/physics/0604214
the 1st world line we found the way to build the reciprocal 2nd world
line. ese 2 lines represent the world lines of the ends of moving and
accelerating stick if it is not deformed. is is the proof that existence of
absolute hard stick does not contradict to SR. e objection to the existence
of a hard stick can come only from dynamics of the stick. is dynamics
will also put limit on the velocity of transfer information along a stick.