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 innitely rigid stick will require that. It was

long time that this question was not answered and it was thought

that innitely 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 diers 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 [1] 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 [2] 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 [3].

By direct dierentiation 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

Copyright: © 2018 Keilman Y. This is an open-access article distributed under

the terms of the Creative Commons Attribution License, which permits unrestricted

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.