Derivation Error Of Albert Einstein

Derivation Error Of Albert Einstein

B. Ravi Sankar

Scientist, MPAD/MDG, ISRO Satellite Centre, Bangalore-560017, INDIA

Abstract

assumed. The aim is to prove

The purpose of this paper is to point out a major derivational error in Albert Einsteins

1905 paper titled ON THE

1 (

2 0

2

) 1 or

ELECTRODYNAMICS OF MOVING

BODIES. An alternate expression for coordinate transformation is derived which shows that the time co-ordinate of the moving clock cannot be expressed in terms of the temporal and spatial co-ordinate of the stationary system.

Keywords Special theory of relativity, on the electrodynamics of moving bodies, coordinate transformation, kinematical part, definition of simultaneity.

1. Introduction

   Albert Einstein published three papers in the year 1905. Among them the paper on photo electric effect yielded him the Nobel Prize. He was not awarded Nobel Prize for his celebrated general theory of relativity or special theory of relativity. The 1905 paper titled ON THE ELECTRODYNAMICS OF

   MOVING BODIES is the source of special theory of relativity. For a quiet long time, people have speculated that there is something wrong in special theory of relativity. The purpose of this paper is not to disprove or dispute the special theory relativity but to point out a major derivational error in the section titled Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation relatively to the Former under the KINEMATICAL PART of the paper ON THE ELECTRODYNAMICS OF THE MOVING BODIES [1].

2. Major derivational error

   1

   1

   The first assumption [1] Einstein made in

   that section is ( ) . The aim of

   2 0 2 1

   this paper is to prove that he has wrongly

   equivalently ( 0 2 ) 2 1 . In order to prove this one need to thoroughly understand the figure 1 as well as the KINEMATICAL PART of the paper ON THE ELECTRODYNAMICS OF MOVING BODIES.

   Let us in Stationary space take two system of co-ordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point (origin O or o). Let the axes of X of the two systems coincide, and their axes of Y and Z be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike [1].

   Now to the origin (o) of one of the two systems ( k ) let a constant velocity v be imparted in the direction of the increasing x

   of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring rod, and the clocks. To any time of stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this t always denotes a time of the stationary system) parallel to the axes of the stationary system [1].

   To any system of values x, y, z, t , which

   completely defines the place and time of an event in the stationary system, there belongs a system of values , , , determining that

   event relatively to the system k , and our task is to find the system of equations connecting these quantities[1]. The foregoing discussion is pictorially represented in figure 1

   From the origin of the moving system k let a ray be emitted at the time 0 along the X- axis to x' and at the time 1 be reflected thence to the origin of the co-ordinates [1], arriving at the time 2 ; we must prove that

   • Green line represents the point at which ray is reflected.

   • O, o Represents origin of stationary and moving system at initial condition.

   • o0 0 Represents the origin of

   1 (

   2 0

   2

   ) 1

   . Before proceeding further,

   moving system when the ray is emitted w.r.to stationary system

   the nomenclature of the figure is discussed.

   • o1 1 Represents the origin of moving system when the ray is received at x'

   • o2 2 Represents the origin of moving

     system when the ray is received back at the origin of moving system.

     • XYZ represents the stationary co- ordinate system.

     • Represents the moving co- ordinate system.

     • Represents the co-ordinate time of

       the uniformly moving system along X-direction.

       1

       1

       Now the stage is set to point out the derivational error. Reminder: the error which we want to point out

       is ( ) . Before proceeding to the

       2 0 2 1

       derivational aspects, the following points are worth mentioning.

     • The ongoing ray travels a distance of

       x' with a velocity of (c v)

     • By the time the reflected ray reaches the origin, the origin has shifted a

   distance of

   x' and hence the

   reflected ray travels a distance lesser

   than

   x' by an amount

   x' ( 2 0 )v . This point has not been noticed by Albert Einstein.

   Figure 1: Pictorial representation of stationary and moving co- ordinate system.

   • Black color co-ordinate represents the

   • 0 is the reading of the clock at the

     time of emission of the ray.

   • 1 is the reading of the clock at the time of reflection of the ray at x' .

     stationary system.

     • 2 is the reading of the clock at the

       Red color co-ordinate represents the moving system.

       time of reception of the ray at the origin.

     • Always represents the time in the moving clock and t represents the time in the stationary clock.

   Referring to figure 1, the following equations are derived.

   x'

   0 1 c v (1)

3. Derivation of A( , c, v)

   The additional term appearing along with

   1 in equation (5) is derived in this section. Substituting equation (3) & (4) in equation (6), one gets the following expression for A( , c, v) .

   2 1

   x'x' c v

   (2)

   A( , c, v)

   x'

   c v

   • x' c v

     x'

     c v

     x' 2 0 v

     x' ( 1 0 )(c v)

     (3)

     (4)

     (8)

     A( ,c, v) (1 0 )(c v) (1 0 )(c v)

     Upon adding equation (1) and equation (2), one gets the following equation.

     ( 2 0 )v c v

     c v

     c v

     0 2

     21

     x'x'

     c v

     x' c v

     (5)

     (9)

     21 A( ,c, v)

     Where

     Upon simplifying the above equation, we

     A( , c, v) x'x'

     c v

     x'

     c v

     get the following expression for A( , c, v) .

     (6)

     Dividing equation (5) by 2, we get the following equation.

     A( , c, v)

     3v 0 2v 1 v 2

     c v

     (10)

     1 (

     2 0

     2

     ) 1

     x'x' 2(c v)

   • x'

   2(c v)

   (7)

   Substituting equation (10) back in equation

   From equation (7), it is clear that

   (5), we get the following expression.

   1 ( ) . It is also clear

   0 2 1

   0 2 1

   c v(3 0 2 )

   2 0 2

   that ( 0 2 ) 2 1 only when v 0 .

   So it is clear form equation (7) that Einstein made a wrong assumption [1-page 6] at the very beginning of his derivation. With this, this section is concluded. An alternate

   (11)

   derivation for the coordinate transformation follows in the following sections. Readers are requested to thoroughly understand this section before proceeding further.

   From the above equation it is clear that ( 0 2 ) 2 1 . It is also clear that ( 0 2 ) 2 1 only when v 0 .

4. An Alternate Derivation of Co- ordinate Transformation

   (c 2v) 0,0,0, t

   (c 2v) x',0,0,t

   x' x'x'

   Upon simplifying equation (11), we get the following expression.

   2c x',0,0, t

   c v

   x'

   c v

   (c 2v) 0

   (c 2v) 2

   2c 1

   (12)

   c v

   (13)

   Before proceeding further,

   x' should be

   Before proceeding further, the following points are worth mentioning.

   • The argument of 0 are 0,0,0, t

     expressed in terms of x' . That is done in the following steps. Subtracting equation (1) from equation (2), the following equation is obtained.

   • The argument of

     1 are

     x' x' x'

     x',0,0, t x'

     2 0 c v c v c v

     c v

     (14)

   • The argument of

     2 are

     ( 2

     0 )

     x'

     c v

     x'

     c v

     • x' c v

       x',0,0, t x'

       x'x' ,

       (15)

       c v

       c v

       Substituting for

       x' from equation (3), the

       above equation simplifies as follows.

       Albert Einstein has overlooked the

       v

       2cx'

       argument of 2

       i.e. he did not notice

       ( 2 0 )1 c v (c v)(c v)

       that the origin of the moving co- ordinate has shifted a distance of

       x' during the rays flight time forth and back. He also overlooked the time t argument of 2 , where he has

       ( 2

       0 )

       2cx'

       (c 2v)(c v)

       (16)

       (17)

       substituted Substituting equation (17) in equation (3),

       t x' c v

     • x' c v

       instead

       the expression for x' is obtained as follows.

       x'

       x'

       of t

       c v

       x'x'

       c v

       . The factor

       x' ( 2

       0

       )v

       2cvx'

       (c 2v)(c v)

       x'

       x' should appear in the time coordinate because the reflected ray

       (18)

       travels

       x' lesser distance than the

       Where is given by the following

       emitted ray.

       Upon substituting the arguments of in

       equation.

       2cv

       equation (12), one gets the following expression [1-page6].

       (c 2v)(c v) . (19)

       The time argument of 2 also contains

       Hence if x' is chosen infinitesimally small

       x' and hence it should be expressed in terms of x' . That is done in the following steps.

       (As claimed by Einstein [1-page 6]), the above equation reduces as follows (upon expanding by Maclaurin series).

       The time argument of 2 is

       t x' x'x'

       from equation (13).

       1

       c v c v

       (c 2 v ) x '

       t 2 c x ' c v

       t

       This x' should be eliminated before proceeding further.

       Let x'

       x'

       c v

     • x'x' . Upon

     c v

     Since (c 2v)

     (24)

     2c

     , the above

     substituting x' from equation (18) in , the

     c v

     expression for simplifies as below.

     equation reduces as follows.

     1 1

     (c 2v) 2c 0

     c v c v c v

     (20)

     x' x'

     0

     (25)

     Upon substituting from equation (19), the above expression reduces as follows.

     x'

     (26)

     2c

     The solution of the above equation is

     (c v)(c 2v)

     (21)

     const Hence Time cannot be co-

     Now the stage is set for deriving the equation of co-ordinate transformation.

     ordinate transferred as claimed by Einstein. There is a peculiar result possible from equation (25).

     Equation (13), reduces as below.

     (c 2v)

     x'

     2c

     x'

     (27)

     (c 2v) 0,0,0, t

     (c 2v) x',0,0, t

     x'

     x'

     x'

     The above equation reduces as follows.

     (c 2v) 2c

     (28)

     2c x',0,0, t

     c v

     x'

     c v

     c v

     Upon substituting in the above equation one gets v c / 2 .

     c v

     (22)

5. Conclusion

The theory of relativity (both special and

general theory of relativity) is proven beyond doubt. The purpose of the manuscript is not to

Further simplifying results in the following

equation.

(c 2v) 0,0,0, t )

(c 2v) x' ,0,0, t x ' 2 c x' ,0,0, t

dispute special theory relativity. The objective of this paper is to prove that the mathematical method employed by Albert Einstein to arrive at his equation is wrong and hence it is

x ' fulfilled.

(23)

c v References

1. Albert Einstein , On the Electrodynamics of Moving

Bodies (Zur Elektrodynamik betwegter Krper), Annalen der physik,1905,

http://www.fourmilab.ch/einstein/specrel/specrel.pdf, pp 5-6.