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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April‐2013 560

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Domain of Relativistic Mechanics.

Pradeep Koshy

Abstract : .It is astonishing to note that ,the Special Relativity Theory Equations are incongruous beyond half the speed of light, by mere Number Theory considerations ,specifically ‘time dilation’ and ‘length contraction’ equations. Nevertheless, there is an exception to this theory,for particles of masses less than

1kg, or time intervals between events which are less than 1s or length contraction less than 300000km ,where it could be absolutely accurate as Einstein predicted and therefore it implies,these could change upto only

13.3%,approximately.

Index terms: Relativity,time dilation,length contraction,mass variance,

number theory,algebraic formulae,Theory of Everything.

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1.Introduction

Einstein's Theory undermined.

A century old theory is being investigated by a mind for 20 years. The result in this case, Einstein’s theory seems to be shattered.
We will prove it in 3 parts.

PART1: A three way related algebraic formula developed by me and its

significance like number theory could

be the basis of even a theory like

Special theory of Relativity. PART2: In the shadow of Einstein.

PART3:Relativistic Mechanics probed to find the domain of it.Note that , part3 might seem independent, but the formula derived in part1 has a

bearing on this.

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Part – 1:

2.Lord Kelvin’s Statement

“I often say, if you can express what you know of something, in numbers, you know something about it. Otherwise, your knowledge is of a meager and unsatisfactory kind. It may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the stage of science, whatever the matter may be”.

3.The bombshell of the 21st century: An algebraic formula developed by me

I was preparing for SAT almost 2 decades back. I studied and observed squares of numbers.

20²=400

21²=441

22²=484

23²=529

24²=576

25²=625

26²=676

27²=729

28²=784

29²=841

30²=900

The difference between the first and last numbers, second and second last numbers and so on are as follows:
500, 400, 300, 200 and 100. This difference looked unbelievable to me. To my observation, it was stunning to observe that the squares of the difference of numbers was falling into a definite pattern.

30² ― 20²= 500 = [26² ― 24²] x 5

29² ― 21²= 400 = [26² ― 24²] x 4

28² ― 22²= 300 = [26² ― 24²] x 3

27² ― 23²= 200 = [26² ― 24²] x2

26² ― 24²= 100 = [26² ― 24²] x1

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At this stage, insight, institution and creativity had been triggered. The pattern, was further checked and studied by extending it to 10‐20 range.

10² =100

11²=121

12²=144

13²=169

14²=196

15²=225

16²=256

17²=289

18²=324

19²=361

20²=400

20²―10²=300=60x5

19²―11²=240=60x4

18²―12²=180=60x3

17²―13²=120=60x2

16²―14²=60=60x1

These observations, were further extensively checked and it led me to the formula.

[N² ― (N―2)² ]X=4(N‐

1)X=[N―(1―X)]²―[N―(1+X)]² where N and X are are two variables. Now, it is the triumph of creativity.It is a three way related formulae and hence ,can be used to pose a question like given an equation, [A² ― B²] C = D= E²― F² where given E and F

to findA,B,C and D. or alternatively posed as given A and C, to find B, D,E and F.where the

Capital letters denote numbers, either Real or Complex.This is a formula so revolutionary

that I would call it miracle equation.It is

God’s gift to me.Note,that my formula is capable

of solving 2 variables in a single

equation,which is conventionally impossible from a mathematical

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point of view.Also,understand that the modified form of special theory of relativity equations contain 3 variables.For instance(t′)²=(t)²―( tv/c)².Using my formula,2 of the variables are determinate. Hence, the only remaining variable, which is the only unknown now could be found out.In effect,in totality,in reality, all the variables can be determined or solved ie meaningful interpretation of the same an be made.From,hereon,we proceed keeping

,this in mind.No longer,is the Special

Theory of Relativity Equations mysterious,inscrutable or perplexing.Armed with a formula like the above,we will pursue the same.

Pythagoreans believed that the number 4 governs the world. It is there in my formula.

Number theory can be the key to the understanding or unlocking the limits of

the special theory of Relativity. It reinforces the maxim “Mathematics is the Mother of all sciences”.

4.VITAL DEDUCTION: The formula developed by me, serves as a way of solving two variables minimum or upto 3, maximum in a single equation relating two sets of difference of squares of numbers. Hence, it is a paradox from mathematical point of view, although ,it is only a particular solution. That can be miraculous, in nature, will be proved later, when considered it is analogous to special relativity equations in number theory form. Something of trivial significance, is mentioned below.

5.Deviation :(Formula has one more use) Let me rearrange the formula:

[√[ {N²―(N―2)²}X]²

+[N―(1+X)]²=[N―(1―X)]² ie of the form A²

+B² = C²

Hence,

Pythagorean triplets can be generated by using this formula.

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6.IN THE SHADOW OF EINSTEIN – Applications in Special Theory of Relativity The formula was invented 2 decades back. It was rejected, or considered trivial among most people I met or submitted, in India. Subsconsciously, feeling always, analogous to Special Relativity-- those issues, were boiling in my mind, for more

than a decade, when at last, it was serving as a catapult to my long standing cherished view, that it could be applied to Special Relativity equations and the domain of Relativistic Mechanics could be identified. I was awestruck at how ,I arrived at it.

The sequence of events:

Knowledge of inequalities(Algebra branch)

is required in order to understand this.

TIME DILATON EQUATION

t = t′

√ (1― v²/c²)

where t is the time measured in S frame is slower than the time t′ measured in inertial frame S′ by time dilation equations.In all these cases, c stands for velocity of light and equal
. 8.CASE1
to it,whereas v is the velocity of the frame S′ with respect to S.
On rearranging the relativistic
equation ,simplifies to( t′c)²=(tc
)²―(tv)²,which finally leads to
(t′)²=(t)²―( tv/c)²
We proceed like this:At this stage,my invented formula’s most simplified form will be adequate in proving my contention. A simpler form of my formula could be treated as ,the formula 4AB= (A+B)² ― (A―B)²,when we apply(N-1)=A and X=B which on further simplification leads to
AB=[(A+B)/2 ]² ― [ ( A ― B)/2]² by rearranging.,will thereby have the merits and advantages of my formula , which is worth mentioning.

Please note also, in reality,

( A+B)/2 > (A―B)/2 for both A and B greater than 0.

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BOARD A/PHYSICS

STEP1 (t′)²=(t)²―(tv/c)²,time dilation equation modified form involving

3 variables,which have their usual meanings. STEP2

Note,that each term of the above equation has the same unit.(second squared)

STEP3

But t>0 and t′ >0 and

t < k,hence maximum of t=k,comparing physics equation divested of its units,with mathematics equation

BOARD B/NUMBER

THEORY

STEP1

AB=[(A+B)/2 ]² ― [ ( A ― B)/2]² is true and has the properties of miracle equation

STEP2

All Equations are analogous to

X² = Y²― Z²,in which X equals squareroot of(AB) or [k(k― 2w)].This is the instance when physics and mathematics are identical.STEP3

Maximum of tv/c=kv/c is the first value of w. STEP4

Equating first value of w with second value,we get kv/c=k/2.Cancelling out k from both sides in it and rearranging,we get maximum of v=c/2 units.

BOARD C

BOARD A/PHYSICS

STEP1 (t′)²=(t)²―(tv/c)²,time dilation equation modified form involving

3 variables,which have their usual meanings. STEP2

Note,that each term of the above equation has the same unit.(second squared)

STEP3

But t>0 and t′ >0 and

t < k,hence maximum of t=k,comparing physics equation divested of its units,with mathematics equation

BOARD B/NUMBER

THEORY

STEP1

AB=[(A+B)/2 ]² ― [ ( A ― B)/2]² is true and has the properties of miracle equation

STEP2

All Equations are analogous to

X² = Y²― Z²,in which X equals squareroot of(AB) or [k(k― 2w)].This is the instance when physics and mathematics are identical.STEP3

Maximum of tv/c=kv/c is the first value of w. STEP4

Equating first value of w with second value,we get kv/c=k/2.Cancelling out k from both sides in it and rearranging,we get maximum of v=c/2 units.

\MATHEMATICS

STEP1

(k)(k―2 w)=(k―w)²― (w)² is an algebraic identity involving 2 vqariables.

STEP2

Further let Z=w= tv/c above.

Let us introduce w= t v/c,then

(k)(k―2 tv/c)=(k―tv/c)²― (tv/c)²

SinceA>0 , B>0& A>B,we get (tv < ck/2) ,

Also(A+B/2)>0 gives(tv<ck), ergo,tv<ck/2 is the deduction.Hence,the maximum value of

tv= ck/2,since even(A- B/2)>0

STEP3

It follows since max. of tv= ck/2,therefore ,max. of tv/c=k/2.This is the second value of w.

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VERY IMPORTANT
In the basic equation [N² ―

(N―2)² ]X=4(N-

1)X=[N―(1―X)]²―[N―(1+X)]²

where N and X are are two variables.,N,(N ― 1) and (N ― 2) ought to be positive which implies k has to be greater than1.

9.CASE2

For values of k<1, the valid equation is

t = t′

√ (1― v²/c²)

,where Einstein’s special theory of relativity be valid and the limitations,I tried to impose upon invalid.Hence v almost could be equal to c

,implies t< 1,which also

implies t′ <1.
.

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10.Conclusion:

Yes, we can reduce a two dimensional multiplication to a squaring operation in a single dimension (though this can raise the question of whether or not higher dimensional spaces can inversely be constructed by operations performed within a single dimension! I think this is true and that the dimensionality of a space arises from quantities and the inherent diversity of qualities in the context of which information is presented)

CASE1

Maximum of v = 2 units in tv<kc/2 , time in frame S′, t′, element of 1 to k seconds

and also t<k seconds,where k stands for any fixed value ,however large in seconds,greater than 1.

In case 1 not only numerical verification is done by number theory developed by me, even if ,units and dimensions are considered as issues,will not hamper the conclusion.
In less than or equal to 1 second time interval between events,mass less than 1 kg or length contraction less than 300000km,Einstein’s special theory of relativity is
100% accurate.Beyond the same,it is only accurate upto half the speed of light,for

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Time dilation , mass variation and Length contraction .Also, Special Relativity is absurd in momentum and energy considerations also in this affected domain.
Ergo, my theory serves to destroy, a century's unabashed approval of Special Theory of Relativity by world's leading scientists , of much, of its miraculous nature. Einstein's view “The distinction between past, present and future is illusory, however tenacious, it may appear to be” does not seem vindicated.
Cosmic speed or speed barrier for particles above 1kg is proved to be half the speed of light.There is no Theory of Everything.
11.References
[1]University Physics by Sears,Zemanski &Young.
[2]The Feynmann Lectures by Richard P.Feynmann,Matthew W.Sands &Robert B. Leighton.
[3]Special Theory of Relativity by Resnick. [4]Fundamentals of Physics by Halliday,Resnick &Walker.
[5]Superstrings and the search for the Theory of Everything by David F. Peat.

Author

Pradeep Koshy, Independent Researcher, Cochin, Kerala, India

Mob:+91-9895854062

Email:koshy68@gmail.com

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