Absolute Theory of Relativity & Parameterized Special Theory of Relativity & Noninertial Multirelativity, by Florentin Smarandache

8/3/2019 Absolute Theory of Relativity & Parameterized Special Theory of Relativity & Noninertial Multirelativity, by Florentin

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Florentin Smarandache

ABSOLUTE THEORY OF RELATIVITY

&PARAMETERIZED SPECIAL THEORY

OF RELATIVITY

&NONINERTIAL MULTIRELATIVITY

B

A s A

B

l l

SOCIETE MAROCAINE

D'IMPRESSION ET D'EOUIPEMENT S.A.R.L

SOMIPRESSSige Sociale : 72, Boulevard Hassan Il

Tl. : 238-68 - FES (v. n. )


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Florentin Smarandache

ABSOLUTE THEORY OF RELATIVITY&

PARAMETERIZED THEORY OFRELATIVITY

&

NONINERTIAL MULTIRELATIVITY

SOCIETE MAROCAINE

D'IMPRESSION ET D'EOUIPEMENT S.A.R.L

SOMIPRESSSige Sociale : 72, Boulevard Hassan IlTl. : 238-68 - FES (v. n. )


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L'auteur autorise la reproduction de ces ides la

seule condition de citer son nom et les references

du prsent ouvrage.

(c) Imprimerie SOMIPRESS, 1982, Edition

Anglaise, FES.Tous droits de traduction , de reproduction etd'adaptation rservs pour tous pays.


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C O N T E N T S

Preface (Many Relativities): 7

Chapter 1.

On Einsteins Thought Experiment

with the Light Clocks: 19

1.1. Einsteins Thought Experiment with the

Light Clocks: 20

1.2. Relativistic Addition of Velocities: 23

1.3. Counter-Intuitive Results of the RelativisticAddition of Velocities: 23

1.4. Contradictory Results of the Relativistic

Addition of Velocities: 24

1.5. Relativistic Addition of Superluminal

Velocities: 26

1.6. Extension of Relativistic Formula for the

Addition of Many Subluminal Velocities: 27

1.7. Generalization of Einsteins Thought

Experiment with the Light Clocks for ArbitraryUltimate Velocity K: 35

1.8. Generalization of the Relativistic Addition

Formula for Arbitrary Ultimate Velocity K: 37

1.9. Generalized Lorentz Factor: 39


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1.10. Generalized Minkowski Norm: 39

1.11. Generalized Time Dilation: 39

1.12. Generalized Length Contraction: 40

1.13. Generalized Relativistic Momentum: 40

1.14. Generalized Energy: 40

1.15. The Generalized Total Energy: 40

1.16. Generalized Kinetic Energy: 41

Chapter 2.

A Hypothesis:

There is No Speed Barrier in the Universe

and

One Can Construct Arbitrary Speeds: 42

2.1. Introduction: 432.2. Entangled Particles: 43

2.3. Scientific Hypothesis: 45

2.4. Open Question: 46

2.5. Future Possible Research: 46

Chapter 3.

Absolute Theory of Relativity (ATR): 48

3.1. Refuting Einsteins Speed of Light

Postulate: 49

3.2. About Inconsistent Theories: 54

3.3. No Relativistic Paradoxes in ATR: 55

3.4. Removing Lorentz Factor in ATR: 56


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3.5. Physics Laws might not be the same in all

Inertial Systems: 58

3.6. Linear Trajectories for Both Observers: 59

3.6.1. Orthogonal Trajectory Vectors: 54

3.6.2. Non-Orthogonal Trajectory Vectors and c

as Ultimate Velocity: 59

3.6.3. Non-Orthogonal Trajectory Vectors and

Arbitrary Ultimate Speed: 63

3.6.4. Non-Orthogonal Trajectory Vectors and

Arbitrary Velocities: 63

Chapter 4.

Parameterized Special Theory of Relativity

(PSTR): 65

4.1. PSTR Equation: 66

4.2. PSTR Elapsed Time Ratio(Parameter): 67

4.3. PSTR Extends STR, ATR, and Introduces

Three More Relativities: 67

4.4. PSTRK Equation and Elapsed Time Ratio

(Parameter) in SKSpace: 69

4.5. PSTRK Extends STRK, ATRK, and

Introduces Three MoreK-Relativities: 70


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Chapter 5.

Accelerated Reference Frames: 72

5.1. Formulas of Kinematics for Constant

Acceleration: 73

5.2. Gravitation and Constant Velocity of the

Rocket: 73

5.3. Gravitation and Constant Acceleration of

the Rocket: 77

5.4. Constant Accelerations and Oblique

Direction Vectors: 80

5.5. Constant Accelerations with Zero Initial

Velocity and Oblique Direction Vectors: 83

Chapter 6.

Noninertial Multirelativity: 85

6.1. Multirelativity with Nonconstant

Acceleration and 3D-Curves: 86

6.2. Research Problems: 87

6.3. Example of Nonlinear 3D-Trajectories of

the Particle and of the Reference Frames: 88

References: 91


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Preface

( Many Relativities )

In the first chapter of this book we discuss

Einsteins thought experiment with atomic

clocks:

A rocket travels at a constant speed v with

respect to the earth. In the rocket, a light pulse is

emitted by a source from A to a mirrorB that

reflects it back to A where it is detected. The

rockets movement and the light pulses

movement are orthogonal. There is an observer

in the rocket (the astronaut) and an observer on

the earth. The trajectory of light pulse (and

implicitly the distance traveled by the light

pulse), the elapsed time it needs to travel this

distance, and the speed of the light pulse at

which is travels are perceived differently by thetwo observers {depending on the theories used

in this book}.


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Bd

A

Fig. (p1)

The Special Theory of Relativity (STR) works

fine in a spacetime Sc where the speed of light c

is the ultimate speed and the relativistic addition

of velocities applies:

1 21 2

1 2

21

c

v vv v

v v

c

+ =

+

(p1)

Several counter-intuitive examples of

formula (p1) are presented. Also, in our opinion

there is neither a real time dilation nor a length

contraction, but an apparent time dilation and an

apparent length contraction.


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Yet, we extend this formula, by induction,

using a recurrent proof, for the relativistic

addition ofn 2 velocities at once.

We show that formula (p1) and itsextension can be straightforwardly extended to a

spacetime SK where the ultimate speed is K>0,

which can be smaller or greater than c.

Therefore, the relativistic addition of two

velocities in SKbecomes:

1 21 2

1 2

21

K

v vv v

v v

K

=

+

+ (p2)

and correspondingly all related formulas from

STR get translated into the SK spacetime by

simply substituting c with K: whence we obtainthe Generalized Lorentz Factor, Generalized

Minkowski Norm, Generalized Time Dilation,

Generalized Length Contraction, Generalized

Relativistic Momentum, Generalized Energy,

Generalized Total Energy, and GeneralizedKinetic Energy.


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In the second chapter we present our hypothesis

that there is no speed barrier in the universe and

one can construct arbitrary speeds from zero to

infinity (Smarandache -1972), thus refuting the

speed of light postulate. We consider that thesuperluminal phenomena do not violate the

causality principle, do not produce time

traveling, and do not necessitate infinite energy

for particles traveling at speeds greater than the

speed of light.

While Einstein considered a relative space

and relative time but ultimate speed of light, we

do the opposite in the third chapter: we redo

Einsteins experiment with atomic clocks by

considering an absolute time and absolute space

but no ultimate speed (according to our previous

hypothesis). Thats why we call our theory

Absolute Theory of Relativity (ATR).

According to this theory, the speed of photon in

the rocket, with respect to the observer on earth,is:

2 2x v c= + (p3)


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which corresponds to the magnitude of the

vectorial addition, i.e. ,cvx

+= since c

and

v

are orthogonal:

v

c cx or x

v

Fig. (p2)

and thusx > c.

Therefore in the ATR we replace Einsteins

Relativistic Addition of Velocities (p1) with

(p3), allowing for superluminal speeds.

Since in Absolute Theory of Relativity there is

no time dilation, in consequence there is no

length contraction, and no relative simultaneity.

Lorentz Factor becomes equal to 1, so it is

useless.

As a consequence, many relativistic paradoxes

are discarded: such as Ehrenfest Paradox (1909),

the Twin Paradox (1911), Bells Spaceship

Paradox (1959), W. Rindlers Paradox about a

man falling into a grate (1961), etc.


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The famous physics formula E0 = mc2

is

questionable: why should it depend on the light

speed?

Similarly:

a) the Michelson-Morley Experiment (1881,1883-1887) of not having detected the

ether, which implied the STR, might have

been erroneous due to imprecise

measurement instruments or construction.

Maybe the ether is very little dense and its

flow hardly perceptible;

b) and the constancy of the speed of light invacuum could be flaw, since the speed of

light might depend on the emitting source

the stronger the emitting source thefaster its light pulse;

because if an inconsistent theory resulted from

some ideas, those ideas must have been

inconsistent too.

Not all physics laws might be the same in

all inertial systems.

As a counter-example lets consider as physics

law the Addition of Velocities in a given inertial

system Si. Lets take the collinear and in the

same direction velocities v1 = 0.8c and

v2 = 0.9c. In STR we get v1+v2 = 0.988372c

while in ATR we get a different result:


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v1+v2 = 1.7c.

ATR generalizes the relativistic addition of

velocities in the following way: instead of a

photon we consider any particle traveling fromA to B with a speed w > 0 such that the angle

between the particles direction vectorw and the

rockets direction vectorv is , with 0 :

Fig. (p3)

w B B

A v

2 2 2 cosx v w vw = + +

or

x v w

= + (p4)


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which is a generalization of(p3).

A computer simulation would better describe

the resultant curve AB[fromFig. (p3)] as seen

by the observer on earth. Herein weapproximate it by a line.

In Chapter 4, we study the general case of

Einsteins thought experiment with atomic

clocks, when we suppose nothing is known

about if the space and time are relative or

absolute or if the speed of light is ultimate speed

or not.

Therefore, we obtain the Parameterized Special

Theory of Relativity (PSTR). Its equation is:

2

2 2 'tx v ct

= +

(p5)

wherex is the speed of the photon as measured

by the observer on earth, 't is the elapsed time

as measured by the astronaut, and t the

elapsed time as measured by the observer on

earth.


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And the PSTRs parameter is

't

t

=

(p6)

where (0, +).

Our PSTR generalizes not only Einsteins STR,

but also the previous ATR, and three other

possible Relativities [3,4, and 5 from below]

that readers can study, as follows:

1. If 22

1

c

v= we get the Special Theory of

Relativity.

2. If =1, we get our Absolute Theory ofRelativity in the particular case when the

two trajectory vectors are perpendicular.

3. If 22

10c

v


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4. If 11 22

' ) with respect to the STR (instead

of 't t < ), and the speedx as seen by theobserver on earth increases even more than

in our ATR.

Then the above definitions and classification of

Relativities are extended in the SK space (space

where the ultimate speed ifK) toK-Relativities.

In Chapter 5, we repeat Einsteins thought

experiment introducing the acceleration. We

consider the particle in the rocket moving at

constant acceleration, while the rocket moving

either at constant velocity or at a constant

acceleration. The trajectory vectors of the

particle and the rocket are either orthogonal or

oblique to each other.


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We used only linear trajectories for both

observers, but at the end we propose, as research

problems, the nonlinear arbitrary 3D-curves

with nonconstant accelerations.

Chapter 6 is the most general, yet to be

investigated.

Noninertial Multirelativity means that the

particle P0 in the reference frame F1 travels on

an arbitrary 3D-curve C0 with a non-constant

acceleration a0 with initial velocity v0. Then the

reference frame F1 moves with respect to

another reference frame F2 on an arbitrary 3D-curve C1 with a non-constant acceleration a1

with initial velocity v1. And so on, the reference

frame Fn-1 moves with respect to another

reference frame Fn (for n 2) on an arbitrary

3D-curve Cn-1 with a non-constant acceleration

an-1 with initial velocity vn-1.

The following research problems are

proposed with respect to the Noninertial

Multirelativity:


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1. How would the particles trajectory curveC0 be seen by an observer in the reference

frameFn ?

2. What would be the particles speed(acceleration) as measured by theobserver from the reference frameFn ?

3. What would be the elapsed time of the particle as seen by the observer in the

reference frameFn ?

4. What are the transformation equationsfrom a reference frame to another?

5. Similar questions for rotating referencesystems.

The author


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Chapter 1.

On Einsteins Thought Experiment

with the Light Clocks


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1.1. Einsteins Thought Experiment withthe Light Clocks.

Lets consider the Einsteins thought

experiment with the light clocks.

There are two identical clocks, one is

placed aboard of a rocket, which travels at a

constant speed v relative to the earth, and the

second one is on earth. In the rocket, a light

pulse is emitted by a source fromA to a mirrorB

that reflects it back to A where it is detected.

The rockets movement and the light pulses

movement are orthogonal. There is an observer

in the rocket (the astronaut) and an observer on

the earth. The trajectory of light pulse (andimplicitly the distance traveled by the light

pulse), the elapsed time it needs to travel this

distance, and the speed of the light pulse at

which is travels are perceived differently by the

two observers {depending on the theories used

see below in this book}.

According to the astronaut:

Fig. 1

B

dA


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2'

dt

c =

(1)

where:

't = time interval, as measured by theastronaut, for the light to follow the path

of distance 2d;

d= distance;c = speed of light.

According to the observer on earth:

Fig. 2

BA s A

B

l l

2l v t=

s = |AB| = |BA|d = |BB| (2)

l = |AB| = |BA|

where t = time interval as measured by theobserver on earth.


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And using the Pythagoras Theorem in the right

triangle ABB, one has

2

2 2 22 2 2

2

v ts d l d

= + = +

(3)

but 2s c t = , whence

2

222

v tc t d

= +

(4)

Squaring and computing for t one gets:

2

2

2 1

1

dt

c v

c

=

. (5)

Whence Einstein gets the following time

dilation:

2

2

'

1

ttv

c

=

(6)


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where 't t > .

1.2. Relativistic Addition of Velocities.

According to the Relativistic Addition of

velocities:

1 2

1 2

21

v vv

v v

c

+=

+

(7)

where:

1v = velocity of the particle inside therocket as measured by the astronaut;

2v = velocity of the rocket as measured by

the observer on earth;

v = velocity of the particle as measured by

the observer on earth.

1.3. Counter-Intuitive Results of the

Relativistic Addition of Velocities.

Some curious additions, in our opinion:

0.999c c c = ; (8)


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0

0c c = = undefined. (9)

1.4. Contradictory Results of theRelativistic Addition of Velocities.

One has not only c v c+ = (for v c< ),butalso c-v = 0, orc+v=c-v, so v=-v ? (10)

Also c c c+ = , and in general

....n times

c c c c+ + + = ; therefore n c c =

for any integer 1n . (11)More general, r c c = for any real

number 1r , (12)

since ifr n a= +

where

[ ]n r= , integer part of r,and

{ }a r= , the fractional part of rthen

( )r c n a c nc ac c ac c = + = + = + = .


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but, having r c c = for any real number 1r looks unrealistic.

If we compute

( )limn

n c c

= = , (13)

but on the other side, because n c c = , we have

( ) ( )lim limn n

n c c c

= = , (14)

which is a contradiction.

Similarly, if 0 v c< < , then

( )1

... ...n times n times

c n v c v v v c v v v

+ = + + + + = + + + + =

1

... ...n times

c v v c

= + + + = = . (15)

Then

( )limn

c n v c v

+ = + = (16)

but, also

( )limn

c n v c c

+ = = , (17)

therefore, again, a contradiction.


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1.5. Relativistic Addition of

Superluminal Velocities.

Of course, for superluminal velocities

Einsteins Relativistic Addition of velocities

does not work. See the following counter-

example:

If 1 2v c= and 2 3v c= , then:

( ) ( )1 2

2

2 3 5 5

2 3 1 6 71

c c cv v c c

c c

c

+ = = = gives:

1 2

1 2

21

v vcv v

c

+

v.


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Therefore now the relativistic factor which

makes the so-called time dilation depends on v

andK.

1.8. Generalization of the Relativistic

Addition Formula for Arbitrary Ultimate

Velocity K.

Actually, we can generalize Einsteins

formula of relativistic addition of velocities in

the following way:

Lets consider a constant 0 K< < and a

physical imaginary space SK, where the ultimate

speed is .K

Postulating, similarly to Einstein, that novelocity overpass K in a space denoted SK, we

define just by substituting " "c by" "K the

addition velocities for an observer on Earth:

1 2

1 2

21

v vv

v v

K

+=

+

(43)

Then, same properties occur:

K v K + = forv K ; (44)


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{ }1 2 1 2max , kv v v v K <

for any 1 2,v v K , with equality in the case

when at least one of 1v or 2v is equal to K.In this imaginary space SK, the ultimate

velocity K can be subluminal, for example the

speed of sound, or superluminal speed (for

example ten times the speed of light).

Multiple addition formula of speeds in SKwill have a similar formula:

1

2

2 12

01 2

22

20

...

n

n

ii

ik k k n n

n

i

ii

sK

v v v

S

K

+

=

=

=

(45)

We prove it by induction upon 2n as

we did before, simply substituting " "c (the

speed of light) by " "K in the previous proof.


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1.9.Generalized Lorentz Factor.

Lorentz factor will also be generalized to

2

2

1 [1, )

1v

K

+

. (46)

1.10.Generalized Minkowski Norm.

The norm of (x, y, z, t) in the Minkowski

spacetime also becomes:

2 2 2 2 2x y z K t + + . (47)

1.11. Generalized Time Dilation

becomes:

2

2

'

1K

v

tt

=

. (48)


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1.12. Generalized Length Contraction

becomes:

2

2

1'. K

v=

. (49)

1.13. Generalized Relativistic

Momentum of an object of mass m, moving

with speed v, becomes:

2

2

1K

v

mvp

= . (50)

1.14.Generalized Energy of an object at

rest, with rest mass m, isE0=mK2. (51)

1.15.TheGeneralized Total Energy of

an object of mass m, moving at speed v,

becomes:

2

2

21

mKE

v

K

=

. (52)


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1.16. Generalized Kinetic Energy of

an object of mass m, moving at speed v,

becomes:

= 1

1

1

2

2

2

K

vmKEk . (53)


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Chapter 2.

A Hypothesis:

There is No Speed Barrier in the Universe

andOne Can Construct Arbitrary Speeds


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2.1. Introduction.

In this short chapter, as an extension and

consequence of Einstein-Podolski-Rosen

paradox and Bells inequality, onepromotes the hypothesis that: There is no

speed barrier in the universe and one can

construct any speed, from zero to infinite

speed (instantaneous transmission).

Future research: to study the composition

of faster-than-light velocities and what

happens with the laws of physics at faster-

than-light velocities?

2.2. Entangled Particles.We recall the following:

- photon is a bit of light, the quantum of

electromagnetic radiation (quantum is the

smallest amount of energy that a system can

gain or lose);

- polarization refers to the direction and

characteristics of the light wave vibration;


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- if one uses the entanglement phenomenon, in

order to transfer the polarization between two

photons, then: whatever happens to one is the

opposite of what happens to the other; hence,

their polarizations are opposite of each other;

- in quantum mechanics, objects such as

subatomic particles do not have specific,

fixed characteristic at any given instant in

time until they are measured;

- suppose a certain physical process produces

a pair of entangled particlesA andB (having

opposite or complementary characteristics),

which fly off into space in the opposite

direction and, when they are billions of miles

apart, one measures particle A; because B is

the opposite, the act of measuring A

instantaneously tells B what to be; therefore

those instructions would somehow have to

travel betweenA andB faster than the speed

of light; hence, one can extend the Einstein-Podolsky-Rosen paradox and Bell's

inequality and assert that the light speed is

not a speed barrier in the universe;


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- even more, one can construct any speed, even

greater than the speed of light (c), by

measuring particleA at various intervals;

- also, the information from particle A and B istransmitted instantaneously (thus, there is no

speed barrier in the universe).

2.3. Scientific Hypothesis.

We promote the hypothesis that: there is no

speed barrier in the universe and one can

construct any speed even infinite (instantaneous

transmission), which would be theoreticallyproven by increasing, in the previous example,

the distance between particles A andB as much

as the universe allows it, and then measuring

particleA.

We consider that the superluminal phenomena

do not violate the causality principle, do not

produce time traveling, and do not necessitateinfinite energy for particles traveling at speeds

greater than the speed of light.


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2.4. Open Question.

Wouldnt it be possible to accelerate a photon

(or another particle traveling at, lets say,

0.999c) and thus to get a speed greater than c(where c is the speed of light)? We dont think it

is needed an infinite energy for this.

2.5. Future Possible Research.

It would be interesting to study the composition

of two velocities v and w in the cases when:

v < c and w = c. (54)

v = c and w = c. (55)

v > c and w = c. (56)

v > c and w > c. (57)

v < c and w = . (58)

v = c and w = . (59)

v > c and w = . (60)


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v = and w = . (61)

What happens with the laws of physics in each

of these cases?


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Chapter 3.Absolute Theory of Relativity (ATR)


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3.1. Refuting Einsteins Speed of Light

Postulate.

Again, we do the opposite of what

Einstein did. Instead of considering the speeds

of the clock light the same for both observers,

while the time intervals different, we consider

the time intervals are the same for both

observers, while the speeds c and respectively

v+c not equal. We refute Einsteins speed oflight postulate according to our hypothesis that

there is no speed limit in the universe and one

can construct arbitrary speeds.

The classical formula

Distance = Speed Time (62)

was distorted in the Special Theory of Relativity

in the following way: Time was increased

(dilated), while Distance was decreased

(contracted).

In order to still keep the validity of this

formula as Einstein did, the Speed had to be

extremely decreased in order to compensate

both the increment ofTime and the decrement of

Distance. The Speed was automatically


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decreased by the fact that it was not allowed to

overpass the speed of light this was the flaw of

Relativity.

In our opinion v+c should be strictly

greater than c for the observer on earth, since to

the speed of light it is added the speed of the

rocket.

Fig. 3

d

We think the time intervals are the same for

both observers in accordance with the commonexperience: an event, which occurs in an inertial

reference frame (in this case: the observer in the

rocket) and has a time interval t , lasts the same

time interval t if it is regarded from another

inertial reference frame (in this case: the

observed on earth); we use the real (absolute)

time interval, not the apparent time interval.

We agree with Einstein that the

trajectories of the clock light are different forthe observers, i.e.

Fig. 4


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for the observer in the rocket,

and

Fig. 5

for the observer on earth, and we also

agree with the mathematics used in the Special

Theory of Relativity to compute their length.

In our opinion there is neither a real time

dilation nor a real length contraction, but an

apparent time dilation and an apparent length

contraction. Surely, we can consider in a

metaphoric way that: time passes faster when

we enjoy it, and slower when we endure it (forexample in prison).

Or, under certain environmental

conditions our biological or psychological

processes could run faster or slower. We have

moments when we can age more or less than

normal. Therefore, our interior clock does not

run constantly. Biologically, it is a chance that

the more active you are (i.e. moving fast), the

less you age (because the brain is more active)

so time dilation, but with respect to the

absolute time you have the same age as

somebody less active but simultaneously born

with you.


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However, time dilation could produce

nice science fiction stories, but it is not fact.

Einstein did not prove that the speed of

light cannot be surpassed, he only postulated it.

Therefore we have the right to question this. He

did a thought not lab experiment. We mean we

dont believe that v+c=c for the observer onearth as Einstein asserted, but we think that

v+c>c for0


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22 2

2 2 ( )( ) 24

v tx t d

= +

(65)

2 2 2 2 2( ) 4 ( )x t d v t = + . (66)

Dividing both sides with2)( t , we get:

2 2 22( )d

x vt

= +

. (67)

We know for the observer in the rocket, that

2 ,d tc

= and thus2 .d c

t=

Therefore:

222 vcx += . (68)Whence the speed of photon in the rocket, with

respect to the observer on earth, is:

,22 ccvx >+= (69)


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which corresponds to the magnitude of the

vectorial addition, i.e. ,cvx

+= since c

and

v

are orthogonal:

v

c cx or x

vFig. 6

3.2. About Inconsistent Theories.

When a theory is based on a set of axioms

(or propositions) and one or more axioms areinconsistent, the theory produces as

consequences new inconsistencies, and so on

Therefore, an anomaly gives birth to other

anomalies

Reciprocally, if an inconsistent theory

results from some assumptions or propositions,

then those propositions on the whole must be

inconsistent too.

a) Therefore, the Michelson-MorleyExperiment (1881, 1883-1887) of not

having detected the ether, which implied

the STR, might have been erroneous due

to imprecise measurement instruments or


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construction. Maybe the ether is very little

dense and its flow hardly perceptible.

b) Similarly the constancy of the speed oflight in vacuum could be flaw, since the

speed of light might depend on the

emitting source the stronger the

emitting source the faster its light pulse.

3.3. No Relativistic Paradoxes in ATR.

Since in Absolute Theory of Relativity

there is no time dilation, in consequence there is

no length contraction, and no relative

simultaneity.

Therefore, many relativistic paradoxes arediscarded:

a) Ehrenfest Paradox (1909) since thereis no length contraction.

b) Twin Paradox (1911) since there isno time dilation and no gravitational

time dilation.

c) Bells Spaceship Paradox (1959) since there is no length contraction.

d) W. Rindlers Paradox about a manfalling into a grate (1961) since there

is no length contraction.

Etc.


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3.4. Removing Lorentz Factor in ATR.

Lorentz Factor

2

2

1

1 vc

(70)

becomes equal to 1 in ATR, because in the

equality'

2

21

tt

v

c

=

(71)

we replace

't t =

Therefore Lorentz Factor has no effect in our

ATR.

As a consequence in ATR we get:

- No time dilation, since'

1

tt

= (72)

- No Lorentz-FitzGerald Length Contraction,

since l = l1 (73)- The Relativistic Momentum of an object of

mass m, moving with speed v, becomes

classical:


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1

mvp = . (74)

- The Total Energy (upon Einstein) of an object

of mass m, moving at speed v, becomes:2

1mcE= (75)

- The Rest Energy (upon Einstein) of an abject

of mass m is

E0 = mc2

(76)

- Whence we obtain the Kinetic Energy of an

object of mass m, moving at speed v, becoming:

2 1 1 01

kE mc

= =

(77)

which doesnt make sense.

Therefore, in our opinion, the famous physics

formula E0 = mc2

is questionable. We

understand that light is electromagnetic energy,

but we dont understand why the energy of an

object should depend on the light speed? We

mean why on the speed?


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3.5. Physics Laws might not be the

same in all Inertial Systems.

If there exist superluminal velocities,

there might be possible that not all physics laws

are the same in all inertial systems.

As a counter example lets consider as physics

law the Addition of Velocities in a given inertial

system Si. Lets take the collinear and in the

same direction velocities v1 = 0.8c and

v2 = 0.9c.If we add them in STR we get:

v1+ v2 =2

0.8 0.9 1.70

0.9883720.8 0.9 1.721

c c c

cc c

c

+=

+

(78)

while in ART we simply get

v1+ v2 = 0.8c + 0.9c = 1.7c. (79)

The results are different.


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3.6. Linear Trajectories for Both

Observers.

We consider the case when the

trajectories seen by both observers are linear.

3.6.1. Orthogonal Trajectory Vectors

and Arbitrary Velocity K.

We can generalize this relationship,

replacing c by any speed K>0 such that v

andK

are orthogonal. Then:

.

22

KvKvx

+=+=

(80)

3.6.2. Non-orthogonal Trajectory

Vectors and c as Ultimate Velocity.

Lets change Einsteins theoretical experi-

ment, and consider d making an angle ,,0 with the motion direction (rockets).

Similarly, as before:

2d tc

= whence2d c

t=

(81)


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Fig. 7 C

s1 s2

A l O l B

In the triangle AOC we apply theTheorem of Cosine (which is a generalization of

Pythagorean Theorem used in the Special

Theory of Relativity):

2 2 2

1 2 .cos( ),s l d ld = +

2

2 2 ( cos( ),2 2

v t v t d d

= +

( )2

2 2 cos4

tv d v t D

= + + .

(82)

Similarly, in the triangle OBC we apply theTheorem of Cosine, and we get:


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2 2 2

2 2 cos ,s l d ld = +

( )2

2 2 os( )4

tv d v d t c

= + .

(83)

But 1 2x t s s = + , then:

( )

( )

2

2 2

2

2 2

. . .cos4

. . .cos

4

tx t v d v t d

tv d v t d

= + + +

+

(84)

Divide by t :

22

22

. . cos4

. . cos4

v d dx v

t t

v d d

vt t

= + + +

+

(85)


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2 2

2 2

. .cos4 4 2

. .cos4 4 2

v c cx v

v c c

v

= + + +

+

(86)

2 2

2 2

12 . os

2

12 cos

2

x v c vc c

v c vc

= + + +

+

(87)

Distance s1 is traveled with the speed2 2 2 cosv c vc + + , while the distance s2 is

traveled with the speed2 2 2 cosv c vc + ,

each of them in the same time interval2

t.

If2

= when v

and c

are perpendicular,

then2 2

,v c= + therefore we get the same

result as in our previous work [section 2.1].

If 0= , then cvcvx ++=2

1

2

1which

means that v

and c

are collinear. Fors1 the


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speed is v+c since v

and c

are in the same

direction, while fors2 the speed is v-c since v

and c

are in opposite directions (like in

Galilean Relativity).

If = , then cvcvx ++= 21

2

1

since

v

and c

are collinear, with opposite directions

on s1 and with the same direction on s2 (again

similarly as in Galilean Relativity).

3.6.3. Non-Orthogonal Trajectory

Vectors and Arbitrary Ultimate Velocity.

We can extend this thought experiment by

substituting c for any speed K (negative, positive, or zero, which can be smaller or

greater than c). Then the speed as measured by

the observer on earth is:

2 2

2 2

12 cos

2

12 cos

2

x v K vK

v K vK

= + + +

+

(88)

3.6.4. Non-Orthogonal Trajectory

Vectors and Arbitrary Velocities.


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We can again generalize the previous

thought experiment by substituting c for anyspeed w (negative, zero, or positive smaller or

greater than c). Then, the speed as measured by

the observer on earth is:

2 2

2 2

12 cos

2

12 cos

2

x v w vw

v w vw

= + + +

+

(89)


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Chapter 4.

Parameterized Special Theory of Relativity

(PSTR)


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4.1. PSTR Equation.

In a more general case when we dont

know the speed x nor the relationship between

't and t , we get:

2

22

2

v tx t d

= +

. (90)

But'

2

c td

= , therefore:

2 2

'22 2

c t v t x t = +

. (91)

Or: ( ) ( )2 22 2'x t c t v t = + . (92)

Dividing the whole equality by t we obtain:

2

2 2 'tx v ct

= +

(93)

which is the PSTR Equation.


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4.2. PSTR Elapsed Time Ratio

(Parameter).

We now substitute in a general case

'(0, )

t

t

= +

, (94)

where is the PSTR Elapsed Time Ratio.

therefore we get another extension of the

Special Theory of Relativity (STR), i.e.:

4.3. PSTR Extends STR, ATR, and

Introduces Three More Relativities.

1. If 22

1c

v= we get the STR, since

replacingx by c, one has2

2 2 2 'tc v ct

= +

,

22 2

2 2

'tc v

c c t

=

,

or


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2

2

'1 [0,1]

t v

t c

=

as in the STR.

2. If =1, we get our Absolute Theory ofRelativity [see Chapter 3] in the particular

case when the two trajectory vectors are

perpendicular, i.e.

2 2x v c= + = | |v c

+ .

3. If 22

10c

v


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69

of 't t < ), and the speedx as seen by the

observer on earth increases even more than

in our ATR.

The reader might be interested in studyingthese new Relativities mathematically

resulted from the above 3, 4, and 5 cases.

4.4. PSTRK Equation and Elapsed Time

Ratio (Parameter) in SK Space

The PSTRK Equation becomes be a simple

substitution ofc withK:

2

2 2 'tx v K t

= +

(95)

while the PSTRK Elapsed Time Ratio

(Parameter) is the same :

'(0, )

t

t

= +

. (96)


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4.5. PSTRK Extends STRK, ATRK, and

Introduces Three More K-Relativities

6.

If

2

2

1v

K =

we get the STRK (Special

Theory of Relativity with K as Ultimate

Speed).

7. If=1, we get the ASTRK (Absolute SpecialTheory of Relativity as derived from STRK,

although there is no distinction between

ASTR and ASTRK) in the particular case

when the two trajectory vectors are

perpendicular, i.e.

2 2x v K = + = | |v K

+ . (97)

8. If2

20 1

v

K< < , the time dilation is

increased with respect to that of the STRK,

therefore the speedx as seen by the observer

on earth is decreased (becomes subluminal)

while in STRK it is c.


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9. If2

21 1

v

K < < there is still time

dilation, but less than the time dilation of the

STRK, but the speed x as seen by theobserver on earth becomes superluminal (yet

less than in our Absolute Theory of

Relativity).

10.If>1, we get an opposite time dilation (i.e.tt > ' ) with respect to the STRK (instead

of 't t < ), and the speedx as seen by the

observer on earth increases even more than

in our ATRK.

The reader might be interested in studying

these new K-Relativities mathematically

resulted from the above 8, 9, and 10 cases

(relativities in the SK space, where the

ultimate speed isK).


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Chapter 5.

Accelerated Reference Frames.


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73

5.1. Formulas of Kinematics for

Constant Acceleration.

We use the following Formulas of

Kinematics for Constant Acceleration:

atvv += 0 (98)

tvvs )(2

10 += (99)

2

0

1

2s v t at = + (100)

asvv 2202

+= (101)

where:

a = constant acceleration;

v0 = initial velocity at time t0 = 0;

v = final velocity at time t;

t= time elapsed since t0 = 0;

s = distance (displacement).

5.2. Gravitation and Constant Velocity of the

Rocket.

Lets modify again Einsteins thoughtexperiment. In the rocket lets consider a rock in

a free fall on a distance d:


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74

Fig. 8 A

d

B

Under the earths acceleration:

a = gv0 = 0

Then2

0

1

2v t at = + becomes:

( )2

10 ( ') '2

d t g t = + ,

or ( )21

'2

d g t= . (102)

The rocket is moving at a constant speed v.

For the observer on earth, the trajectory of the

rock isAB(notAB as seen by the astronaut).


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75

AFig. 9

d

B l B

( )21

'2

B d g t = =

(103)

But the distance BB , traveled by the rocket in

the elapsed time t ,is:

'BB l v t = = . (104)

Using the Pythagorean Theorem in the triangle

ABBwe get:

2 2'B d l = +

( ) ( )2

2 21 '2

g t v t = +


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76

( )2

4 2 2' ( )4

gt v t= + (105)

But ( )

24 2 2 21

' ( ) 0 ( ) ,4 2 ag

t v t t x t + = +

(92)

where ax = constant acceleration as seen by the

observer on earth.

We get:

( )

24 2 2 21

' ( ) ( )4 2a

g

t v t x t + = (106)

If we consider an absolute time as in our

ATR, then we take ,' tt = and it results:

( )2

4 2 2 2 41( ) ( ) .4 4

a

gt v t x t + = (107)

Divide by

4

)( t :

44

222ax

t

vg=

+


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77

or

2

2

2 4 axt

vg =

+ , (108)

whence:

.2

2

2 gt

vgxa >

+=

(109)

The observer on earth perceives a greater

constant acceleration of the rock than the

astronaut does.

It is remarkable to know thatt

v

2is equal to

the constant acceleration of a rocket that would

start to move in the same moment when the rock

starts to freely fall and whose final velocity at

time t(when the rock reaches the rocket floor)

is 2v.

5.3. Gravitation with Constant

Acceleration of the Rocket.


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We consider the previous thought

experiment: a rock in fall free in the rocket, but

the rocket having an initial speed v, and at the

moment when the rock begins to fall the rocket

starts to accelerate with constant acceleration a.

Similarly: ( )21

'2

B d g t = =

A

Fig. 10

d

B l B

The new trajectory AB is greater than the

previous trajectoryAB.

The distanceBB is:

( )2

"2

aBB l v t t = = +

. (110)

2 2"B d l = +


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79

( ) ( )2 2

2 21'

2 2

ag t v t t

= + +

(111)

But supposing that ax is the constant accelera-

tion of the rock travelling on the trajectoryAB,

as seen by the observer on earth, and using the

distance formula with respect to the accelera-

tion, we also have:

21" 0 ( )2

aB t x t = + ,(112)

whence:

( ) ( )2

42 2 21 1' ( ) ( )4 2 2

a

ag t v t t x t

+ + =

(113)

Multiplying with 2)(2t

we get:


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80

,2

2

2 gat

vgxa >

+

+= (114)

and also .axa >

The observer on earth sees the rock falling with

a constant acceleration greater than both the

gravitationgand the rockets constant accelera-

tion a.

Instead of a rocket, lets suppose a train

with initial velocity v=0. Then

22 agxa += . (115)

5.4. Constant Accelerations and Oblique

Direction Vectors.

In the rocket a particle travels fromA toB

with a constant acceleration a1 starting at initial

speed v1 in the elapsed time t1 with respect to

the observer in the rocket.


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81

A

d

B B

Fig. 11

2

1 1 1 1

1( ) ( )

2d v t a t = + (116)

The rocket travels with a constant accelerationa2 starting at initial speed v2 in the elapsed time

t2 with respect to the observer on earth. The

angle between the two direction vectors is ,

0 .

2

2222 ).(2

1).('" tatvBB += (117)

Using the Cosine Theorem in the triangleABBwe get:


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cos.'".2'"'"222

BBABBBABAB +=

(118)

( )

( )

( )( ) cos2

.

2

..2

2

.

2

.

222

11121

2

222

2

2

2

111

2

1

+

+

+

+

+=

tav

tavtt

tavt

tavt

( ) ( )2

222 33 2

1

+= txtx av

(119)

where3v

x is the initial speed and3a

x is the

constant acceleration of the particle on

trajectoryABas seen by the observer on earth

(gravitation not included).

This is the general equation, without anyassumption about the relationships between the

elapsed times t1 and t2, nor about the


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acceleration3a

x (with initial velocity3v

x ) as

seen by the observer on earth.

Now, in particular case when we consider the

absolute time (therefore t1=t2) we get, after

dividing by ( ) ( ) ( )2

2

2

1

2ttt == :

( )2

22

11

2

22

2

11

33 2

1cos

2

.

2

..2

2

.

2

.

+=

+

+

++

+

txxta

vta

v

tav

tav

av

Divide again by ( ) :2

t

2

2211

2

22

2

11

3

3

2

1

cos22

.222

+

=

+

+

+

+

+

a

vx

t

x

a

t

va

t

va

t

va

t

v

(120)

5.5. Constant Accelerations with Zero

Initial Velocity and Oblique Direction

Vectors.


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In the previous case, we set:

0321

=== vxvv .

Therefore:

4cos

4

2

44

2

21

2

2

2

1 3axaaaa

=+ ;

whence the constant acceleration of the particle

as seen by the observer on earth on the path

ABis:

cos.2 2122213 aaaaxa += .

(121)


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Chapter 6.

Noninertial Multirelativity


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6.1. Multirelativity with NonconstantAcceleration and 3D-Curves.

In a 3D-Euclidean space for location and in

an 1D-oriented Euclidean space for time weconsider a reference frame F1 with respect to

which a particle P0 travels with a nonconstant

acceleration a0 on a 3D curve C0 in an elapsed

time t0.

Then we suppose the reference frame F1 is

moving with nonconstant acceleration a1 on a

3D curve C1 with respect to another reference

frame F2. Similarly, the reference frame F2 ismoving with a nonconstant acceleration a2 on a

3D curve C2 with respect to another reference

frameF3.

And so on: the reference frame Fn-1 is moving

with a nonconstant acceleration an-1 on a 3D

curve Cn-1 with respect to another reference

frameFn (where n>2).

We call this a Noninertial Multirelativity, i.e.

the most general case.


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6.2. Research Problems.1. How would the particles trajectory curve

C0 be seen by an observer in the reference

frameFn ?2. What would be the particles speed

(acceleration) as measured by the

observer from the reference frameFn ?

3. What would be the elapsed time of the particle as seen by the observer in the

reference frameFn ?

4. What are the transformation equationsfrom a reference frame to another?

5. Similar questions for rotating referencesystems.

Particular cases would be helpful in

starting such research, for example studying

particles or reference frames travelling on

linear curves, or on special curves, with

constant speeds or constant accelerations, in

reference frames that have one, two, or threeparallel coordinate axes. Then later trying to

generalize the results.


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6.3. Example of Nonlinear 3D-Trajectories ofParticle and Reference Frames.

Since each constant speed v can be considered

a constant zero-acceleration with initial velocityv, we treat the general case (i.e. the constant

acceleration).

Lets consider in the reference frame F1 a

particleP0traveling on a curve C0 fromA toB:

Fig. 12

B B

C0

A

with a constant acceleration a0 and initial

velocity v0. Lets take into consideration the

earths gravity g too that influences the

trajectory.


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F1 (which has the Cartesian system X1Y1Z1) is

moving with a constant acceleration a1 with

initial velocity v1 in the positive direction of the

X1-axis (the OY1- and OZ1-axes are parallel

respectively with OY2 and OZ2) with respect tothe frame F2 (whose Cartesian system is

X2Y2Z2).

The arclength ofAB is noted by d.

From an observer inF2 the trajectory B

of the

particle P0 in F1 is seen as a 2D- or 3D-curve

'B

.

The curveABis described inF2by a function

( )0 1 0 2 2 20 1( , , , , , , , , ) ( ), ( ), ( )a v a v g C A B x t y t z t =

(122)

i.e.

[ ] [ ] [ ]2 2 2

2 2 2

0

( ') '( ) '( ) '( ) ( ', )

t

rcLength AB x t y t z t dt L t t

= + + (123)


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90

where x2(t), y2(t), z2(t) are respectively the

derivatives of x2(t), y2(t), z2(t) with respect to t,

and L(t,t) is a notation to mean that the

arclength L, from A to B, depends on

t andalso on d, but ddepends on t.

The distance traveled by the reference frame F1

in telapsed time is

( ) ( )21112

1tatvs +=

(124)

Supposing that particles traveling is seen as aconstant acceleration by the observer inF2, then

we have:

( ) ( )200 2

1),'( txtxttL av += (125)

where0v

x = the initial particles velocity as seen

by the observer in F2,

and0a

x = the particles constant acceleration as

seen by the observer inF2.


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We know that in F1:

( ) ( )200 '2

1' tatvdAB +== . (126)

Depending on the suppositions regarding heconnections between tand t(in an absolute

time reference frame they are equal), or on the

supposition about the acceleration of the particle

0ax and

0vx we get particular cases in formula

(125).

The reader can repeat this thought

experiment for the case when the accelerationsa0 and a1 are not constant, and the reference

frameF1 is moving with respect to the reference

frameF2 on an arbitrary 3D-curve.

References:

Bell, J. S. "On the Einstein-Podolsky-RosenParadox,"Physics 1, pp. 195-200, 1964.

Bohm, D. "The Paradox of Einstein, Rosen, and

Podolsky," Quantum Th., pp. 611-623, 1951.


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Einstein, A.; Podolsky, B.; and Rosen, N. "Can

Quantum-Mechanical Description of Physical

Reality Be Considered Complete?" Phys. Rev.

47, pp. 777-780, 1935.

Smarandache, F., There is No Speed Barrier in

the Universe, mss., Liceul Pedagogic Rm.

Vlcea, Physics Prof. Elena Albu,1972.

Rindler, W., "Length Contraction Paradox,"Am.

J. Phys., 29(6), 1961.

Einstein, A., Zur Eletrodynamik bewegter

Krper,Annalen der Physik, Vol. 17, pp. 891-921, 1905.


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In this book we present the hypothesis

that there is no speed barrier in the universe -

thus refuting the speed of light postulate.

While Einstein considered a relative

space and relative time but the ultimate speed

of light, we do the opposite: we consider an

absolute time and absolute space but no

ultimate speed, and we call it the Absolute

Theory of Relativity (ATR).

We then parameterize Einsteins

thought experiment with atomic clocks,

supposing that we know neither if the space

and time are relative or absolute, nor if the

speed of light is ultimate speed or not. We

obtain a Parameterized Special Theory ofRelativity (PSTR). Our PSTR generalizes not

only Einsteins Special Theory of Relativity,

but also our ATR, and introduces three more

possible Relativities to be studied in the

future.

Afterwards, we extend our research

considering not only constant velocities but

constant accelerations too.

Eventually we propose a Noninertial

Multirelativity for the same thought

experiment, i.e. considering non-constant

accelerations and arbitrary 3D-curves.

Documents

Absolute Theory of Relativity & Parameterized Special Theory of Relativity & Noninertial Multirelativity, by Florentin Smarandache