Armenian Transformation Equations For Relativity

Transcription

1 Armenian Transformation Equations For Relativity Robert Nazaryan an Haik Nazaryan 00th Anniversary of the Special Relativity Theory This Research one in Armenia , Translate from the Armenian Manuscript Yerevan State University (Armenia), Byurakan Astrophysical Observatory (Armenia) Abstract In our article, we erive a new transformation of relativity for inertial systems using the following guielines:. We use vector notations to obtain new transformation equations in a general form.. In the process of eriving the general new transformation equations in vector form, we keep the term v r as well.. We a one more postulate to the existing two, to resolve ambiguity problems in the orientation of the inertial coorinate systems axes. 4. Newly obtaine transformation equations nee to satisfy the linear transformation funamental laws. 5. Aition of velocities we calculate in two ifferent ways: by linear superposition an by ifferentiation, an they nee to coincie each other. If not, then we force them to match for obtaining the final relation between coefficients. After using the above mentione general guielines, we obtain straight an inverse transformation equations name the Armenian transformation equations an they are expresse in two notations: by absolute relative velocity v an by local relative velocity v. Armenian transformation equations are the replacement for the Lorentz transformation equations. Contents Introuction to the Armenian Transformation Equations for Relativity Part - Pure Mathematical Approach. - Three Dimensional Species Transformation. - Derivation of the Armenian Transformation Equations. - Expressing the Armenian Transformation Equations by Real Time.4 - Expressing the Armenian Transformation Equations by Local Time Part - A Physical Approach. - Movement of the Origin Inertial System S Expresse by Local Time.. - Relative Velocity v is the Same as Local Velocityv.. - Relative Velocity v is not Equal to the Local Velocityv. - Movement of the Origin Inertial System S Expresse by Real Time. - Calculation of Local Velocities an the Important Consequences.4 - Superposition an Transformation of the Absolute an Local Velocities.4. - Successive Transformations to Derive Superposition Formula for Absolute Velocities.4. - Differentiation Metho to Derive Superposition Formula for Absolute Velocities.4. - Aition, Subtraction an Transformation of the Absolute Velocities Aition an Subtraction of the Local Velocities.5 - Summary of the Armenian Transformation Equations References

2 Introuction to the Armenian Transformation Equations for Relativity The Lorentz transformation equations, as we know them, in two imensional time-space t, x or in four-imensional time-space t, r, when the inertial systems move at a constant relative velocity v along one of the chosen axis, are linear orthogonal transformations. In these cases Lorentz transformations are a group an satisfy the funamental linear transformation rules: LvLu Lu. Where the resultant transformation is a Lorentz transformation as well with the resultant velocity u v u / vu. In general, when the relative velocity v of the inertial systems S an S have an arbitrary irection, then the Lorentz transformation is not a group an are therefore less iscusse as a case. Only a few brave authors, 6, 7, 9 an 0 mention an iscuss this general case (axes of the inertial systems they take parallel to each other as usual). The main linear transformation law LvLu Lu fails for the general Lorentz transformation. Since, however, a resultant transformation must be a Lorentz transformation as well, physicist nee therefore to a an extra artificial transformation calle the Thomas precession, to compensate for the error. This is the Achilles heel in the Lorentz transformation equations an more precisely in all special an general theory of relativity. Therefore it is an imperative, that the Lorentz transformation equations be replace by new ones, which must be consistent with linear transformation funamental laws an have a common sense in respect to reality. Here we shall erive these New transformation equations, using pure mathematical logic without any limitations an the following three postulates:. All physical laws have the same mathematical(tensor) form in all inertial systems.. There exists a bounary velocity, enote as c, between micro an macro worls, which is the same in all inertial systems.. The simplest transformation equations of the moving particle between two inertial systems S an S / we have only when relative velocity, measure in two inertial systems, satisfy the relation v v. These first two postulates are almost the same as the Special Relativity Theory postulates, but the thir postulate is quite new an necessary for receiving the simplest transformation equations without ambiguity problems in orientation of the inertial systems axes. All authors that I know, erive the Lorentz transformation equations using two Cartesian coorinates t, x or as a general way using four Cartesian coorinates t, x, y, z. Noboy (that I know of) uses vector notations to erive general transformation equations for relativity. Many authors artificially construct general Lorentz transformation equations in vector form using special Lorentz transformation equations an therefore those generalize results cannot be correct. The laws of logic tell us, that we nee to go from the general case to the special case. That s why we erive our New transformation equations using the most general consierations an aapting vector notation. The great merit of the vectors in the theoretical an applie problems is that equations escribing physical phenomena can be formulate without reference to any particular coorinate system, without worry that coorinate systems axes are parallel to each other or not. However, in actually carrying out the calculations we nee to fin a suitable coorinate system (our thir postulate) where equations can have the simplest form. Therefore to receive the correct transformation we nee to use only vector notations an focus on it entirely. Using this new promising approach an one aitional postulate we erive truly correct transformation equations in the most general an simplest form. The other question can arise - why are we calling our newly receive transformation equations the Armenian Transformation Equations? The answer is very simple. This research was one for more than 0 years in Armenia by an Armenian an the manuscripts were written in Armenian. This research is purely from the min of an Armenian an from the Holy lan of Armenia, therefore we can rightfully call these newly erive transformation equations the Armenian Transformation Equations an the theory the Armenian Relativity Theory or ART. Part - Pure Mathematical Approach In this part we will use a purely mathematical approach to erive transformation equations for relative movement, without using physical realities. In part we will exercise the receive New transformation equations using a physical approach to fin the relation between coefficients an eventually to obtain the final form of the Armenian transformation.

3 Armenian Transformation Equations. - Three Dimensional Species Transformation We nee to use a imensional approach which will best escribe our existence. Let us escribe an event of the moving particle in the inertial system S by t, r an in the uniform moving inertial system S by t, r, where t an t are the local times an r an r are the raius vectors of the moving particle. Relations between these raius vectors with respect to two inertial systems in general can be represente in the following way. r r v c v r r r v c v r.- Three vectors v, r an v r or v, r an v r are not coplanar, therefore the resolution of raius vectors r an r in those three irections are unique. Transformations of the local time t an t in general can be constructe from infinite scalar terms as showing below: t t vr t 4v 5r 6 c 4 vr... t t c v r t 4 v 5 r 6 c 4 v r... Since the local time an space in the inertial systems are homogeneous an isotropic, therefore time-space transformation must be linear in respect to the local time an raius vector, an that can only happen when the first power of time an raius vectors are involve. Besies that, the other terms that contain the power of v vanish as well, because we assume that the inertial systems origin coincie ( r r 0 ) at t t 0. Therefore in the local time transformation equations. we keep only the first two terms. To calibrate the transformation equations coefficients, we use a universal constant - bounary velocity c, which by the secon principle of relativity has the same value for all the inertial systems. Finally, as a imensional species, we nee to solve the following system of equations. t t vr r r v c v r t t c v r r r v c v r Accoring to the first principle of relativity, all laws of physics, incluing transformation laws, must be the same in all inertial systems. Therefore all numerical coefficients in the transformations. must be equal to each other..-.- an.-4 Coefficients an are not numerical quantities an the first principle of relativity oesn t apply for them an therefore they are not equal to each other. We nee to fin a transformation law for them as well. These time-like coefficients we will call real time an for the simplicity an physical-meaning purposes we enote them as:.-5 Finally, accoring to the thir principle of relativity, we can obtain the simplest transformation equations when: v v.-6 Substituting prime values from. 4,. 5 an. 6 into the transformation equations. we get: Straight transformation t t vr r r v c v r.-7 Inverse transformation t t vr r r v c v r.-8

4 We nee to be very careful, because in these transformation equations the real timesan are not the physical local times of the inertial systems. In general, we nee to assume, that they can be a function from both local times t an t an therefore we nee to fin those relations as well. All other coefficients can be constants or they nee to be epenent on relative velocity v only. To fin the limitations of this coefficients, we nee to approximate the New transformation equations. 7 an. 8 when v c, an then they nee to coincie with the Galilean transformation equations as shown below. Approximation of. 7 Galilean Transformation t t r r v t t r r vt.-9 An Approximation of. 8 Galilean Transformation t t r r v t t r r vt.-0 Transformation coefficients accoring to. 9 an. 0 nee to satisfy the following limitations. 0 0 an lim v0 lim v0 an lim t v0 lim t One more thing is clear, if we are looking for a new time-space transformation, ifferent from the Galilean transformation, then on top of the coefficient limitations., they must also satisfy the following conition: v0.- an.- To erive the New transformation equations, first we nee to use only, as we have mentione alreay, pure mathematics without limitations an then, if necessary, to use realms of the physical realities.. - Derivation of the Armenian Transformation Equations. Inserting values t an r from. 7 into the expression of t in. 8 t t c vr t c vr c v r v c v r t vr v vr c c c t v c c vr From the above calculation for the real timewe get: v c t vr.- c. Inserting values t an r from. 8 into the expression of t in. 7 t t c vr t c vr c v r v c v r t vr v vr c c c t v c c vr From the above calculation for the real time we get: v c t vr.- c. Inserting value r from. 7 into the expression of r in. 8 r r v c v r r v c v r v c v r v c v r r v c v r v c v r c v v r r v c vr v v c r From the above calculation we obtain a vector relation. v c r vr v 0.- c 4

5 Armenian Transformation Equations 4. Inserting value r from. 8 into the expression of r in. 7 r r v c v r r v c v r v c v r v c v r r v c v r v c v r c v v r r v c vr v v c r From the above calculation we obtain a vector relation. v c r vr v 0.-4 c Equating. an. 4 null vectors components to zero we obtain the following: Relation between coefficients 0 an real time transformation equations v or v.-5 vr vr.-6 Equations. an. are expressing the relations between real timesan into local times an raius vectors in the same inertial system, an vice versa relation as well v t vr v t vr or t v v vr t v v vr.-7 Substituting values vr from the first equation of. 8 an vr from the first equitation of. 7 into c c the first system of equations. 7 we obtain relations between real timesan into local times t an t only. v t t c or v t t c v v t t v v t t.-8 Remark() Not all receive equations are inepenent. For example, equations. 7 can be erive from the equations. 6 using transformation equations. 7 an. 8, also relation. 5. For our convenience we nee to enote v v.-9 Remark() When we have only one relative velocity, for simplicity purposes instea of using v we use just. When we have multiple inertial systems we then use v, u, u, an so on. If we substitute an in the secon system of equations. 8 an then using the thir system of. we obtain the following limitations between coefficients: lim v0 v lim 0 v0 v.-0 5

6 . - Expressing the Armenian Transformation Equations by Real Time Inertial systems with real times, which transforme accoring to. 6 an raius vectors r, r which transforme by. 7 an. 8, are the genuine components of the event. Also if we enote the real zero components of the event in a four-imensional time-space as: h c h c.- Then the New transformation equations expresse by the real time or by real zero components become: Straight transformation vr r r v or c v r h h c vr r r c vh c v r.- Inverse transformation c vr r r v or c v r h h c vr r r c vh c v r.- From transformations. an. we obtain an invariant expression calling it real interval an enoting it as s R an we can express it by a real time or by a real zero component as shown below. s R r r or s R h r h r.-4 Definition() Any set of components a 0, a which transforms in a similar way to. an., also the istance between two events whichepenontimeanonraiusvectoronly an expresse by. 4, will be calle afour-vector. Remark() From the above efinition it follows that the real interval s R in. 4 can be epenent on time an raius vector of the event only, if the coefficient is a constant.thisisthefirst crucial result. Coefficient is a truly universal time-space constant characterizing the meium of time-space or the moving particle. Const.- 5 Another important consequence following from the real interval expression. 4 is that it is a monotonic increasing value, or more correctly, a strictly monotonic increasing value. Because, if there is no spatial movement at all, the real time is flowing nonstop an therefore the real interval s R only increases. Therefore, we can implement an efine the iea of absolute time or universal time for the event. Definition() Using a real interval expression. 4 we efine the absolute time of the event, enoting it as, which is "flowing" by the bounary spee c an it is also invariant an strictly a monotonic increasing value. Absolute time an it s ifferential is shown below. c sr r r c sr r r.-6 Absolute time oesn t have a singularity problem like the Lorentz transformation proper-time, but it has a real time characteristics. From. 6 we can obtain the expression for real time an it s ifferential: 6

7 Armenian Transformation Equations r r or r r Expressing the Armenian Transformation Equations by Local Time To express the New transformation equations by local time t an t, we nee to insert real time values anfrom. 7 into raius vector expressions. 7 an. 8, also using notations. 9 we get. Straight transformation equations become: t t vr Inverse transformation equations become: r r vt v v vr v c v r.4- t t vr r r vt v v vr v c v r.4- Transformation equations expresse by local time. 4 an. 4 in case of approximation of v c nee to coincie with the Galilean transformation equations, as we ve alreay state, an that can only happen, when in aition to the conitions.,. an. 0 also exists the conition: 0.4- From the transformation equations. 4 or. 4 we get the invariant expression for the event, which we will call the local interval an enote as s L s L t v r t v r.4-4 By four-vector efinition in section., the interval must epen only on time an raius vector of the event, an will never be the function of the relative velocity. Therefore, the coefficient in the formula of local interval. 4 4 nee to satisfy the following conition: Therefore the invariant local interval expression. 4 4 becomes: v q Const.4-5 s L t qr t qr.4-6 From the conition. 4 follows that the coefficient q in. 4 5 has a sign of an in general,we assume, that it is a real number. Because at this moment we o not know the sign of the coefficient q, therefore we cannot talk about absolute time erive from s L. Here we can talk only about proper-time, enoting it by L. This proper-time an it s ifferential is: L c sl t q r t q r L c sl t q r t q r.4-7 7

8 Part - A Physical Approach In this part, in aition to the pure mathematical methos, we will try to use some physical realities as well, to obtain the final form of the New transformation equations. In kinematics we are ealing with two basic physical quantities - istance an time. Quantity istance is very real, we can see it an we can touch it, which we can t say about the quantity of time. Time, as such, is a very mysterious quantity an in our article we are using many types of time. In the section. we alreay efine local time t an real time. In the sections. an. 4 we also efine absolute timean proper time L. Therefore the isplacement rate of the raius vectors of the moving particle in respect to these ifferent times generate ifferent types of velocities, as shown three of them below, for arbitrary movement an for uniform movement separately. Local Velocity or Newtonian Velocity u r t Real Velocity u R r Absolute Velocity u A r or or or u r t u R r u A r On top of those velocities in A, at the beginning of our article, we alreay use relative velocity v to erive New transformation equations an we nee to ientify that velocity as well. The type of relative velocity v is unknown for now, because we on t know in respect to which time we ve mae our calculation to fin the isplacement rate of the inertial systems. Therefore Type of the Relative Velocity v Is Unknown -B -A. - Movement of the Origin Inertial System S Expresse by Local Time In this section we will illustrate only, how the scientific community, from the birth of Special Relativity Theory until now, hasn t even imagine the existence of two types of velocities - local velocity an absolute velocity. Of course, in SRT there exists the iea of the Minkovsky velocity, which is not a real physical quantity, but just a hany instrument for mathematical calculation. Therefore, scientists on t istinguish these two velocities, have felle an "unknown trap". Existence of this entrapment, which brings us to a ea en, we like to show here. Lets calculate the velocity of the origin of Inertial system S with respect to the inertial system S expresse by local time. If we put r 0 in. 4 an using velocity efinitions A, then for uniform local velocity of the origin of the inertial system S we get: S. If r 0 then t t r vt v therefore v r t v v.- From. we obtain a relation formula between local an relative velocities of the moving origin inertial system v v v or v v v.- In relative velocity formula. we still nee to fin coefficients an. To o that we have two choices: ) v v or ) v v First Choice - Relative Velocity v is the Same as Local Velocity v. v v.-4 This first choice seems very intuitive an har to eny, which is easily one by the founing fathers of Special 8

9 Armenian Transformation Equations Relativity Theory. But this is the trap which I am trying to escribe here, an we nee to avoi it. Before moving further let s expose how this trap works. Inserting value v from. 4 into the relative velocity formula. we get the following relation between coefficients: v.-5 Solving. 5 in respect to an remembering. we take only the positive root of. v v Also inserting value from. 5 into. 4 5 we get the expression for q..-6 q.-7 Because, accoring to., is positive, then from. 7 it follows that an q have the same sign which we alreay mentione in the section. 4. Substituting value from. 6 into. 7 we obtain the following equation. q v v Solving. 8 with respect to an substituting obtaine into. 7 we get the value as well. q q v an q v After all of this preparation we can finally prove the contraiction of the first choice. 4. For that purpose, we nee to recall the secon limitation of. 0 an compare coefficients from. 9 an from. 9. Here there is the possibility of only two assumptions. a) If we assume a) 0 or b) 0.-0 then q v v.- Solving equation. with respect to q we get the expression for q epening on the relative velocity v q v.- Conclusion() Expression. is contraictory to the fact of. 4 5, which manifests that coefficient q must be constant an oesn t epen on the relative velocity. Therefore, the first assumption is incorrect. b) If we assume 0.- Then inserting value from. 5 into transformation equations. 4 an. 4 we obtain Straight transformation equations. t t vr r r vt v vr v c v r.-4 Inverse transformation equations. 9

10 t t vr r r vt v vr v c v r.-5 Even in this "entrapment choice" which appears as though we are following a logical path, the receive New transformation equations. 4 an. 5 are very ifferent from the Lorentz transformation equations. Conclusion() Transformation equations. 4 an. 5 are not looking ba. However, when we calculate successive transformations, it oes not satisfy the linear transformation funamental laws an therefore, accoring to the general guieline, we regar it as an incorrect transformation. Therefore, the first intuitive choice v v has faile. In the section. we prove that relative velocity v in fact is an absolute velocity.. - Secon Choice - Relative Velocity v is Not Equal to the Local Velocity v. v v.-6 To analyze this secon choice, we nee first, to calculate the movement of the origin of the inertial system S in respect to the inertial system S using transformation equations. an., which we will iscuss in the next section.. - Movement of the Origin Inertial System S Expresse by Real Time Let us fin now the velocity origin of the inertial system S in respect to the inertial system S expresse by real time an then we can make important conclusions from that. For that purpose, if we put r 0 into. an also recalling velocity efinitions A, then for uniform real velocity of the origin of the inertial system S we get: If r 0 then r v therefore v R r v.- An absolute time expression. 6 when r 0 becomes: r.- Now, in the real velocity expression., iviing the left fraction numerator an enominator to the absolute time an inserting also the value from. 9 into it, we get: r v v From the efinition of velocities in A following, that absolute relative velocity v A of the origin inertial system S is: From. we can calculate the expression for an inserting the value r.- r v A.-4 r v A from. 4 into it, we get:.-5 Now substituting values r from. 4 an from. 5 into. we obtain a vector equation: v A v A v v Vector equation. 6 will have only one solution, an that is:.-6 v v A.-7 0

11 Armenian Transformation Equations Conclusion() Thisistheseconcrucialresult. From. 7 we obtain the final proof that relative velocity vwhich we use to erive our New transformation equations is in factnotalocal velocity but an absolute velocity. Remark(4) This is a very important point in the Armenian Theory of Relativity. Therefore, in the rest of this article we will omit inex "A" enoting absolute velocity. Using fact. 7, efinition of velocities A, also. an system of equations. 7 we can efine zero components of the absolute velocity of the moving particle in two inertial systems in the following way: u 0 h c c r u 0 h c c r c u c u cu cu.-8 Also, the same way, we can efine a zero component of the absolute relative velocity v as: v 0 c v cv.-9 From. 8 an. 9 it follows that components of the absolute velocity of the moving particle or the absolute relative velocity satisfy the invariant relation. u 0 u u 0 u v 0 v.-0 Remark(5) In the subsection we will prove that components of the absolute velocity of the moving particle form a four-vector.. - Calculation of Local Velocities an the Important Consequences Lets calculate local velocities of the moving particle in two inertial systems S an S using the efinition for velocities from A. u r t u r t r t r t u t u t To continue calculation. we nee to fin ifferentials t an. For that purpose substituting notations t. 9 into. 7 an then ifferentiating local times t an t by absolute timean inserting also values an from. 8 into it we get: Substituting expressions t v t v t an t v v u v v v vu.- v v u v v v vu.- from. into. we get the expression for local velocities. u v v v u v v v u u v v v vu u u v v v vu Expressions in. are not aition formulas for the velocities of the moving particle but it is a local velocity conversion formula in the same inertial system. Therefore local velocity of the particle nees to be epenent only on the absolute velocity of the particle in the same inertial system an must be inepenent of the relative velocity v. But that only can happen when coefficients in. satisfy the following conitions..-

12 v v v Const.-4 Inserting values from. 4 into the first equation of the limitation in. 0 lim v0 v lim v0 v Finally from. 5 we obtain a relation between two constant coefficients an. lim v Conclusion(4). 4 an. 6 are thethircrucialresults. Inserting value an from. 4 an. 6 into. 4 5 we fin a value for constant coefficient q as well. Finally we obtain all necessary relations between the New transformation equation coefficients an they are: q Const v.-7 Substituting coefficients. 7 into. 4 an. 4 we obtain the final form of the New transformation equations, which we will call the Armenian transformation equations an they are: t v t vr r v r vt c v r t v t vr r v r vt c v r.-8 Using. 9 an inserting values of coefficients from. 7 into real times expressions. 8 we obtain t t.-9 Remark(6) Relation. 9 manifest thatrealtimeisthelocaltime. Therefore, from now on we are omitting all inexes "R" an "L" escribing variables as real or local respectively. Particularly interval expressions. 4 an. 4 6 coincie to each other an we will call it Armenian interval of the event an enote it by s. s t r t r.-0 Substituting coefficients from. 7 into. we obtain expressions for the ifferentials t an t. t u u an t u u.- Also substituting coefficients. 7 into. we obtain a conversion formula for any arbitrary velocities. u u u u u u u u From. it follows that for any arbitrary velocity we obtain a value for expresse by absolute velocity or by local velocity. u u u.- u.- Using notation for from. we rewrite the velocity conversion formula. for any arbitrary velocity. u u u u u u.-4 From local an absolute velocity conversion formula. follows their limits.

13 Armenian Transformation Equations 0 u c 0 u Superposition an Transformation of the Absolute an Local Velocities In this section we nee to erive an aition formula for absolute an local velocities of the moving particle with respect to two inertial systems S an S. Following our general guieline, we will obtain it in two ifferent ways - by linear superposition an by ifferentiation, an they nee to coincie each other. This is the important criteria, which asserts, that our new receive transformation equations passes the mathematical test to be correct. To prove this, we nee o the following:. Use two successive transformations of the Armenian transformation equations. 8 to obtain relations between absolute velocities of the moving particle in two inertial systems, using a linear superposition law by the scheme: Where operator A is the Armenian transformation matrix AvAu Au.4-. Use a ifferentiation metho to obtain relations between absolute velocities of the moving particle in two inertial systems, using Armenian transformation equations. 8. Lets now consier three inertial systems S, S an S, where the absolute relative velocities are: v is absolute relative velocity of the inertial systems S an S u is absolute relative velocity of the inertial systems S an S.4- u is absolute relative velocity of the inertial systems S an S.4. - Successive Transformations to Derive Superposition Formula for Absolute Velocities We nee to use Armenian transformation equations. 8, where coefficients, as we ve alreay mentione, is constant an therefore oesn t epen on absolute relative velocities of the inertial systems. Lets make two successive transformations. Straight an inverse Armenian transformation equations between three inertial systems as you can see below. Straight Armenian Transformation Inverse Armenian Transformation S S t v t c vr r v r vt c v r S S t v t c vr r v r vt c v r.4- Straight Armenian Transformation Inverse Armenian Transformation S S t u t c u r r u r u t c u r S S t u t c u r r u r u t c u r.4-4

14 Straight Armenian Transformation S S t u t c ur r u r ut c u r Inverse Armenian Transformation S S t u t c ur r u r ut c u r.4-5 Implementing two successive straight transformations(it oesn t matter if we use straight or inverse transformations). 4 4 an. 4 5 by scheme S S S. After eliminating S we can get transformation S S. We can calculate resultant velocity u in two ways - using local time transformation equations or raius vector transformation equations. In both ways we are getting the same result. Therefore it is reasonable to use here a more simple one, which is the local time transformation equations. Substituting t an r from. 4 into the expression t in. 4 4 we obtain: t u t c u r u v t c vr c u v r vt c v r v u t u vr v u r vu t c u v r v u c vu t v u u v c v u r Now equating the above result to the time t expression in. 4 5 we get: t v u c vu t u v v u c v u r u t ur.4-6 c The equality. 4 6 can only happen when the following relations exist for the resultant absolute velocity u. u v u vu u v u u v c v u.4-7 Two equations of. 4 7 are consistent with each other an one can prove the correctness of the first equation using the secon equation. From the aition formula. 4 7 we can obtain a subtraction formula for absolute velocities, just by exchanging primes an absolute relative velocity v with v Differentiation Metho to Derive Superposition Formula for Absolute Velocities For this purpose we nee to use the inverse Armenian transformation equations. 4 an ifferentiate local time t an raius vector r by the absolute time. Also inserting ifferentials t an t from. into it, we obtain the superposition formula for absolute velocities. t v t r v r v r c v t c v r u v u vu u v u u v c v u.4-8 Remark(7) Resultant absolute velocity expressions are receive in two ifferent ways - by successive transformation. 4 7 an by ifferentiation. 4 8 coincie to each other. That means the Armenian transformation equations pass the mathematical test to be correct Aition, Subtraction an Transformation of the Absolute Velocities Using values for from. we can represent aition an subtraction formulas for absolute velocities. 4 7 the following way: Aition formula for absolute velocities symbolically enote as u v 4

15 Armenian Transformation Equations u v u vu u u v u v v u c v u Subtraction formula for absolute velocities symbolically enote as u v u v u u v u v u vu v u c v u One can prove that the aition formula for absolute velocities given by the secon equations of. 4 9 satisfy the associative law. u u u u u u.4- Multiplying the first equations of the systems. 4 9 an. 4 0 by c an using expressions. 8 an. 9 we can express aition an subtraction formulas by zero an spatial components of the absolute velocities only, which is the transformation equations for absolute velocities. u 0 c v0u 0 c vu u c v0u c u 0 v an c v u u 0 c v0u0 c vu u c v0u c u0v.4- c v u Using absolute velocity transformation equations. 4 an making a straight calculation also recalling formulas. 9 an. 0 we obtain the following invariant relation: u 0 u v 0 v u 0 u u 0 u.4- Remark(8) The aition an subtraction formulas. 4 also invariant relation. 4 clearly shows, that absolute velocity transforms as a four-vector Aition an Subtraction of the Local Velocities Diviing the secon equations of the. 4 9 an. 4 0 into the first equations an using velocity conversion formulas. we obtain aition an subtraction formulas for local velocities. u u v c v u c vu or u u v c v u c vu.4-4 Remark(9) Local velocity or Newtonian velocity u r t four-vector. oesnottransformsaspartofa Using a straight forwar calculation from. 4 4 we can obtain the expressions for local velocities similar to the first equations of. 4 9 an u v u c vu an u v u c vu.4-5 The formulas. 4 5 we can obtain also from the first equations of. 4 9 using relations. an.. Multiplying together the first an secon equations of. 4 5 we receive an interesting ientity between local velocities. vu vu v.4-6 5

16 .5 - Summary of the Armenian Transformation Equations In Armenian transformation equations. 8 substituting absolute relative velocity value v v v, we can express by local relative velocity v as well. Straight Armenian transformation equations expresse by absolute relative velocity v or by local relative velocity v: t v t vr r v r vt or c v r t v t vr r v r vt c v r.5- Inverse Armenian transformation equations expresse by absolute relative velocity v or by local relative velocity v: t v t c vr r v r vt or c v r t v t vr r v r vt c v r.5- Where Invariant Armenian interval: s t r t r.5- v v v v.5-4 An coefficient is a real number an also it s a constant value characterizing the moving particle or time-space-meium. Conclusion(5) Armenian transformation equations expresse by. 5 an. 5 are the replacement for the Lorentz transformation equations. References. Aharoni J., The Special Theory of Relativity, (Clarenon Press, Oxfor, 959), pp. 4-47, Clye Davenport, A complex Calculus with Applications to Special Relativity, 99. Davi Bohm, The Special Theory of Relativity, 965, pp Lorentz H. A., Electromagnetic phenomena in a system moving with any velocity less than that of light, (Dover, 9) 5. Louis Bran, Vector an Tensor Analysis, (John Wiley & Sons, Inc., New York) 6. Moller C., The Theory of Relativity, (Oxfor University Press, 97,977) 7. Robertson H.P., Thomas W. Noonan, Relativity an Cosmology, (W. B. Sauners, Lonon, 968), pp Robert Resnick, Davi Halliay, Basic Concepts in Relativity, (Macmillan Publishing Company, New York, 99), pp Rosser W.G.V., An Introuction to The Theory of Relativity, (Butterworths, Lonon, 964), pp Stephenson an Kilminster, Special relativity for Physicists, (Lonon, 958). Thomas L. H., The Kinematics of an Electron with an Axis, (Philosophical Magazine an Journal of Science, Lonon, 97). Yuan Zhong Zhang, Special Relativity an Its Experimental Founations, (Worl Scientific), pp