Lorentzian SRT-transformation factors as solutions of oscillation- equations

Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillationdifferential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass. key-words: undamped oscillation; SRT; k-factor;Differential-equation of second order;EinsteinLorentz;Amplitude-analogy;damped oscillation; developed SRT of fourth order I. Lorentz-Einstein SRT k-factor as an amplitude solution of undamped oscillatonequation 1.Introduction: It is obvious to remark the similiarity between the amplitude curves of an undamped oscillation and of k-factor of SRT given by Lorentz and Einstein for velocities with are smaller than light on the one hand and by Feinberg for FTL on the other [1.],[2.],[3.],[5],, if both are described and drawed together.Trough this similiarity there can be tried to get the lorentzian k-factor not only from pure kinematic examinations like in [1.],[2.] and [6.], but as an exact solution of an oscillation equation as is demanded in [4.]. If the oscillation equation of second order is set in the following form there can be derived the lorentz-k -Factor as an solution resp. an interpretation for amplitude of the oscillating system. 2.Calculation: There is the ansatz for the following differential equation, which can be interpreted as an oscillation equation for undamped states, φ̈ + c 2 r 2 ⋅φ= Ā 2⋅e i ( r⋅t ) (1.) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 May 2021 doi:10.20944/preprints202105.0242.v1 © 2021 by the author(s). Distributed under a Creative Commons CC BY license. where r is an unknown constant length, which value and meaning has to be discussed (maybe planck-length or the oscillation wavelength or as rotation radius), v the velocity of a moving body or particle in local inertial frame of flat Minkowski-Space and c the invariance velocity by Lorentztransformations, which occurs here in interpretation as the eigenfrequency-velocity of local spacetime. Also is set: φ (t )=A 2⋅e i( r⋅t ) (2a.) as an ansatz für the solution of this equation. Then there is derivated: φ̈=−v 2 r 2 ⋅A 2⋅e i ( r⋅t ) (2b.) If (2a.) and (2b.) are set into (1.), there follows the equation: A2⋅c 2−v 2 r 2 ⋅e i vr= Ā⋅e i ( r⋅t ) (3.) which gives the following relation: c 2−v 2 r 2 = Ā 2 A ⋅e (4.) If now the terms are separated seen as a realterm R and an imaginary term I , there is set: c 2−v 2 r 2 =R= Ā 2 A 2 ⋅cos (θ) (5a.) and 0=I= Ā 2 A ⋅sin (θ) (5b.) This term means, that θ=0 ° . There is no phase shifting in angle for classical SRT-term which leads to the barrier of invariance-velocity c for undamped local spacetime-states in tangent-space. Therefore follows with theorem of Pythagoras sin(θ)+cos (θ)=1 the relation of: A1,2,3,4=±±i Ā √ c 2−v 2 r 2 (6.) Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 May 2021 doi:10.20944/preprints202105.0242.v1

key-words: undamped oscillation; SRT; k-factor;Differential-equation of second order;Einstein-Lorentz;Amplitude-analogy;damped oscillation; developed SRT of fourth order I. Lorentz-Einstein SRT -k-factor as an amplitude solution of undamped oscillatonequation

1.Introduction:
It is obvious to remark the similiarity between the amplitude curves of an undamped oscillation and of k-factor of SRT given by Lorentz and Einstein for velocities with are smaller than light on the one hand and by Feinberg for FTL on the other [1.], [2.], [3.], [5],, if both are described and drawed together.Trough this similiarity there can be tried to get the lorentzian k-factor not only from pure kinematic examinations like in [1.], [2.] and [6.], but as an exact solution of an oscillation equation as is demanded in [4.].
If the oscillation equation of second order is set in the following form there can be derived the lorentz-k -Factor as an solution resp. an interpretation for amplitude of the oscillating system.

2.Calculation:
There is the ansatz for the following differential equation, which can be interpreted as an oscillation equation for undamped states, where r is an unknown constant length, which value and meaning has to be discussed (maybe planck-length or the oscillation wavelength or as rotation radius), v the velocity of a moving body or particle in local inertial frame of flat Minkowski-Space and c the invariance velocity by Lorentztransformations, which occurs here in interpretation as the eigenfrequency-velocity of local spacetime.
Also is set: ) as an ansatz für the solution of this equation.
Then there is derivated: If (2a.) and (2b.) are set into (1.), there follows the equation: which gives the following relation: If now the terms are separated seen as a realterm ℜ and an imaginary term ℑ , there is set: This term means, that θ=0°. There is no phase shifting in angle for classical SRT-term which leads to the barrier of invariance-velocity c for undamped local spacetime-states in tangent-space.
Therefore follows with theorem of Pythagoras sin(θ) 2 +cos (θ 2 )=1 the relation of: If now is taken the positive real sign-term of the whole term (6.) which is chosen as A 1 and also is set: then follows from (7a): and from (7b): These are the classical Lorentz-Einstein-terms and the Feinberg-term for moving bodies in local inertial frames of flat space-time in classical SRT.

3.Discussion:
The similiarity between Einstein-Lorentz-Feinberg k-factors and the amplitudeterm of the model of an undamped oscillation with the given bounding conditions may be coincidentally of a mere pure mathematically analogy without any physical evidence.But this derivation may throw a new light into the interpretation of local space-time-conditions.Specially the role of the supposed constant length-term r has to be discussed further.Also the phase-angle θ must be discussed. For undamped state analogy this angle is equal to zero. Therefore can be concluded, that for phaseangles with other values there can be derived a developed SRT-theory for damped states as worked out in [4.], which may unify the broken symmetry of both Einsteinian and Feinberg kterms.

4.Result:
In analogy to I. there is calculated the differential equation of second order for damping oscillation processes.
The result of calculating this equation is given in short form (without further algorithm processes): It is easy shown, that the equation for these processes must have the form: with the boundary conditions for the solution-function and the variables: So there is finally the form of oscillation-equation for advanced SRT of fourth order as: and there is the term of amplitude given for this equation as solution: which ist the right form of developed K-factor for advanced SRT in fourth order if the positive real sign is chosen because in the case a≡0 this leads to classical k-term of Lorentz-Einstein-SRT.
The factor a is interpreted as the damping velocity of the relativistic system which can be coupled with an equivalent circle-frequency ω .
The result for the phase angle forms to: Also here is seen: for a≡0 there is the phaseangle θ a (v )=0 and no phase shifting in classical SRT-term which agrees in line with I but there is a special phase angle for the bothside limit: So the classical Lorentz-Einstein-factor can be interpreted as a solution of an oscillation equation with phase-angle θ equivalent to zero.For the advanced system there are phase angles θ different from zero, so there are given phase-shifted states in the fundamental function of the system

5.Conclusions and summary:
If differential-equations of second order for undamped or damped oscillating systems are set with special bounding conditions like proposed in [4.] and solved, then the Lorentz-Einstein-k-Factor from Special Relativity Theory (SRT) can be derived from this equation for undamped oscillation as a solution. Also the advanced Lorentz-factor for SRT of fourth order can be derived from the equation for damped oscillation states.