Difference between the Time Intervals in the Special Relativistic Theory Calculated with the Aid of an Invariance Property between the Mechanical Expressions

In the relativistic mechanics, we obtain a difference between the time scale of a one-dimensional motion having a larger velocity and the time scale of a similar motion with a lower velocity. This calculation does involve usually also the differences of parameters other than time. Basing on the invariance of a pair of the mechanical parameters, it can be shown that the difference of two scales of time can be attained independently from the differences of the other physical parameters.


Introduction
In principle, any calculation of a change of the scale of some mechanical parameter in the special relativistic theory is accompanied by an insight into the scale changes belonging also to other parameters than that taken into account [1]. But recently an invariance of the differences of parameters belonging to different mechanical scales has been pointed out [2]. Our aim became to demonstrate that this situation allows us to estimate the size change of a single mechanical parameter alone. The scale of time has been chosen as our basic example.
In the first step, we present the Lorentz transformation of the variables pair , t x into the pair , t x ′ ′ [2]. Next the intervals t t ′ − and x x ′ − are calculated in terms of t and x. Finally the quanta of the electric capacitance in the hydrogen How to cite this paper: Olszewski, S. (2021) Difference between the Time Intervals in the Special Relativistic Theory Calculated with the Aid of an Invariance Property between the Mechanical Expressions.
The result in (3) indicates the invariant behaviour of the difference upon the Lorentz transformation formulae in (1) and (2). From (3a) it comes out that on condition we put and Our aim is to point out that the above properties concerning the transformed and non-transformed parameters of t and x can lead to a simple reference between t' and t.

The Difference of t' and t Represented as a Function of Some Remainder Parameters
We multiply Equation (1) by t and Equation (2) by x. A simple rearrangement of The result in (8) is dictated by the property (3a). The calculations done farther in the paper refer to the non-primed coordinates system Another approach to the rearrangement of terms in (8) gives: By taking into account (4a) and (4b) we have: but in virtue of (9): The left-hand side of (8) can be transformed into indicating a positive value of (11a) on condition t x > .
An effect of (4a) and (4b) leads together with the relation (3) to the formula: so the last step of (12) gives

An Equation for the Interval Δt and Its Properties
Because of (11a) we have ( ) (13) and (14a): We divide (16) by and get the equation: We divide both sides of (19) by t: 1 We see from (21) that the ratio t t ∆ is a sum of a constant dependent solely on v c increased by the ratio x t multiplied by another constant term depend solely on v c . The next step concerns solution of the equation for x ∆ . We obtain from This gives The solution of (22) gives: .
Assuming x ∆ to be a small number the square root in (28) is taken next only with a positive sign. We obtain ( ) ( ) On the basis of the result in (21) we obtain for the right-hand side of (29) the expression dependent solely on t and x: The term ( ) 2 t ∆ can be readily obtained from (21) as a function of t and x.

The Calculation of
We do it by taking into account the relation (9):

Example of Distance x Representing the Quanta of the Electric Capacitance and Their Change
An example of x and Δx can be presented by considering the electric capacitance C of the condenser. A reference to C and the charge q of the condenser is given by the formula [3]: where V is the potential difference. On the other side, when the potential difference (37) is applied to the resistance R, the current intensity i produced by V and R becomes Evidently the lowest quantum of the capacitance n C corresponds to the orbit 1 n = : In this way we find that cm .
A relativistic change of x modifies the x entering (46) into x + Δx.

Euler-Lagrange Equation Applied in Examining the Particle Mass
In [2] an attempt is done to apply the Euler-Lagrange equation in examining the behaviour of the particle mass m. A special application may concern the force F acting on a moving particle when it is reduced to a single term [3]: