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DOI: 10.14738/aivp.86.9268

Publication Date: 10th November, 2020

URL: http://dx.doi.org/10.14738/aivp.86.9268

Analysis of Correct Use of Time Transformation in Physics

1

Libor Neumann 1

Prague, Czech Republic;

Libor.Neumann@email.cz

ABSTRACT

The paper deals with the analysis of mathematically correct use of time derivative in coordinate

systems with time transformation. It combines the use of time derivative in different areas of physics

with the results of this analysis. The results are used to verify the most well-known theories using time

transformation. A general limit of the use of time transformation in physics is formulated and proved.

The analysis is supplemented by specific examples which illustrate the consequences of using a

nonlinear (curved) time transformation. They show the formation of a resistor in the LC circuit as a

result of time transformation, energy generation or loss, tilting of high rock walls, change in distance

between the Earth's centre and the Earth's surface, twisting of Earth's axis, astronomical paradoxes,

unknown acceleration properties. The internal consistency of both theories of relativity is also

analysed, including the equivalence principle.

The result of the analysis shows that the use of a nonlinear (curved) time transformation is a

dangerous tool of contemporary physics that decomposes its internal consistency. This also applies to

the time transformations used in both theories of relativity.

Keywords: Time transformation; Time derivative; Coordinate system; Coordinate transformation;

Special and General theory of relativity; Space-time.

1 Introduction

Physics often works with a time derivative. The use of time derivative is not limited to one area of

physics. It is a common mathematical apparatus of experimental and theoretical physics.

Theoretical physics works with hypotheses or theories based on the transformation of time between

different coordinate systems.

The text deals with the general analysis of the impact of the time transformation to time derivative of

general physical quantities.

The results of the general analysis are used to analyse selected time transformations, namely the time

transformations included in Special theory of relativity (STR) [1] and General theory of relativity (GTR)

[2].

A general limit of the use of time transformation in physics is formulated and proved.

2 General Analysis of Time Transformation

Consider two coordinate systems with generally different time. The relationship between time in one

system and another system is described by a time transformation.

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Libor Neumann. Analysis of Correct Use of Time Transformation in Physics. European Journal of Applied

Sciences, Volume 8 No 6, Dec 2020; pp: 113-134

URL: http://dx.doi.org/10.14738/aivp.86.9268 114

Time tt is used in the reference system, time τ is used in the second system. Time transformation is

described by general dependence τ = τ(tt) or tt = tt(τ) where τ(tt) and tt(τ) are the time

transformation functions.

We have a general physical time dependence ff of a physical quantity (e.g. spatial coordinate, voltage,

electrical current, energy, momentum, temperature) such that in both coordinate systems the time

dependence is identical for the corresponding pair of times tt and τ .

Writing: ff̂(τ) = ff(tt) = ff̂(τ(tt)) = ff(tt(τ)). This means that the measured values of the physical

quantity are the same at the same place and moment in both coordinate systems. Emphasize that this

is of such physical quantities which can be measured objectively.

2.1 Types of Time Transformations

Let’s divide time transformations into two types:

o Linear time transformation

o Non-linear (curved) time transformation

2.1.1 Linear Time Transformation

Let us define a linear time transformation with a condition dd2tt(τ)

ddτ2 = 0 = dd2τ(tt)

dd 2 valid throughout the

scope of the time transformation validity. This means that the second derivative of the time-to-time

transformation is zero at all points in space and for all times.

From this condition we can calculate (by double integration) a general description of linear time

transformation

τ = τ(tt) = AA ⋅ tt + BB or tt = tt(τ) = AÂ ⋅ τ + BB� (1)

where AA, BB, AÂ and BB� are constants. Physical meaning of linear time transformation is a change of

units of time and/or the time offset. Such a transformation of time is independent of the position in

the space and hence the time changes equally in all movements in space.

An important feature of the linear time transformation is that it is stable over time. It is the same at

any moment.

2.1.2 Non-linear Time Transformation

Let us define a non-linear (curved) time transformation as a complement to a linear time

transformation, thus a condition dd2tt

ddτ2 ≠ 0 or

ddτ2tt

dd 2 ≠ 0 is valid in at least one point in the domain of

validity of the time transformation. This means any other time transformation that cannot be

described as a linear time transformation in at least one point of validity scope (at least one time

and/or in any area of space). Physically it is a case of time-varying time transformation, signifying the

curvature of the time transformation. It is irrelevant for what reason the time transformation is

changed, whether it is directly included in the analytical notation of the time transformation, or

whether it is the result of another physical phenomenon, e.g. the displacement of a body between

places with different time transformation.

Generally, objects can move in space between arbitrary places at any time. If the time transformation

were different in different places of space, then there would be a time change of the time

transformation if the body was moved between places with different time transformation. Therefore,

invariance of time transformation in space is a necessary condition for invariance of time

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European Journal of Applied Sciences, Volume 8 No. 6, December 2020

Services for Science and Education, United Kingdom 115

transformation in time. Thus, a sufficient condition for the non-linearity of the time transformation is

the dependence of the time transformation on the position in space.

2.2 Transformation of Time Derivative

Now we analyse the behaviour of the first and second time derivative of the general function ff in

coordinate systems with a general time transformation. In the second coordinate system, the function

ff̂ is a composite function and therefore we calculate the derivative according to the derivative rules

of the composite function. When using Leibniz notation,

we get: ddff̂(τ)

ddτ = dd (tt)

dd .

dd (τ)

ddτ (2)

dd2ff̂(τ)

ddτ2 = dd2ff(tt)

dd 2 . (

dd (τ)

ddτ )2 + dd (tt)

dd .

dd2tt(τ)

ddτ2 (3)

Let us pay attention to the second derivative (3). The second derivative contains the sum of two

components, where the first component is the second time derivative of the analysed physical

function ff multiplied by the square of the first time derivative of the transformation function.

The second component is the product of the first time derivative of the analysed function ff and the

second time derivative of the transformation function. This means that this second term will exist or

disappear depending on the type of time transformation.

For a linear time transformation, i.e. the time transformation described by (1), this term disappears

and has no physical meaning. For non-linear time transformation, i.e. for all other time

transformations, the term is non-zero. Let us analyse the physical impacts of this option in the

following text.

2.3 Example LC Circuit

For further analysis we will use an example of LC circuit. The LC circuit is described in the reference

coordinate system using the generalized coordinate QQ(tt) (capacitor charge) and the generalized

velocity II(tt) = dd (tt)⁄dd (capacitor current).

QQ̈+ 1

LLLL QQ = 0 thus dd2QQ(tt)

dd 2 + 1

LLLL QQ(tt) = 0 (4)

Figure 1: LC circuit in reference coordinate system

In the second coordinate system using time transformation τ = τ(tt) , the LC circuit is described by a

transformed second time derivative according to (3).

We substitute the transformed second derivative of the function QQ�(ττ)

dd2QQ(ττ)

ddττ2 . (

dd (tt)

dd )2 + dd (ττ)

dd .

dd2ττ(tt)

dd 2 + 1

LL QQ(tt(ττ)) = 0 and adjust to the standard shape

dd2QQ(ττ)

ddττ2 +

dd2ττ(tt)

dd 2

(

dd (tt)

dd )2 ⋅

dd (ττ)

dd + 1

LLLL⋅(

dd (tt)

dd )2 QQ(tt(ττ)) = 0 (5)