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**Deyssenroth H ^{*}**

Holzgasse 28, 79539 Loerrach, Germany

- *Corresponding Author:
- Deyssenroth H

Senior Researcher

Holzgasse 28 79539

Loerrach, Germany

**Tel:**49-762187175

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**Received Date:** May 16, 2017; **Accepted Date:** May 31, 2017; **Published Date:** June 05, 2017

**Citation: ** Deyssenroth H (2017) Symmetry Experiment to the Lorentz Transformation. J Phys Math 8: 230. doi: 10.4172/2090-0902.1000230

**Copyright:** © 2017 Deyssenroth H. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The result of the Lorentz Transformation on a classical formula is that an observer in reference frame A observes the same phenomena in reference frame B like an observer in reference frame B observing the phenomena in reference frame A. Is that true? In this article I demonstrate at the Doppler effect that this is valid even for ‘absolute’ velocities belonging to a reference frame outside of A and B. I suggest to test whether the geometric means of frequencies belong to the frames A and B or to a reference frame outside of A and B by an experiment at CERN. In the latter case this would have significant implications in space-time modelling

Lorentz-transformation; Doppler effect; Frequency; Galilean transformation

The Lorentz Transformation (LT) applied to the classical Doppler Effect at a Lorentz boost provides the formula for the Relativistic Doppler Effect (RDE), e.g. if sender and receiver move to each other in x-direction [1]:

(1)

with β= v/c and

The observer in A gets the same frequency as the observer in B.

The RDE is a symmetric formula concerning the observations of A and B. But one can see at one glance that the formula of RDE is a geometric mean of the two classical Doppler Effect formulas for the observer A in rest and B moving to A [2]

f_{B} = f_{0}/(1 - β) (2)

and the system B in rest and A moving to B

f_{A} = f_{0} (1 + β) (3)

where f_{B} and f_{A}

are not symmetric to each other.

This is no accident. In my former paper I demonstrated the deduction of Lorentz-transformation by three completely different methods, each having the same result: The transformed variable is a geometric mean as a result of simultaneous movement of frames A and B in opposite directions [3].

The geometric mean might be useful if the observers in A and B
cannot measure their own velocities related to a fixed point but only
the relative velocity between A and B. However, in this case formula (1)
gets another meaning because the velocities v_{A} and v_{B} referring to an
outside reference frame of A and B (e.g. the air for sound) determine
the results of (2) and (3). One can get the relative velocity v in all
variations of

v = v_{A} - v_{B } (4)

e.g. v = 900 – (- 100) = 1000, or v = 800 – (- 200) or v = 500 – (- 500) = 1000 km/h.

Therefore, there is no classical formula of Doppler effect with the relative velocity between A and B.

If |v_{A}| ≠ |v_{B}| there is no more symmetry in observations though v
has the same value in all cases and the physical meaning is different as
well, because (2) can only be derived from a rest frame C outside of A
and B, where the light propagates isotropically with constant speed c
(also valid without a medium). An observer in A or B cannot derivate
the classical Doppler effect formulas from their own system because
he/she cannot see the changed wavelength by the velocity of the sender.
This is possible only for an observer in C outside of A and B where
the velocities v_{A} and v_{B} to a common fixed point in this frame can be
measured and only such a person can anticipate which frequencies can
be measured in the systems A and B.

But there is a problem: To which system should the reference frame C belong to? To the surface of Earth, to the center of Earth, to the space over the sun, to the space over our galaxy or galaxy clusters, or to the space of the Universe?

From experiments we know that atomic clocks go slower in a
system that moves with high speed. If e.g. v_{A} = 1000 km/h and v_{B} = -100
km/h, an atomic clock in A goes slower than in B. This was detected
in the Hafele-Keating experiment (by the way: without symmetry
between the times of clocks in the aircrafts and the ground clock),
where the reference frame was the center of Earth and not the clock on
the ground [4]. But the formulas of the STR determine the time to go
slower by the same factor γ in the observed system that moves with the
relative speed v to the own system. An experiment could perhaps reveal
which interpretation of formula (1) is correct.

Let’s regard the situation with an example where A and B move to each other at the same altitude and take aim with a laser beam at each other. If an observer in B measures the frequency shift he/she will get in the first step by the classical Doppler-formula

f_{B} = f_{0} (1 + β_{B}) / (1 – β_{A}), and an observer in A: f_{A} = f_{0} (1 + β_{A}) / (1 – β_{B}) (5)

It is evident, that these formulas are not symmetric, except for
|v_{A}| = |v_{B}|. On the other hand, we should take account of the Hafele
Keating experiment. The results show that the time t = N/f and therefore the frequency must be corrected by a γ-factor. But in this case
we don’t know the amount, because we don’t know to which system
the reference frame C is allocated to. We get, as (5) is the product of the
frequencies (2) and (3),

f_{BA} = f_{0} [γ_{B} /γ_{A} ] (1 + β_{B}) / (1 – β_{A}) (6)

In B the atomic clock goes more slowly. Therefore, B gets in its own time – observed from the space over the North Pole- more impulses from A and hence a higher frequency. Then, the correction must be the multiplication with γ. In A, the atomic clock goes more slowly too. Therefore, the sender sends less impulses and hence a lower frequency. Then, the correction must be the division by γ. This is a new ansatz where classical physics is combined with the experimentally found effects at high velocities in which it is unproved whether the gammas really follow the formula

If an observer in A measures the frequency-shift, he/she will get

f_{AB} = f_{0} [γ_{A} /γ_{B} ] (1 + β_{A}) / (1 – β_{B}) (7)

If we assume that (inserted in (7)),

(7a)

At low speeds (<1000 km/h) they would show the same value
as f. This gamma-correction results in the geometric mean of the
frequencies, measured in A and in B (which move to each other) as
well, like at formula (1). If v_{B} = 0 then f_{BA} = f, the Lorentz-transformed
formula (1) of the RDE.

This consideration shows that the Doppler frequencies coming from the opposite system (B or A) have the same value for an observer in A and B. The symmetry of observations is also valid for a reference frame outside of A and B. Hence, an experiment for testing the asymmetry of observations doesn’t make sense. One could assume that the STR is verified by this symmetry but on the other hand: Formulas (7) and (1) are different because the geometric means belong to different reference frames.

At which velocity v_{B} can the difference to formula (1) be detected?
The simplest case would be to set v_{B} = v_{A} and v = 2v_{A}, because the
smaller v_{B} , the more f_{BA} is related to f. This is really interesting: In order
to detect the difference to the Lorentz-transformed Doppler-effect,
v_{B} = v_{A} seems to be the best approach. In this case the gammas are
canceled. Now we compare the two formulas (7) and (1)

(7b)

The rotation of Earth and the other velocities of Earth might increase the gammas (in the STR by the Transversal Doppler effect). But we don’t know to which amount as we don’t know where the reference system is allocated to. Actually, the gammas must be evaluated by experiments.

The measurements can be done by the frequency-comb technique.

Let’s calculate these values with an example. The formulas (7) show
that a difference between f and f_{BA} cannot be found at low speeds of
frames A and B. If e.g. β = 0.000 01 (which corresponds a velocity of
about 11 000 km/h) the difference of f and f_{A} is just about 2 Hz at the
15^{th} digit.

For the ISS, flying with 27.576 km/h = 7.66 km/s and a satellite with the same speed in the opposite direction at the same altitude, we get with the relative velocity v = 15.32 km/s and c = 300 000 km/s = >

β = 5.1066666666666666666666666666667e-5. Inserted in (1):

(10)

Blue laser light (e.g. 445 nm) has the frequency in vacuum

f_{0} = 674 157 303 370 786.516 Hz (11)

In this case A and B observe according to formula (1) the same frequency

f = 674 191 731 216 158.748 Hz (12)

The direct comparison of frequencies in this case is according to formulas (5) expected to be (without the effect by earth rotation or other velocities)

f_{A} = f_{B} = f_{0} (1 + β_{A}) / (1 – β_{B}) = f_{0} (1 + β_{B}) / (1 – β_{A})

= 1.0000255333333333333333333333333 / 0.99997446666666666666666666666667 =

= 1.000051067970602182709065171464

f_{A} = f_{B} = 674 191 731 216 136.302 Hz (reference frame: center of
Earth) (13)

f = 674 191 731 216 158.748 Hz (Lorentz-transformed DE)

This experiment is very challenging, because one could detect a
difference between f and f_{B} = f_{B} at the 14^{th} digit only. For the case that
this small difference can be detected reliably the STR is falsified from
the mathematical view point.

A better experiment could be performed at CERN where ions
and electrons are accelerated to e.g. β_{A} = 0.1 and are merged for
recombination. Could this light ionize other recombined ions which
fly in the opposite direction, similar to the experiments of G. Gwinner?
By the way: G. Gwinner assumes that there is a preferred cosmological
reference frame [5].

Here the difference could be recognized at the third digit already. If
the experiment shows that the result fits better to f_{A} then the principle of
relativity is disproved and the space-time-modelling must be modified.

The proposed very basic experiments above (which are very difficult to do) will hopefully bring clarity into this issue. The experimental results of the Hafele-Keating experiments show that the symmetry of observations is also to be expected if the Doppler-effect is interpreted as a geometric mean of frequencies observed from a reference frame outside of A and B. The gamma-correction results in each case in a geometric mean of frequencies observed in A and B. In this case the mathematical basis of the Theories of Relativity is wrong. Anyway, this experiment should be done to find out if there is a reference frame where the light propagates isotropically with constant speed. If this is found, the assert that the light speed is constant in all frames, would be wrong. Then it would be time to think about physical mechanisms e.g. in the Yokto range (10-24 m) with the goal to explain gravitation and quantum phenomena by physical mechanisms (which also can be described mathematically) rather than to describe them mathematically in space-time. In this case the implications on physics and philosophy would be eminent.

- Einstein A (1905) Electrodynamic moving body. Wiley Online Library.
- Andrade C (1959) Doppler and the Doppler Effect. Einstein Online
- Deyssenroth H (2016) Alternative Interpretation of the Lorentz-transformation. J Phys Math 7: 1-7
- Hafele JC, Keating RE (1972) Around the world atomic clocks: Predicted relativistic time gains. Science 177: 166-168.
- Hafele JC, Keating RE (1972) Gwinner G (2007) Test of relativistic time dilation with fast optical atomic clocks at different velocities. Nature Physics 3: 861-864.

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