A treatise on the relative optical velocity of the moving observer against light
Takashi Kubota
(Electronics technological writer)
Prologue
It was decided theoretically that the velocity of light in free space is a fixed value c (about 299,792,458 m/sec) as a fact on the measurement as well.Therefore, it is the fact known again by the Doppler effect and so on for the moving observer that a fluctuation arises in the relative optical velocity, too.
However,is being taken that an optical velocity is c with Einstein's theory of special relativity by the assumption of "the principle of a universal constant" as to any observer and any coordinate.
It is decided to be mentioned that there is a mathematical defect in this assumption on this paper.
The matter that "the MichelsonMorley experiment " and "the phenomenon of Bradley's aberration " are verified theoretically by "the relative optical velocity of the moving observer against light" which it could get from another approach ,and,which depended is made the purpose of this paper.
The mathematical defect of "the principle of a universal constant c "
Einstein's theory of special relativity is very precise to know well, and it is being built up by the ingenious mathematical descriptions.
But, this is expanded in the wellknown fact in the spacetime coordinate four dimensions, and theory is being built up in the starting point of those contents as for "L, vt, ct, the rightangled triangle of the light" shown in the next figure 1 and the figure 2 ( "L, Vt, ct, the rightangled triangle of the light") being foundation.
It is decided to be shown next that there is a mathematical defect in this light triangle.
We refer to systems "rest" and "motion" showed figure 1.The motional system moves in the direction of the +X axis with respect to rest system with velocity v. A motional system corresponded in the origin O, with t=0 as for the rest system.
And a figure 1 showed the position of the motional system after a time ｔ_{１}.
Now,Einstein's theory mentioned that the light reaches D after a time ｔ_{１} when light is shot in the C direction of the Y axis from the origin O (A' of the motional system) at t=0.
Why the light flew like this to side direction, because Einstein thought so that the light might move as well as the movement of an object.
The thought to be so is called "Einstein's special principle of relativity" with "Galilean relativity" with the case of "an object".
In other consideration, Einstein thought so that the velocity of light might be a universal constant by reason that the motional observer insists that the light flew to D from A',so the velocity of light is c in the rest coordinate (therefore ＯＤ＝ｃｔ_{１ }), and also same value c in the motional coordinate (Ａ’Ｄ＝Ｌ＝ｃｔ’ ).
This is the foundation of the special theory of relativity with the rest system and the motional system to say that it is different "the time".
If this thought is right, it is decided in the same way that a figure 2 can exist, too.As for the figure 2, all others are the same conditions as the figure 1 by the thing which is V>v as to both the way of taking the scale of the coordinate axis and the transition of the time.These figures become right both because the relative velocity (in these figures, v and V ) of the rest system and the motional system is liberal with a special theory of relativity.
But, a figure is seen, and it knows that it isn't right .ＯＤ＝ｃｔ_{１ } of the figure 1 and ＯＤ＝ｃｔ_{１ } of the figure 2 are because obviously length is different.As more obvious mathematical wrong descriptions are next formulas.
（ｃｔ_{１ }）^{２}＝（Ｌ）^{２}＋（ｖｔ_{１ }）^{２} from the figure 1
（ｃｔ_{１ }）^{２}＝（Ｌ）^{２}＋（Ｖｔ_{１ }）^{２} from the figure 2
Why did we have such a wrong matter ?There are two reasons in that.
One light is surely the socalled "Two coordinates of the rest system and the motional system are connected with ct." benefit to reach D after a time ｔ_{１}.This is the mathematical foundation not to do it.
One more is that "the theorem of 3 square" is applied to the rightangled triangle of the coordinate that it is surrounded in the rest system and the motional system. We must formulate an equation within one coordinate,that is mathematical foundation for the theorem of 3 square,too.
The general concept of the relative optical velocity
Well, how should we interpret a figure 1 and a figure 2 ?
What it can think about first is in the shoot direction of the light.The apparatus is adjusted so that the light may be catch in D after a timeｔ_{１} (the figure 3 ), after a time ｔ_{２} (the figure 4 ), and it is probably to shoot it in the OD direction.It can guess this from the famous "MichelsonMorley experiment" and "the Ring laser gyro", and so on.
Then, as for saying ct' of the figure 1 and the figure 2, meaning is stopped.In other words, it becomes a "Such a thing is not here.".It is because an optical way is only one of OD's.That is shown in the figure 3 and the figure 4.
Next, it is the fact of "The velocity of light in free space is a fixed value c." to be important.
Though it is "an inertial coordinate" with a special theory of relativity, as for saying the rest system of the figure 1 and the figure 2, the rest system of the figure 3 and the figure 4 is "the coordinate that the velocity of light is c".
This didn't consider "free space" a rest system, by the thing which defined a coordinate from the origin which light was fired at.
The velocity of light in the rest coordinate (basis coordinate) defined by the light like this is "c", and they are ＯＤ＝ｃｔ_{１ }(figure 3) and ＯＤ＝ｃｔ_{２ }(figure 4).
A motional system can be calculated when it is moving in the velocity v (figure 3) and the velocity V (figure 4) to this.
When it proceeds with the consideration here, it is decided that the optical velocity in the motional system is not c. It is decided that only cosθ takes an influence in the relative optical velocity to understand from the figure.
They are
ｃ−ｖcosθ_{１ } by the figure 3,
ｃ−Ｖcosθ_{２ } by the figure 4.
Incidentally, as for the figure 3, a motional system is an example to be moving in about 44% of the velocity c, about 132,000km/s, and a relative optical velocity is about 240,000km/s.
As for the figure 4, it is an example to be moving in about 58% of the velocity c , about 174,000km/s , and a relative optical velocity is about 200,000km/s.
It is based on the light which came out from the sun, and θ is about 89.994゜from the easy calculation when the earth is made a motional system D.
And, though θ isn't clear, it is probably about 90゜in the experiment of MichelsonMorley.
Application
The general concept of the relative optical velocity is applied to "the MichelsonMorley experiment " and "the Bradley's formula of aberration ".
(1) Application to the experiment of MichelsonMorley
The experiment of MichelsonMorley splashed light in the eastwest direction and the northsouth direction to know well, and it checked whether it had that time lag.
Although the apparatus and the way of the calculation of the experiment is on the textbook of many special theory of relativity, I show the calculation which is different from before here though it .
As for the time when going and returning of the light in the eastwest direction take
As for the time when going and returning of the light in the northsouth direction take
It is shown that becomes a (1) ≒ (2), so the fraction shift should not be observed.
Where, is a distance "to the mirror which reflects it from the light source ", and different from L of the figure 1 and the figure 2. V isn't clear at a displacement velocity against the light ray in the MichelsonMorley experiment . And θ is the angle that an optical axis and a mirror make it.
But it is shown that there was a small shift of some fringe patterns even in the actual experiment to understand from a (1) ≒ (2).
And, this calculation consists of a free direction.An experiment can be made in the free direction even if an interferometer isn't turned to the eastwest direction and the northsouth direction.The apparatus is moved precisely at right angles, and we have only to confirm the number of fraction shifts .
To be natural, we expect that the experiment is varying in the testing ground place more because it comes to detect the rotation of the earth, too, as becoming precise measurement more.
(2) Application to the Bradley's formula of aberration.
We try to calculate the famous formula sinβ＝（ｖ／ｃ）sinα theoretically.
In the figure 5, it is the general concept of the relative optical velocity to be cvcosθ in the motional system (the earth K' ), and the basis velocity of light is c in free space K .
In ΔABC,
ＢＣ＝（ｃ−ｖcosθ ）ｔ・tanβ＝ｖｔ・sinθ
Therefore,
ｃ−ｖcosθ ＝ｖsinθ／tanβ ＝ｖcosβsinθ／sinβ
ｃsinβ ＝ｖcosβsinθ＋ｖcosθsinβ ＝ｖsin（β＋θ）
＝ｖsin（π−∠ＳＢＡ）＝ｖsinα
Hence, we obtain a next Bradley's formula.
sinβ＝（ｖ／ｃ）sinα
To the hereafter
Though "the principle of a universal constant c " and "Einstein's principle of special relativity" were the foundation to build up that theory as for Einstein's special theory of relativity, which both show that there is a defect mathematically in this paper,and was mentioned the importance of the general concept of the relative optical velocity.
From now on, we will expect to develop more by the excellent scientist.
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