_{1}

Goal of this experiment is basically measuring the velocity of light. As usual we will measure two-way velocity of light (from A to B and back). In contrast to the similar experiments we will not assume that speeds of light from A to B and from B to A are equal. To achieve this we will take into account Earth’s movement through the space, rotation around its axis and apply “least squares method for cosine function”, which will be explained in Section 9. Assuming that direction East-West is already known, one clock, a source of light and a mirror, is all equipment we need for this experiment.

Observe the planet Earth. The Earth orbits the Sun. For this motion we will join the vector v_{1}. Sun orbits the center of the Milky Way. For this motion we will join the vector v_{2}. In relation to the center of the Milky Way, we can join to the Earth movement sum of vectors

It is also known that our Galaxy is moving relative to other galaxies (or to a point in the space outside the Milky Way Galaxy). Similarly, to this motion we could join the vector v_{3}.

Denote by v the sum of all these vectors

At the end of the sum three points are left, because eventually there may be some other movements.

In the period of 24 h vectors v_{2}, v_{3} can be taken as constants, while the vector v_{1} by making a certain error could also be taken as constant.

Thus for the Earth’s motion through the space within 24 h, we can join the constant vector v.

The speed and direction Earth orbits the Sun are known, and let v_{0} represent its avarage speed.

Suppose that some approximate values for vectors v_{2} and v_{3} are known as well. On the basis of these values, let suppose that we have inequality

Suppose that an arbitrary point A is given. Earth rotation axis will be taken as the z coordinate, and as the plane xy we will take the plane passing through point A and perpendicular to the z axis. In this case it is natural to take section of the plane xy and z axis as the center of the coordinate system. In addition to point A let the points B and D are given. Line AB lies in the plane xy and parallel to the direction of the Earth’s rotation. Distance AB will be marked with L. For the x axis, at some initial time t_{0}, we will take the line in the plane xy, parallel to AB. The projection of the vector v in the plane xy denote by v_{xy}. Due to the Earth’s rotation the direction of AB will be changed, so that it will be changed the angle, marked by Φ, between the x axis (which remained fixed) and the line AB. Let at point A we have a clock and some source of light. Suppose that speed of light in the direction AB is given by equation

Point D will be chosen so the line AD is parralel to direction South-North. Distance AD is marked by L_{1}. Angle between line AD and z axis we will denote by j. Angle j actually represents Latitude of point A on the Earth’s surface, thus it remains unchanged during the experiment.

The projection of the vector v on z axis denote by v_{z} (actually v_{2} + v_{3}, because v_{1} is perpendicular on z axis). Assume that the speed of signal in the direction AD is given by equation

where c represents “velocity of light in vacuum for a body at rest”. Our aim is to find the constant c, vectors v_{xy} and v_{z}.

In some moment T_{0} we will send signal from point A to point B. The angle between the axis x and v_{xy} is marked by Θ.

Once the signal arrived at point B it will be reflected back to point A.

Difference between the time when the signal was being sent from point A, and the time when the signal reached to the point A is denoted by t_{0}.

At the same time we will send signal from point A to D and return back to point A. Difference between the time when signal was being sent and reached to point A we will denote by t_{0}.

The same procedure will be within 24 h repeated N (N > 4) times, whereas the time between the two sets of consecutive procedure to be same and equal to 24 h/N.

In that way we will get the series {t_{i}} and {t_{i}}

To the each t_{i} we can join an angle a_{i} between x axis and line AB.

In that way we get the series

By assumption (3.1) the speed of the signal c_{i} in the direction AB is equal to

and in opposite direction BA

It follows that

If we swap the roles of the points A and B, we would get the same formula as in (6). Therefore it is completely irrelevant whether direction of the vector v_{xy} is equal to direction AB or BA.

We assume that

for

It would be in principle our experiment.

In this section we will deal only with the measurements in direction East-West.

Let t_{i} is given by (3.6) and

denote the average speed c_{i} (from point A to point B and back to A).

It follows that

where e_{i} represents some experimental error. Replacing

we get

in short form

The coefficients A, B and Q will be chosen so the sum of squares

has a minimum value.

To acheive our goal we are going to apply Theorem 1 for k = 2.

For the sake of simplicity we’ve only considered cases when

Thus we have

We’ll make a small digression. From Lemma 1 it follows

In the similiar way we can get

Generally we have

Function Atan () takes values at interval (−P/2, P/2).

If we consider A_{0} as function of

From (6) it folows that between the values Θ_{1} and Θ_{2} we have to choose that one for which A_{0} > 0.

From (5) and (6) we can derive values for c and

We don’t know exact direction of vector v_{xy}, thus positive and negative value are assigned to_{ }

In this section we will make comparison between “the least squares method” and “the least squares method for cosine function”.

Let consider

Let c_{m} represents the mean value of serial

If we apply Least squares method, Variance V_{1} is given by

and standard deviation s_{1} by

Suppose that to the each c_{i} we joined the time when measurement took place, or rather the angle between the direction of AB and vector v_{xy}. Expected value E_{2}(a_{i}) for “The Least squares method for cosine function” is given by

where

Denote a_{i} by

Let us find Variance V_{2} for this method

Standard deviation s_{2} for this method is given by

From (7) _{ }

If standard deviation s_{2} is bigger then some expected value it means either our measurement are not accurate enough or our method (curve) doesn’t suit to our data.

In this chapter we will deal with the series

Just to remind that t_{i} represents time it takes for signal to travel from A to D and back to A in direction South- North.

Let

denote the average speed g_{i}. In that way we get the series {g_{i}}

where e_{i} represents some experimental error.

Since angle j kept constant value during the experiment we could apply Least squares method to the series given by (4).

Let denote g_{m} by_{ }

mean value of the series {g_{i}}.

We can calculate Variance V_{1}

and standard deviation s_{1}

If standard deviation s_{1} is bigger then some expected value we should declare the experiment failed.

Combining equations (4) and (5) we get

We don’t know exact direction of vector v_{z}, thus positive and negative value were assigned to_{ }

From (5.13) and (7.8) it follows that length of vector v is given by

while vector v is given by

Recall (from 2.1) that vector v can be written also as

Suppose that during one year the same experiments have been repeated 2*K times. In that way we will get the series

where

Let

Suppose also that origins of vectors

Mean value

Depending on

1)

In other words _{ }

In this case we have to reject hypothesis given by (3.1) and declare that velocity of light is not effected by Earth’s movement through the space.

This results is consistent with some other experiments, for example with Michelson-Morley experiment.

2)

During the experiments in period of one year v_{1} is changing, while v_{2} + v_{3} is keeping the constant value.

Recall that vector v_{1} is perpendicular to z axis.

Denote vector u by

If we replace

(

From (9) and (10) we can get approximate value for

We can form serial

Mean value

Let find standard deviation s_{1} for serial (13).

If s_{1} is bigger then some expected value we have to decline our hypothesis (2.1) and declare the experiment failed.

where

For serial (15) mean value u_{z} is given by

Let standard deviation for serial (15) is marked by s_{2}.

If s_{2} is bigger then some expected value we have to decline our hypothesis (2.1) and declare the experiment failed.

Otherwise hypothesis given by (3.1) holds and we can conclude that velocity of light depends on Earth’s movement through space. In other words velocity of light depends on the direction in which has been measured, what would be in contradiction with Michelson-Morley experiment [

The speed that Solar system moves in the space in this case is given by equation

Note that while performing the experiment we committed some mistakes.

It was not taken into account the speed of Earth’s rotation. This problem can be solved by conducting an experiment at place closer to the Earth’s poles, and thus the speed of Earth’s rotation taken as small as we want. On other hand this would be counter-productive to our conditions for South-North measurement. Ideally, E-W experiment should be performed on the North/South Pole and S-N experiment at some place on equator.

In addition, within 24 h the Earth changes its direction and the speed at which it revolves around the Sun. We can’t solve this problem but we can assume that this speed is relatively small comparing to total speed at which Earth moves through the space.

If N, k are natural numbers (1 < N, 0< k < N) and Q an arbitrary angle then

Proof.

where

Q.E.D.

Suppose we are given the series {c_{i}}, c_{i} > 0, _{p} <> c_{q}

Let take arbitrary coefficients B, A, Q and form equations

, (1)

Define function g(B, A, Q) by

We will prove that in case

where

Proof.

Let B, A and Q have arbitrary values

thus we get

In that way we can reduce function g() from function of three variables to fuction of two variables A and Q, keeping coefficent B fixed and equal to c_{m}.

Now we can write the function g() in the form

In order to find minimum for function g(), first we have to find partial derivates with respect to A and Q and critical point (A_{0}, Q_{0})

Let us find the first partial derivatives

1)

In this case we would have

It’s easy to prove that g() has minimum at

2)

Let us look at the Equations (10) and (12)

For

1)

From (12) it follows A = 0. We will reject this posibility because

2)

From (10) it follows

3)

From (10)

From (12)

Now we have to find the second order partial derivatives of g() with respect to A and Q.

Equations given by (13) and (18) are sufficient conditions for minimum.

Q.E.D.

Miloš Čojanović, (2016) The Impact of the Earth’s Movement through the Space on Measuring the Velocity of Light. Journal of Applied Mathematics and Physics,04,1168-1178. doi: 10.4236/jamp.2016.46121