[David Cooper]

You want immediately to use the principle of relativity. But historically there was still no theory of relativity. So, we have an absolute ether. How do you want to proceed then?

Lorentz developed one theory of relativity, while Einstein developed another. Both use the same maths, but with different explanations, and both were developed after MMX. Length contraction resolved part of the puzzle and gave us a clue as to how to solve the rest. Imagine a ring with lasers on it pointing outwards. Move the ring at high speed and it length-contracts into an ellipse, changing the alignment of most of the lasers. Trace out the paths of the laser light by calculating how photons will move within the moving lasers at c relative to the aether, and what you end up with is a distribution of light that conforms to the headlights effect. This illustrates why a moving room (a cube) has all six walls equally brightly lit by a lamp in the middle of the room regardless of how fast the room is moving through the aether. That light exerts a force upon the walls, and that force is unchanging, so this is an illustration of the mechanism by which one kind of force automatically adjusts to maintain the illusion that the system is stationary. Note that all the lasers are still hitting the same targets in the moving system as they were when the system was stationary. The frequency of the laser light (and lamp) is also reduced due to the slowed functionality of the system producing it, that slowed functionality being due to movement through the aether and increased cycle times due to anything cycling having a greater distance to travel through the aether to complete each cycle, and this tunes the wavelength to the distance the light has to go to reach the same target such that there are the same number of crests and troughs in each between source and target as before. No matter how you emit light from a device, it will be flung out in ways that follow the same rules, concentrating it forwards more strongly (and thereby avoiding breaking the rules of momentum). Other forces are clearly adjusted in the same way, hiding the movement of the system through the aether - they too are concentrated forwards, so the measurements of force are unchanging.

Think about it that the vector of velocity and the vector of field strength are physically different. You can not somehow unite them, e.g. by an addition.

The vector of velocity and field strength are perpendicular in both cases, and when the system is moving, they are measured as being at the same angles as when the system is stationary, even if those angles are not the true ones. There is no observation that changes for a co-moving observer making measurements - he always sees the light going north, and measures the field strength force as aligned perpendicular to that. You spoke about all of this applying to a single point, but you can't measure the alignment of anything from a single point - you need to use another point, and as soon as you do so, you have synchronisation issues polluting your measurements.

[worlov (OP)]

Just show mathematically how it works. How can two second-order factors, length contraction and time dilation (~ [1 - v²/c²]^1/2, for v << c =>  ~ 1 - [v/c]²/2), completely compensate one first-order factor, the inclination of the field strength (for v << c, ~ v/c) ?

[David Cooper]

Just show mathematically how it works. How can two second-order factors, length contraction and time dilation (~ [1 - v²/c²]^1/2, for v << c =>  ~ 1 - [v/c]²/2), completely compensate one first-order factor, the inclination of the field strength (for v << c, ~ v/c) ?

They don't compensate it - they merely mask it and prevent you from measuring the actual angles. Show me how you intend to measure those angles while co-moving with the system and I'll show you why that won't measure the true angles.

[worlov (OP)]

... the actual angles

And you know that the information about the current position of the emitter can not be transmitted instantly over long distances? The maximum transmission speed is known to be the speed of light.

[David Cooper]

And you know that the information about the current position of the emitter can not be transmitted instantly over long distances? The maximum transmission speed is known to be the speed of light.

And you know that the slowed functionality of moving clocks ensures that the transmission speed appears to be the same to the moving system as it does to the stationary system.

[worlov (OP)]

If the wave reaches the receiver,

My opinion: From the point of view of the recipient, this is inclined and point to its place of origin.

Your (and Lorentz&Einstein) opinion: From the point of view of the recipient, this is directly and point to actual position of the emitter. 

Meanwhile I thought about what the experiment might look like (see figure below). The receiver consists of two dipole antennas, these are perpendicular to each other and at an angle of 45 ° to the emitter. If the wave front passes the receiver flat, the signals in the antennas balance out. Then SR or LET is valid. And on the contrary, both are invalid.

[David Cooper]

If the wave reaches the receiver,

My opinion: From the point of view of the recipient, this is inclined and point to its place of origin.

Your (and Lorentz&Einstein) opinion: From the point of view of the recipient, this is directly and point to actual position of the emitter.

See my diagram. The red box is an emitter, the green box is a receiver, the ones on the left are the same ones as on the right, showing them in two different locations as they move from left to right. The co-moving black things are a shutter which blocks the radiation such that it can only reach the receiver if it passes through the hole in the middle. The Path the emitter thinks it's sending the radio signal along is shown in light blue (cyan) because it wrongly assumes it is stationary. The path the radio signal actually goes along is shown in dark blue. The path the receiver thinks the radio signal came along is shown in purple (magenta). The radio signal clearly came through the hole in the shutter, so the receiver has proof that it followed the purple path. But its proof is faulty.

Meanwhile I thought about what the experiment might look like (see figure below). The receiver consists of two dipole antennas, these are perpendicular to each other and at an angle of 45 ° to the emitter. If the wave front passes the receiver flat, the signals in the antennas balance out. Then SR or LET is valid. And on the contrary, both are invalid.

As for your diagram, when do you imagine the induced signals reach the middle of the antennas? The one in the blue antenna moves through it more slowly than the one in the red, but when you take into account the synchronisation issue with leading clocks running behind trailing ones, both take the same length of recorded time (by a co-moving observer to the system) to pass along each antenna. All differences are masked.

[David Cooper]

Oh, and I meant to say something about the direction the signal moves through the antenna that slopes from top left to bottom right. As the signal comes from further and further left (with the apparatus moving to the right), you might think there would come a point where the signal starts to run through it from the top left end instead of the bottom right end, thereby running though it in the wrong direction, but that doesn't happen because of length contraction - the X is squashed more and more as the speed goes up, ensuring that the signal always hits the same end of the antenna first.