Genady

Area of a sector with angle a in radians and unit radius is a/(2p)*(p*r^2)=a/2 for any a.

a=1.226819...

I don't need to read your book because I know algebra. If your universe allows FTL, and if causality is not violated in your universe, then algebra shows that special relativity does not apply in your universe. Then your universe is not the universe we live in. The End.

Galactic evolution is very far from having a well established mainstream theory. There are several models, each one having open questions. The new data set limits to these models. What is sensational is delivery of the new findings by popular media.

New and unpredicted, yes. Sensational and contradicting, no. This is the purpose of this instrument - to deliver new data which were unknown before.

The causality violation does not happen to frames. It happens to observers. It does not prevent causality violation.

I'm curious to see some examples of these rare cases.

The "ship" can be just another planet with people on it. Then causality is violated.

Talking of wishful thinking that has been mentioned earlier in this thread. I don't see why we can accept inevitability of our personal death, while have a problem accepting inevitability of death of humanity. Or inevitability of our cosmic isolation until that happens. Perhaps, this question belongs to a thread in a different forum though.

Any instant transmission in one frame violates causality in other frame. There is no physical difference between frames.

A transmission that is instantaneous in the Earth frame goes parallel to the Earth's space axis, the horizontal axis on the diagram. A transmission that is instantaneous in the ship frame goes parallel to the ship's space axis, the sloped dashed line on the diagram.

I don't see why or how.

It is fundamental for theories and tested in experiments and observations on all levels of physics: quantum field, standard model, high energy, astrophysics, gravitation, cosmology, etc. AFAIK, there is no hint that it will or might go away. When they compare the order of events between the Earth frame and the ship's frame on the same diagram, they apply Lorentz transforms implicitly.

They are diagrammatic representations of Lorentz transformations, which connect spacetime coordinates between inertial frames moving relative to each other.

Straight from special relativity.

If you mean that the diagram is drawn in the Earth's, and the Vega's, reference frame, then yes.

The speed, in the units of c, is equal to tan of the angle between the ship's worldline and the Earth time axis. It is less than 450, so its speed is less than c. There is no error.

Let one object fall straight down and at the same time throw another object horizontally. While free falling at the same rate, they will be moving relative to each other.

This is incorrect. They move relative to each other as they all free fall in a uniform gravitational field. So, the causality violation happens as is shown on the diagram, and there is no way to "extrapolate" it away.

It is free falling.

No, this will not help them to know that they are moving. Remember that Earth and Vega also move, relative to some other stars. Everything moves relative to something. There is no difference between frames that are in free move.

They don't have a way to know that they are moving. All they know is that Earth and Vega move relative to them.

There is no error. The STL space axis is not the same as the space axis of Earth and Vega because the ship moves relative to Earth and Vega.

Let's look at this diagram from the video, before STL ship receives or sends any messages - just what they "think" happens, i.e., the order of events in their reference frame: Vega receives the signal, step 1 above, before Earth sends it, step 3 above. The effect occurs before the cause. Isn't it a causality violation?

They are moving. They don't send any transmission. They only receive the transmission from Earth.