Genady

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.

There is no FTL transmission from a small moving object in my scenario. Only from the Earth.

The string theory is one of better known such attempts. These mathematical singularities make it impossible to analyze physics near the center of BH and at the very early stages on the universe. Anything that gets close to Planck scale.

In his scenario, yes, the FTL transmission goes from STL ship to Earth. But this is not necessary. The paradox can be demonstrated also without such transmission. Instead, assume that there is a "train" of STL ships with synchronized clocks between all the "cars" (ships.) The "conductor" of the train made an arrangement that whatever ship passes the Earth at 9h, blows off the FTL transmitter on Earth. To be specific, let's say that the train moves with 3/5 speed of light. This speed makes a relativistic time dilation factor equal 5/4. At 9:45h on the Earth clock, the transmitter sends a FTL signal to the train engine. Let's say it arrives to the engine at 10h on the Earth clock. Because of the time dilation, it is 10*4/5=8h on the train clock. At 9h on the train clock, i.e., one hour later, a car that passes the Earth blows off the transmitter. Because of the time dilation, it is 9*4/5=7h on the Earth clock. The transmitter gets blown off before a transmission is sent (before 9:45h on the Earth clock). But the engine already has received a transmission. It received a transmission that has never been sent. This violates causality.

There are many phenomena moving with any arbitrary speed. They all can't carry energy or information.

One needs to take in account a teacher/student relative weight.

In principle, anything that has energy-momentum affects the curvature. Practically, a few million particles will not have a measurable effect. They are all fields, i.e., machines that get spacetime coordinates in and produce a scalar / spinor / vector / tensor out. They all have Lagrangians and obey stationary action principle. But gravity does not appear in interaction terms of Lagrangians the way other fields do. Thus, it is entirely distinct. (These are short preliminary answers. Markus will reply with much more depth, I expect.) No. All charged particles interact with EM (photon) field, but neutral particles don't. Quarks and gluons interact with gluons. All particles interact weakly. All massive particles interact with Higgs (not sure about neutrinos.) Seems, there are more interactions than fields.

4:1

Their interactions are described by interaction terms in the Lagrangians. They operate in the curved spacetime. A particle and its anti-particle are excitations of the same field. No.

I am not sure about the particles. Take electron neutrino and electron anti-neutrino, for example. They have the same 1), 2), and 3), but they are different particles.