Is the Lorentz transformation one of the key factors for unsolvable elements in elementary physics?

Initial situation

Albert Einstein postulates that every object that is currently not experiencing any acceleration can be viewed as stationary. From this still view, objects in the environment move in relation to this object. The speed of other objects is calculated using the Lorentz transformation.

A relevant example

Objects moving at speeds below 1% of light speed (below 3'000 km/s) only get slightly modified by the Lorentz transformation. Therefore I choose for this reflections neutrinos that get emitted by stars and move at 99.99% the speed of light away from their stars.

Neutrinos emitted by stars

From the point of view of an observer, the sun is approximately stationary and the emitted neutrino is moving.

Proxima Centauri is 4.2 light years away.

Neutrino₁, moving at 99.99% the speed of light, will reach Proxima Centauri in = 4.2004 years.

Transformation

The Lorentz transformation ensures that no object moves faster than light, neither by itself nor in relation to other other objects:

This is reached with the Lorentz transformation formula: 

A) Transformation of the star – neutrino movements:

• V_Sun ~ 0

• V_Neutrino2 = 0.9999 * c

This corresponds to the expectation: from a resting point of view, the other object takes over your speed if you consider yourself to be at rest.

It also corresponds to the Galilean transformation:

B) Transformation of the neutrino – neutrino movements:

Two neutrinos move towards each other, Neutrino₁ emitted by the sun, Neutrino₂ emitted by Proxima Centauri.

The two neutrinos are expected to meet in the middle of the 4.2 light year distance after = 2.1002 years.

If one neutrino is considered to be at rest, at what speed does it see the other neutrino approach?

We would expect that they see each other approach at v₁ + v₂ = 0.9999c + 0.9999c = 1.9998c.

The Lorentz transformation forbids this (v_rel.1+ v_rel.2 > c) and gives us a different result:

After the Lorentz transformation,

• the resting Neutrino₁ sees Neutrino₂ approaching at the speed of light.

• the resting Neutrino₂ sees Neutrino₁ approaching at the speed of light.

The result corresponds to the expectation for the Lorentz transformation: nothing moves faster than light, not even relatively.

A major question remains: when will they reach each other? They were supposed to encounter in 2.1 years. But Neutrino₁, at light speed, would now need 4.2 years to reach Neutrino₂.

Trying to answer this question is hard. At least it seems very contradictory.

But there is a far more obvious fact: If we put the two stars into the neutrino picture (Lorentz transformed, as shown above, from the resting view point of Neutrino2), then we get this result:

It is impossible that the sun and Neutrino₁ (which the sun emitted) both reach Neutrino₂ at the ~same time.