Proposal: Time paradox in Special Relativity Theory.

That is correct, but irrelevant.
Specifically, the Lorentz transform operates as a linear transform on vectors, thus it is an analog of multiplication, and we have in one dimension (for velocities parallel or anti-parallel): \(\Lambda( a \hat{x} )\Lambda( b \hat{x} ) = \Lambda( \frac{a + b}{1 + \frac{a b}{c^2}} \hat{x} )\).
And for arbitrary directions:
\(\Lambda( \vec{a} )\Lambda( \vec{b} ) = { \huge \Lambda } \left( \frac{ \left( 1 + \frac{\vec{a} \cdot \vec{b}}{c^2 + c \sqrt{c^2 - \vec{a}^2}} \right) \vec{a} + \sqrt{ 1 - \frac{\vec{a}^2}{c^2}} \vec{b}}{1 + \frac{ \vec{a} \cdot \vec{b} }{c^2}} \right) {\huge R } \left( \frac{ \left( \left( 1 + \frac{\vec{a} \cdot \vec{b}}{c^2 + c \sqrt{c^2 - \vec{a}^2}} \right) \vec{a} + \sqrt{ 1 - \frac{\vec{a}^2}{c^2}} \vec{b} \right) \times \left( \left( 1 + \frac{\vec{a} \cdot \vec{b}}{c^2 + c \sqrt{c^2 - \vec{b}^2}} \right) \vec{b} + \sqrt{ 1 - \frac{\vec{b}^2}{c^2}} \vec{a} \right) }{ \left(\vec{a} + \vec{b}\right)^2 - \left(\frac{ \vec{a} \times \vec{b} }{c}\right)^2 } \right) \)
which is just one of the ways the Lorentz transformation is famously related to rotation.

Need to double check the above for sign errors.

Since only the square of the velocity enters into proper time calculations in an inertial frame, the direction is irrelevant.

You have assumed that either a or b may be taken as motionless in some inertial frame, and that contradicts your assumption that both a and b describe curved trajectories in some inertial Cartesian coordinate system. Thus neither of your cases applies.

General relativity does not save you since due to the flatness of space time assumed when you specified that you were going to describe a special relativity "paradox" means that in no coordinate system may the curved trajectories you depict can fairly represent geodesics of space time

Again you run afoul of formal debate rules. Because you had only a vague idea as to what physical situation you wish to describe as a "paradox" we still don't have a topic of debate many iterations in. This is not the thread to debate in -- this is the thread in which to propose a formal debate where I would be motivated to get my signs right in the above rotation matrix.

 

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Mod note: A Proposal thread is not for general discussion and debate. It’s for proposing and agreeing on the rules for a formal debate which subsequently takes place in another thread. Since this still hasn’t been done after 61 posts, I am closing the thread.

 

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