General relativity breaks the weak equivalence principle

Literature says that the weak equivalence principle acted as a guide for Albert Einstein when he designed his field equations. But we were not able to find any proof that general relativity actually satisfies the weak equivalence principle. The reason for the lack of a proof seems to be that general relativity does not satisfy the weak equivalence principle!

The weak equivalence principle

https://en.wikipedia.org/wiki/Equivalence_principle

"The local effects of motion in a curved spacetime (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime, without exception."

Above we have one of the most common definitions of the weak equivalence principle.

                                v_h

                                     <- • m

                                        /    v

                                      v       

                             o         g = acceleration of

                            /|\              gravity

                             /\   

              ---------------------------

                        planet

Let us have an observer on a planet. A test mass m approaches him, obeying the Schwarzschild metric.

https://en.wikipedia.org/wiki/Schwarzschild_geodesics

As the mass m comes to a lower altitude, its proper time τ slows down relative to the proper time of the observer. Then dφ above must be smaller, too. The observer sees that the horizontal velocity v_h of the test mass m slows down.

                                v_h

                                     <- • m

                                        /    v

                                      v    

                                     

                             o         g = acceleration of

                            /|\              rocket

                             /\             

               ---------------------------    

                        rocket

Let us repeat the experiment in a rocket. The rocket is accelerated upward. As the velocity of the rocket increases, the proper time of the observer slows down relative to the global Minkowski time. The observes sees the horizontal velocity v_h to increase.

Could it be that this is a "tidal" effect and the discrepancy is removed if we reduce the width of the configuration? No. The accelerations stay the same regardless of the width.

Conclusions

Theories of gravity must satisfy a very rudimentary equivalence principle which states that all objects moving at a slow velocity are accelerated downward at approximately the same rate. This is an empirical fact observed by Galileo Galilei and others.

On the other hand, more complex interactions with a gravity field generally are not similar to what happens in an accelerated frame in Minkowski space.