By analyzing the Einstein-box thought experiment with the principle of relativity, it is shown that Abraham’s light momentum and energy in a medium cannot constitute a Lorentz four-vector, and they consequentially break global momentum and energy conservation laws. In contrast, Minkowski’s momentum and energy always constitute a Lorentz four-vector no matter whether in a medium or in vacuum, and the Minkowski’s momentum is the unique correct light momentum. A momentum-associated photon mass in a medium is exposed, which explains why only the Abraham’s momentum is derived in the traditional “center-of-mass-energy” approach. The EM boundary-condition matching approach, combined with Einstein light-quantum hypothesis, is proposed to analyze this thought experiment, and it is found for the first time that only from Maxwell equations without resort to the relativity, the correctness of light momentum definitions cannot be identified. Optical pulling effect is studied as well.