To say that a physical quantity or an object "transforms sensibly" under Lorentz transformations means that it undergoes a consistent and well-defined transformation when transitioning between different inertial frames of reference related by Lorentz transformations.
Lorentz transformations are mathematical transformations that relate the coordinates and time measurements of events observed in one inertial frame to those observed in another inertial frame moving at a constant velocity relative to the first frame. These transformations are a fundamental part of special relativity and preserve the structure of the theory.
When we say that a physical quantity or object transforms sensibly under Lorentz transformations, it means that its properties or components change in a way that respects the laws of special relativity. In other words, the quantity or object should obey the same physical laws in all inertial frames of reference.
For example, in special relativity, the spacetime coordinates of an event are combined into a four-vector called the four-position. A sensible transformation under Lorentz transformations means that the components of the four-position transform according to the Lorentz transformation equations. Similarly, other quantities such as four-momentum, energy, electric and magnetic fields, and the wavefunctions of particles are required to transform consistently under Lorentz transformations.
By ensuring that physical quantities transform sensibly under Lorentz transformations, we maintain the relativistic covariance of the theory, which means that the laws of physics have the same form in all inertial frames of reference. This is a crucial aspect of special relativity and is essential for constructing consistent and valid physical theories in the relativistic regime.