Final answer:
If the transformation law for contravariant tensors gives an incorrect result, it may be due to improper application, particularly in the context of Lorentz transformations that involve time as well as space. The metric in space-time behaves differently from purely spatial metrics, affecting how displacements are measured under such transformations.
Step-by-step explanation:
The transformation law for contravariant tensors may sometimes give the wrong transformation if it is not applied correctly, especially in the context of Lorentz transformations in space-time. Lorentz transformations are generalizations of spatial rotations that include the time dimension, and thus have to consider the effects on the space-time interval, which remains invariant under these transformations.
The metric, which is the rule for measuring displacements labeled Ar and As, behaves differently under Lorentz transformations than under pure spatial rotations. In three-dimensional spatial rotations, axes remain perpendicular and the length scale is preserved. However, a Lorentz transformation involving the time axis does not guarantee that these features will remain the same due to the nature of the space-time metric that combines space and time differently.
Correct application of the transformation laws is crucial to ensure that the transformed quantities accurately reflect the new frame of reference, taking into account the Lorentz contraction and time dilation effects that are hallmarks of the special theory of relativity.