  The problem of covariance of physical quantities has not been solved
fundamentally in the theory of relativity, which has caused a lot of confusion
in the community; a typical example is the Gordon metric tensor, which was
developed almost a century ago, and has been widely used to describe the
equivalent gravitational effect of moving media on light propagation,
predicting a novel physics of optical black hole. In this paper, it is shown
that under Lorentz transformation, a covariant tensor satisfies three rules:
(1) the tensor keeps invariant in mathematical form in all inertial frames; (2)
all elements of the tensor have the same physical definitions in all frames;
(3) the tensor expression in one inertial frame does not include any physical
quantities defined in other frames. The three rules constitute a criterion for
testing the covariance of physical laws, required by Einstein's principle of
relativity. Gordon metric does not satisfy Rule (3), and its covariance cannot
be identified before a general refractive index is defined. Finally, it is also
shown as well that in the relativistic quantum mechanics, the Lorentz
covariance of Dirac wave equation is not compatible with Einstein's mass-energy
equivalence.
