  It was generally believed that, in general relativity, the fundamental laws
of nature should be invariant or covariant under a general coordinate
transformation. In general relativity, the equivalence principle tells us the
existence of a local inertial coordinate system and the fundamental laws in the
local inertial coordinate system which are the same as those in inertial
reference system. Then, after a general coordinate transformation, the
fundamental laws of nature in arbitrary coordinate system or in arbitrary
curved space-time can be obtained. However, through a simple example, we find
that, under a general coordinate transformation, basic physical equations in
general relativity do not transform covariantly, especially they do not
preserve their forms under the transformation from a local inertial coordinate
system to a curved space-time. The origination of the violation of the general
covariance is then studied, and a general theory on general coordinate
transformations is developed. Because of the the existence of the
non-homogeneous term, the fundamental laws of nature in arbitrary curved
space-time can not be expressed by space-time metric, physical observable and
their derivatives. In other words, basic physical equations obtained from the
equivalence principle and the principle of general covariance are different
from those in general relativity. Both the equivalence principle and the
principle of general covariance can not be treated as foundations of general
relativity. So, what are the foundations of General Relativity? Such kind of
essential problems on General Relativity can be avoided in the physical picture
of gravity. Quantum gauge theory of gravity, which is founded in the physics
picture of gravity, does not have such kind of fundamental problems.
