On General Covariance

If the invariance of light speed is absolutely universal in covariant space rather than contra variant space, the following researches and conclusions on general covariance would be so far as to catch up with very high probabilities and reliabilities. As has been verified, it is. General covariance is tested controversial after the investigations on gravitational redshift and acceleration. Further inspections on differential geometry indicate the opportunities of inequality of mixed derivatives of bases for the transformations between Riemannian spaces that will then lead to the inequality of Christoffel symbols of alternative sub-index and then the failure of the classical equations of Christoffel symbols. That is one of the reasons that causes controversies on general covariance. Even the negative form of time metric could be proved to be a false setting in that space transformations do nothing with the negative sign inherited from that of Minkowski space. Nevertheless, after discussions on transformations between original spherical space, distance expressed spherical space and Cartesian space, it has been seen that the distance factors for angular coordinates of a spherical space are improper to be employed the metrics in general relativity, instead of that, the concept of gravitational metrics were suggested. In fact, Christoffel symbols and base derivatives both are valid methodologies for analysis in a non-Euclidean space. The concept of trajectory derivatives was carried out to define the derivations on trajectory of matter motions that could help to revise those equations and calculations. Measurable experiments on gravitational redshifts and accelerations have been carried out to support the theoretical results. Conclusions have been drawn that light speed keeps general covariance in gravitational fields but light energy momentum would not, may as well, the motions of massive matters in gravitational fields do not perform general covariance thoroughly. It is impossible to geometrize the gravity effects of massive matters with position depending metrics in that the variable velocities cannot be eliminated thoroughly. Consequently, inferences on kinematics and relativistic release were carried out, which might have been forcefully verified in applications. With the concept of gravitational metrics, the so-called geodesic equations have been falsified to be kinematic equations anymore. Velocities, mass energy and momentum all should have their conservative forms. To seek for these conservative forms is the critical route to create kinematic or dynamic equations of motions of light rays and massive matters. The conservativeness of light angular momentum has been discovered in most surprising form. Renovated solutions for light rays as well as massive matters have been carried out that forcefully impact the traditional methodologies on kinematics of trajectory and time delay because they are the correct interpretations of the realities. It should be highlighted that the renovated mass equation, the general mass equation for free motions, could completely demonstrate energy variations, especially the variations in gravitational fields, that help to create more general dynamic equations, so as to cause an irrelativistic solution of planet perihelion precession. As a proof by contradictions, the traditional solution of that must be involved with errors. Newtonian ballistic method was put forward to make numerical analyses on close-to-light-speed motions, and the invalidity of that method on light propagation could be seen as another support to the conclusions and inferences. Another forceful falsification on energy momentum conservativeness carried out in the discussions on the traditional treatment of time delay of close-to-light-speed particles, has thoroughly exposed the essential mistakes in the way of methodologies. Dynamic models of fluid planet rings were founded to interpret the evolutions of accretions of quasars and active galactic nuclei and the mechanism of relativistic release. It is predicted that the peak release of an inflow is at 1.5 of gravitational radius and the peak luminosity of an accretion may locate at the position of about 1.33 of gravitational radius. Relativistic frequency shift interprets the mechanism of giant redshifts that predict the probability of observational redshifts might be up to a particularly higher level with respect to that have been observed in the past years. The width equations of emission and absorption lines indicate the mechanism and the positions of broad line regions and narrow line regions. It could be imagined that the relativistic emissions and relativistic absorptions with relativistic redshifts would have been involved with fantastic mystery of intrinsic structures of matters that we know less.