is actually baseless since these singularity theorems have been proven to be unrealistic in physics. As pointed out by Einstein [2], his equation may not be valid for very high density of field and matter. In short, the singularity theorems show only the breaking down of theories of the Wheeler-Hawking school, which are actually different3) from general relativity.
The theories of this school, in addition to making crucial mistakes in mathematics as shown in this paper (see also [11,28]), differ from general relativity in at least the following important aspects:
1) They reject an anti-gravity coupling7), which is considered as highly probable by Einstein himself.
2) They implicitly replaced Einstein's equivalence principle in physics3) with merely the mathematical requirement of the existence of local Minkowski spaces [5,6].
3) They, do not consider physical principles [9-11,28] (see also Section 5), such as the principle of causality, the coordinate relativistic causality, the correspondence principle and etc. of which the satisfaction is vital for a physical space, which models reality, such that Einstein's equivalence principle can be applicable.
Thus, in spite of currently declaring their theories as the development of general relativity, these theories actually contradict crucial features that are indispensable in Einstein's theory of general relativity. More importantly, in the development of their so-called "orthodox theory," they implicitly violate physical principles that took generations to establish. As a result, Einstein's theory has been unfairly considered as irrelevant in the eyes of many physicists.
Of course, the exact form of t(g)(( is important for the investigation of high density of field. However, it seems, the physics of very high density of field and matter is not yet mature enough at present to allow a definitive conclusion. For instance, it is unclear what influence the discovery of quarks and gluons in particle physics would have on the evolution of stars. It is known that atomic physics supports the notion of white-dwarf stars, and that nuclear physics leads to the notion of neutron stars.

5. Physically Invalid Unbounded "Gravitational Waves" and the Principle of Causality
"To my mind there must be at the bottom of it all, not an equation, but an utterly simple idea. And to me that idea, when we finally discover it, will be so compelling, so inevitable, that we will say to one another, 'Oh, how beautiful. How could it have been otherwise?' " -- J. A. Wheeler [32].
It seems, the principle of causality2) (i.e., phenomena can be explained in terms of identifiable causes) [9,10] would be qualified as Wheeler's utterly simple idea. Being a physicist, his notion of beauty should be based on compelling and inevitability, but would not be based on some perceived mathematical ideas. It will be shown that the principle of causality is useful in examining validity of accepted "wave" solutions.
According to the principle of causality, a wave solution must be related to a dynamic source, and therefore is not just a time-dependent metric. A time-dependent solution, which can be obtained simply by a coordinate transformation, may not be related to a dynamic source8) [33]. Even in electrodynamics, satisfying the vacuum equation can be insufficient. For instance, the electromagnetic potential solution A0[exp(t - z)2] (A0 is a constant), is not valid in physics because one cannot relate such a solution to a dynamic source. Thus, as shown in Section 4, a solution free of singularities may not be physically valid.
A major problem in general relativity is that the equivalence principle has not been understood adequately [11,34]. Since a Lorentz manifold was mistaken as always valid, physical principles were often not considered. For instance, the principle of causality was neglected such that a gravitational wave was not considered as related to a dynamic source, which may not be just the source term in the field equatio

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