What the covariant and ordinary divergences of the tensors in Einstein's field equation tell about Newton's apple when it hits the ground

Abstract

The covariant divergence of each tensor in Einstein’s field equation is zero as a mathematical necessity (and hence as a tacit presupposition), whereas the ordinary divergence is not necessarily so. The principle of energy conservation is thus observed in Einstein’s field equation, but only if all kinetic energies obtained by gravitational accelerations (and hence by “forces” which are no real forces) are thrown out. This leads to an apparent dilemma for the principle of energy conservation when Newton’s apple hits the ground, where the kinetic energy generated by gravitational acceleration converts into thermal energy and can thus no longer be disregarded. The dilemma can be solved. The solution sheds light on the disputed hypothesis according to which the gravitational field does not carry any energy at all. It also provides a surprising insight into the nature of dark energy – not as a result of speculations, but as a mathematical consequence of the covariant divergence of all tensors in Einstein’s field equation being zero. In addition, the solution sheds light on the disputed concept of the world as a spatially four-(or more) dimensional brane.