Most people would answer this question ad hoc: What a redundant question, of course a negative energy density cannot exist. But all these “ad hoc “-people oversee that the gravitational field around a mass indeed has a negative energy density. The next ad hoc remark is then, what is the sense of this question if the negative energy density of gravitational fields is fact.
After this remark we get closer to the point. The only possibility to avoid a negative energy density around a mass is the existence of a gravitational field with positive energy density, which overlays all gravitational fields around masses.
The existence of such a gravitational field, omnipresent, with a positive energy density, and nearly homogenous in the whole universe, now has consequences, which we consider with the following points.
(1) The existence of such a field requires a closed finite universe. A positive energy density cannot fill an infinite volume.
(2) A gravitational field with a positive energy density X allows the propagation of gravitational waves with a wave pressure p and an energy density E with p=√(EX). This is a result of wave theory.
(3) Every grain of mass, including atomic nuclei, electrons, and photons moves a small dent through the homogenous background field with energy density X. This movement leads to post pulse oscillations. “Generation of post pulse oscillations” is a wave excitation mechanism completely different from “quadrupole radiation”, which currently is proposed as mechanism for gravitational wave generation.
(4) The post pulse mechanism is much more efficient. It leads to the general effect that any mechanic or electrodynamic activity generates gravitational waves. This means that X in a closed universe, is continuously growing.
(5) X as a medium, which supports wave propagation has a density, a wave pressure component, and a kinetic component. The pressure and the kinetic component are the result of wave excitation.
(6) What is the current amount of X? This amount is presumably above the range of the gravitational field around a neutron star. The Tolman-Oppenheimer-Volkoff-limit determines the value.
(7) Black holes slightly exceed this limit. But they cannot really exceed it because there remains nothing to be exceeded.
(8) Black holes become clouds of matter in a strange condition surrounded by a zone of zero gravity.
(9) The zone of zero gravity shields the cloud of matter from the oscillating background field with energy density X.
(10) The field X penetrates all kinds of matter except the extremely dense matter inside black holes. The energetic interaction of field X with normal matter including neutron star matter is a small one-way street towards X. But the matter inside a black hole is in a much denser state, susceptible for excited matter X with a pressure- and a kinetic component.
(11) The excitation of field X with gravitational waves shrinks the zone without gravitation around black hole matter. If the field X reaches the matter, the port of energy flow from X to the black hole matter is open.
(12) The consequence is that after a few million years the black hole matter becomes transformed into a widespread cloud of atomic hydrogen which provides new star fuel for the next life cycle of the universe.
If nature does not allow a negative energy density, the points (1) to (5) above are mandatory consequences. The points (6) to (12) are speculations about the amount of the gravitational energy density which surrounds us. However, the most remarkable point is that the “nonexistence of negative energy density” leads to a new model of the universe, fundamentally different to the big bang theory.