Do gravitational fields have negative energy?
The negative energy is not yet discovered. We still don't know the presence of negative in the energy in gravitational field.
Gravitational fields do not possess negative energy.
One constant of the universe is that energy is conserved. There is no evidence that negative energy even exists.
When an object falls into a gravitational well it converts gravitational potential energy into kinetic energy. So, an object in a gravitational field has exchanged potential energy for kinetic energy.
To escape a gravitational field requires energy to restore the potential energy it requires. This can be misinterpreted as the gravity field having negative energy.
The question is where is the potential energy stored? It was thought that it was stored in the gravitational field itself. This is impossible. Consider an asteroid falling to Earth from a great distance. Where did its potential energy come from?
The answer is complex and requires an understanding of General Relativity. An object in a gravity field actually has less mass. Lifting an object against gravity increases its mass. The energy is explained by
Gravity is not a force. It is the consequence of matter curving space time. Gravity fields have no energy at all. Only particles have energy and negative energy particles have never been observed.