Gravitational Field Energy

I meant to talk about how I am thinking about the gravitational field energy. My first thought experiment is to imagine a single photon of sufficient energy (and frequency) is converted to matter. That matter falls into a gravitational well, adding to the mass that is already there. As it fell it also gained kinetic energy.

The total amount of energy gained by matter falling into a gravity well is the energy of assembly. But since it gained energy by assembling into one big mass, the energy of assembly for gravity should be negative.

But I realized that time-dilation and gravitational acceleration are proportional. Now imagine that instead of the matter colliding with the planet surface, it is converted back into a single photon and reflected back into space. The new photon starts with a frequency higher than the original photon, corresponding to the gain in kinetic energy. But as it travels back out of the potential well it is red-shifted back to the frequency of the original photon: total conservation of energy.

This means that the kinetic energy gained through gravity only has a relevant meaning within the gravitational well. From outside observers, there is no change of energy at all. Gains of kinetic energy through gravitational collapse are exactly matched by a time-dilation factor, which for observers away from the well, there is no net change of frequency (or energy) at all!

A second thought experiment: imagine two massive bodies well separated with total mass M. The two bodies then collapse gravitationally. In the gravity well energy is gained increasing causing the energy of the single new body to be higher than the original bodies. Since the new body is 2M in mass, it also experiences a higher time dilation. However, from outside observers the total energy of the system is still only 2mc^2. The extra thermal energy is folded into the total energy. If the thermal energy is radiated away (out of the gravity well), observers will actually see a total mass less than 2M, even though observers inside the well still see 2M worth of mass.

This now brings me to black holes. If a massive body simply collapses without any interruptions, and ignoring radiation taking away thermal energy, the total mass of a body will remain the same for outside observers even though the internal thermal energy increases. As the radius of the body approaches the Schwarzschild radius (although the definition of that radius seems a bit wonky), the time dilation goes to zero, and the internal energy approaches infinity. But because of the near infinite time-dilation, outside observers still see the same total mass, energy, and temperature no matter how much the body collapses under gravity.

So what does this mean for field energy? In a way, the energy is taken from the time dimension. Object gains energy falling, but then causes time-dilation which makes it appear as total energy of the matter is unchanged. If the total energy is unchanged, then there is no need to invoke the idea of a field energy.

What is really wonky is the perspective of the rest of the universe from inside a gravity well.