The Coulomb electric field of a charge q is
E = k q / r²
where k is the Coulomb constant and the energy density of the field is
D = 1/2 * 1 / (4 π k) * E².
The gravity field of a mass m is
g = G m / r²,
and we claim that the energy density of a weak gravity field is
D = -1/2 * 1 / (4 π G) * g².
For a strong field, the formula is probably more complex.
At what radius R does the negative energy of a weak gravity field of a mass m match its positive energy? We can integrate from R to infinity to obtain the energy:
∞
m c² = ∫ 1 / (8 π) G m² / r⁴ * 4 π r² dr
R
= 1/2 G m² / R.
We get R = 1/2 G m / c², which is 1/4 of the Schwarzschild radius of m.
The energy of the gravity field has to be negative because when masses come together, energy is freed, and at the same time the field grows stronger.
The negative energy of the gravity field is the viewpoint of a Minkowski observer who believes that the "true geometry" is flat. General relativity interprets that the gravity field has no energy which should appear in the stress-energy tensor. In the Einstein-Hilbert action we can interpret that the geometric part calculates the energy of the gravity field.
The newtonian expanding ball mimics the FLRW universe perfectly
Valerio Faraoni and Farah Atieh in their 2020 paper review various solutions of general relativity for a newtonian expanding ball of dust.
Mashhoon and Partovi (1980) among others have calculated that the FLRW metric inside, and the Schwarzschild metric outside is the general relativity solution of the newtonian expanding ball.
The cosmic microwave background inside the ball is just like in the FLRW universe as long as our line of sight only sees the hot gas of the 380,000 years old universe. Eventually, the CMB will disappear when we start to see the emptiness of the Minkowski space.
Dark energy in a newtonian expanding ball
In an asymptotically Minkowski space, Birkhoff's theorem and possibly the ADM formalism enforce energy conservation. A positive energy field, which grows with a newtonian expanding ball, is prohibited.
We need to find out if the accelerated expansion of the universe might be caused by emission and absorption of light inside the ball.
In the FLRW metric, a photon in the expanding universe seems to lose its energy through a redshift. In a newtonian ball the energy loss is only an illusion which is caused by the high velocities of the observers. Nevertheless, photons exert some pressure on matter. We have to figure out a way to calculate this effect.
In the interior Schwarzschild metric, pressure is able stop the contraction of the uniform, incompressible, fluid ball if its radius is at least 9/8 of the Schwarzschild radius.
Suppose that our newtonian expanding ball is not inside 9/8 of the Schwarzschild radius. Then positive pressure probably would accelerate the expansion.
A low gravitational potential slows down light to a crawl in the Minkowski view of things. An analogy is a block of glass which slows down photons which are emitted from the lamp inside glass. In a sense, the photons cause an outward pressure on the glass.
If we put two observable universes side by side, then their newtonian gravity force is ~ 10⁴⁴ N, while the force from pressure is only ~ 10³⁸ N. We conclude that the radiation pressure cannot explain dark energy.
Dark matter which turned into radiation might explain the accelerated expansion
Let us assume an asymptotically Minkowski geometry and a newtonian expanding ball.
Let us assume that most of the matter in the universe is of a new type of dark matter. Initially it was particles which had mass. They decayed into radiation some 4 billion years ago.
The dark radiation has a similar mass-energy as the hypothetical dark energy field. The dark radiation has a similar pressure as the dark energy field, except that the pressure is positive.
If we are not inside 9/8 of the Schwarzschild radius, then the positive pressure will speed up acceleration.
Our model has the advantage that it conserves energy. The dark energy field of standard cosmology creates energy from nothing - that is not allowed in quantum mechanics or classical mechanics.
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