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Einstein's field equation have had great success in weak gravitational field. However, there is a problem in strong gravitational field where the energy of the gravitational field must be taken into account.

1. The fundamental principle of general relativity states that “all energy is a source of gravity”. However, the field equation created by Einstein did not fully realize this principle

The energy of the gravitational field must also function as a gravitational source. Einstein was also aware of this, and for over two years, beginning in 1913, he worked to formulate a field equation that included the energy-momentum of the gravitational field. However, because it was difficult to define the energy of the gravitational field in general relativity, Einstein could not complete the field equation including the gravitational action of the gravitational field. So, we have a field equation involving only the energy-momentum tensor of matter. Because of the omission of the energy-momentum of the gravitational field, the singularity problem and the dark energy problem came into existence.

1)Gravitation and Spacetime (Book)

(1) According to (48) matter acts on the gravitational field (changes the fields), but there is no mutual action of gravitational fields on matter; that is, the gravitational field can acquire energy-momentum from matter, but nevertheless the energy-momentum of matter is conserved ({\partial _\nu }{T^{\mu \nu }} = 0). This is an inconsistency. (2) Gravitational energy does not act as source of gravitation, in contradiction to the principle of equivalence. Thus, although Eq. (48) may be a fair approximation in the case of weak gravitational fields, it cannot be an exact equation. The obvious way to correct for our sin of omission is to include the energy-momentum tensor of the gravitational field in T^{\mu \nu }. This means that we take for the quantity T^{\mu \nu } the total energy-momentum tensor of matter plus gravitation: T^{\mu \nu }} = T_{(m)}^{\mu \nu } + {t^{\mu \nu }}

(1) According to (48) matter acts on the gravitational field (changes the fields), but there is no mutual action of gravitational fields on matter; that is, the gravitational field can acquire energy-momentum from matter, but nevertheless the energy-momentum of matter is conserved ({\partial _\nu }{T^{\mu \nu }} = 0). This is an inconsistency.

(2) Gravitational energy does not act as source of gravitation, in contradiction to the principle of equivalence. Thus, although Eq. (48) may be a fair approximation in the case of weak gravitational fields, it cannot be an exact equation.

The obvious way to correct for our sin of omission is to include the energy-momentum tensor of the gravitational field in T^{\mu \nu }. This means that we take for the quantity T^{\mu \nu } the total energy-momentum tensor of matter plus gravitation:

T^{\mu \nu }} = T_{(m)}^{\mu \nu } + {t^{\mu \nu }}

2)Explanation of GRAVITY PROBE B team

Do gravitational fields produce their own gravity? Yes. A gravitational field contains energy just like electromagnetic fields do. This energy also produces its own gravity,

Do gravitational fields produce their own gravity?

Yes. A gravitational field contains energy just like electromagnetic fields do. This energy also produces its own gravity,

2. As the mass increases, the ratio of negative gravitational potential energy to mass energy increases

Wikipedia, Gravitational binding energy

Two bodies, placed at the distance R from each other and reciprocally not moving, exert a gravitational force on a third body slightly smaller when R is small. This can be seen as a negative mass (Negative mass - Wikipedia) component of the system, equal, for uniformly spherical solutions, to: M_binding=-(3/5)(GM^2)/(Rc^2)

Two bodies, placed at the distance R from each other and reciprocally not moving, exert a gravitational force on a third body slightly smaller when R is small. This can be seen as a negative mass (Negative mass - Wikipedia) component of the system, equal, for uniformly spherical solutions, to:

M_binding=-(3/5)(GM^2)/(Rc^2)

-M_gs = Equivalent mass of total gravitational potential energy (gravitational self energy) of an object. Mass energy is proportional to M, whereas the gravitational self-energy is proportional to -M^2/R.

Moon’s -M_gs = (-2.0 x 10^-11)M_Moon Earth's -M_gs = (- 4.17 x 10^-10)M_Earth Sun's -M_gs = (- 1.27 x 10^-4)M_Sun Black hole's -M_gs = (- 3 x 10^-1)M_Black-hole

It can be seen that as the mass increases, the ratio of negative gravitational self-energy to mass energy increases. Therefore, as the mass increases, the gravitational action of the gravitational field must be taken into account.

So, now the question we have to ask is, "What value would the negative gravitational potential energy in the case of a universe with a greater mass?"

3. In the universe, if we calculate the gravitational self-energy or total gravitational potential energy The universe is almost flat, and its mass density is also very low. Thus, Newtonian mechanics approximation can be applied. Since the particle horizon is the range of interaction, if we find the Mass energy (Mc^2) and Gravitational self-energy ((-M_gs)c^2) values at each particle horizon, Mass energy is an attractive component, and the equivalent mass of gravitational self-energy is a repulsive component. Critical density value p_c = 8.50 x 10^-27[kgm^-3] was used.

At particle horizon R=16.7Gly, (-M_gs)c^2 = (-0.39M)c^2 : |(-M_gs)c^2| < (Mc^2) : Decelerated expansion period At particle horizon R=26.2Gly, (-M_gs)c^2 = (-1.00M)c^2 : |(-M_gs)c^2| = (Mc^2) : Inflection point (About 5-7 billion years ago, consistent with standard cosmology.) At particle horizon R=46.5Gly, (-M_gs)c^2 = (-3.04M)c^2 : |(-M_gs)c^2| > (Mc^2) : Accelerated expansion period

Even in the universe, gravitational potential energy (or gravitational field’s energy) must be considered. And, in fact, if we calculate the value, since gravitational potential energy is larger than mass energy, so the universe has accelerated expansion. Gravitational potential energy accounts for decelerated expansion, inflection point, and accelerated expansion. If we could create a new field equation that perfectly reflects the principle(All energy is a source of gravity, so, the energy of the gravitational field is also a source of gravity) of general relativity and find a solution to it, the dark energy problem will be solved.

The model of Gravitational Potential Energy or Gravitational Field's Energy

1) has a clear origin, 2) is a physical quantity required by the principle of general relativity, 3) has decelerated expansion, inflection point, and accelerated expansion period, 4) does not require fine-tuning, 5) does not have the problem of coincidence.

Since dark energy term is presented as a function of time, it is verifiable.

Λ(t) = (6πGR(t)ρ(t)/5c^2)^2

The sources of dark energy are gravitational potential energy (or gravitational field's energy) and the expansion of the particle horizon.

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