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[Solution of the Singularity Problem of Black Hole]

1. Understanding of the problem : Generation of singularity

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We assume that the solution for this singularity consists in quantum mechanics. Though exact explanation is not available because quantum gravity theory in integration of quantum mechanics and gravity has not been completed yet.

This writing will prove that an object of positive energy has the minimum size for its existence and that since this size is in proportion to that of energy, there is no singularity with infinite density.

2. Solution of the problem

2-1. Gravitational potential energy with negative values.

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2-1-2. All energies are a gravitational source.

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2-1-3. (except r=0) is considered to provide consequently the right explanation for all points.

Teachers and professors have explained that it is alright to set the randomly reference point for gravitational potential energy because, since the variation of gravitational potential energy has caused kinetic change in the problem under review, there was no problem in dealing with only the variation of gravitational potential energy.

From the equation K + U = const. we obtain such equation as , which can explain motion with variation, but this neither means that all observers in the same inertial system may set randomly reference point at random nor confirms that U is an object with an optional value.

Let's consider the following case that the value of gravitational potential energy has been fixed for the distance of 0 to infinity from gravitational source.

Even though, as above, gravitational potential energy has the value of energy defined for r= 0 to r= infinity from gravitational source, we can obtain the right result in a problem in which its variation matters.

2-1-4. Effect of mass defect in atomic scale caused by binding energy

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2-2. Gravitational self-energy or Gravitational binding energy

The concept of gravitational self-energy is the total of gravitational potential energy possessed by a certain object M itself. Since a certain object M itself is a binding state of infinitesimal mass dM, it involves the existence of gravitational potential energy among these dMs and is the value of adding up these.

Fig06. Since all mass M is a set of infinitesimal mass dMs and each dM is gravitational source, too, there exists gravitational potential energy among each of dMs. Generally, gravitational potential energy by infinitesimal mass that consists of an object itself is reflected on the mass of the object itself. Mass of an object measured from its outside corresponds to the value of dividing the total of all energy into .

Gravitational self-energy or Gravitational binding energy() in case of uniform density is given by:

( : gravitational self-energy)

2-3. For black hole or singularity, never fail to consider gravitational self-energy

In the generality of cases, the value of gravitational self-energy is small enough to be negligible, compared to mass energy . So generally, there was no need to consider gravitational self-energy. However the smaller R becomes, the higher the absolute value of . For this reason, we can see that is likely to offset the mass energy in a certain radius.

Thus,looking for the size in which gravitational self-energy becomes equal to rest mass energy by comparing both,

This equation means that if mass is uniformly distributed within the radius , gravitational self-energy for such an object equals mass energy in size. So, in case of such an object, mass energy and gravitational self-energy can be completely offset while total energy is zero. Since total energy of such an object is 0, gravity exercised on another object outside is also 0.

Comparing with , the radius of Schwarzschild black hole,

This means that there exists the point where gravitational self-energy becomes equal to mass energy within the radius of black hole, and that, supposing a uniform distribution, the value exists at the point , a 30% level of the black hole radius. Even with kinetic energy and virial theorem applied only the radius diminishes as negative energy counterbalances positive energy, but no effects at all on this point: "there is a zone which cannot be compressed anymore due to the negative gravitational potential energy (a sort of the binding energy)")

Since this value is on a level not negligible against the size of black hole, we should never fail to consider "gravitational self-energy'' for case of black hole.