Abstract

Based on the cosmological model without singularity, this paper demonstrates that there is an upper limit of the mass of stable black holes. An unstable black hole could become quasar or an orphan quasar. When the mass of an unstable black hole increases to large enough so that the temperature of its central zone comes up to the highest temperature, the highest symmetry will be realized and then inflation will occur. Consequently, the black hole will transform into a new v-cosmic or a s-cosmic island inside the original s-cosmic island. This v-cosmic island seems to be a huge orphan quasar, and this s-cosmic island seems to be a huge quasar or white hole. This is because there is only a repulsive force between the s-particles and v-particles, and when the temperature is greater than the critical temperature, the s-particles and the v-particles must significantly transform from one to other. The more massive the black hole is, and the higher the density and temperature of the black hole are, the faster the transformation is. It is seen that the critical temperature and the highest temperature of the universe determine the mass of a stable black hole and evolution of an unstable black holes.

1. Introduction

Based on the cosmological model without singularity ¹, this paper demonstrates that there is an upper limit of the mass of stable black holes, and unstable black holes could become quasars or, as their mass grows, they could inflate and become a new cosmic island.

Model ¹ puts forward a new basic hypothesis that the universe is composed of s-matter and v-matter which are completely symmetric before symmetry breaking and the symmetry is and whose contributions to Einstein tensor are opposite to each other. There is only the coupling of the s-Higgs field and the v-Higgs field The masses of and are all very large and the coupling coefficient is positive. Thus, it can be concluded that there are two sorts of symmetry breaking, i.e., s- and v-symmetry breaking. When s-breaking occurs, i.e. v-symmetry must remain, i.e. , and vice versa. Thus, when the s-elementary particles must acquire their corresponding masses, and forming visible matter and dark matter; However, the masses of the v-elementary particles must be zero, forming color singlets of a large unified group, e.g., the color singlets, at low temperatures. In addition to the known universal gravitation, there is no other interaction among these color singlets so that they cannot form atoms, molecules or block masses, but can only diffusely distribute in space as the dark energy, and they have the effect of dark energy on the evolution of the universe. It is impossible to detect the v-color singlets in the universe with s-breaking. Therefore, this hypothesis is compatible with known theories and experiments.

Based on this model, the following results have been obtained.

A. The premise of hawking theorem is no longer valid in this model, so this model has no singularity.

B. There is the highest temperature in the universe.

C. The cosmological constant and the effective cosmological constant are both zero, and based on this, the evolution of the universe is explained ²;

D. The covariant definition of localized energy conservation in the general relativity is given ³;

E. The universe consists of infinite v-cosmic islands, infinite s-cosmic islands and transition zones. Each cosmic island is made up of huge holes and galaxies. It is impossible to communicate any information between a s-cosmic island and a neighboring v-cosmic island, so that every observer of a cosmic island thinks that his own cosmic island is the whole universe ⁴. The cosmological principle holds firm for the universe as a whole, but it is not strictly valid for a cosmic island.

F. The gravity between two distant galaxies predicted by this model will be less than that predicted by general relativity ¹, because there is a lot of v-color singlets between the two galaxies.

G. S-huge voids in a s-cosmic island are not empty, but full of v-color singlets. Thus a s-void has a repulsive effect on s-matter ¹. S-voids have the following characteristics.

a. S-matter and s-galaxies are very rare in a s-void;

b. A s-void has concave lens effect for s-light passing through the s-void.

c. The redshift of a s-galaxies in a s-voids is less than that of galaxies which are not in s-voids at the same distance. This is due to the repulsive effect of the v-color singlets on the s- photons.

d. The s-voids in a s-cosmic island is taking up more and more space than the s-galaxies.

e. The s-voids exerts pressure on the neighboring s-galaxies. This effect is combined with the gravitational effect of dark matter inside the s-galaxies, both are not easily distinguishable.

2. The Upper Limit of the Masses of Stable Black Holes and Evolution of Unstable Black Holes

When the distribution of s-matter around a s-black hole is constant, the s-energy absorbed by the s-black hole is equal to the released v-energy. The explanation for this definition is as follows:

When the s-mass of a s-black hole is large enough, the temperature of its central zone can be higher than a critical temperature (see below). In this case, the s-particle in the central zone can significantly transform into v-particles. It is seen that a sufficiently massive s-black hole is composed of a large number of s-particles and a small number of v-particles. S-black holes do not emit s-particles, butthey emit v-particles. The larger the density and the higher the temperature in the central zone of the black hole, the more s-particles transform into v-particles, more v-particles are radiated. A s-black hole with its s-mass absorbs s-energy from the s-matter around it, transforms into v-energy , is radiated out. In this way, the total energy and the s-energy of the s-black hole do not change. It is called a stable black hole. The temperature in present paper is the conventional thermodynamic temperature.

2.2. S-particles Significantly Transform into v-particles when

According to model ¹, there are the Higgs fields and , and are respectively dimensional representations of the group, are the generators ^{5, 6}. Temperature will rise when space contracts.When the temperature effect is considered, the Higgs potentials become.

(1a)

(1b)

(1c)

(1d)

¹ has proven that there is the highest temperature at which the expectation values of all Higgs fields are equal to zero, i.e., , and . Consequently, s-particles and v-particles can significantly transform from each to other so that . In this case, , space inflation occurs. Then the highest symmetry will break to or (finally or in low temperatures ^{5, 6}), and the reheating process occurs. Let a cosmic island is in s-breaking, i.e. there be symmetry

Considering the temperature effect, the one-loop correction of the gauge fields and in the S-breaking, ignoring the one loop corrections of the fermions and the Higgs fields, the terms proportional to and the term s irrelevant to and only considering and the breaking components of for brevity ( when is considered as well, the following inferences are still qualitatively valid), let and , here and are the gauge coupling constant of the gauge fields and the Boltzmann constant, (here ), respectively, we get ^{7, 8, 9}.

(2a)

(2b)

Let It is seen from that there are the three critical temperatures and in the S-breaking in which satisfies

(3a)

(3b)

(3c)

(3d)

(3e)

satisfies

(4)

(5)

When , there is the relative minimum of and is the absolute minimum of When , there is no relative minimum of and is still the minimum of When the vacuum expectation values of all Higgs fields are zero so that the highest symmetry realized and inflation will occurs ¹.

Thus, when but is very small, In this case, the masses of all s-fermions, v-fermions, s-gauge particles and v-gauge particles are zero.

(6)

(7)

(8)

(9)

It is easily seen from (6)-(9), that the masses of the Higgs particle are very small and and the masses of all gauge particles and all fermions are equal to zero when Consequently, s-particles and v-particles can significantly transform from one to other, so that The higher the temperature is, the faster the transformation is. When , the transformation of s-particles and v-particles from one to other may be ignored.

When although s-particles and v-particles can significantly transform from one to other, the s-total mass is still more than the v-total mass inside the black hole. This is because inside and outside the s-black, and there is the repulsive force on the v-particles coming from s-matter so that the v-particles can easily get out of the black hole. In contrast with the v-particles, it is hard that v-particles to fly out of a black hole the, because and the strong gravitation coming from s-black hole on the s-particles, here and are the expectation values of the s-Higgs fields outside and inside the s-black hole, , respectively.

Let there be a s-black hole in s-cosmic island, and the distribution of s-matter around be determined. is spherical and the density of s-matter is uniform in the and angular directions. It is obvious that the temperature in the centre of the hole is the highest in the s-black hole. The zone in which the temperature is defined as the central zone. Let the radius of the central zone be the total s-mass of the s-black hole be , the s-mass and the v-mass in the central zone be and respectively. The s-black hole absorbs the s-matter around it so that its mass to increase. will decrease because v-particles can easily escape from the s-black hole. and will change because s-particles and v-particles can transform from one to other. Taking these factors into account, the changes of and can be obtained,

(11)

(12)

Here is the effective gravitational mass, the coefficient is determined by the gravitational constant and the distribution of s-matter encircling so that is a function of time. is such s-matter which transform into v-matter; is such v-matter which transform into s-matter. is the repulsive force coming from the repulsive force causes the v-particles in the central zone to fly out of the black hole. The coefficients will increase as and the density ; will all increase as and the density . When the heat balances, and monotonously increase as and monotonously increase as increase. The parameter is a is positive and undetermined. Here the transformation of s-particles and v-particles from one to other outside the central zone is ignored.

When and the distribution of s-matter encircling is determined, the transformation of s-particles and v-particles from one to other may be ignored, so that may be ignored. Thus, (10) is simplified as so that monotonously increases. and will increase as increases.

According to the conventional theory about black holes, the mass of a black hole monotonously increases always. In contrast with the conventional theory, according to the model ¹, when is large enough so that , s-particles and v-particles can conspicuously transform from one to other. In the case, the change of and are described by (10)-(11). When the distribution of s-matter is determined, is determined. According to the definition of stable black holes, In order to , it is necessary . In this case, (10)-(11) is simplified as

(12)

(13)

From (12) - (13) we have

(14)

When and let for simplicity, we have which satisfies (12) -(13) when is called the mass of a stable s-black hole and is denoted by .When the heat balances, and determine , and and from this determine and When is not a stable black hole.

It is possible that , and is large enough, or is small, so that . can decrease because s-matter encircling decreases. Higher is, larger is. When conspicuously transforms into so that decreases. When is no longer a black hole. On the other hand, there is the strong repulsive force between and so that the gravitation of the black hole on the s-particles. Thus, emits a lot of s-particles so that becomes a quasar. Simply put, this is what happens when a black hole loses its gravitational constraints.

It is possible that is small enough so that the gravitation of the black hole is small enough and . In the case, becomes an orphan quasar.

When and the s-matter around is plentiful enough, , is possible. In the case, will increase so that and increase. Consequently becomes a huge black hole.

When continues to increase so that the s-particles and the v-particles are perfectly symmetrical, i.e. the highest symmetry will be restored, and inflation will occurs. Consequently, the s-black hole will transform into a v-cosmic or s-cosmic island. This is because the v-particles can easily escape out from the s-black hole so that Thus, the state with the highest potential and symmetry will possibly jump to the state in v-breaking and forms a small v-cosmic island in the s-cosmic island, even though is inside s-cosmic island. The small v-cosmic island does not attract s-celestial bodies, but can emit a lot of s-particles. Consequently, the small v-cosmic island seems to be an huge orphan quasar. When with the highest temperature jumps to the state with s-breaking, becomes a s-cosmic island which seems to be an huge quasar or white hole.

The discussion above is only qualitative. These coefficients , and require further study and astronomical observations to determine.

3. Conclusions

Based on the cosmological model without singularity, this paper demonstrates that there is an upper limit of the mass of stable black holes. An unstable black hole could become quasar or an orphan quasar. When the mass of an unstable black hole increases to large enough so that the temperature of its central zone comes up to the highest temperature, the highest symmetry will be realized and then inflation will occur. Consequently, the black hole will transform into a new v-cosmic or a s-cosmic island inside the original s-cosmic island. This v-cosmic island seems to be a huge orphan quasar, and this s-cosmic island seems to be a huge quasar or white hole. This is because there is only a repulsive force between the s-particles and v-particles, and when the temperature is greater than the critical temperature, the s-particles and the v-particles must significantly transform from one to other. The more massive the black hole is, and the higher the density and temperature of the black hole are, the faster the transformation is. It is seen that the critical temperature and the highest temperature of the universe determine the mass of a stable black hole and evolution of an unstable black holes.

Acknowledgements

MMy sincere thanks to professor Zhanyue Zhao for his helpful discussion and strong support!

This work is supported by National Natural Science foundation of China 11075064. We are very grateful for the fund’s support.

References