part0016

6.Hidden in plain sight2: From white dwarfs to Black Holes

A precise and consistent quantum theory of gravity has not yet been proved, not even by the self proclaimed geniuses of this time. We are aware and satisfied that classical General Relativity is the most precise description of gravity due to its predictable nature. The left hand side of Einstein field equation represents the metric of space time curvature while the right hand side represents the matter - energy content of the classical matter fields of pressure and energy density. It is known that quantum mechanics plays an important role in the behaviour of the matter fields but has no place in the Einsteins field equations.According to S.W.Hawking (1975), one therefore has a problem of defining a consistent scheme in which the space time metric is treated classically but is coupled to the matter fields which are treated quantum mechanically.

In this book we propose that, in order to estimate stellar parameters to a high degree of accuracy for both microscopic and macroscopic descriptions of white dwarfs and black holes one has to treat the right hand side of Einstein field equation quantum mechanically as,

, where

is the total pressure,

is the gravitational force,

is the electric force , G is the gravitational constant, c is the constant speed of light,

is the reduced planck constant and

is the mass of an Hydrogen atom.

Proof of the Chandrasker Mass Limit and the Lowest Principal Quantum Number from a New Approach to Quantum Gravity

Although in the Bohr theory of an hydogen atom orbit quantization doesnot permit a lower orbit than the bohr radius of

, this section sets out to show that this is not the case with white dwarfs due to the state of a hydrogen atom under high pressure.

We know from the Chandrasker derivations that, the equation governing the hydrostatic equilbrium of a star is given by

o r

Where P denotes the total pressure,

is density, and M(r) is the mass interior to a sphere of radius r.

We could however write the same equation in a different form given by

, where

(24)

The total Gravitational Binding Energy of a Star

The electric potential energy

as we know it can be can be deduced from (24) and is given by,

where

is the gravitational potential binding energy given by

Using the principle of energy equipartition, we assume that the electric binding energy is of order the discrete energy of an hydrogen atom from Bohrs theory as,

where,

is the Coulomb constant, e is the charge on an electron and n is the principal quantum number.

From the above assumptions the gravitational binding energy is given as,

(25)

where

is the fine structure constant

In Table 1 we list the values of

for several values of n-the principal quantum number. From this table it follows in particular, that the higher the principal quantum number, the higher the gravitational binding energy of a star.

The total gravitational binding energy of a star

n(Principal quantum number)	(Joules)	Remarks
0.003212
0.0345		White Dwarf
1

What do we conclude from the foregoing calculation? We conclude that equation (25) is at the base of the equilbrium of actual stars in relation to the energy state and binding energy of the Hydrogen atom. It differs from the Chandrasker calculation by the introduction of a natural fine structure constant, providing the energy of proper magnitude for the measurement of stellar energies and therefore proving to be a better theory for stellar structure.This could be elaborated in detail by flowers original words,

"The Black-dwarf material is best likened to a single gigantic molecule in its lowest quantum state. On the Fermi-Dirac statistics, its high density can be achieved in one and only one way, in virtue of a correspondingly great energy content. But this energy can no more be expended in radiation than the energy of a normal atom or molecule. The only difference between Black-dwarf matter and a normal molecule is that the molecule can exist in afree state while the black dwarf matter can only so exist under high external pressure.

The Theory of White -Dwarf Stars and Black Holes; The Limiting Mass at the Lowest Principal Quantum Number

The gravitational energy is known to be of order

, M being the mass of a star. Then equating this to equation (25) we obtain the radius of a star as,

(26)

while the above equation states that the radius is proportional to the square of it's mass, the Chandrasker analysis is in disagreement, stating that r is inversely proportional to the cube root of the mass.

But at a point where r equation (26) approaches the schwarzichilds radius

We obtain an upper limit to the mass of,

(27)

Now consider equating the original solution of Chandrasker mass limit to our newly developed formula (27) , we have

, is a constant connected with the solution to the lane-Emden equation, and

, average molecular weight per electron,then

In the table below we list the values of M and r for several values of n-the principal quantum number, including the one calculated above.

The Mass limit and radius limit of a star

n(Principal quantum number)	M (Kilograms)	r (meters)	Remarks
		34.153
0.0345		3944.601	Chandrasekar mass limit
1			Maximum mass of a white dwarf

What do we conclude from the foregoing calculation? We conclude that the formation of a white dwarf star or any other stellar structure will never exceed the Schwarzichild's radius of 34.153m, this will only happen at the most lowest quantum principal number of

. For example, at the principal quantum number the size of the fine structure constant

, the mass obtained will be of

and r=176.443m. Therefore under high external pressure the minimum mass of a last star that is formed is of order

and this only occurs at r=34.153m under the lowest energy state below the known Bohrs radius of

What is Wrong With Hawking Temperature

In his paper "Particle creation by Black holes" Hawking pointed out that "In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature

". However this is not the case when the assumptions given in the first sections of this book are taken into account. For example, we know that, the electric potential energy is given by,

But treating the particles in the process General relativisticaly (at the Schwarzichild radius), the gravitational potential energy will be of order

, giving the electric energy as,

Now the thermal energy is given by

, where k is the Boltzmann constant.

By the principal of Equipartition

(28)

Note: in a limit where

is the planck mass

, equation 28 above for the temperature of a black hole reduces to the hawking temperature formula

For conditions at the centre of the Sun,

which is in disagreement with the Hawking temperature of

. This is left for the reader to analyse.

Entropy of a Black Hole

For derivations which i will not show here, I am led to the total energy of a Black Hole given by,

where, A is the surface area of the event horizon

But since the entropy is energy per unit temperature,

Remember that temperature is given by equation(28),

Then the entropy will be given by,

This is in agreement with the Bekenstein-Hawking area entopy law

On the Development of a Quantum Gravity-Hydrostatic Equation and its Implication to Physics-Minimum Black hole mass

It is known that the equation governing the hydrostatic equilibrium of a star is given by,

(29)

Where P denotes the total pressure,

is density, and M(r) is the mass interior to a sphere of radius r.

what if we rewrite the above formula in a form given by,

(30)

Where

is the gravitational force,

is the electric force, c is the speed of light,

is the reduced Planck constant and G is the gravitational constant. Let the pressure be,

, this means that pressure is dependent on the product of the gravitational and electric forces in a quantum-relativistic realm. Therefore in simple terms, we can write (30) in its simplest form as,

and to include the density, we have

where

is schwarzichild's radius. thus at

, the star will form a black hole.

To differ from (29) we have formulated one of the first quantum gravity -hydrostatic equation. From (30) we can write the electric potential energy as,

where,

is the gravitational potential energy

at a point where the potential gravitational energy is in equilibrium with the potential electric energy the total energy is that of the Planck energy by,