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OVERGENERALIZED THEORY

Feb. 13th, 2008 | 03:47 pm

 

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Makerere University, Faculty of technology, P.O.Box 7062, Kampala, UGANDA
 
Email balungifrancis@yahoo.com , bfrancis@tech.mak.ac.ug


  1. INTRODUCTION
 
In this research I try to present what I have come to call the theory of everything. I deduce some of the fundamental laws and theories of physics from the one equation, and finally give out the new forces of physics not known in nature.
 
It all starts out when carrying out a calculation that combines quantum mechanics with general relativity, which I come out with a precise and general equation that combines gravity with electromagnetism such that when the two forces in question cancels I remain with the coupling constant.
 
Consider the equation given by,
 
[8πG/ c4][GM2][Ee] = n2ћc                                                  1
 
Where
G – gravitational constant ,c– speed of light ,M –mass, ћ-Dirac constant, n- quantum number, E- electric field, e – elementary charge.
 
As you can see Ee is the electromagnetic force, and 8πG/ c4 is the gravitational force at the swcharzichild’s radius.
          
 The equation given above gives out the predictions of general relativity(Black holes), as follows,
 

2. THE TEMPERATURE OF A BLACK HOLE

 
On arranging equation one to get the random transilational kinetic energy, we obtgain
 
[GM/ c2]Ee = n2 c3 ħ /8π GM =kT
 
Where k is the boltzmann’s constant and T is the temperature.Hence at n=1,
 
T = c3 ħ /8π GMk                                                                      2
 
3. TIME TAKEN BY A BLACK HOLE TO EVAPORATE
 
On dividing through eqn1 by the momentum Mc we obtain, the time t given by
 
t= mc/Ee = 8πG2M3/n2 ħ c4 
 
Such that when n=0.03953
 
t= 5120πG2mo3/ ħ c4                                                                       3
 
4. ENTROPY OF A BLACK HOLE

Squaring both sides of equation 1 and arranging we generate the intensity as
 
W/tA = E2e2/2nh = n3 c10 ħ /256π3 G4M4                                        4
 
Where A is the area on which the radiations fall, W is energy, and t is time
But entropy is energy divided by temperature Eq so then
 
W/T = S = (n3 c10 ħ /256π3 G4M4 )(tA/T)
 
Since t is known from Eq3 and T from Eq2 then at n= π
 
S = Akc3 /4G ħ
 
5. THERMAL PROPERTIES OF SOLIDS (Wiedemann Fanz law)        
 
From the intensity equation4,

E2e2/2nh =  c10 ħ /256π3 G4M4
 
Arranging the above equation to introduce in the transilational kinetic energy obtained above [kT= c3 ħ /8π GM], we have
 
          π M2G2E2/3c 4 =( π2/3e 2)(c3 ħ /8π GM) 2=( π2/3e 2)T2k2
 
Dividing both sides by T we obtain on the left hand side of the equation the ratio of thermal conductivity K to electric conductivity δ as
 
K/ δ =( π2/3)(k/e)2 T
 
This is true
 
 6. STEFAN'S RADIATION LAW

Still from the intensity Eqn4 we can arrange the expreession on the left hand side of the equation, to read as
 
W/tA = E2e2/2nh = (1.875n3/ π2)( π2/60h3c 2)(c3 ħ /8π GM) 4
 
This is the same as Eqn4 only that it is arranged to predict something
 
But  n=1 and (c3 ħ /8π GM ) =kT, hence the rate at which energy is radiated is given by
 
 
W/t = A(1.875/ π2)( π2/60h3c 2)(Tk) 4  = 0.19σAT4  
 
 
Where σ = π2/60h3c 2  is the Stefan boltzmann’s constant
 
This is Stefan’s radiation law.
 
7. CONCLUSION

As you can see from the deduction, the theory developed can't fail to deduce a new theory or law so long as it exists in nature. This theory also has the capability of deducing theories describing both small and large objects hence it becomes a ultilmate theory of matter and nature as well. In the coming paper soon to be published we shall see how it deduces the two new forces forming a unified field theory and hence the final theory.   

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Another form of general relativity and its new predictions

Feb. 5th, 2008 | 11:53 am

Makerere University, Faculty of technology, P.O.Box 7062, Kampala, UGANDA
 
 
 
Abstract
 
This study addresses the issue of analyzing galactic features and the big bang whose portfolio is related to the age and density of the universe. The age of the universe and its features that is density, pressure, temperature and its epoch are all formulated from first, considering the curved nature of space and time. And second modifying the first law of thermodynamics to include the features describing the large objects and small particles, It is then seen that the area occupied by heavenly bodies in space divided by a dimensionless constant determines the tidal force acting on that body and that Charles’s and Boyles law of an ideal gas when applied to black holes gives its entropy with a gravitational coupling constant determining the strength of the gravitational field.
 
 
 

1.      Introduction
 
The development of general relativity followed a publication of acceleration under special relativity in 1907 by Albert Einstein. In his article he argued that any mass will "Distort" the region of space around it so that all freely moving objects will follow the same curved paths curving toward the mass producing the distortions. The questions raised by the principle of equivalence and general relativity are intimately related to the questions of the origin, size, and structure of the universe. Is the universe infinite or finite? How old is our solar system and galaxy? How were they formed? How many other galaxies are there and how are they distributed? Where did they come from? What was the universe like before these galaxies were formed? The field of physics that deals with these questions is called cosmology, a very fast moving field.
 In 1916, Schwarzschild found a solution to the Einstein field equations, laying the groundwork for the description of gravitational collapse and, eventually, black holes. In 1917, Einstein tried to describe a static universe, where he added cosmological constant to his original field equations for that purpose. With Hubble’s observations in 1929, on the movement of galaxies which predicted an expanding universe, Lemaître formulated the earliest version of the big bang models.
General relativity uses a complex mathematical equation that makes it so hard for people to master the theory. This paper gives out a simple and accurate mathematical formulation of space and time is some what a similar fashion to that of general relativity. This paper deviates from the theory in that for it takes into account the description of space and time for both small (quantum effects) and large particles. This paper also explains the features of cosmology (black holes) and the Big bang (the earliest period of the universe).
Finally, there have been various attempts through the years to find modifications to general relativity. The most famous of these are the Brans-Dicke theory and Rosen's bimetric theory. Both of these proposed changes to the field equations, and both suffer from these changes permitting the presence of bipolar gravitational radiation. As a result, Rosen's original theory has been refuted by observations of binary pulsars. As for Brans-Dicke the amount by which it can differ from general relativity has been severely constrained by these observations. It is generally held that one of the most important unsolved problems in modern physics is the problem of obtaining the true quantum theory of gravitation, that is, the theory chosen by nature, one that will work at all energies. Discarded attempts at obtaining such theories include supergravity, a field theory which unifies general relativity with supersymmetry. In the second superstring revolution, supergravity has come back into fashion, with its as yet undefined quantum completion rebranded with a new name: M-theory.
 
2.      Materials and methods
 
 
a)      The movement of a particle in a curved path and their associated forces
 
Since gravity increases in inverse proportion to volume, any quantity of matter that is sufficiently compressed will become a black hole. When a large enough amount of mass is present within a sufficiently small region of space, all paths through space are warped inwards towards the center of the volume, forcing all matter and radiation to fall inward. I formulated a new solution to Einstein field equation which describes black holes, and is given by;
 
 
Volume = ABRd = (1/ Fe)(h2/m)                                                                              [1]
 
Where AB is the area of the small region of space, Fe is the tidal force (An object in any very strong gravitational field feels a tidal force stretching it in the direction of the object generating the gravitational field.) Near black holes, the tidal force is expected to be strong enough to deform any object falling into it, even atoms or composite nucleons; this is called spaghettification. The strength of the tidal force depends on how gravitational attraction changes with distance, rather than on the absolute force being felt. This means that small black holes cause spaghettification while infalling objects are still outside their event horizons, whereas objects falling into large, supermassive black holes may not be deformed or otherwise feel excessively large forces before passing the event horizon.
 
 Rd = Ae2/ Rs3 is the radius of that region of space, Ae= hc/ Fe is the area occupied by each particle experiencing the tidal force ( in other words area of the object). Rs= Gm/c2  is Schwarzschild radius, It is the radius for a given mass where, if that mass could be compressed to fit within that radius, no known force or degeneracy pressure could stop it from continuing to collapse into a gravitational singularity, h is the Planck constant, m is the mass of the object, c is the speed of light and G is the gravitational constant. It should be noted that as the volume Rs3increases the radius Rd reduces and as it reduces the radius increases. Therefore Rd is the radius of a region of space that is changed when ever the volume occupied by a compressed mass in that region changes.
 
Therefore the area is given by,
 
AB= ( Rs3/ Ae2Fe)(h2/m)=G3m2 Fe /c8                                                                                                      [2]
 
The equation obtained shows how the force depends on the area where the mass is concentrated. The above force differs from Newton’s gravitational law in that it is directly proportional to the area but inversely proportional to the square of the mass of the body. Hence Fe = AB c8/ G3m2  = N AB/ m2           where N= c8/ G2                                                                                                                                                                                              
 
 
  1. Results
 
The area gives the forces
 
Since in Eq2 the force is related to the area we can then use it to obtain the force on any object occupying any given area. If we take a square of the Schwarzschild radius to be the area where if a mass could be compressed to fit within that area, no known force or degeneracy pressure could stop it from continuing to collapse into a gravitational singularity, then the following is obtained
 
For a black hole of area Rs2 , the force is Fe = c4/ G
 
For two particles separated by a distance R and within an area Rs4/ r2, Newton’s gravitational force is Fe = Gm2 / r2                                                                                                              
 
For particles probing the big bang , the areas are Rs2/ αs and Rs2/ αg (where αg = Gm2/ ћc is the gravitational coupling constant and αs= ke2/ ћc is the fine structure constant, k is coulomb constant and e is elementary charge) the following forces are obtained respectively ,
 
 FG= Eo2 /ke2 and FE = Eo2 /Gm2
 
the energy Eo =√ћc5/G where ћ is the Dirac constant h/2π. This is the energy describing the scale of the energy that the universe had in its early formation. Therefore substituting the value of FE in Eq2 we obtain AB1 = ћG/c3 =2.60624×10-70m2And for FG = Fe = Eo2 /ke2 we obtain AB2= AB1 (Gm2/ ke2). This means that the force FE only becomes comparable to Fe at the Planck length scale. And FG = Fe doesn’t achieve the correct scale when the forces are compared, it approaches the scale but with an effect Gm2/ ke2 which indicates that the gravitational force cannot be compared to the electromagnetic force.                                                                                          
This therefore states that FG is the gravitational force and FE is the electromagnetic force at the big bang scale.
 
The cosmological pressure and temperature
 
From Eq2 the pressure P is simply a tidal force on an object per unit area occupied by the object in a region of space, hence,
 
P = Fe / AB= c8/ G3m2                                                                                                              [3]
 
And finally the temperature from Eq1 is
 
T = (RdFe /k) = (1/ AB)(h2/mk)                                                              [4]
 
 
Where k is gas constant per molecule in joules per Kelvin
 
 
 
b)      The entropy of a black hole and the first law of thermodynamics
 
Entropy of a black hole
 
Keeping the volume constant, the pressure of a gas is directly proportional to its absolute temperature that is P α T, hence from Eq3 and Eq4
 
P/T = (β2 AB / Rs)(k/ h2 )                                                                                         [5]
 
Where β is the rate of change of mass equal to c3/G
 
For an ideal gas, keeping the temperature constant, the volume of a gas will vary inversely proportional to its pressure that is V α 1/P, hence
 
P R3s /T = (β2 AB R2s)(k/ h2 )                                                                                 [6]
 
The above equation is the entropy of a black hole derived from the properties of a gas and therefore it can be expressed as 
 
Entropy = P R3s /T = (kAB /4AB1)(2β R2s /π h ) = (kAB /4AB1)αg                            [7]
 
Where αg =(2β R2s /π h ) is the gravitational coupling constant.
 
 
The first law of thermodynamics
 
The sum of the kinetic energy and potential energy of all the individual particles making up the system is the internal energy given by
U = ∆Q + ∆W
Where ∆Q is the heat flow into the system and ∆W is the work done by the system. Basing on the results obtained
 
∆Q = √ћc5/G = c√ βћ = Eo and ∆W= -(β2 AB R2s)(kT/ h2 ) = - P R3s.
 
Hence the internal energy is formulated as
 
U = (βћ/ Eo)(c2- R2sEo[kT / h2 ]  )                                                                              [8]
 
Letting [kT / h2 ]  = 4 π 2 / ћ τ where τ is the time
 
We obtain
 
U = (βћ/ Eo)(c2- R2sEo[4 π 2 / ћ τ]  )                                                                             [9]
 
as a result when U =0 , Rs = 1.61414×10-35 m , and Eo= 1.9605×109 J. the time
τ = 2.1238×10-42s, which is the earliest period of the universe is obtained.
 
Still from Eq8 we find that the quantity (βћ/ Eo) represents mass which is given by Mp = 2.1765 × 10-8 kg. Multiplying this mass throughout we generate a principle equation
 
U = Mpc2 - Mp (Rs2 [EokT / h2 ] )                                                                                    [10]
 
 
This equation gives us a mechanism of combining the laws governing small particles (quantum mechanics) with those governing heavenly bodies (General relativity). The appearance of the Schwarzschild radius Rs which explains galactic bodies, the appearance of the random energy kT that describes small particles, the appearance of the Planck mass Mp and energy Eo which describe the Planck epoch in the early universe are all evidence of the generalized formulation of the combined theory of quantum and gravity hence obtaining a quantum gravity theory of nature.
 
From which we get the speed of particles in the early universe given by 
 
υ =√ (Rs2 [EokT / h2])
 
For Eo= kT the speed is got as υ = RsEo / h = 0.4773×108 m/s which is smaller than the speed of light by only (υ/c = 0.1591)
 
  1. Discussion
 
From the results obtained it is studied that the forces acting on heavenly objects depend on the areas of space in which these objects occupy. These areas are also related to the “Schwarzschild area” Rs2. any area in space will have this effect and it will only change  when Rs2 is divided by a dimensionless constants, for example,  the Newtonian gravity will depend on the dimensionless constant Rs2 /r2 , the cosmological features depend on a unity dimensionless constant and  For forces describing the big bang the dimensionless constants will be the coupling constants determining the strength of the gravitational and electromagnetic fields. When all these dimensionless constants are equal to unity it means that the area occupied by one object in the universe corresponds directly to that occupied by other objects and that the effect of the force to one object is the same to all other objects, therefore implying that the forces will then be unified into one fundamental force.
 
Forces probing the big bang are directly related to the square of the Planck energy, and it is known that at the Planck scale the description of subatomic particle interactions in terms of quantum field theory breaks down, but since both forces have energies at the Planck scale it means that the two are comparable to the other forces and when they are equated the result is a dimensionless constant which is unity Gm2/ ke2 =1.
 
The thermodynamic laws that describe gases here on earth are seen to be the same laws that govern the particles found in our galaxy. It is seen that Boyle’s and Charles laws can be applied to heavenly bodies, the result of this is that the entropy of these bodies is directly proportional to the product of the area of the event horizon of the body and the gravitational constant that determines the strength of the gravitational force.
 
The earliest period or time line of the big bang is studied. The random energy of particles kT forming matter during that time was in equilibrium with the energy of the photons, the time when this happened was 2.1238×10-42s, every particle during this time moved at a speed of light c=3×108 m/s, particles moving at a speed closer to that of light where produced when kT was equal to Eo= 1.9605×109 J and when calculated had a speed υ = RsEo / h = 0.4773×108 m/s which varies directly with the Schwarzschild radius.
 
  1. Conclusion
 
 Successfully I have analyzed a method of combining elementary particle physics with astrophysics. It is now possible to apply the laws governing small particles in the description of the nature of large particles hence the possibility of combining quantum mechanics with general relativity has been given out in detail. The equation for the first law of thermodynamics has also been generalized to U = Mpc2 - Mp (Rs2 [EokT / h2 ] ) = Mpc2 - Mp (R2sEo[4 π 2 / ћ τ])    where τ defines the life time and Mp = 2.1765 × 10-8 Kg is the Planck mass. These results therefore show a clear future for the formulation of the unified law of all of physics.
 
 
 
 
 
 
  1. References
 
    1. Abhay Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett., 57, 2244-2247, 1986
    2. Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593-622, e-print available as gr-qc/9411005
    3.  Castelvecchi, Davide; Valerie Jamieson (August 12 2006). "You are made of space-time". New Scientist (2564
    4. Researchers Look Beyond the Birth of the Universe", Eberly College of Science, 12 May 2006. 
    5. Smolin, Lee. "The case for background independence". hep-th/0507235
    6. http://en.wikipedia.org/wiki/Loop_quantum_gravity
    7. C.L.Chin and C.R.Westgate (Editors), The Hall Effect and Its Applications,” Plenum Press,New York, 1979, p.535.
    8. Eddington, A. S., The Internal Constitution of the Stars (Cambridge University Press, England,1926), p. 16
    9. E. Kolb and M. Turner, The Early Universe (Addison-Wesley, Reading, MA,1990).
    10.  W. Garretson and E. Carlson, Phys. Lett. B 315, 232(1993); H. Goldberg, hep-ph/0003197
    11. Eddington, A. S., The Internal Constitution of the Stars (Cambridge University Press, England,1926), p. 16
 
 
 

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THE SIMPLIFICATION OF THE NATURE AND STRUCTURE OF PARTICLE PHYSICS

Feb. 5th, 2008 | 11:48 am

 
BY BALUNGI FRANCIS
Makerere University, Faculty of technology, P.O.Box 7062, Kampala, UGANDA
 
                                                           
Abstract
 
The uniformity of all interactions is here by explained by the method of comparing forces and examining the coupling constants that determine the strength of the interaction. The physical constants such as the speed of light among others are normalized to unity and the Planck scale at which quantum mechanics is comparable to gravity studied. Mass (m) corrections are also examined under the method of comparing forces and thereafter described by an expansion in powers of m at which the fine structure constant power series equation is deduced. The nature of the symmetry that exists at the Planck scale is therefore deduced through the methods of generating the relationship between elementary particle physics and astrophysics.
 
Mass-Energy; 1kg is equivalent to 5.6096×1026GeV, Time; 1year = 3.156 ×107s
KEYWORDS: super symmetry, grand unified theories, quantum mechanics, interactions, and Planck scale.
 
 
 
 
 

1.0. Introduction
 
Physicists have argued out that the more elegant and symmetrical the theory is, the more it is beautiful. The elegancy of any physical theory is suspected at a level to which it holds well with other theories , that is ,the capability of the theory to conform with the well known laws of nature at all levels.
In this paper we examine the mechanism through which quantum mechanics becomes comparable with gravity and the scale to which this occurs. At the Planck scale all interactions (the weak interaction, strong interaction and electromagnetism) are assumed to merge into a single interaction that alone occurs at very high energies of about 1TeV. The equations that do describe this phenomenon are not yet found and therefore require one’s deep effort to capture the reality of this entire puzzle.
 
To capture interest in these interactions we need to know first, their strength and second the range in which they occur. The strength defines the coupling constants and the range defines the attractions, on the other hand the coupling constant determines the strength of any interaction and therefore is a number in a sense that it is a dimensionless constant. A coupling constant is a very important quantity in dynamics, for example, in the motion of a large lump of magnetized iron, the magnetic forces are more important than the gravitational forces because of the relative magnitudes of the coupling constants.
The Standard Model is a theory of three fundamental forces — electromagnetism, weak interactions and strong interactions; however, these three forces are not tied together. Howard Georgi and Sheldon Glashow discovered that the Standard Model particles can arise from a single interaction, known as a grand unified theory. Grand unified theories predict relationships between otherwise unrelated constants of nature in the Standard Model. Gauge coupling unification is the prediction from grand unified theories for the relative strengths of the electromagnetic, weak and strong forces and this prediction was verified at LEP in 1991 for supersymmetric theories.
In particle physics, supersymmetry (often abbreviated SUSY) is a novel symmetry that relates elementary particles of one spin to another particle that differs by half a unit of spin and are known as superpartners. Since the particles of the Standard Model do not have this property, supersymmetry must be a broken symmetry allowing the 'sparticles' to be heavy.
One of the main motivations for SUSY comes from the quadratically divergent contributions to the Higgs mass squared. The quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and unless there is an accidental cancellation, the natural size of the Higgs mass is the highest scale possible. This problem is known as the hierarchy problem. Supersymmetry reduces the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions. If supersymmetry is restored at the weak scale, then the Higgs mass is related to supersymmetry breaking which can be induced from small non-perturbative effects explaining the vastly different scales in the weak interactions and gravitational interactions. The failure of experiments to discover either supersymmetric partners or extra spatial dimensions, as of 2006, has encouraged loop quantum gravity researchers.
 
 
 
1.1. Materials and methods
 
1.2. The determination of the strength of the forces
 
We assume a model that explains everything on the length scales, the best scale so far we are familiar with is the Planck length scale, however in this model we don’t associate our selves in knowing this scale and there fore develop new scales that alone are combined together to lead to some observable phenomenon describing the forces involved in the interactions. The equation describing the model is developed and given by;
 
( v2/c2 + n2βQo )      =   8πβgEo                                                                                 (1)
 
 
Where βQo is a length ratio given by lQ/lo, in this case  lQ =ħc/W,  ħ is Dirac constant, c is the speed of light and W is the energy.Also  βgEo = lgE/ lo where lgE= 8πGMgE2/W, G is the universal gravitational constant , MgE is the mass of a particle in the combined fields given by PgE /c where PgE =Gm2ke2/R2c2 ,is the momentum for an elementary particle of mass m and an elementary charge e , k is the coulomb constant and R is the distance between any two particles. The equation here addresses the problems in form of length scales simply because it is at these scales that quantum mechanics seem to be comparable to gravity. The momentum PgE  is a momentum of a particle experiencing the strength of the electromagnetic fields and gravity. The strength is determined by a very small coupling constant as we shall see later. The smaller the distance between elementary particles, the higher the momentum and vice versa is true.
 
The exchange of photons between an electron and a proton in an atom is explained by Quantum Electrodynamics (QED), with a coupling constant determining the strength of the electromagnetic force. The equation of the interaction responsible for QED on the length scale, which is the Compton length, is given by the equation
 
 
∑ψ2fiti     =  2πβRCE                                                                                                                                           (2)
 
 
The expression ∑ψ2fiti  is the force changer where ψfi = fiR2/ke2 and t={fi2ke2/R2}/Fn3,
βRCE ,remains a constant given by lc / lRE ( lc is the Compton length ħ/mc and lRE=ke2/mR2c2 ).
 
On multiplying both sides of Eqn1 by a quantity ∑ψ2fiti we obtain,
 
(v2/c2 + n2βQo ) ∑ψ2fiti     =  8πβgEo∑ψ2fiti                                        
 
We then examine the condition for which βQo will be a maximum and minimum. It is found out from relativity that βQo is maximum when the lorentz factor γ= (1-v2/c2)-1/2 is very small that is,
 
γ= 1/n√ βQo or when the velocity v = c √( ∑ψ2fiti  - n 2βQO)
 
We hence obtain a general interaction equation as,
 
∑fy3ψ2fiti     =  2πβRCE∑Fn3 /ξβgEo  , n= 1,2,3……..                                              (3)
 
The following conditions are then taken into account
 
 
1)   For   lo = lx = mc2/Fp,     βgEo=βgEx =lgE/lx .
 
2)   For   lo = lc ,    βgEo=βgEQ =lgE/lc , and
 
3)   For    lo = ls = Gm/c2,    βgEo=βgEs =lgE/ls , which gives
 
 
∑fy3ψ2fiti     = F13 + F23+ F33 =  2πβRCE ( Fp3 /8πβgEx + Fp3 / 256π 3βgEQ+ 2FB3 / π βgEo )  (4)
 
Where Fp= c4/G is the Planck unit force and FB3= m 2c 3/ ħ  is the force required for strong and weak interactions to take place.
 
Again setting a condition,
 
For for  lo =lz =ke2 /mc2,     βgEo=βgEz =lgE/lz .
 
∑fy3ψ2fiti   = F43 =  2πβRCE ( Fz3 /32π3βgEx)
 
Where Fz3= m 2c 4/ke 2 ,
 
Also for lo =lN = ħ2 m3G2/k3e6, βgEo=βgEN =lgE/lN , we obtain,
 
 
 
∑fy3ψ2fiti  = F53 =  2πβRCE ( Fz3 /2π3βgEN)                                                                 (5)
 
 
Measuring the value of the strong, weak and electromagnetic coupling constants gives us away through which we can determine supersymmetric levels. From supersymmetry and grand unification of elementary particles the couplings agree to 1%. The relationships of the sum of the cubes of the forces to each individual cube of the force, and that of the sum of the square of masses with each known mass squared casts much information about the masses and couplings of the supersymmetric particles as shown below, when Eqn4 is divided through respectively by the cubes of the forces F13 , F23 and F33 the following equations are obtained,
 
∑Fn3 /F13        = 1 +16 αg3 + 1/32π2αg                                                                                                                (6)                                                                         
 
 ∑Fn3 / F23    =1+32π2αg + 512 π2 αg4                                                                                                               (7)                                                                                   
 
∑Fn3 / F33     =1+ 1/6 αg3 +1/512 π2 αg4                                                                                                           (8)
 
 ∑Fn3 /F43        = 1 + β2 (4 π2 + 1/8αg) + 64 π2αs2αg                                                                                  (9)                                                                     
                                                                               
Where β = ke2/Gm2 is the ratio of the fine structure constant αs to the gravitational coupling constant αg, given respectively as αg = Gm2/ ħ c and αs =ke2/ ħ c .
 
 
Now equating   F4 = F5  , F5= F3   , F5= F1    we obtain; m1, m2 , m3and  m4   respectively,                                                                                                                              Adding the squares of the masses we obtain,
 
∑mn 2= m12+ m22+ m32+ m42                                                                                                                     (10)                                                                                                                           
 
Which  gives the sum per unit mass as,
 
∑mn 2/m12= 1 + 16π2αs4 + (8π/ αs) ½ + 4/(128 αs 4) 1/5                                           (11)
 
∑mn 2/m22= 1 + 1/16π2αs4 + (1/8π αs9) ½ + (1/4π2)(128 αs 24)                                              (12)
  
 
 The equations generated so far give a basis for the nature and type of supersymmetry exhibited by a particle experiencing forces at both the Planck and grand unified scales. It is thus shown here that the electromagnetic coupling constant is a result   of mathematically summing the squares of the masses generated and then dividing through by the square of the mass in the summation while the gravitational coupling constant is the result of summing the cubes of the forces and then dividing through by the cube of the force in the sum. This idea at its best is taken to be the basis for symmetric theories as we shall see in the results obtained.
 
 
1.3. Results
 
  • The unification of coupling calculations
 
At equal forces that is F1 = F2= F3 = Fp the massMp = ( ћc/8π G)1/2  = 2.1765 × 10-8 kg, is obtained  which is the Planck mass  for which the Schwarzschild radiusWhen Eq4 is divided through by F16and F26 we obtain equations of the form; is equal to the Compton length divided by π.
 ∑Fn3 /F16 =Ω/ Fp3                                                                                             (13)
 
 ∑Fn3 /F26   = €/ Fp3                                                                                                                                          (14)
 
Where,Ω=4m2/mp2+1/π +m8/32π2mp8,
        
            €= m6/8πmp6 +16πm4/mp4 +2m12/πmp12
 
The mass relations equations obtained above indicate the scale at which gravity may be strong and weak. Obtaining these results on the Planck force and mass scale is evidence for the existence of the theory of quantum gravity. The values Ω and € represent a series equation defined by increasing powers in the mass ratio (m/mp). The mass m is assigned to any particle and the mass mp is assigned to the Planck scale defining quantum gravity.
 
The unit of energy is MPc2; the unit of electric charge is √hc/k, where k is coulomb constant and so forth. On the other hand, one cannot form a pure number from these three physical constants. Thus one might hope that in a physical theory where ћ, c, and G were all profoundly incorporated, all physical quantities could be expressed in natural units as pure numbers. Within its domain, this paper has achieved it for example, imagining that there were just two quark species with vanishing masses. Then from the two integers 3 (colors) and 2 (flavors), ћ, and c (without mass parameters), the spectrum of hadrons with mass ratios and other properties close to those observed in reality, emerges by through calculation (Ω and €) as indicated from Eqn13 and Eq14 shown above. The overall unit of mass is indeterminate, but this ambiguity has no significance within the theory itself. The results obtained show an ideal Planckian theory that alone does not contain any pure numbers as parameters. Thus, for example, the value me/mp=10-22 of the electron mass in Planck units is obtained from a dynamical calculation. This ideal might be overly ambitious, yet it seems reasonable to hope that significant constraints among physical observables will emerge from the inner requirements of a quantum theory which consistently incorporates gravity. The model therefore provides; first, the unification of couplings calculation. second, it points to a symmetry breaking scale remarkably close to the Planck scale (though apparently smaller by 10-2 to 10-3), so there are pure numbers with much more 'reasonable' values than 10-22 to shoot for. Third, it shows quite concretely how very large scale factors can be controlled by modest ratios of coupling strength, due to the logarithmic nature of the running of couplings (so that 10-22 may not be so 'unreasonable' after all).
 
 
While the above result is based on the study of the strength of the gravitational force, we now look for ways in which we can examine the strength of the electromagnetic force depending on the mass. This is done by dividing the sum of the squares of the masses (Eqn10) by the fourth power of the individual masses hence,
 
∑mn 2/m24 =ω/ mG2                                                                                             (15)
 
 ∑mn 2/mE4 =λ/ mp2.                                                                                             (16)
 
Where ω=1/ 16π4αs9+1 /4π2αs5 +1/ (512π8 αs19 )1/2
         
 λ=128π3αs6+8192π5αs10 +128√π3αs11+128(π15αs26)1/5
 
 
mE[(1/ 8π Ke2)( ћ 3c 3/G)1/2] is the mass obtained when F43 = F33,and
MG = (Ke2 / G)1/2 is the mass obtained when the electromagnetic force is equal to the gravitational force.
 
It can now be theorized that the strength of the electromagnetic force is determined by Eqn15 and 16 at which a series power equation in the fine structure constant defined by ω and λ is a constant.
 
 
  • The length scales at which the masses predicted by the standard model survive
 
The mass of the Wand Z bosons (MW , MZ ),Higgs particle (MH ) and the mass scale at the grand unification (MGUT) are generated. We multiply a coupling constant µ with the force F33, of which we equate to F43 that is;
 
µ F33 = F43
 
From which
         
      µ =RB2/ Ro2,
 
Where Ro is the length scale determined experimentally and
RB =(8 πGke2/c4) ½=6.9101×10-36 m ,which is greater than the Planck length.
 
So the equation that produces the different masses at Ro will be given by the square of the mass as,
 
 M2 = mp2/8 πµ αs 2  
 
 Where αs = 1/137,is the electromagnetic coupling constant.
 
To obtain the masses, we need to find the length Ro , theoretically we develop the  lengths given by; 1.03741×10-39 m, 8.3182×10-54m, 9.4334×10-54m,     and 1.2345×10-53m.
Following the given lengths we respectively obtain the masses;
MGUT=1016GeV,  MW =80.18GeV , MZ =90.82GeV, and MH =119GeV respectively.
 
But at RB = Ro, the mass MB =6.661×1019GeV is obtained. And at Ro =2.529×10-37m, the Planck mass is obtained (that is M= mp).Therefore it is found out that the W and Z boson particles survive in length of 10-54m . The Higgs particle survives to a length greater than that of the W boson ≥10-53m. And finally particles at the grand unified scale will survive at 10-39 m.
 
 
  • The big bang acceleration and proton decay
 
For proton decay the intensity P is used such that  at Schwarzschild radius R and Planck   mass scale mp the life time of the proton as explained by SUSY is seen to agree so well with the equation T(time) =α2 m5p R/ 4096 π 3mk4 ħ such that at mk = 7.96×10-29GeV, T =1035yrs. We have obtained the lifetime of protons and the mass of a particle produced during the decay process. The mass of the particle obtained is very small and can therefore be taken to be a neutrino.
 
The force F3 can be expressed in the form,
 
F3 = a3(m35 /16π2mp2)1/3
 
Where a3 is the acceleration, this acceleration at a Planck scale will be given by
 
a3 = (c11 / ħ G2m) 1/3 = 2.4772×1052m/s2
 
 This is quite a very large acceleration and therefore defined as the acceleration of particles during the early formation of the universe.
 
 
 
1.4. Discussion
 
The results obtained describe super symmetry which is a theory required for the unification of everything we know about the physical world into a theory of everything. Significantly a larger enterprise of the theory is to produce a theory of quantum gravity which is required for the unification of general relativity with the standard model, which explains the other three basic forces in physics (electromagnetism, the strong interaction, and the weak interaction), and provides a palette of fundamental particles upon which all four forces act. Theoretically the results obtained (Eqn11and Eqn12) show a huge correction to the particles' masses, which without fine-tuning will make them much larger than they are in nature. The problem of the unification of the weak interactions, the strong interactions and electromagnetism is solved mathematically, through the comparisons of the cube of the forces in a ratio that generates the gravitational coupling constant power equation.
The Planck mass is the mass of a black hole whose Schwarzschild radiusF1 , F2 ,F3 and Fp ) are equal, the forces are then taken to be related to the origin of the universe simply because at those high energies that formed the dense soup of the universe the forces were equal and the masses probing the Planck mass scale that is black holes were produced, hence those four forces a significant in that they play a crucial role in the formation of black holes. The intensity P on the other hand explains a phenomenon that occurs at the cosmic scale, for example it explains the nature of Black holes and the age of the universe. The acceleration obtained is so large that it is the acceleration that the universe had at the instant after the big bang. Obtaining this acceleration is the possibility of studying the rate of expansion of the universe at large, the accelerating universe is therefore the observation that the universe appears to be expanding at an accelerated rate. multiplied by π equals its Compton wavelength. The radius of such a black hole is roughly the Planck length, which is believed to be the length scale at which both general relativity and quantum mechanics simultaneously become important. In accordance with the results obtained it is seen that the Planck mass is the mass at which the four forces (
At the Planck scale the descriptions of subatomic particle interactions in terms of quantum field theory breaks down. Also at the same scale, the strength of gravity is expected to become comparable to the other forces, mathematically all the fundamental forces are unified at that scale. The results obtained explain both the weak and strong interactions that at a length between 10-37m and10-35 the Planck scale is attained also at lengths10-39m , the grand unified scale becomes relevant , but for lengths10-53m and
10-54m, the standard model holds on well. We have therefore attained a unification that increases from about 10-59m (standard model) to10-35m (quantum gravity).The paper there fore gives out the relationship between elementary particle physics and astrophysics at a large scale.  
 
 
1.5. Conclusion
 
Basing on the results obtained, it is now clearly justified that gravity can be integrated with quantum mechanics at the Planck scale. And therefore the success of the “standard model” which includes both the electroweak theory and quantum chromodynamics can now be regarded as successful in providing accurate descriptions of the fundamental particles and their interactions.
 
 
1.6. Acknowledgement
 
I would like to thank the many people who contributed in various ways to improving this manuscript. Among these are all i@Mak members and Nantubwe Florence for her financial support.
Any errors that remain are fully my responsibility and therefore welcome corrections for future manuscripts.
 
 
1.7.  References
 
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6)      S. Weinberg, The Quantum Theory of Fields, VolumeIII: Supersymmetry (Cambridge University Press, Cambridge,UK, 2000).
 
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The earliest period of time in the history of the universe

Feb. 5th, 2008 | 11:44 am

 
BY BALUNGI FRANCIS    
Makerere University, Faculty of technology, P.O.Box 7062, Kampala, UGANDA
 
 
 

Abstract
 
A simple mathematical model is used to describe the relationship between the forces at the Planck epoch and the grand unification epoch. This model explains the problems concerned with what the accelerations and wave length of the particles would have been in the early universe. The forces are found to depend on the time at any given scale in the epoch and when these times are equal the fine structure constant is generated.
 
KEYWORDS: Planck epoch, Big bang, quantum gravity.
 
 
 

  1. Introduction
 
This paper represents a brief history of the universe, that is from its past to present. Observations have suggested that the universe began 13.7billion years ago. The universe was so hot with particles having a very high energy, in its earlier phase. The evolution then proceeded with this energy forming the first protons, electrons and neutrons, then nuclei and finally atoms. The microwave background was also emitted during the formation of the neutral hydrogen. Finally the structure of the universe was formed when matters aggregated into the first stars and quasars and on large scale clusters of galaxies and super clusters were formed.
In cosmology, the Planck epoch , named after Max Planck, is the earliest period of time in the history of the universe, from zero to approximately 10-43 seconds, it is at this time that quantum effects of gravity were significant. At this period approximately 1.37×1010 years ago all fundamental forces were unified. The state of the universe during the Planck epoch was unstable, tending to evolve and giving rise to the familiar manifestations of the fundamental forces through a process known as symmetry breaking. It is currently believed that the Planck epoch inaugurated the Grand unification epoch, and that symmetry breaking quickly led to the era of cosmic inflation, the Inflationary epoch, during which the universe greatly expanded in scale over a very short period of time.
The age of the universe, in Big Bang cosmology, refers to the time elapsed between the Big Bang and the present day. Current observations suggest that this is about 13.7 billion years, with an uncertainty of about +/-200 million years. Extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. This singularity signals the breakdown of general relativity. How closely we can extrapolate towards the singularity is debated—certainly not earlier than the Planck epoch. The early hot, dense phase is itself referred to as "the Big Bang", and is considered the "birth" of our universe. Based on measurements of the expansion using Type Ia supernovae, measurements of temperature fluctuations in the cosmic microwave background, and measurements of the correlation function of galaxies, the universe has a calculated age of 13.7 ± 0.2 billion years.[21] The agreement of these three independent measurements strongly supports the ΛCDM model that describes in detail the contents of the universe.
The aim of this paper is to examine the Planck epoch and the grand unification epoch and therefore find out the true scale that explains the earliest period of the universe.
 
  1. Methodology
 
I construct a mathematical model to study the relationship between the ratio of the wavelength and the ratio of the accelerations. In the model I refer to different particles probing different scales of time, I refer to the Planck scale and the quantum gravity scale. Assuming two particles, one under the influence of gravity and the other the influence of quantum fields, when the two particles are set to fall through the fields each falls with an acceleration describing the fields that is gQ and gG respectively. The length scales describing the falls are λe and λc, so the equation below will fully describe the model,
 
 
λe / λc =( gQ / gG )1/2                                                                                                                                               [1]
 
 
 
where, λe = e/2λpEε ,is the wavelength of an electron depending on the wave length of a photon λp. The higher the photons wavelength the smaller the electrons wavelength and the smaller the wavelength of the photon the higher that of an electron, this takes place on the assumption that the electric field E, the charge e, and the permittivity of free space ε, are all constants. λc= ħ/mc, is the Compton wavelength of an electron with mass m , here ħ is Dirac constant and c is a constant speed of light.   gQ = e2f/2 ħε  describes the acceleration of a particle with a frequency f in the quantum field and , gG = Gm/R2 is the acceleration due to gravity and G is the universal gravitational constant.
 
for ,λp=c/f=2πR, E=e/4πεR2 where R is the distance between any two particles.
 
We compare the forces that come as a result of the motion. There are two forces each occurring at a different scale length. First there is a force describing the Inflation, baryogenesis at the grand unification transition. Second there is another force describing the quantum gravity barrier at the Planck epoch. Both of these forces will have a similar characteristic, which is they will depend on the time t at any level.
 
Therefore from eqn1 we multiply through by Gc5 and obtain a relationship of the forces given by,
 
 
F = tg (c7/32πmG2 ) = tu (c7/16πG2m)                                                                    [2]
 
Where
 
t u= e2ħ/2εGm3c2 and tg =A/cR , A is the area.
 
 
  1. Results
 
At Planck scale mass mp = √ħc/G , t u = (e2/ε)√G/ ħc7= 4.932×10-45s and at the Planck length scale Lp = R =(√ ħG/8π c3 ) , A ~ L2p =ħG/8π c3 and  tg=√ħG/c5 , then equating t u to t g the fine structure constant is generated as (e2/16π2εħc).
 
  1. Discussion
 
The results obtained show exactly the required mechanism responsible for the description of the time line of the big bang. The reduction of the fine structure constant from the theory shows that there exists a relationship between quantum mechanics and gravity, that at the two times t u and t g the force of gravity was strong and that there was a possibility for the unification of all the fundamental forces of nature. It can now be theorized that there exists a scale that when merged with the Planck scale the result is the earliest period of the universe at which all of physics problems can be solved. Therefore both scales are needed to explain the origin of the universe from the big bang.
 
  1. Conclusion
 
Ignoring quantum effects means that the universe starts from a singularity with an infinite density. This hypothesis however can change when quantum gravity is taken into account. The works of String theory , Loop quantum gravity, Noncom mutative geometry and other fields of physics holds promise for our understanding of the very beginning.
However; the more we understand about how matter forms, the more precisely we will be able to interpret what we learn from astrophysical data, and from other sources.
 
  1. Acknowledgement
 
I would like to thank the many people who contributed in various ways to improving this manuscript. Among these are Nantumbwe Florence for her financial support.
 Any errors that remain are fully my responsibility and therefore welcome corrections for future manuscripts.
 
  1. Reference:
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ON THE GENERALISATION OF LOOP QUANTUM GRAVITY

Feb. 5th, 2008 | 11:38 am


 
 
BY BALUNGI FRANCIS
Makerere University, Faculty of technology, P.O.Box 7062, Kampala, UGANDA
 
 
 
 
Abstract
 
The paper models out a way of combining quantum mechanics with gravity by employing a four dimensional (4D) space which follows Schwarzschild radius in the form Rs ,Rs2 and Rs4. The generalized equation for the quantization of angular momentum is then used to form general expressions of energy and mass equations, from which for example the energy equation is used to express the Planck scale and its cause, examine the thermodynamics of Black Holes, study the kind of momentum possessed by particles describing the quantum Hall Effect and finally examine the line separating the quantum theory from the theories of condensed matter.
 
 
KEYWORDS: Quantum gravity, loop quantum gravity, renormalization, Hall Effect and superconductivity.
 
 

1.0 .Introduction
 
Quantum gravity is the field of theoretical physics that tries to unify quantum mechanics with general relativity. Quantum mechanics describes the three fundamental forces of nature while general relativity is a theory of the fourth fundamental force: gravity. The goal every one is waiting for to emerge from this unification is a "theory of everything", or "Grand Unified Theory" (GUT). So many researches have been conducted in line with the theory, for example in 1986, Abhay Ashtekar reformulated Einstein's field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. He was able to quantize gravity using gauge field theory. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport and a coordinate frame known as a vierbein at each point. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states.
Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose's spin networks. In this context, spin networks arose as a generalization of Wilson loops necessary to deal with mutually intersecting loops. Mathematically, spin networks are related to group representation theory and can be used to construct knot invariants such as the Jones polynomial.
Quantum field theory depends on particle fields embedded in the flat space-time of special relativity. General relativity models gravity as a curvature within space-time that changes as a gravitational mass moves. Historically, the most obvious way of combining the two (such as treating gravity as simply another particle field) ran quickly into what is known as the renormalization problem. In the old-fashioned understanding of renormalization, gravity particles would attract each other and adding together all of the interactions results in many infinite values which cannot easily be cancelled out mathematically to yield sensible, finite results. This is in contrast with quantum electrodynamics where, while the series still don't converge, the interactions sometimes evaluate to infinite results, but those are few enough in number to be removable via renormalization. Being closely related to topological quantum field theory and group representation theory, LQG is mostly established at the level of rigour of mathematical physics. While confirming that quantum mechanics and gravity are indeed consistent at reasonable energies, this way of thinking makes clear that near or above the fundamental cutoff of our effective quantum theory of gravity a new model of nature will be needed. That is, in the modern way of thinking, the problem of combining quantum mechanics and gravity becomes an issue only at very high energies, and may well require a totally new kind of model.
The general approach taken in deriving a theory of quantum gravity that is valid at even the highest energy scales is to assume that the underlying theory will be simple and elegant and then to look at current theories for symmetries and hints for how to combine them elegantly into an overarching theory. One problem with this approach is that it is not known if quantum gravity will be a simple and elegant theory (that resolves the conundrum of Special and General Relativity with regard to the uniformity of acceleration and gravity, in the former case and space time curvature in the latter case).
1.1. Objectives
The need for the paper is to understand those problems involving the combination of very large mass or energy and very small dimensions of space, such as the behavior of black holes, and the origin of the universe.
The aim of the paper is to answer questions like; How can the theory of quantum mechanics be merged with the theory of general relativity  and remain correct at microscopic length scales? What verifiable predictions does any theory of quantum gravity make?
To study the level at which wave mechanics becomes inapplicable, and there fore explain the nature of condensed matter physics using a simple model.
 
 
 
1.2. Materials and methods
 
1.3. The formula for the quantization of quantum gravity
 
The model is based on separating the gravitational field into the sum of two components; that is the background and the quantum field. The background left is one for all our calculations. But because loop gravity ignores the back ground space as a lost entity that does not occur in space, there fore the need to reconstruct quantum field theory from scratch without a background space is taken into account. I therefore suggest that the calculation should be performed by summing all possible space-times.
 
Quantum field theoryAssuming a spherical symmetric object that space time is of dimensions increasing from 1, 2, 3, 4...N, where N is the nth term of the dimensions. To quantize space and time is to create a space in which all of physics is quantized. The nature of the curved space surface is described by increasing powers in the Schwarzschild radius Rs = Gm/c2, Hence describing the dimensions of space. Quantum mechanics explains the existence of discrete energy states in an atom, in away that the angular momentum of the atom must be quantized, which is also the case for quantum gravity. The equation for the quantization of the loop quantum gravity can then be written as, depends on particle fields embedded in the flat space-time of special relativity. General relativity models gravity as a curvature within space-time that changes as a gravitational mass (m) moves.
 
 
 
ηRs +βRs2+μRs4+…………..+δRsN = nħ                                                                   [1]
 
Where η= √Beh, is the momentum of a particle probing another form of quantum mechanics, ħ=h/2π, where h is Planck constant, β= 8πBe, e is the elementary charge, B is the magnetic field and finally μ=256π3P/c2, where P is the intensity and c is the constant speed of light.
 
1.4. The energy equation
 
What changes is the form of the equation the rest remaining constant. The principle behind this is that eqn1 can be changed to any form simply for purposes of calculating complex phenomenon. The energy to which we are concerned here is expressed as a general expression describing the energy scales forming smaller and larger matter entities in the universe. The energy will thus be given by;
 
ηc +βcRs+μcRs3+…………..+δcRsN-1 = nħc/Rs                                                         [2]
 
Note: the background space described by the Schwarzschild radius has changed, thus the above equation in any case can be used to calculate the basic properties of Black holes. Remember the Schwarzschild radius is the radius for a given mass where, if that mass could be compressed to fit within that radius, no known force or degeneracy pressure could stop it from continuing to collapse into a gravitational singularity.
 
1.5. The mass equation
 
Having explored the energy scale we now form general equation that describes well the mass scale. This is also done the same way as eqn2 and therefore generate,
 
 
η/c +βRs/c+μRs3/c +…………..+δRsN-1/c= nħ/cRs                                                    [3]
 
 
 
 
1.6. Results
 
         i.            The Planck scale and the gravity magnetic field
 
Assuming that the energy W= βcRs , from eqn2 is equal to the energy W= mc2, we hence obtain the magnetic field as, B = c3/8πGe = 1.0054×1053 N/Am. using this magnetic field in the energy equation, W= ηc we get the energy in the form W = (c2/2) √ħc/G where the quantity √ħc/G is the Planck mass Mp at an energy of 6.119×1018 GeV.
 
       ii.            2. Time taken by a black hole to evaporate and its entropy
 
The energy required here is given in Eqn2, it is at this, that the intensity P = W/AΔt, (where A is the area and t is the time) is used. We take the energy W = μcRs3 (from Eq2) as our interest from which we obtain the time as Δt =256π3 Rs3/Ac. But with black holes the area will become exactly equal to the square of the Planck length as A ~ L2p=ħG/8π c3 hence the change in time is given by Δt =63500.86π G3m3/ħ c4.
 
For entropy we set the energy to kT, where k is Stefan’s-Boltzmann’s constant and T is the temperature of the body. Now for kT = μcRs3, since Δt is known the entropy is thus given by S= W/T = 78.96Ak c3/ π ħ G ~ A/4. In conclusion we state that the entropy of a black hole is proportional to the area of the event horizon.
 
     iii.            3. The quantum Hall Effect
 
For this effect the momentum η is used. From Eqn2 we set, ηc = nħ / Rs which gives the magnetic flux as  4 π Rs2B = nh/e, from which the resistance is given by ζ = 4 π Rs2B /e = nh/e2. for n= 1,2,3,4 the resistance is of a value 25833.8Ω.
 
 
      iv.            4. Symmetry breaking at the Planck scale
 
Using eqn3 in this case, since B is known and P got from μRs4 =nħ ; as P =ħc 2/256π3 Rs4, we hence obtain, Mp /2 + m + Mp /m = Mp /m, which gives Mp+2m =0, and for identical mass M =0, which is true. The intensity at the planck length that is for Rs = Lp is
P=c8/πħ G2
 
 
1.7. Discussion
 
The results obtained signify the comparisons between quantum mechanics and gravity that at the known Planck scale both theories are combined into one theory. The deduction of black hole thermodynamics implies that the theory is at its best a true quantum gravity theory. The Planck scale is what it is, not because it expresses natural units but because of the magnetic field of about 1053 N/Am through which matter interacts with energy to produce particles probing the Planck scale. The success of this theory implies the production of features resembling the standard model, for this I will regard my self to only condensed matter physics as discussed below;
For Compton and photoelectric effect to occur particles with momentum p = h/λ, which describe the wave properties of matter at a wave length λ, are required but for the quantum Hall Effect to occur the particles must have a momentum p = √Beh at which the magnetic flux and resistance hold on well. This is a darling new idea describing particles probing the wave particle duality model, and therefore addresses a phenomenon referred to as the “exact quantization”. The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductance takes on  quantized values. The momentum η on the other hand indicates a move from a probabilistic theory of quantum mechanics to a complete theory of mechanics. For example comparing this momentum with that of De Brogile you find out that the magnetic flux is in terms of wavelength and given by B λ2 =h/e, this is known as flux quantization.Flux quantization occurs in Type II superconductors subjected to a magnetic field. The possibility of obtaining such a result from simply combining the different momentums must have an implication. The uncertainties brought by the quantum theory can be foregone by introducing in “quantum mechanics” a new mechanics that states that the outcome of an ideal measurement of a system is deterministic. Therefore the momentum is taken as the separating line between quantum mechanics and condensed matter physics.
 
1.8. Conclusion
The recent observation of naked singularities and doubly special relativity as part of loop quantum cosmology show a clear future for the significance of the theory to all of physics researchers, but as for now there is no experimental observation to show whether loop quantum gravity makes the predictions not made by the standard model or general relativity however the successful prediction and calculation of the Bekenstein – Hawking formula S=A/4 in this paper is evidence to show that loop quantum gravity is a theory that must be generalized regardless of it’s past mistakes.
Successfully I have made a new mechanics that is different from that of deBrogile but makes observable results when combined with the deBrogile hypothesis. Thus a mechanics for condensed matter has been finally developed.
1.9. Acknowledgement
 
I would like to thank the many people who contributed in various ways to improving this manuscript. Among these are Nantumbwe Florence for her financial support.
 Any errors that remain are fully my responsibility and therefore welcome corrections for future manuscripts.
 
 
 
 
 
 
 
 
 
2.0.Reference:
 
    1. Abhay Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett., 57, 2244-2247, 1986
    2. Carlo Rovelli and Lee Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys., B442 (1995) 593-622, e-print available as gr-qc/9411005
    3.  Castelvecchi, Davide; Valerie Jamieson (August 12 2006). "You are made of space-time". New Scientist (2564
    4. Researchers Look Beyond the Birth of the Universe", Eberly College of Science, 12 May 2006. 
    5. Smolin, Lee. "The case for background independence". hep-th/0507235
    6. http://en.wikipedia.org/wiki/Loop_quantum_gravity
    7. C.L.Chin and C.R.Westgate (Editors), The Hall Effect and Its Applications,” Plenum Press,New York, 1979, p.535.
    8. Epstein. M., et al, “Principals and Applications of Hall-Effect Devices”, Proceedings of the National Electronics Conference, 1959, Vol.15, p.241.
    9.  Final Engineering Report on Hall Effect Device Investigation”, Device DevelopmentCorporation, Weston 93, Massachusetts, Contract No. NOBsr-72823, July 1, 1958 toFebruary28, 1959, pp.12-17
    10.  

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THE SPECTRUM OF THE ATOMIC UNIVERSE

Feb. 5th, 2008 | 11:17 am

                                  
Makerere University, Faculty of technology, P.O.Box 7062, Kampala, UGANDA
 
 

Abstract
 
Quantum mechanics or quantum theory is a physical science that is concerned with the behaviors of matter and energy at a scale of atoms and subatomic particles/waves.
Quantum mechanics also acts as the basis through which we can study, analyze and explain very large objects such as stars and galaxies, and cosmological events such as the big bang. To describe the atomic universe fully we need both quantum mechanics and gravity. This is achieved through the study of the accelerations of the particles leading to the radiations of electromagnetic energy, and predicting that all matter is unstable. It is then theorized that there appears two accelerations, ga and gb whose ratio explains the formation of two or more lines close together in the Hydrogen spectrum which is known as the fine structure. Comparing this model with Bohr’s model of the Hydrogen atom produces very precise results for cosmological events hence the atomic universe.
 
Keywords: quantum mechanics, cosmology, general relativity.
 
 
 

  1. Introduction
 
In the early 20th century, Ernest Rutherford experiments established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, it was quite natural for Rutherford to consider a planetary model for the atom, the Rutherford model of 1911, with electrons orbiting a sun-like nucleus. This model was a difficulty. The laws of classical mechanics predict that the electron will release electromagnetic radiation as it orbits a nucleus. Because the electron would be losing energy, it would gradually spiral inwards and collapse into the nucleus. This is a disaster, because it predicts that all matter is unstable.
To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. The model's key success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, but it provided a justification for its empirical results in terms of fundamental physical constants.
This paper looks at the model in a very different way than that of Bohr. The fact that all accelerated particles do emit electromagnetic radiations is taken into account and therefore the acceptance for the unstableness of all matter is considered in due respect. In fact Bohr’s ideas never required classical mechanics simply because it could not conform to the experimental observations of the spectrum of the Hydrogen atom that were obtained by Rydberg using his formula.
To merge gravity with Planck’s quantum theory by then was also a problem at hand and therefore Bohr had to forego the problem by introducing in his theory adhoc postulates, and this could have been the reason why Einstein found problems in merging gravity with electromagnetism in what is called “The Grand unified field theory”, of which he had to question the problem with the quantum theory and therefore request for a complete quantum theory. From Bohr’s model many theories have been formed each building from the ideas of the model, but a certain point is reached where the theories can not conform well to the known laws of nature and therefore regarded as failures, which of course in their judgments is true. The problem is seen to come from exactly the roots of quantum mechanics.  
The aim of this paper is therefore to produce a generalized theory of atomic structure that incorporates in it gravity and quantum mechanics. In other words a theory that takes the laws of classical mechanics into consideration.
 
  1. Methodology
 
The Hydrogen atom exists in certain stationary states of discrete energies. The acceleration due to gravity of an electron in orbit around the nucleus will cause the atom to emit radiations (radiate energy) and thus make the atom unstable. The acceleration (g) falls off with time t provided the radius of orbit of the electron R is a constant thus the acceleration due to gravity is given by;
 
g= R/∆t2                                                                                                              (1)
 
The rate of change of energy P radiated as a result of the above acceleration will depend on the constants c (speed of light) and G (universal gravitational constant), hence;
 
P= c5/G                                                                                                               (2)
 
The power and time must be re- quantized in units of ћ = h/2π where h is Planck constant, hence
 
P∆t = n2ћ                                                                                                             (3)
 
Where n= 1,2,3…….. is the principle quantum number.
 
But the total energy of the atom in the various energy states is W= -ke2/R where k is the Coulomb constant and e is the elementary charge. Since Δt2 is known from Eqn1 and P from Eqn2 then using Eqn3 the radius is given by
 
R =n2Gg ћ /c5                                                                                                                                                          (4)
 
From which the total energy is given by,
 
W= - ke2c5/ n2Gg ћ                                                                                                (5)
 
From the Bohr-Einstein frequency (f) condition, applied to a transition from a level with n =ni to a level with n = nf, The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy levels
 
hf= Ei – Ef
 
Finally we get since frequency f = c/λ, where λ is the wavelength
 
1/ λ = [ke2c4/ 2πGћ2][1/g][1/ nf 2-1/ ni2]                                                                   (6)
 
The equation obtained above shows some how a great significance of gravity in the quantum theory. So far it states that regardless of the levels in the transitions of an atom the acceleration due to gravity of the particles in the atom do greatly affect the nature of its spectrum.
 
  1. Results
 
The quantity [ke2c4/ 2πGћ2] is the inverse of the square of time t and therefore
1/t2 = [ke2c4/ 2πGћ2], from which the time is obtained as t = 1.58873 ×10-42s.
 
Comparing Eq6 with Bohr’s model, here we shall equate the Rydberg constant [k2e4m/4πcћ3], where m is the mass of the particle, to the constant [ke2c4/ 2πGћ2][1/g]. Doing this generates an acceleration given by ga= [ 2ћc5/ke2 Gm] from which we obtain a general equation of forces given by [8πG/ c4][gm][ke2 /R2] = 16πћc/R2.where [gm] is the gravitational force and [ke2 /R2] is the electromagnetic force.
 
At the Schwarz child’s radius R=Gm/c2 the acceleration is gb = c4/Gm which gives an equation for the spectrum as 1/ λ = [/ 2π ao][1/ nf 2-1/ ni2] where ao is the first Bohr radius [ћ2/ mke2] = 5.28 ×10-11m.
The interesting part of it is that the ratio gb / ga =[ke2/2ћc] the fine structure constant. This result therefore explains the fine structure shown by the Hydrogen spectrum and thus suggests that an electron describes an elliptical orbit. Now using the acceleration ga, the radius aoand the mass m the energy W of a particle will be given by W =gaaom =β/m, where β is a constant given by [ 2ћ3c5/k2e4G] =1.64367 ×106Jkg. For two different masses m and mo we have the equation for the product of the masses as, mmo = [ 2ћ3c3/k2e4G].
 
 
 
 
  1. Discussion
 
The results obtained give out a clear image for the description of the atomicity of both large and small particles. Firstly the time t obtained is the is the earliest period of time in the history of the universe from zero to approximately 10-43 seconds , during which quantum effects of gravity were significant. At this period all the fundamental forces of physics were unified. The state of the universe during this epoch was unstable, tending to evolve and giving rise to the familiar manifestations of the fundamental forces through a process known as breaking. Symmetry breaking quickly led to the era of cosmic inflation, the Inflationary epoch, during which the universe greatly expanded in scale over a very short period of time.
Secondly, the accelerations ga and gb led to different spectrums of the Hydrogen atom. Where ga produces the Rydberg equation for the spectrum of the hydrogen atom, that is incorporated in it the Rydberg constant [k2e4m/4πcћ3], the acceleration gb produces a different equation which instead of a Rydberg constant, it has the inverse of the first Bohr radius ao.These differences in the spectrum of the hydrogen atom with the former producing a single line and the latter two or more lines of the spectrum close together, imply that the electron moves in an elliptical orbit as those of the planets in orbit around the sun, hence the ratio of the accelerations gb / ga will generate a fine structure constant describing the closeness of the spectrum lines produced by the hydrogen atom. Finally the energy obtained using the acceleration ga and the first Bohr radius ao has an impact on the way we express the energy of large and small particles. For example a body of one kilogram mass (1Kg), will have an energy of 1 .64367 ×106Joules (1 .64367 ×106J) which is a very high energy. This energy is independent of the speed of a body or particle in question, and thus gives the energy to a particle regardless of it’s speed. We very well know that the speed of light is a constant and therefore doesn’t change and that with relativity such a body of 1kg will have energy approximately 1016J which is lager than the first one by 1010. For smaller particles say an electron we have from the result equation the energy as 1.80425×1036J, but for relativity it is ~10-15J. These differences in   energies imply that without knowing the speed of the particle we can obtain it’s energy depending on it’s mass since some particles tend to move at a speed greater than that of light.
 
  1. Conclusion
 
In conclusion the results produced successfully indicate that without gravity, quantum mechanics can not survive and without quantum mechanics, gravity cannot survive. Therefore the two theories are needed to explain the atomic universe fully.
 
  1. Acknowledgement
 
I would like to thank the many people who contributed in various ways to improving this manuscript. Among these are Nantubwe Florence for her financial support.
Any errors that remain are fully my responsibility and therefore welcome corrections for future manuscripts.
 
  1. Reference:
 
 
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