In this research I try to present what I have come to call the wide theory. 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. The idea here is to create the one equation or theory that must be capable of explaining other theories of physics, In this paper I try to derive all the properties of black holes in the theory known as general relativity, and also deduce, Widemann Franz law ( thermal properties of solids) and Stefan’s radiation law as we shall see,
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 ћ /8π GM =kT
Where k is the boltzmann’s constant and T is the temperature, Hence at n=1,
T = c3 ћ /8π GMk 2
This is the temperature of the black hole.
3. TIME TAKEN BY A BLACK HOLE TO EVAPORATE
Under the assumption of an otherwise empty universe, so that no matter or cosmic microwave background radiation falls into the black hole, it is possible to calculate how long it would take for the black hole to evaporate
On dividing through eqn1 by the momentum Mc we obtain, the time t given by
t= mc/Ee = 8πG2M3/n2ћ c4
Such that when n=0.03953
t= 5120πG2m3/ ћ c4 3
This is the time taken by a black hole to evaporate.
4. ENTROPY OF A BLACK HOLE
If black holes carried no entropy, it would be possible to violate the second law of thermodynamics by throwing mass into the black hole. The only way to satisfy the second law is to admit that the black holes have entropy whose increase more than compensates for the decrease of the entropy carried by the object that was swallowed.
Squaring both sides of equation 1 and arranging we generate the intensity as
W/tA = E2e2/2nh = n3 c10 ћ /256π3 G4M4 4
Where A is the area on which the radiations fall, W is energy, and t is time
This might be one of the great discoveries of physics so far, this is the equation of the intensity of the universe before and after the big bang. We shall discuss about it as a new branch of physics in the coming papers.
But entropy is energy divided by temperature Eq so then
W/T = S = (n3 c10 ћ /256π3 G4M4 ) (tA/T)
Since t is known from Eq3 and T from Eq2 then at n= π
S = Akc3 /4G ћ
Without the use of statistical mechanics I have successfully derived the entropy of the black hole, which states that the entropy of a black hole is proportional to it’s area A.
5. THERMAL PROPERTIES OF SOLIDS (Weidman- Franz law)
In physics, the Weidman- Franz law states that the ratio of the thermal conductivity (K) to the electrical conductivity (σ) of a metal is proportional to the temperature (T). Although the law is right but it’s proof is not quietly given in physics, which is why I decided to look for the one theory that I can use to derive it from first principles such that every one can witness it’s existence as being for real.
From the intensity Eq4,
E2e2/2nh = c10 ћ /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 ћ /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 truly the law given in physics books but with no derivation of it.
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 ћ /8π GM) 4
This is the same as Eqn4 only that it is arranged to predict something
But n=1 and (c3 ћ /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 an ultimate 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
BY
BALUNGI FRANCIS
Makerere University, Faculty of tchnology, P.O.Box 7062, Kampala, UGANDAN
Email balungifrancis@yahoo.com
, bfrancis@tech.mak.ac.ug
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
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
- 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 thermodynamic
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)
- 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.
- 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.
- Acknowledgment
I would like to thank Nantubwe Florence for her financial
support.
- References
- Abhay Ashtekar, New variables for classical and quantum gravity, Phys.
Rev. Lett., 57, 2244-2247,
1986
- 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
- Castelvecchi, Davide; Valerie
Jamieson (August 12 2006). "You are made of space-time". New Scientist (2564
- Researchers Look Beyond the Birth of the Universe",
Eberly College of Science, 12 May 2006.
- Smolin, Lee. "The case for background independence".
hep-th/0507235
- http://en.wikipedia.org/wiki/Loop_quantum_gravity
- C.L.Chin
and C.R.Westgate (Editors), The Hall Effect and Its Applications,” Plenum
Press,New York,
1979, p.535.
- Eddington,
A. S., The Internal Constitution of the Stars (Cambridge
University Press, England,1926), p. 16
- E.
Kolb and M. Turner, The Early Universe (Addison-Wesley, Reading,
MA,1990).
- W. Garretson and E. Carlson, Phys. Lett.
B 315, 232(1993); H.
Goldberg, hep-ph/0003197
- Eddington,
A. S., The Internal Constitution of the Stars (Cambridge
University Press, England,1926), p. 16
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