THEORY OF QUANTUM GRAVITY
A Theory of Quantum Gravity Based on the Ideas of Quantum Electrodynamics
The gravitational strength between objects is extremely small: it is a force that is weaker, by 30 to 40 orders of magnitude, than the electrical force between two electrons. While electrical forces hold atoms together, gravitational forces hold planets, galaxies, and perhaps the cosmos together. Few experiments are possible and only observations can be made of gravitational effects. There are, however, a number of theories of gravitation that involve "gravitons" as the intermediaries between objects but no theory has the precision of a quantum theory of gravitation to explain the effect of gravity. So not only have we no experiments with which to check a quantum theory of gravitation, we also have no reasonable theory. This work intends to provide a reasonable theory of gravitation based on the ideas of Quantum Electrodynamics (QED).1
The need for a QED emerged from the problem of the interaction of light and matter: Maxwell's theory of electromagnetism had to be changed to be in accord with the new principles of quantum mechanics that had then been developed and was a tremendous success in explaining all of chemistry and properties of substances. It was known that electrons interact with light but the computations were increasingly impossible beyond a certain accuracy. QED now fully developed offers no significant difference between experiment and theory.2 The need for a quantum theory for gravitation is less justified because we have no reasonable dynamic theory for gravitation and only a handful of experiments and observations to proceed with.
QED rests on the idea that charged particles (e.g., electrons and positrons) interact by emitting and absorbing photons, the particles of light that transmit electromagnetic forces. These photons are virtual; that is, they cannot be seen or detected in any way because their existence violates the conservation of energy and momentum. The particle exchange is merely the "force" of the interaction, because the interacting particles change their speed and direction of travel as they release or absorb the energy of a photon. Photons also can be emitted in a free state, in which case they may be observed. The QED theory states that the more complex the process (i.e., the presence of additional virtual photons), the smaller the probability of its occurrence. For each level of complexity, a factor of (1/137)2 decreases the contribution of the process, and thus, after a few levels the contribution is negligible. This factor, symbolized by a, is called the fine-structure constant and serves as a measure of the strength of the electromagnetic interaction. It equals e2/c, where e is the electron charge, is Planck's constant divided by 2, and c is the speed of light.
QED is often called a perturbation theory because of the smallness of the fine-structure constant and the resultant decreasing size of higher order contributions. This relative simplicity and the success of QED have made it a model for other quantum field theories. Finally, the picture of electromagnetic interactions as the exchange of virtual particles has been carried over to the theories of the strong, weak, and gravitational forces. The present work rests on the idea that masses interact by emitting and absorbing gravitons, particles of matter that transmit gravitational forces. These gravitons are virtual; that is, they cannot be seen or detected in any way and perhaps because their existence violates the conservation of energy and momentum. The exchange of gravitons is merely the "force" of the gravitational interaction, because the interacting masses change their speed and direction of travel as they emit or absorb a graviton. However, in the present theory, gravitons are not only virtual, they also have imaginary proper masses and, therefor, they cannot exist in a free state. The present Quantum Gravity Theory (QGT) states that the more complex the process (i.e., the presence of additional virtual gravitons), the smaller the probability of its occurrence. For each level of complexity, a factor decreases the contribution of the process, and thus, after a few levels the contribution is negligible. This factor, symbolized by ag, is called the gravitational fine-structure constant and serves as a measure of the strength of the gravitational interaction. The factor is Gm/c, where G is the Newtonian constant, m is mass, is Planck's constant divided by 2, and c is the speed of light.
In what follows, I first summarize the well known Quantum Electrodynamics (QED)3 and then I set forth my Quantum Gravity Theory (QGT).
A relativistic wave equation can be obtained by writing the relativistic energy equation
E2 = p2 c2+ mp2c4 . . . . . . . . . .p2 =px2 +py2 +pz2
and replacing the total energy and the momentum components by the associated operators by
E . . . . . px . . . . . py . . . . . pz
and then allowing the operator equation thereby obtained to operate on the function . The result is
. . . . . . . . . .
which is called the Klein-Gordon equation. It plays an important role in the quantum electrodynamics. For instance, for mp=0 it reduces to the classical wave equation
for photons, the quanta of the electromagnetic field.
The classical wave equation has a static solution of the form
. . . . . . . . . . r>0
as can easily be verified by substitution, using the relation
for =(r). For mp the Klein-Gordon equation has a static solution of the form
. . . . . . . . . . r>0 . . . . . . . . . .
as can also easily be verified by substitution. Since the solution to the wave equation for zero rest mass quanta gives the Coulomb interaction potential for the electromagnetic field, the solution for non zero rest mass quanta gives the interaction potential for the Yukawa potential which has application to the meson field.
The constant g2 determines the strength of the Yukawa potential, just as the constant e2 (the square of the electron charge) determines the strength of the coulomb potential. The dimensionless quantity g2/c has a value about 15 whereas the dimensionless quantity e2/4pec (the fine structure constant) has the value about 1/137. This is an indication of the strength of the nuclear force compared to the electric force.
Quantum Gravity Theory
For mp=imi imaginary and for identical interacting masses m, the Klein-Gordon equation has a static solution of the form
. . . . . . . . . . r>0 . . . . . . . . . .
as can also easily be verified by substitution. The solution for imaginary rest mass quanta gives the interaction potential of the gravitational field. Unlike the Newtonian potential, as r the gravitational interaction potential spatially oscillates between +Gm2/r and -Gm2/r with a period 1/ri'=2 Since the value for mi is not known, values for the distance ri' and spatial wavelength - .are unknown. The interaction force for the static gravitational field is obtained as F=- which is a modified oscillatory Newtonian force.
It is known that the Newtonian force is attractive to distances of about 4.32 million light years, the low end of Hubble's linear law. Beyond that distance, the observed cosmic expansion suggests a repulsive force, perhaps as indicated by the modified oscillatory Newtonian force. The foregoing suggests that on the local, planetary and galactic scales r<<ri', and the modified Newtonian force is written as
F= . . . . . . . . . . ri'>>r>0 . . . . . . . . . .
which says that a constant repulsive force +Gm2/2ri'2 must be added to Newton's force. While the value of Gm2/2ri'2 is presently unknown, its theory opens the possibility of making experiments and observations at local, planetary and galactic distances to obtain its magnitude, and values for ri' and mi. The existence of an oscillatory Newton's law of gravitation for all other values of r/ri' predicted by the QGT must be left to be determined by observations on the cosmic scale.
For masses m of interacting electrons, the dimensionless quantity Gm2/c has a value about 10-27 which indicates the extremely small value of the gravitational force compared to the electric and nuclear forces discussed previously. However, while all forces fall off with distances, the electric and nuclear forces are short range forces but the gravitational force alone increases as the product of interacting masses increases.
The virtual particle which interacts between masses is the graviton. Its speed is v>c. This can be seen by recasting the relativistic energy equation in terms of its equivalent relativistic mass equation and setting mp=imi imaginary. While the form of the relativistic energy equation remains unchanged, the resulting relativistic mass equation is valid when v>c. A graviton has momentum energy Pc greater than its total energy E. In the static case, gravitons have zero total energy and infinite speed. In the dynamic case, gravitons have finite total energy E<Pc and v>c but less than infinite speed.
In the foregoing, the sole discussion concentrates on static gravitation, i.e., the static solution of the Klein-Gordon equation. A static solution means that gravitons travel at infinite speed. For dynamic gravitation, i. e., for a dynamic solution of the Klein-Gordon equation, gravitons travel at speeds v>c but less than infinite speed. Theories of dynamic gravitation 4 and quantum gravity 5 have been proposed by many theorists, including theories based on Einstein's theory of gravitation and theories based on Maxwell's theory of electromagnetism. Such theories have serious defects, namely, they predict that gravitons travel at the speed of light and that virtual or free state gravitons can be detected. In contrast, the present QGT predicts that gravitons travel at speed v>c and that virtual gravitons cannot be detected. And, since gravitons have imaginary rest mass mp=imi, they cannot be emitted or absorbed in a free state. Looking for gravity waves with energy detectors is both a waste of time and money. Gravity effects can be detected only with spatial distortions such as tides and, perhaps, anomalous orbits which deviate from predictions using Newton's law of gravity.
Consider briefly the evidence for interactions between gravitons and the three other virtual or free state particles. We have no information that gravity acts at atomic and nuclear distances. However, as shown by the classical redshift, deflection of light and advance of perihelion paradigms, we know that photons and planets interact with the Sun's mass. In these paradigms, the free state photons and planets (and the source mass Sun) change their energy and momenta. But does the QGT predict interactions between free state photons or planets and the Sun? Using the relativistic energy equation, the interaction between objects 1 and 2 produces the following equality
which, in many cases, can be reduced to the form of a relativistic energy equation, and corresponding Klein-Gordon equations can be obtained. Note that the conservation of energy and momentum is violated in these equations.
In summary, the relativistic energy equation tells us that detectable mass travels at speeds less than the speed of light v<c and that detectable light travels at speed equal to light v=c. QED tells us that, based on the relativistic energy equation and quantum theory, undetectable virtual mass (rest mass mp) and undetectable virtual light (rest mass mp=0) interact between nuclear and atomic objects at speeds vc but violate the conservation of energy and momentum. QGT tells us that, based on the relativistic energy equation and quantum theory, undetectable virtual mass (imaginary rest mass mp=imi) interacts between masses at speeds vc and pending experiments may violate the conservation of energy and momentum.
The identities of energy, mass and frequency, E=mc2=, with their numerical constants c and h are the basis of all laws of physics. They are the points of departure, not the results of any theory.6 The relativistic wave equation, QED and QGT are theories based on these identities which are beyond our comprehension. Apparently, the sole criterion of a theory is whether it works. The fact that a theory introduces virtual objects (QED and QGT) or both virtual and imaginary objects (QGT) is no bar to its success. The difference between QED and QGT is that QED is abundantly confirmed by experiment which QGT presently falls short. The strongest evidence for QGT is the observed action at a distance of the Newtonian force.
1 Quantum Electrodynamics at http://www.britannica.com/nobel/micro/489_26.html
2 Richard Feynman, QED Princeton University Press pages 5-7.
3 R. Eisberg and R. Resnick, Quantum Physics, John Wiley and Sons 1974 page 690.
4 Dynamic gravity http://en.wikipedia.org/wiki/Generalized_theory_of_gravitation
5 Quantum Gravity at http://www.weburbia.com/pg/qugrav.htm
6 L. Brillouin, Relativity Reexamined, Academic Press 1970 page 34.
Copyright © 2006 by James Constant
By the same author: http://www.coolissues.com/gravitation/sameauthor.htm