Variable Planck Scale Model

VARIABLE PLANCK SCALE MODEL By: Paul Hoiland Special thanks to the online yahoo group Stardrive former known as ESAA, Fernando Loup, and many others who have aided this creative research. : ABSTRACT: Cosmological theories and theories of fundamental physics must ultimately not only account for the structure and evolution of the universe, the physics of fundamental interactions but also an understanding of why this particular universe follows the physics that it does. Such theories must lead to an understanding of the values of the fundamental constants themselves. Moreover, the understanding of universe has to utilize experimental data from the present to deduce the state of the universe in distant regions of the past and also account for certain peculiarities or coincidences observed. The prevalent view today in cosmology is the big bang, inflationary evolutionary model. Although certain problems have remained, e.g. the need to postulate cold, dark matter in amounts much larger than all the observable matter put together, dark matter not detected so far in the laboratory or the recent need to re-introduce the cosmological constant, the big bang cosmology has, nevertheless, achieved impressive results (Silk 1989). There have also been recent observational evidence hinting at (Barrow and Magueijo 1998) has recently been found which seems to be consistent with a time-varying fine structure constant α = e2/(hc). A varying speed of light theory (with h α c) has also been proposed by Albrecht and Magueijo(1998). Added to this we are confronted with a Pioneer slowdown that seems to require some modification to General Relativity and evidence of higher amounts of certain elements than the standard model can account for. These mixed messages could be found via one common model that stems from both the Dutch Equation[1] and Fernando Loup’s model for hyperdrive[2] that was based upon that equation. BACKGROUND: Fernando Loup has in several published articles offered a possible theoretical method of star travel via hyperspace out of modern brane theory. Part of this model involves the usage and implications of a certain Dutch equation in which the Planck scale is seen as a variable as far as size goes. A compact extra dimension has a completely different effect on the Newtonian Force law. In a D-dimensional space with one dimension compactified on circle of Radius R with an angular coordinates that is periodic with period 2p, the line Element becomes

The force law derived from the potential that solves the Laplace equation Becomes

noncompact space dimensions, then D=4, but D-2=2, so the force law is still an inverse square law. The Newtonian force law only cares about the number of noncompact dimensions. At distances much larger than R, An extra compact dimension can't be detected gravitationally by an altered force law. But you might be asking how this extra space manages to act like its horizon is set at the horizon of our universe? Part of the answer lies in its own local velocity of light. If that velocity crosses its own universe in 1 second then in essence as you shrink that universe in volume size one still has a lightcone extending far further than our own. When you try to compare both these frames even though C is a constant in any one frame the velocity of C remains different from each other. The result is any information carried from our space-time through it seems to transfer non-local due to differences in our measuring rod, while information transferred from hyperspace to here is forced to remain local so that we only get a fraction of the total information. This is where the difference between quantum derived expectation values for the ZPF and observed values comes into play. Quantum Theory deals with the Planck scale. By nature it measures value from this external frame of reference and derives answers that do not equal those based upon observation. If one knows the actual velocity of C within hyperspace one can reduce those answers back to our observed ones simply by division of those answers by that value for C there. That leads one to assume that the local velocity of C is some 120 powers higher in hyperspace than here. Such a large velocity as far as localized lab experiments go would seem infinite. But if we could perform quantum information transfer via entanglement over a very large distance then one could detect that actual local value for C in hyperspace. Dirac waves transfer through hyperspace the same as they do here using the model I have proposed. The difference is in the wavelength spread due to the much faster local velocity of C. The energy spectrum is simply spread out to the point that we can only measure a small fraction of its total energy per Planck unit here. That’s why we observe an energy for the vacuum some 120 powers smaller than theory predicts. Its actual energy is the higher value. But we only see part of the picture due to the wave function spread. The only thing required to solve this quantum problem is the acceptance of a two reference frame system instead of one. The effect of adding an extra compact dimension is more subtle than that. It causes the effective gravitational constant to change by a factor of the volume 2pR of the compact

dimension. If R is very small, then gravity is going to be stronger in the lower dimensional compactified theory than in the full higher dimensional theory. So if this were our Universe, then Newton's constant that we measure in our noncompact 3 space dimensions would have a strength equal to the full Newton's constant of the total 4-dimensional space, divided by the volume of the compact dimension. The actual volume internal for hyperspace is set by its lightcone horizon. In hyperspace all four forces (strong, weak, EM, and Gravity) are equal. But their transfer into our noncompact 3 space dimensions alters these forces to all look different. This leads then to the issue that quantum information is different from normal information, yet, it in its own frame it is the same. In theory, normal information could be sent through hyperspace. But to get the correct picture of that information so as to restore it correctly we’d have to measure the return over a far longer time period. What we’d get is just bits of the information that we’d have to add together to get the whole message. In essence every EM signal ever sent out has traveled through hyperspace. But we only get the results back in a limited fashion here because of the frame difference. In essence those signals traveled ahead in time all the way to their course end in a fraction of a second there. But we only arrive at that point here in a much slower time rate. Consider a 5-dimensional space-time with space coordinates x1,x2,x3,x4 and time coordinate x0, where the x4 coordinate is rolled up into a circle of radius R so that x 4 is the same as x4+2pR

Suppose the metric components are all independent of x4. The space-time metric can be decomposed into components with indices in the three noncompact directions (signified by a,b below) or with indices in the x4 direction:

The four ga4 components of the metric look like the components of a space-time vector in four space-time dimensions that could be identified with the vector potential of electromagnetism with the usual field strength Fab

The field strength is invariant under a a reparametrization of the compact x4 dimension via

which acts like a U(1) gauge transformation, as it should if this is to act like electromagnetism. This field obeys the expected equations of motion for an electromagnetic vector potential in four space-time dimensions. The g44 component of the metric acts like a scalar field and also has the appropriate equations of motion. In this model a theory with a gravitational force in five space-time dimensions becomes a theory in four space-time dimensions with three forces: gravitational, electromagnetic, and scalar. But the idea that Dirac waves can carry through in hyperspace also brings up in itself that there is more than 1 extra dimension at play here. The solution for the scalar field φ smoothly interpolates between the two attractor solutions, the function A(r) is singular. It behaves as log |r| at |r| → 0. Metric near the domain wall is given by: ds2 = r2dxμdxνημν+ dr2 This implies the existence of the curvature singularity at r = 0, which separates the universe into two parts corresponding to the two different attractors each with their own respective space-time. The relevant equation of motion for the interpolating scalars in the background metric is: φ”+(4A’+g,φ/g)(φ’)φ’+6g-1 P,φ=0 where at the critical points P,φφ is positive. If we assume that the solution of this equation asymptotically approaches an attractor point φcr at large |r| > 0, so that g and g,φ are then constant, A’ becomes negative constant, and φ’ gradually vanishes at large |r|. Then the deviation δφ gradually vanishes at large |r|. Then the deviation δφ of the field φ from its asymptotic value φcr at large |r| satisfies the following equation: δφ”− 4|A′|δφ’= −6|g−1P,φφ | δφ, This is equation for a harmonic oscillator with a negative friction term −|A′| δφ’. Solutions of this equation describe oscillations of δφ with amplitude blowing up at large | r|. But I think the solution to this runaway inflation at large |r| is exactly that found with the variable Planck scale where at large |r| the universe simply recycles finding itself back in only the enlarged Planck scale state it started with because the far side of the harmonic oscillator at large |r| equals the initial stage of rebound in the first place. As a further solution to the problems presented above we will first look at the Planck scale itself. The Planck scale can be written as a function of some very well known constants for which its expression was obtained by a research group at the University of Amsterdam Holland[1]. In the Dutch equation R=4Ρie20Gh-cross2m0/e0

G=6.67 * 10-11Nm2/Kg2, h-cross=6.626/2Pi * 10-34J s, e=1.6 * 10-19C, m0=4P * 10-7H/m, and e0=8.854157817 * 10-12F/m to yield the known present vacuum state. Allow that the value of e0 has varied higher over time during the history of the cosmos one finds that the Planck scale would become larger as one went backwards in time, small at the BB stage, increasing with time as the universe expands and local energy density begins to lower, and eventually becoming large again to start the whole cycle over. At the BB stage the effect here would be the same in a forward time fashion as the compacting process of String Theory making the Planck scale itself equal to this hidden extra dimensional set. An interesting comment can be made about the properties of the excitations around the two gauged theory vacua. The gravitino mass near one critical point is positive Mgrav=Z−> 0 and the one near the second critical point is negative M+grav = Z+ < 0 since its value at each critical point is the value of the central charge. This would lead to an explanation of the different values of C in hyperspace compared to our 3-brane. This includes a change of γμ matrices into − γμ, as well as of the representation of the little part SO(4) of the Lorentz group. We need both versions of the theory to make acceptable not only the vacuum state but also the excitations around each vacuum. We also need that rebound energy state from out of LQFT to account for why inflation took place and why there would never be a singularity in our model. It’s the large |r| and the local 3-brane decrease in energy density that allows the Planck scale to increase in volume and force a recycle stage prior to runaway inflation and deflation in either side of the oscillator. It is also the divergence at large |r| that accounts for the accelerated expansion seen by observation. But the above also supplies a solution the Pioneer slowdown problem. If C can vary as the Planck scale varies from region to region then it is not General Relativity that needs to be modified at all. It would also account for why this slowdown seems to be pointed sunward since the Sun has the most mass density in our local area. THE MODEL: The Friedmann-Lemaître-Robertson-Walker (FLRW) metric: ds2FLRW=-dt2+a2(t)[(dr2/1-kr2)+r2(dΘ2+sin2Θdφ2)] describes a homogeneous and isotropic universe. Here τ is cosmological time, (r, θ, ϕ) are comoving coordinates, a is the scale factor and k = 0, ±1 the curvature index. The proper radial distance is defined as ar. FLRW branes with k = 0 and brane cosmological constant Λ, embedded symmetrically. The bulk is the Vaidya-anti de Sitter space-time with cosmological constant , and it contains bulk black holes with masses m on both sides of the brane. The black hole masses can change if the brane radiates into the bulk. An ansatz comparable with structure formation has been advanced for the Weyl fluid m/a4 for the case when the brane radiates, m = m0aα, where m0 is a constant and α = 2, 3. For α = 0 the Weyl fluid is known as dark radiation and then the bulk space-time becomes Schwarzschild-anti de Sitter. The brane tension and the two cosmological constants are inter-related as

2Λ=K2λ+k2Λ’ The Friedmann equation gives the Hubble parameter to Λ, m, the scale factor a and the matter energy density ρ on the brane: H2=Λ/3+(K2 ρ/3)[1+(p/2λ)]+(2mo/a4-α) It is normally assumed that in the matter dominated era the brane is dominated by dust, obeying the continuity equation ρ+3Hρ which gives ρ ~ a3. But given the modification a variable Planck scale would add to such a model the brane may be dominated by dust and vacuum pressure differences at the same time. With α =<> 0, the Weyl fluid is itself a variable and then the bulk space-time becomes a variable from any of the Schwarzschild de Sitter types. In this case the universe is no longer a homogeneous and isotropic universe. It in fact would be a composite whose general global pattern tends to fit the homogeneous and isotropic universe type with Schwarzschild-anti de Sitter the global normal on the bulk space-time. If you add in the similar aspect from out of Loop Quantum Gravity of no singularity point and a recycle from this model then some matter may be present from other cycles which would tend over several cycle histories to push the cosmos eventually towards a dust dominated model which is either flat or collapsing due to a rise in matter/energy density over that history. At that future point given all the variables no one can predict with accuracy which it will end up in even though I would tend to wager towards the latter. Given all the above I would see our cosmos is somewhere in the early cycle succession stage based upon aspects from observable cosmology at present. REFERENCES: 1,) Stefan Kowalczyk, Quinten Krijger, Maarten Van Der Ment, Jorn Mossel, Gerben Schoonveldt, Bart Verdoen, Contraints on Large Extra Dimensions (pp12 eq. 14) 2.) Fernando Loup , Paulo Alexandre Santos, and Dorabella Martins da Silva Santos: Can Geodesics in Extra Dimensions Solve the Cosmic Light Speed Limit General Relativity and Gravitation 35(10) p.1849-1855 October 2003 AUTHOR”S NOTES :

1.) This model I am using has similar properties to the one used under Double Special Relativity (see: Jerzy Kowalski-Glikman, Sebastian Nowak, Noncommutative space-time of Doubly Special Relativity theories, Int.J.Mod.Phys. D12 (2003) 299-316). But the two frames system here is different from the one employed there owning to the PV nature of this model. The actual model basis employed and its implications can be found at: Hyperspace a Vanishing act, http://doc.cern.ch//archive/electronic/other/ext/ext-2004109.pdf, Implication of the Dutch Equation Modified PV Model, http://doc.cern.ch//archive/electronic/other/ext/ext-2004-115.pdf, and Why Quantum Theory does not fit observational data, http://doc.cern.ch//archive/electronic/other/ext/ext-2004-116.pdf 2.) The strongest basis for assuming that C stays constant in hyperspace is from observations of the CMB itself. However, it is possible that C may also vary in hyperspace. The implications of such have not been worked out in this model to date. Also to be noted the K=0 in the original paper assumes that value for hyperspace itself before inflation took place. The best fit currently with our space-time is that K would equal 1. The usage of K=0 was to simplify the modeling. In reality I suspect that K=1 for both space-time frames. At one time I had played with modeling K as a variable with extra values to simulate a PV model where one can account for the Pioneer Problem and an older PV based problem where C ought to increase instead of observational evidence that it either stays constant or slows with time. 3.) Aside from the variable planck scale idea one is left with these solutions: 1. The equation of state w may differ from -1 by an observable amount, or may change rapidly with time. 2. The dark energy may be a scalar field, coupled to matter in such a way as to cause time variations in fundamental constants, or to violate the gravitational equivalence principle. 3. The cosmic acceleration may be caused by a deviation from general relativistic gravity on cosmological scales, rather than by a dark energy. None of the above alternatives seems viable at this time. 4.) DIFFERENT BRANE MODELS:

This figure shows how different theories vary from each other as far as classification goes. All modeling is based upon the Friedmann-Lemaître-Robertson-Walker (FLRW) metric later modified by Loop Quantum Gravity. 5.) Using Λ=(β/β-3) 4ΠGρ from a non-viscous model (Arbab 2002). This form is interesting since it relates the vacuum energy directly to the matter content in the universe. Hence, any change in ρ will immediately imply a change in Λ. This would apply both locally and globally which is also consistant with the variable Planck scale model: ( Metric used here comes from Arbab I. Arbab The Universe With Bulk Viscosity, Chin. J. Astron. Astrophys. Vol. 3 (2003), No. 2, 113–118 ( http://www.chjaa.org or http://chjaa.bao.ac.cn with a different equation of state) In a flat Robertson Walker metric ds2=dt2-R2(t)(dr2+r2dΘ2+r2sinΘ2dφ2) Einstein’s field equations with a time-dependent G and ¤ read (Weinberg 1971)

Rμν -1/2(gμνR)=8ΠG(t)Tμν+Λ(t)gμν, Now cosmologists believe that Λ is not identically, but very close to zero. They relate this constant to the vacuum energy that first inflated our universe, causing it to expand. From the point of view of particle physics, a vacuum energy could correspond to a quantum field that is diluted to its present small value. However, other cosmologists dictate a time variation of this constant in order to account for its present smallness. The variation of this constant could resolve some of the standard model problems. Like G, the constant Λ is a gravity coupling and both should therefore be treated on an equal footing. A proper way in which G varies is incorporated in the Brans-Dicke theory (Brans & Dicke 1961). In this theory G is related to a scalar field that shares the long range interaction with gravity. Considering the imperfect-fluid energy momentum tensor Tμν=(p+p’)uμuν-p’gμν, this yields the two independent equations, 3(Ř/R)=4ΠG(3p’+p)+Λ, and 3(Ř/R)2=8ΠGp+(Λ/8ΠG)R, Elimination of Ř between the first and the second differentiated form of the equation gives 3(p’+p)Ř=-((Ĝ/G)p+p’+ Λ/8ΠG)R, where a dot denotes differentiation with respect to time t and p’=p-3ηH, η being the coefficient of bulk viscosity, H the Hubble constant. The equation of state relates the pressure (p) and the energy density (p’) of the cosmic fluid: p=1/((γ-1)p’), where γ=constant. Vanishing of the covariant divergence of the Einstein tensor and the usual energy-momentum conservation relation Tμνν=0 leads to 8ΠGp+Λ=0 and p’+3(p++p)H=0

One finds that the bulk viscosity appears as a source term in the energy conservation equation. Now if we consider the very special form (Arbab 1997), Λ=3βH2, β=constant, and η=ηopn, ηo≥0, n=constant. This is equal to writing Λ=(β/β-3)4ΠGp for a non-viscous model (Arbab 2002). This form relates the vacuum energy directly to the matter content in the universe or in any local region. Hence, any change in p will immediately imply a change in Λ, i.e., if p varies with the cosmic time then Λ also varies with the cosmic time. If local matter content varies so will the vacuum energy density. Now what makes the difference is weather we have a decaying mode of the energy density or an increasing mode of energy density globally and locally. From an expansion of the cosmos perspective the mode is decaying while locally it will vary in mode. This yields a stress energy tensor that also varies locally which leads to C being a variable itself which indirectly implies that certain values of the Dutch Equation are a variable. That translates to the Planck scale as a variable. While the above equations are just one way we could express the general idea from a cosmological perspective the example does point out that the general idea of expanding a bubble of hyperspace around a craft is possible. I would also suggest this same modeling offers a possible solution to some of our own sub-light methods of propulsion. One other implication of this is the system has to always have some local energy density present even at the start or one is left with a runway inflation of hyperspace. This issue is solved by the fact that at large expansion R the Planck scale has a near infinite probability of having virtual particles borrow enough energy to become real particles which starts the collapse of the planck scale towards a second rebound point guaranteed by aspects of Loup Quantum Gravity. This implies that not only may some matter transfer over from other cycles accounting for more of certain elements than the Standard model can account for. It also implies some elementary particles have always been present even if the rebound point during collapse tends towards a high enough temperature to wipe a lot of the entropy history clean. The actual Dutch Equation is: R=4Π2oGh-cross2mo/eo G=6.67 * 10-11Nm2/Kg2, h-cross=6.626/2Pi * 10-34J s, e=1.6 * 10-19C, m0=4P * 10-7H/m, and e0=8.854157817 * 10-12F/m to yield the known present vacuum state.

DISCUSSION AND CONCLUSIONS The existence of horizons of knowledge in cosmology, indicate that as a horizon is approached, ambiguity as to a unique view of the universe sets in. It was precisely these circumstances that apply at the quantum level, requiring that complementary constructs be employed (Bohr 1961). Today we stand on another horizon that seems to be displaying ambiguity. This if forcing us as Scientists to rethink established dogma and cosmological views. It has been my attempt here to offer just such a rework of an older model into something that fits the observational evidence a bit closer. Further testing of this idea could come via study of signals from more than one probe outside of our solar system. It could come from experimenting with some of Fernando Loup’s ideas. It could come from ways I have not even thought of at present. At present the Universe is considered a general relativistic Friedmann space-time with flat spatial sections, containing more than 70% dark energy and at about 25% of dark matter. Dark energy could be simply a cosmological constant Λ, or quintessence or something entirely different. There is no widely accepted explanation for the nature of any of the dark matter or dark energy (even the existence of the cosmological constant remains unexplained). This has been my attempt to come up with a solution that fits the observational mixed signals at present.