Finding the Planck Length Independent of Newton s Gravitational Constant and the Planck Constant The Compton Clock Model of Matter

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1 Finding the Plank Length Independent of Newton s Gravitational Constant and the Plank Constant The Compton Clok Model of Matter Espen Gaarder Haug Norwegian University of Life Sienes September 9, 08 In modern physis, it is assumed that the Plank length is a derived onstant from Newton s gravitational onstant, the Plank onstant and the speed of light, = G 3 This was first disovered by Max Plank in 899 We suggest a way to find the Plank length independent of any knowledge of Newton s gravitational onstant or the Plank onstant, but still dependent on the speed of light (diretly or indiretly) Key words: Plank length, Plank onstant, Newton s gravitational onstant, Compton frequeny, Shwarzshild radius I INTRODUCTION In 899, Max Plank [, ] introdued what he alled the natural units : the Plank mass, the Plank length, the Plank time, and the Plank energy He derived these units using dimensional analysis, assuming that Newton s gravitational onstant, the Plank onstant, and the speed of light were the most important universal onstants The Max Plank formula for the Plank length is given by = G 3 () In other words, it seems that we need to know Newton s gravitational onstant G and the Plank onstant and the speed of light to find the Plank length It has therefore been assumed that the Plank length is a derived onstant and that Newton s gravitational onstant is a more fundamental onstant, a view we will hallenge here II PLANCK LENGTH INDEPENDENT OF G AND First, find the redued Compton wavelength of an eletron by Compton sattering The redued Compton frequeny is then given by f C = λe () where f C stands for the redued Compton frequeny, and λ e is the redued Compton wavelength of the eletron Further, the Cylotron frequeny is linearly proportional to the redued Compton frequeny Conduting a ylotron experiment, one an find the redued Compton frequeny ratio between the proton and the eletron For example, [3] measured it to be about λ P λ e = f C,P f C,e = (76) (3) In fat, [3] measured the Proton eletron mass ratio this way, but the redued Compton frequeny is only a deeper aspet of mass, that has reently also been more or less onfirmed by experimental researh Theoretially, it is no surprise that f C,P f C,e = m P m e We simply laim that a simpler and deeper way to express mass is through the redued Compton frequeny in matter; see [4] Next, from the Shwarzshild metri [5, 6] solution of the Einstein field equation [7], we have the well-known formula for the Shwarzshild radius r s = GM (4) Haug [4] reently pointed out that the Shwarzshild radius is also equal to twie the redued Compton frequeny of the gravity objet over a time period of one Plank seond, multiplied by the Plank length That is, we must have r s = GM = f C t p = λ = l p λ (5) This means that the Shwarzshild radius ontains the Plank length Still, is there a way to extrat this without knowing the mass and Newton s gravitational onstant? Haug [8] reently showed that the Shwarzshild radius an be measured in a series of ways with no knowledge of the mass size or Newton s gravitational onstant For example, the Shwarzshild radius of the earth an be found from the gravitational aeleration field, the speed of light and the radius of the earth

2 r s = gr (6) where g is the gravitational aeleration and R is the radius of the gravity objet For smaller size objets, we an find the Shwarzshild radius using a Cavendish apparatus; it is given by r s = L4π R θ T (7) where L is the distane between the small balls and R is the distane, enter to enter, between the small and large ball T is the natural resonant osillation period of a torsion balane, θ is the defletion angle of the balane This is somewhat different than the known use of a Cavendish apparatus used to find the weight of the Earth or to find G, where one needs to know the mass size (weight) of the lead balls Again, we do not need to know the mass size in the normal sense nor do we need to know anything about Newton s gravitational onstant to find the Shwarzshild radius So, in pratie, take a lump of matter, divide it in two and make two idential balls of it Plae the two balls in the Cavendish apparatus In addition, one needs two muh smaller balls By measuring T, L, R and θ, we know the Shwarzshild radius of the large balls in the apparatus Assume that we measure a Shwarzshild radius of meters Next, we find the redued Compton frequeny of an eletron: it is f C,e = λe (8) Next, we use the ylotron to find the redued Compton frequeny of a Proton f C,P = λe (9) Next, we break the large balls up into piees and ount the number of protons in it Or, alternatively, we an pak a known amount of atoms together in a ball, and keep trak of the number of protons This is, in theory at least, possible Assume that we ount approximately protons in the mass, and we next divide it into two The redued Compton frequeny is additive for a larger mass, so based on the proton ount the redued Compton frequeny in eah of the large balls must be approximately f C = (0) Remember that the Shwarzshild radius that we measured was m To find the Plank length, we an use the following formula This gives us r s = f C r s = () f C m This also means that the Plank length is always the square root of half the Shwarzshild radius times the redued Compton wavelength of the mass = r s λ () The whole method requires no knowledge of traditional mass size measures, whih also means no knowledge of the Plank onstant, and no knowledge of Newton s gravitational onstant G III THE PLANCK TIME AND THE PLANCK MASS After we have found the Plank length, ompletely independent of the Plank onstant and the gravitational onstant, we an naturally easily find the Plank time as it is simply the Plank length divided by the speed of light However, the Plank mass and all masses at the deepest level an be desribed as Compton frequeny per Plank seond This means that the Plank mass in this view is ˆm p = f C t p = = (3) That is, the Plank mass is one in terms of frequeny per Plank seond That mass is related to Compton time has also been supported in reent experimental researh Dole and Perali [9] onlude that the rest-mass of a partile is assoiated to a rest periodiity known as Compton periodiity Additionally, [0] onluded that This diretly demonstrates the onnetion between time and mass In our view, the redued Compton frequeny hidden inside matter is the key to really understand how to extrat the Plank length independent of the Plank onstant and to understand mass and gravity from a deeper perspetive

3 3 IV AN ALTERNATIVE WAY THAT IS INDEPENDENT OF G BUT DEPENDENT ON KNOWING At a deeper level then, the Plank onstant must somehow ontain an embedded redued Compton frequeny for one kg The redued Compton wavelength for kg is λ = kg = The redued Compton frequeny of kg is therefore f C,kg = kg = kg (4) (5) That is, is simply the frequeny of kg Next, let us make two balls with weight kg and also two muh smaller balls to use in a Cavendish apparatus Next, we measure the Shwarzshild radius using the apparatus There is no need to know the Plank onstant or the weight of the balls to find the Shwarzshild radius It is when we are going to extrat the Plank length from the Shwarzshild radius that we, in this method, will utilize the Plank onstant This is beause the Plank onstant is needed to extrat the redued Compton frequeny in the matter when working with kg In the setion above, we had to ount the number of protons, but if working from kg instead we an use the Plank onstant to extrat the redued Compton frequeny The Plank length from the measured Shwarzshild radius in kg is simply equal to = r s,kg f C m (6) s It is also worth mentioning that the Plank frequeny is given by the redued Compton frequeny of the matter in question multiplied by the Plank length divided by half the Shwarzshild radius f C,p = f C r s = (7) V RELATION TO MASS AS KG In the above setions, we have shown that we an find the Plank length independent of any knowledge of Newton s gravitational onstant We have also laimed that the Plank mass ould simply be desribed by its Compton frequeny It is only if we want to onvert the Plank mass frequeny per Plank seond into kg that we really need the Plank onstant Any mass in kg an be found if we know the redued Compton wavelength of the mass in question m = λ (8) We have found the Plank length independent of G and, but need the Plank onstant to find the Plank mass in terms of kg This gives m p = = kg (9) The redued Compton frequeny of kg is, as we laimed earlier in this paper, f C,kg = kg = (0) This means the Plank onstant is equal to = () Now let us replae this into the mass equation m = λ = λ = λ = f C f C,kg () That is, any mass in kg form is simply the redued Compton frequeny of the mass in question divided by the redued Compton frequeny of kg So, the Plank onstant is simply needed to make our mass unit into a kg measure/definition of mass An alternative would be to desribe mass as the redued Compton frequeny over one Plank seond This means that all rest-masses an now be simply desribed as ˆm = f C t p = λ l = p λ λ (3) Any mass smaller than a Plank mass will then somehow be probabilisti as it means a frequeny of less than one per Plank seond This ould explain why quantum mehanis is neessary for the quantum world, but not in the marosopi world; see [4] for more details along these lines

4 4 VI WHY NEWTON S GRAVITATIONAL CONSTANT IS A COMPOSITE CONSTANT Haug has, in a series of papers [ 3], suggested that Newton s gravitational onstant is a omposite onstant of the form G = l p 3 (4) whih is basially the same as the MCulloh [4] onstant m sine m p = p Here, we will list some reasons that seem to support the fat that Newton s gravitational onstant is a Composite onstant rather than a deep fundamental onstant Already, when looking at the output units of Newton s gravitational onstant m 3 kg s, one should beome suspiious that this must be a omposite onstant It is highly unlikely that the universe invented something fundamental that has some omplex units Muh more likely is that the Plank length is something very fundamental Everyone an intuitively understand, to a ertain extent, what a length is One an also, to a large degree, understand what the speed of light is: it is how far the light travels in a given time period The Plank length is the shortest possible redued Compton wavelength The Plank length an be found ompletely independent of any knowledge of Newton s gravitational onstant; it an also be found independent of the Plank onstant, as shown in this paper The Plank onstant in the Max Plank formula, in our view, is atually needed in that formula to anel out the Plank onstant embedded in Newton s gravitational onstant By re-formulating G as a omposite of the form G = l p 3, a long series of Plank units is dramatially simplified and beomes more logial; see [] G 5 The Plank time desribed as t p = gives minimal intuition We may ask, what is the logial meaning of 5 and what is the deep logi behind the gravitational onstant? First, when replaing G with its omposite form, we see that the Plank time is simply t p = lp, so the time it takes for light to travel the Plank length is naturally known All the Plank units seem mystial when alulated from Newton s gravitational onstant and the Plank onstant Only when we have an output in kg, do we need the Plank onstant We do not question that Newton s gravitational onstant is a universal onstant, but it is, in our view, learly a omposite onstant that onsists of more fundamental entities VII CONCLUSION We have, for the first time, shown how the Plank length an be measured without any knowledge of Newton s gravitational onstant and the Plank onstant It seems that the Compton frequeny of matter is what is important for understanding mass at a deeper level Further, the Shwarzshild radius is the redued Compton frequeny over the Plank time period multiplied by the Plank length All we need to extrat the Plank length is to know the redued Compton frequeny of matter and the Shwarzshild radius The Shwarzshild radius also seems to play a entral role for gravity We laim that it is the redued Compton frequeny in matter, over the shortest time intervaossible, multiplied by the shortest length possible This view on the Shwarzshild radius seems to possibly be key to understanding gravity at a deeper level This should enourage further investigation, theoretially and experimentally [] Max Plank Naturlishe Maasseinheiten Der Königlih Preussishen Akademie Der Wissenshaften, p 479, 899 [] Max Plank Vorlesungen über die Theorie der Wärmestrahlung Leipzig: JA Barth, p 63, see also the English translation The Theory of Radiation (959) Dover, 906 [3] RS Van Dyk, FL Moore, DL Farnham, and PB Shwinberg New measurement of the proton-eletron mass ratio International Journal of Mass Spetrometry and Ion Proesses, 66 [4] E G Haug Revisiting the derivation of Heisenberg s unertainty priniple: The ollapse of unertainty at the Plank sale preprintsorg, 08 [5] K Shwarzshild Über das gravitationsfeld eines massenpunktes nah der einsteinshen theorie Sitzungsberihte der Deutshen Akademie der Wissenshaften zu Berlin, Klasse fur Mathematik, Physik, und Tehnik, page 89, 96 [6] K Shwarzshild über das gravitationsfeld einer kugel aus inkompressibler flussigkeit nah der einsteinshen theorie Sitzungsberihte der Deutshen Akademie der Wissenshaften zu Berlin, Klasse fur Mathematik, Physik, und Tehnik, page 44, 96 [7] Albert Einstein Näherungsweise integration der feldgleihungen der gravitation Sitzungsberihte der Königlih Preussishen Akademie der Wissenshaften Berlin, 96 [8] E G Haug Gravity without Newton s gravitational onstant and no knowledge of mass size preprintsorg, 08 [9] D Dole and Perali On the Compton lok and the undulatory nature of partile mass in graphene systems The European Physial Journal Plus, 30(4), 05 [0] S Lan, P Kuan, B Estey, D English, J M Brown, M A Hohensee, and Müller A lok diretly linking

5 5 time to a partile s mass Siene, 339, 03 [] E G Haug The gravitational onstant and the Plank units a deeper understanding of the quantum realm wwwvixraorg , 06 [] E G Haug Plank quantization of newton and einstein gravitation International Journal of Astronomy and Astrophysis, 6(), 06 [3] E G Haug Newton and Einstein s gravity in a new perspetive for Plank masses and smaller sized objets International Journal of Astronomy and Astrophysis, 08 [4] M E MCulloh Gravity from the unertainty priniple Astrophysis and Spae Siene, 349(), 04