The Mass Gap, Kg, the Plank Constant and the Gravity Gap The Plank Constant Is a Composite Constant One kg Is 85465435748 0 36 Collisions per Seond The Mass Gap Is.734 0 5 kg and also m p The Possibility of EmDrive? Espen Gaarder Haug Norwegian University of Life Sienes April 4, 07 Abstrat In this paper we disuss and alulate the mass gap. Based on the mass gap we are redefining what a kilogram likely truly represents. This enables us to redefine the Plank onstant into what we onsider to be more fundamental units. Part of the analysis is based on reent developments in mathematial atomism. Haug [, ] has shown that all of Einstein s speial relativity mathematial end results [3] an be derived from two postulates in atomism. However, atomism gives some additional boundary onditions and removes a series of infinite hallenges in physis in a very simple and logial way. While the mass gap in quantum field theory is an unsolved mystery, under atomism we have an easily defined, disrete and exat mass gap. The minimum rest-mass that exists above zero is.734 0 5 kg, assuming the observational time window of one seond. Under our theory it seems meaningless to talk about a mass gap without also talking about the observational time-window. The mass gap in one Plank seond is the Plank mass. Further, the mass gap of just.734 0 5 kg has a relativisti mass equal to the Plank mass. The very fundamental partile that makes up all mass and energy has a rest-mass of.734 0 5 kg. This is also equivalent to a Plank mass that lasts for one Plank seond. In this paper we are not trying to solve the Millennium mass gap problem in terms of the Yang-Mills theory. We think the world is better understood by atomism and its reent mathematial framework. Whether or not a link between these two theories exists, we may leave up to others to find out. Keywords: Composite onstant, kg, mass gap, Plank mass, relativisti mass, atomism, partile frequeny. Introdution The Plank onstant was introdued by Max Plank [4] in 900. The Plank onstant is linked to the idea that energy omes in quanta and plays a entral role in all of quantum mehanis. The Plank onstant is one of the fundamental onstants that have been most aurately measured, in ontrast to Newton s gravitational onstant G, for example, where there still is onsiderable unertainty in what its exat value should be. See [5, 6, 7], for example. Reent researh related to the Watt balane also makes the Plank onstant very entral in relation to possibly redefining the kilogram, see [8]. for example. In this paper we suggest that the Plank onstant is a omposite onstant and that by breaking it down into what it truly represents we are able to better understand the Plank onstant. This in turn helps us to understand the mass gap and also to redefine the Plank onstant, find its exat value, and (extending that finding) to redefine a kg with an exat value. Any fundamental partile mass an be written as m = e-mail espenhaug@ma.om. Thanks to Vitoria Teres for helping me edit this manusript. ()
where is the redued Plank onstant, is the redued Compton wavelength of the partile in question, and is the speed of light The output unit is then in units of kg. The speed of light is simply the distane light travelled for a given time period. The speed of light is typially given in meters per seond. We all know roughly what a meter is and what a seond is; they are something we an all relate to. Further, the redued Compton wavelength is a length again, we an relate to a length. On the other hand, the redued Plank onstant in terms of SI units is given as kg m /s. I think few if any of us an relate to what this represents exatly, kg times meters squared per seond. What kind of exoti animal is that? This omplex notation alone seems to give a hint that the Plank onstant is a omposite onstant that we an break down in far simpler and more intuitive fundamental onstants. The Mass Gap and the Kg We will assume that at the very depth of reality there only exists one type of fundamental partile, namely indivisible partiles, always moving at the speed of light; see [, ]. In this model we will have a binary system of energy and matter. We have matter with rest-mass when two indivisible partiles ollide, and we an all the indivisible partiles energy when they not are olliding. We assume these indivisible partiles always move at the speed of light. An exeption is in the very ollision point when two indivisible partile ounter-strike (ollide) and hanges their diretion of movement. We will laim a single ollision between two indivisible partiles is equal to a mass of.734 0 5 kg, if the observational period is one seond. This is what we will all the mass gap. This is the minimum mass we an observe within a seond; in other words, the mass gap within a seond. If we look away from units for a moment this is the same value as given by the redued Plank onstant divided by :.734 0 5 () The units here are not in kg, something we soon will get bak to. Under atomism any known subatomi partile is rapidly flutuating between mass (the mass gap) and internal energy. Based on atomism (see [, ]) mass is simply ounter-strikes between indivisible partiles. When two indivisible partiles ounter-strike (ollide), we define this as mass, and when they do not ounterstrike, they are internal energy. An eletron, for example, an simply be thought of as two indivisible partiles traveling bak and forth, eah over a distane equal to the redued Compton wavelength of the eletron. Based on this senario, an eletron has the following number of internal ounter-strikes per seond e 7.76344 0 0 ounter-strikes per seond (3) The eletron is rapidly flutuating between energy and mass 7.76344 0 0 times per seond. At eah ounter-strike we have a mass of.734 0 5 kg. That is to say the total rest-mass of an eletron is m e = e m g = e.734 0 5 9.094 0 3 (4) where m g is the mass gap. The mass gap.734 0 5 kg is interestingly also equal to the mass of one Plank mass for one Plank seond. Looking away from units for a moment, we have m g = = m pt p.734 0 5 (5) However, is not the notation of kg, but rather kg s. In our atomist model if we have a time window of one seond then the maximum redued Compton wavelength we an have in a mass in order not to reah zero mass is g = = 9979458 m (6) m g where m g is the mass in kg of the mass gap. We suggest that there is a very important redued Compton wavelength equal to the distane light travels in one seond (when we operate with the speed of light in terms More preisely the round-trip speed of light, or the one-way speed of light as measured with Einstein-Poinaré synhronized loks.
of meters per seond). The redued Compton wavelength with a distane equal to the distane the light travels per time unit hosen is oneptually important, as it is linked to the mass gap, in our view. Solved with respet to the mass gap, m g, we get m g = g = 9979458.7337 0 5 kg (7) This is equal in value to, but one of the s is atually the redued Compton wavelength, and now the output of the mass gap is in kg. This an be used to better understand what one kg truly represents at a deeper level. If one ounter-strike is equal to the mass gap, then one kg must be equal to the following number of ounter-strikes per seond One kg in terms of number of ounter-strikes = m g =.7337 0 5 8.547 0 (8) This shows that one kg is an enormous amount of ounter-strikes between indivisible partiles per seond. One kg is related to 8.547 0 ounter-strikes between indivisible partiles per seond. Based on this observation, we an also better understand the relationship between kg and fundamental partiles suh as eletrons. The eletron is ounter-striking =7.76344 0 0 times per seond. As a fration of the number e of ounter-strikes in one kg we find that an eletron mass is 7.76344 00 m e = 8.547 0 =9.0938 0 3 fration of the number of ounter-strikes in one kg (9) Any mass in kg is simply a fration of the number of ounter-strikes that exist in a kg. Further, based on this, the redued Plank onstant is simply = 8.547 0 (0) And any fundamental partile, as a fration of the number of ounter-strikes in one kg, is given by m = 8.547 0 = 8.547 0 () That is to say, any type of fundamental partile is simply the internal frequeny of ounter-strikes per seond multiplied by the mass gap. The mass gap in terms of the fration of kg ounter-strikes is m g = 8.547 0.7337 0 5 fration of the number of ounter-strikes in one kg () 3 The Mass Gap as a Funtion of the Observational Period It is important to understand that if we observed a point di erent than one seond the mass gap ould be smaller or larger than.7337 0 5 kg. If we use an observational time window shorter than one seond, the mass gap will be larger than this, and if the observational time window is larger than one seond, then the mass gap will be smaller than this. For a two-seond period the mass gap is mg and for a three-seond observation period the mass gap is mg 3. The mass gap is simply one-ounter-strike, but to talk about the mass in terms of kg we must ompare one ounter-strike with the number of ounter-strikes in a kg during the same time period. And per our definition, one kg will be approximately 8.547 0 ounter-strikes per seond and naturally 8.547 0 per two seonds. One kg is always one kg, but the mass gap hanges with the time window. Hypothetially, the shortest time window that is likely possible is one Plank seond. In one Plank seond, one kg is approximately 459459.3 ounter-strikes. And the mass gap is always one ounter-strike, and one ounter-strike as a fration of the number of ounter-strikes in one Plank seond for a kg is m g = 459459.3.765 0 8 = m p (3) That is to say, the mass gap is one Plank mass for one Plank seond. This is onsistent with Haug s atomist model for anything with rest-mass, where he has laimed all known subatomi partiles onsist of Plank masses that lasts for one Plank seond and this yle is repeated many times per seond based on 3
the subatomi partile frequeny. This also means that any mass with a mass larger than the mass gap (as measured in one Plank seond) annot be one single fundamental partile, but must onsist of two or more of the most fundamental partile. In this view, one fundamental partile most likely onsists of two indivisible partiles. However, there ould be some modifiations here without altering the main onept of our theory. This does not mean that one kg has a lower mass the shorter the time period over whih we measure. Using kg is simply making use of a standardized referene for a given amount of matter. If we observed the number of ounter-strikes in one kg over half a seond, then the number of ounter-strikes in that kg would be 8.547 0. Aordingly, the number of ounter-strikes in any known subatomi partile would also be redued in half ompared to what they ahieve in one seond. The relative mass is invariant to what time frame we look at. However, the mass gap will vary for di erent time windows. This is beause the mass gap always is only one ounter-strike. 4 Frequeny Summary The table summarizes how mass for any subatomi partile or even omposite matter an be desribed as a number of ounter-strikes per seond. One kg is the enormous amount of 8.547 0 ounter-strikes per seond and for any subatomi mass, if we want to onvert it to kg, we an find the value by dividing the subatomi partile frequeny with the number of ounter-strikes that represent one kg. Mass as frequeny Mass as kg Counter-strikes per seond frequeny ratio Mass gap for one seond m g = a m g = 8.547 0.734 0 5 kg Eletron m e = e 7.76344 0 0 m e = 7.76344 00 8.547 0 9.0938 0 3 kg Meson m m = m.04949 0 3 m m =.04949 03 8.547 0.4048 0 8 kg Muon m M = M.6053 0 3 m M =.6053 03 8.547 0.88353 0 8 kg Plank mass m p = l p.8549 0 43 m p =.8549 043 8.547 0.765 0 8 kg Proton mass m P = P.4549 0 4 m P =.4549 04 8.547 0.676 0 7 kg One kg 8.547 0 8.547 0 8.547 0 =kg One kg 459459.3 (per Plank seond) 459459.3 459459.3 =kg Mass gap for one Plank seond m g = b m g = 459459.3.765 0 08 kg Table : The table shows how subatomi partile masses an be expressed as Clok frequenies and that kg simply an be seen as a standardized frequeny ratio. a The mass gap ould be less than one ounter-strikes per seond, the mass gap is ounter-strike per any time window we hoose to measure. This simply means the minimum mass above zero simply is one ounter-strike between two indivisible partiles. b The mass gap ould be less than one ounter-strike per seond; the mass gap is ounter-strike per any time window we hoose to measure. This simply means the minimum mass above zero simply is one ounter-strike between two indivisible partiles. 5 The Redued Plank Constant and Kg Exatly Defined By defining a kg as an integer number of ounter-strikes per seond we an get an exatly defined kg measure and also an exatly defined Plank onstant. We ould define one kg as an exat number of ounter-strikes, for example One kg in terms of ounter-strikes per seond = 8.5465435748 0 (4) We ould all 85465435748 0 36 the one kg onstant. The re-defined redued Plank onstant would then be defined as exatly = 85465435748 0 36 = 9979458 85465435748 0 36 (5) 4
6 Two Faes of the Mass Gap Haug [, 9, 0, ] has reently introdued a new maximum veloity for subatomi partiles (anything with rest-mass) that is just below the speed of light given by s v max = where is the redued Compton wavelength of the partile we are trying to aelerate and l p is the Plank length, []. To observe a photon we laim the photon has to interat with something. If we want to observe a single photon within one seond, then we will laim we need a ollision between two photons (that is two indivisible partiles). This means the two indivisible partiles moving towards eah other an be seen as a mass with redued Compton wavelength equal to g if they have one ollision within one seond. And only if there is at least one ollision in observed time-window we have a mass gap. This means the maximum veloity of two photons that we atually observe within one seond is s l p v max,g = g 0.99999999999999999999999999999999999999 l p (6) 999999999999999999999999999999999999999999999999855 (7) That it is slightly below and has to do with the fat that the two photons ollided. Interestingly, the relativisti mass of the mass gap is the Plank mass m p = q m g v max,g = m g g l p = g g = l p l p This naturally also means that the rest-mass of a Plank mass partile (whih is equal in value to the Plank mass times the Plank time) is m g = m p r (8) v max,g (9) If we observe one photon to photon ollision in one seond, then eah indivisible partile (photon) has a relativisti mass equal to half of the Plank mass. At the very instant when two light partiles ollide we an onsider the veloity to be zero. Even if a single indivisible partile has a relativisti mass equal to half of the Plank mass, its rest-mass is just equal to half of the mass gap. However, the mass gap always onsists of two indivisible partiles olliding and is m p if observed in one Plank seond and.7337 0 5 kg if it is observed in a di erent time window. 5.7337 0 This means the photon has rest-mass, the rest-mass of a single photon (indivisible) partile is kg. However, this rest-mass an never observed alone, but an only be observed when the photon ollides with 5.7337 0 another indivisible partile. Both make up kg of the potential observable mass. This means the smallest mass we an observe within a seond above zero is.7337 0 5 kg. We an say eah photon (that is eah indivisible partile) has a relativisti potential mass of half the Plank mass, but a rest-mass of half the mass gap. The relativisti mass of a photon is di erent than the relativisti mass of any other partile. The relativisti mass of the photon is a mass than never omes into play, so we ould just as well laim it has no relativisti mass. For all pratial purposes photons have no relativisti mass, but only rest-mass. And this rest-mass they only have at ounter-strike with other partiles. 5.7337 0 Their minimum rest-mass is kg per seond observational window and m p for one Plank seond observational window. We think modern physis got it partly wrong. The photon has rest-mass at ollision lasting for an instant, but no relativisti mass. However, a relatively stable system of two or more indivisible partiles going bak and forth ounter-striking will also have a relativisti mass when viewed as an objet moving relative to the observer frame. 5
7 Time Dependent Mass Gap Impliations: The Possibility of the EmDrive? If the mass gap is dependent on the observational time window, then this has impliations for how we look at mass and energy. This might explain why the so- alled EmDrive (RF resonant avity thruster) seems to works. The one shape of the EmDrive will make free indivisible partiles ounters-strike more frequently in the narrow end than they do in the wide end. At eah strike, an indivisible partile has a rest-mass of.7337 0 5 kg and also an energy of m g = g 5.786 0 35 J. What modern physis alls a single photon is under atomism atually a whole frequeny of indivisible partiles traveling after eah other (Muh like what has been suggested by Newton). The so-alled wavelength is the distane between eah indivisible partile in the photon, although that is not so important in this disussion. What is important is that indivisible partiles that the photon onsists of when they are not trapped in a stable ounter-strike pattern (stable matter, matter that last onsiderably longer than a Plank seond) will have an observable energy that is dependent on number of ounter-strikes, and the number of ounter strikes is dependent on the time window. The time window an be manipulated by setting up mirrors with di erent distanes between them. The EM one drive an be seen as a series of loks laying next to eah other. At the narrow end of the one is a lok tiking frequently (the indivisible partiles are bouning more often bak and forth there), while at the wide end the lok is tiking more slowly. Eah tik in the lok has a fore of 5.786 0 35 J. The EmDrive is likely to ome into onflit with energy mass onservation in the way modern physis looks at energy and matter. Under modern physis we do not know muh about the mass gap, nor do we have a theory that shows the mass gap is dependent on the time window. This means the energy in a beam of light hitting a surfae and bouning o is following the rules of modern physis, while a beam that is bouning bak and forth between some mirrors possibly not is that well understood. The EmDrive does not seem to be in onflit with atomism, but rather is onsistent with what one ould expet from atomism. However, we are assuming that the walls in the EmDrive one are rigid and not deformed by the ounter-strikes from the indivisibles hitting them. The walls are part of a omplex system ontaining an enormous number of subatomi partiles, that under atomism onsist of indivisible partiles trapped in relatively stable systems, moving bak and forth ounter-striking with eah other. There ould be additional fators that need to be taken into aount. We do not blindly endorse the EmDrive, but think it ould be interesting to see it in the ontext of atomism. 8 The Gravity Gap We suspet our theory of the mass gap (that must be seen in onnetion to my other work on atomism) also gives us what we an all the gravity gap. The gravity gap is the smallest amount of gravity we an even hypothetially observe above zero. That is even with the finest possible instruments of the future. The gravity gap is linked to the rest-mass of the indivisible partiles. As with the mass gap, the time window for the gravity gap is important. To obtain the gravity gap for an observational time period of one seond we will use Newton s gravitational formula and get F = G mm r = G m g m g.5559 0 30 N (0) g where m g is the mass gap and is the redued Compton wavelength of the mass gap for a one seond observational time period, that is 9979458 meter. Haug [0, 3, 4] has reently suggested that big G is a universal omposite onstant that an be written in the form G = l p 3 () This formula an naturally be found by simply rewriting the Plank length formula with respet to big G. However, Haug [0] has also derived this formula from dimensional analysis as well as from Heisenberg s unertainty priniple, using his newly-introdued maximum veloity formula for matter [5]. The rewritten form of big G gives us the gravity gap from more fundamental units In value terms this is simply equal to. 6
F g = l p 3 g g g = 4 g l p g.5559 0 30 N () The gravity gap for one Plank seond should be related to the mass gap we have for one Plank seond. F g = G mm r = G m p m p 8.7948 0 44 N (3) g Unlike the mass gap, we laim that the gravity gap is linked to probability where the gravitational oupling fator l p then l p e is a onditional probability as suggested by [6]. In the speial ase of eletrons we have = e and is the dimensionless gravitational oupling onstant. We think it is better to all it a dimensionless gravitational oupling fator (or a quantum gravity probability fator), sine the redued Compton length is di erent for di erent fundamental partiles.. At the subatomi level we think gravity is linked to onditional probabilities of gravity shielding. So this is the gravitational gap we will get on average measurements from a tremendous number of observations of the gravity shielding between two indivisible partiles. Personally we do not think this gravity gap ever an be measured due to tehnial di ulties, but we think it ould take us further in unifying gravity with the quantum world. Possibly we may be able to measure it indiretly and find that this is likely the gravity gap. If the two indivisibles are only separated by the Plank length, whih is the minimum separation they an have, we find that the gravity fore is F g = G m p m p lp 4 lp 3.0585 0 43 N (4) However, this strong gravity will in our model only last for one Plank seond. 9 Summary We have suggested that the mass gap is.7337 0 5 kg per seond observational period, or 85465435748 0 36 fration of the number of ounter-strikes of one kg. In the atomism model, the mass gap is one single ounterstrike between two indivisible partiles. The mass gap an also oneptually be seen as a subatomi partile with a redued Compton wavelength equal to the distane light travels in one seond if we are interested in the mass gap for a one-seond time period. Further, based on this perspetive, one kg of mass an be redefined as 85465435748 0 36 ounterstrikes between indivisible partiles per seond. This leads us to a redefinition of the redued Plank onstant, whih an be represented as = 8.5465435748 0. And every type of fundamental partile an be represented as a fration of the number of ounter-strikes in one kg. That is to say, any mass is given as a partile frequeny divided by the number of ounter-strikes in one kg. m = ounter-strike fration of one kg (5) 85465435748 036 Further, the mass gap for one Plank seond is one Plank mass. One kg has approximately 459459.3 ounter-strikes per Plank seond. This means that the minimum mass one an observe in one Plank seond as a fration of the ounter-strikes in one kg is 459459.3.765 0 08. We think that our theory ould be useful for deiding on an exat definition of the kilogram and thereby also the Plank onstant. The speed of light is already exatly defined, so is one meter, and here the Plank onstant and the kilogram ould be as well. The unertainty in many measurements would then lie in how long a seond is. More importantly, if one studies Haugs full theory on atomism, then one may see that many of the mysteries in physiists an be redued to very simple logi. It is lear that the Plank onstant and the gravitational onstant are omposite onstants. When deomposing these onstants into what they likely truly represent we are able to develop very simple and logial explanations for mass, energy, time, and muh more. Further, we will laim that the photon has rest-mass at ollision lasting for an instant, but no relativisti mass. The time dependent mass gap seems to be a possible explanation for why the EmDrive an work. 7
We have suggested that there likely is a gravity gap that also is related to the time window of observation. This gravity gap is.5559 0 30 N for a one seond observational window. I enourage people interested in this theory to read my full theory on atomism to better grasp this paper. Referenes [] E. G. Haug. Unified Revolution: New Fundamental Physis. Oslo, E.G.H. Publishing, 04. [] E. G. Haug. The Plank mass partile finally disovered! Good bye to the point partile hypothesis! http://vixra.org/abs/607.0496, 06. [3] A. Einstein. On the eletrodynamis of moving bodies. Annalen der Physik, English translation by George Barker Je ery 93, (7), 905. [4] M. Plank. Ueber das gesetz der energieverteilung im normalspetrum. Annalen der Physik, 4, 90. [5] B. Fixler, G. T. Foster, J. M. MGuirk, and M. A. Kasevih. Atom interferometer measurement of the Newtonian onstant of gravity. Siene, 35, 007. [6] S. Shlamminger. A fundamental onstant: A ool way to measure big G. Nature,, 04. [7] G. Rosi, F. Sorrentino, L. Caiapuoti, M. Prevedelli, and G. M. Tino. Preision measurement of the Newtonian gravitational onstant using old atoms. Nature,, 04. [8] M. Stok. The watt balane: determination of the Plank onstant and redefinition of the kilogram. Philosophial Transations of the Royal Soiety, 369:3936 3953, 0. [9] E. G. Haug. A new solution to Einstein s relativisti mass hallenge based on maximum frequeny. http://vixra.org/abs/609.0083, 06. [0] E. G. Haug. The gravitational onstant and the Plank units: A simplifiation of the quantum realm. Physis Essays Vol 9, No 4, 06. [] E. G. Haug. The ultimate limits of the relativisti roket equation. The Plank photon roket. Ata Astronautia Vol 36, 07. [] M. Plank. The Theory of Radiation. Dover 959 translation, 906. [3] E. G. Haug. Plank quantization of Newton and Einstein gravitation. International Journal of Astronomy and Astrophysis, 6(), 06. [4] E. G. Haug. Plank quantization of Newton and Einstein gravitation for Plank masses and smaller size objets. www.vixra.org 60.038 06, 06. [5] E. G. Haug. A suggested boundary for Heisenberg s unertainty priniple. http://vixra.org/abs/70.0497, 07. [6] E. G. Haug. Unifiation of gravity and eletromagnetism: Gravityeletromagnetism: A probability interpretation of gravity. www.vixra.org, 06. 8