A Modified Newtonian Quantum Gravity Theory Derived from Heisenberg s Uncertainty Principle that Predicts the Same Bending of Light as GR

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1 A Modified Newtonian Quantum Gravity Theory Derived from Heisenberg s Unertainty Priniple that Predits the Same Bending of Light as GR Espen Gaarder Haug Norwegian University of Life Sienes Marh 6, 208 Abstrat Mike MCulloh has derived Newton s gravity from Heisenberg s unertainty priniple in a very interesting way that we think makes great sense. In our view, it also shows that gravity, even at the osmi (marosopi) sale, is related to the Plank sale. Inspired by MCulloh, in this paper we are using his approah to the derivation to take another step forward and show that the gravitational onstant is not always the same, depending on whether we are dealing with light and matter, or matter against matter. Based on ertain key onepts of the photon, ombined with Heisenberg s unertainty priniple, we get a gravitational onstant that is twie that of Newton s when we are working with gravity between matter and light, and we get the (normal) Newtonian gravitational onstant when we are working with matter against matter. This leads to a very simple theory of quantum gravity that gives the orret predition on bending of light, i.e. the same as the General Relativity theory does, whih is a value twie that of Newton s predition. One of the main reasons the theory of GR has surpassed Newton s theory of gravitation is beause Newton s theory predits a bending of light that is not onsistent with experiments. Key words: Heisenberg s unertainty priniple, Newtonian gravity, Plank momentum, Plank mass, Plank length, gravitational onstant, Lorentz symmetry break down. The Rest-mass of Light Partiles and Introdution to Gravitational Bending of Light The mainstream view in modern physis is that light has no rest-mass. However, to alulate Newton s bending of light preditions, we must assume that light has a hypothetial rest-mass. This is normally obtained by taking the light energy and dividing by 2. The Newton bending of light was first derived by Soldner in 88 and 884, [, 2]) S = 2Gm () 2 r In 9, Einstein obtained the same formula for the bending of light when he derived it from Newtonian gravitation; see [3]. The angle of defletion in Einstein s general relativity theory [4] is twie what one gets from Newtonian gravity GR = 4Gm (2) 2 r The solar elipse experiment of Dyson, Eddington, and Davidson performed in 99 onfirmed [5] theidea that the defletion of light was very lose to that predited by Einstein s general relativity theory. In other words, the defletion was twie what was predited by Newton s formula. First of all, it must be lear that Newton never derived a mathematial value for his gravitational onstant. It is a onstant that needs to be alibrated to gravitational observations to make the theory work. The gravitational onstant was first indiretly measured by Cavendish [6] in798usingaso-alledcavendishset-up. Bearinmind, however, that the Cavendish set-up inluded only traditional masses and no light; in short it onsisted of four lead balls. espenhaug@ma.om. Thanks to Vitoria Teres for helping me edit this manusript.

2 2 Over the last years Haug has derived an extensive mathematial theory from atomism that gives all the same mathematial end results as Einstein s speial relativity theory. However, Haug s work develops ideas around upper boundary onditions that are linked to the Plank length and the Plank mass. In this theory, all matter and mass are built from indivisible partiles and empty spae. This indivisible partile has a diameter equal to the Plank length. The indivisible partile always moves at the speed of light and has no rest-mass when it is moving. Only when olliding with another indivisible partile, does it stand still for one Plank seond. This ollision point is what we all mass. The onept that photons interating with other photons an reate matter in so-alled photon ollisions has long been predited, but not definitely proven in experiments yet. However, the idea that photon-to-photon ollisions will reate mass has reently reeived inreased attention in the siene ommunity; see [7, 8], for example. This ollision point between two indivisible partiles (light partiles) is, in our theory, the Plank mass. However, this Plank mass only lasts for one Plank seond. And again, this mass onsists of two indivisible partiles, so eah indivisible partile has a rest-mass (at ollision) of half the Plank mass. We also get a hint about the lifetime of a Plank partile from the Plank aeleration, a p = 2 l p m/s 2.ThePlank aeleration is assumed to be the maximum possible aeleration by several physiists; see [9, 0], for example. The veloity of a partile that undergoes Plank aeleration will atually reah the speed of light within one Plank seond: a pt p = 2 l p l p =. However, we know that nothing with rest-mass an travel at the speed of light, so no normal partile an undergo Plank aeleration if the shortest possible aeleration time interval is the Plank seond. The solution is simple. The Plank aeleration is an internal aeleration inside the Plank partile that within one Plank seond turns the Plank mass partile into pure energy. This also explains why the Plank momentum is so speial, namely always m p, unlike for any other partiles, whih an take a wide range of veloities and therefore a wide range of momentums. We an write the mass of any elementary partile in the form m = ~ (3) However, we rarely see disussions on why elementary masses are dependent on the speed of light? The fat that their journey from energy to mass and from mass to energy is dependent on the speed of light speed is well-known, of ourse, but why elementary masses an be written as a funtion of the speed of light has not been disussed in depth. Under atomism, the speed of light in the elementary partile formula is related to the onept that any normal mass is assumed to onsist of a minimum of two light partiles, that is to say, two indivisible partiles. These partiles are eah moving bak and forth over the distane of the redued Compton wavelength of the partile at the speed of light. Eah indivisible partile is massless when moving and will attain half of the Plank mass for one Plank seond when olliding with another indivisible partile. If eletrons ultimately onsist of light, we suggest that suh ollisions happen internally inside the eletron an tremendous number of times per seond, namely e (4) Eah of these events is omprised of a Plank mass times one Plank seond, whih is the mass gap in the atomism theory, and this gives an eletron mass of e kg (5) This means all elementary partiles have a Plank mass element to them and it also explains why modern physis sometimes uses the formula m = ~ to desribe elementary masses. This also indiates why the speed of light appears in the elementary mass formula. The Plank mass is onsidered large and there has been a signifiant amount of speulation on where and how the Plank mass partile ame into being: did it only exist at the beginning of the Big Bang, or it is hidden away somewhere as a miro blak hole? The answers are not lear yet, but we laim the the Plank mass partile is diretly linked to the Plank seond, whih is a very natural progression. This ould explain why the Plank mass partile is basially everywhere inside normal matter, but so hard to measure. 2 Heisenberg s Unertainty Priniple, Newton s Theory of Gravitation, and MCulloh s Derivation In this setion we will follow the approah that MCulloh has used to derive Newtonian gravity [] relying on Plank masses, and the Heisenberg unertainty priniple; see [2, 3]. However, we will assume that light partiles are linked to half the Plank mass (half the mass gap), while matter (beause it onsists of minimum two light partiles) is linked to the Plank mass (the mass gap). Heisenberg s unertainty priniple [4] is given by

3 3 p x ~ (6) Next MCulloh goes from momentum to energy simply by multiplying p with and gets E x ~ (7) As reently pointed out by Haug, this approah is atually only valid as long as the momentum is related to the Plank mass partile. This is a key point. While the Plank mass, Plank length, Plank time, and Plank energy were introdued in 899 by Max Plank himself [5, 6], the Plank mass partile was likely first introdued by Lloyd Motz, when he was working at the Rutherford Laboratory in 962; see [7, 8, 9]. Modern physis states, often without arguing exatly why, that m an be applied to light, but then light is assumed to be mass-less, so it should not apply. However, we think we have a sound explanation. The momentum formula that holds for any partile is the relativisti momentum formula given by Einstein mv p = q v 2 2 (8) Multiplying this by learly does not give us the kineti energy formula, or the rest-mass energy formula. So how does the Plank momentum fit in here? If it also follows Einstein s [20] relativisti priniples, then we must have p p = q mp = m p (9) v 2 This is only possible if v = 0. That is to say, the Plank mass partile must always stand still as pointed out by Haug in a series of papers [2], even as observed aross di erent referene frames. In other words, this indiates that there is a breakdown of Lorentz symmetry at the Plank sale, something that several quantum gravity theories also laim [22]. Our point is that the way MCulloh gets to his relation E x ~ likely only an be done in relation to Plank mass partiles, or at least Plank masses. Next, just as MCulloh does, we will use the following relationship E x = NX i nx (~) i,j (0) where m i is the potential rest-mass of photons given by its energy divided by 2 ; this is the traditional way of finding a hypothetial photon mass. And P N i is the number of half Plank masses in the light beam m i.this mp 2 is due to our hypothesis that light is linked to the Plank mass partile (the mass-gap) the way desribed in the introdution setion, and it is the only di erene with the MCulloh derivation, but it is a very important one, as we soon will see. In addition, P n j is the same as in the MCulloh derivation, namely the number of Plank masses in a larger mass M m p. The idea is simply that we need a minimum two light partiles to ollide to reate mass. This gives us the equation j Further, we an divide by E x = E = 2 ~ x on both sides and this gives m i M ~ 2 mp m p m im x () E x = F = 2 ~ m im (2) ( x) 2 From Haug [23], we have that x l p,sowethink x an be set to any larger value, for example the radius of the Earth, or any other radius r l p,andthisgivesus F =2 ~ m im (3) r 2 The formula we have derived above looks very similar to the MCulloh formula, whih is idential to the Newton gravitation formula exept that here we have a omposite gravitational onstant of 2 ~ numerial output from the MCulloh onstant is approximately versus ~.The ~ m 3 kg s 2 (4)

4 4 This is basially the same as the value of the Newtonian gravitational onstant, and in the same units. But when we are working with gravity between light and standard matter (suh light and the Sun), we laim that light is linked to half the Plank mass (mass-gap) and that other masses are linked to the full mass gap (the Plank mass over the Plank time), based on the atomism hypothesis. In 979, Motz and Epstein [24] suggested that a key to understanding gravity ould be linked to a half Plank mass partile and that that there ould be a disontinuity in Newton s gravitational onstant for speial situations: We believe that introduing a disontinuity in G is suh an ad ho hypothesis and that a full understanding an only ome from an analysis of the physial operational meaning of the onstant. MotzandEpstein979 Our theory is one step forward from Motz and Epstein s brilliant ad-ho hypothesis. Here we have shown that the omposite gravitational onstant indeed likely is di erent when working with light and matter than when working only with matter against matter. Sine our gravitational onstant is twie of Newton when working with light and matter, versus matter and matter, we will get a light bending predition equal to that of GR, but based on a speial property of the photon rather than based on bending of spae. Others are also working in this area; Sato and Sato [25], for example, have suggested that the 2 fator (double of Newton) in observed light defletion likely ould be due to an unknown property of the photon, rather than the bending of spae-time. This is exatly what our theory strongly indiates as well. Consider also for a moment that it was very easy to measure light bending from the Sun or even the Earth at the time of Newton (the Earth has suh a weak gravitational field that we have not even been able to do it today). Then assume that light bending was the first type of gravity experiment that had been used to find and alibrate the Newton gravitational onstant to fit experiments. If that was the ase, then Newton s gravitational onstant would have twie the value of today. There are some assumptions that are hard to test in our theory. However, at least it is a very simple quantum gravity theory that orretly predits the bending of light and other gravity phenomena. We have not looked into the Perihelion of Merury yet, whih is supposedly another key observation supporting GR, but as pointed out by several researhers it is not ompletely lear that it is inonsistent with Newtonian gravity, as is often assumed; see [26, 27, 28, 29, 25], for example. In a series of reently published papers, [30, 3, 32, 33] Haug has suggested and shown strong evidene for the idea that Newton s gravitational onstant must be a omposite onstant of the form G = l2 p 3 (5) ~ This has been derived from dimensional analysis [30], but one an also derive it diretly from the Plank length formula. MCulloh s Heisenberg-derived gravitational onstant G = ~ is naturally the same as the Plank mass and is diretly linked to the Plank length, m p = ~. l p Many physiists will likely protest here and laim that we annot find the Plank mass before we know Newton s gravitational onstant. However, Haug [32] has shown that one easily an measure the Plank length with a Cavendish apparatus and thereby know the Plank mass without any knowledge of the Newton gravitational onstant. Further, Haug has shown that the standard measurement unertainty in the Plank length experiments must be exatly half of that of the standard unertainty in the gravitational onstant, whih is very likely a omposite onstant. See also [34]. 3 Summary In this setion we will summarize our theory in bullet point style: We suggest that the minimum ordinary mass is linked to the ollision of a minimum of two photons, soalled photon-to-photon ollisions that reently have reeived inreased attention by the physis ommunity. From this we laim that a minimum mass an be reated (the mass-gap) that is linked to the Plank mass and lasts for one Plank seond. Sine this requires a minimum of two photons, it means that the building bloks of photons only have half of that as a potential rest-mass that only will be realized at ollision with other matter or photons. That is the rest-mass of photons are linked to half the Plank mass (over one Plank seond), so an inredibly small, mass, and the half mass an never be observed, only the mass-gap that is the ollision of two photons. MCulloh has already a gravity theory that output-wise is the same as that of Newton, but indiretly suggests that Newton s gravitational onstant is a omposite onstant. MCulloh has been relying on Plank masses and Plank momentum to arrive at this theory. There are some additional hallenges here, i.e. that one first would have to know the mass of the Sun before one ould find the big G, forexample,butforillustrativepurposesweandisregardthatfornow.

5 5 When ombining ordinary matter with light, we get a gravitational onstant with twie the value of the ordinary Newton formula, and this therefore gives an extended quantum Newtonian formula that gives the orret bending of light, the same as GR, but based on muh simpler priniples. Our theory simply is a Newtonian quantum gravity theory derived from Heisenberg s unertainty priniple where the gravitational onstant takes two di erent values, one value when working with standard matter against matter, and twie the normal value when working with light against matter. The omposite gravitational onstant is of the form G h = l2 p 3 = G when dealing with matter against ~ matter and G h =2 l2 p 3 = 2G when dealing with light against matter. The Plank length and thereby ~ the Plank mass an be measured with no knowledge of the Newtonian gravitational onstant using a Cavendish apparatus. One ould laim this is a simple attempt to fudge the Heisenberg priniple into beoming ompatible with gravity. However, we think it is muh more than that. Even before he started to work with gravity, Haug has laimed that light must be linked to a half Plank mass partile. Previously, researhers like Motz and Epstein have also suggested that a half Plank mass partile likely was essential to understanding gravity from a quantum perspetive and they suggested that the gravitational onstant had to be disontinuous for speial situations. Light against matter is preisely suh a speial situation and by ombining key onepts from Plank and Heisenberg priniple with our hypothesis about light versus matter, we seem to get a simple theory of quantum gravity that is onsistent with experiments. 4 Conlusion By ombining the MCulloh Heisenberg derivation of gravity with key ideas on matter from atomism we have derived a new quantum gravity theory that is the same as Newton when working with matter against matter, but gives twie the value of Newton s gravitational onstant when working with light and matter. Our quantum Newtonian gravity should give the same light bending as general relativity theory, whih is also twie of that of Newton. The gravity in this theory is quantized. It seems to be a muh simpler gravity theory than Einstein s theory of general relativity theory, and yet it predits similar results to those of GR, while also appearing to be ompatible with MCulloh s new and promising onept of Quantized Inertia. If so, it deserves further onsideration as a gravity theory that is onsistent with galaxy rotations [35], without the need to resort to onepts like dark matter to explain what we observe. Referenes [] L. Soldner. On the defletion of a light ray from its retilinear motion, by the attration of a elestial body at whih it nearly passes by. Berliner Astronomishes Jahrbuh:, 88. [2] L. Soldner. On probability theory and probabilisti physis axiomatis and methodology. Foundations of Physis, 3(4), 884. [3] A. Einstein. Über den einfluss der shwerraft auf die ausbreitung des lihtes. Annalen der Physik, 4, 9. [4] A. Einstein. Näherungsweise integration der feldgleihungen der gravitation. Sitzungsberihte der Königlih Preussishen Akademie der Wissenshaften Berlin, 96. [5] F. Dyson, A. Eddington, and C. Davidson. A determination of the defletion of light by the sun s gravitational field, from observations made at the total elipse of may 29, 99. Philosophial Translation Royal Soiety, 920. [6] H. Cavendish. Experiments to determine the density of the earth. Philosophial Transations of the Royal Soiety of London, (part II), 88,798. [7] O. J. Pike, F. Makenroth, E. G. Hill, and R. S. J. A photon photon ollider in a vauum hohlraum. Nature Photonis, 8, 204. [8] B. King and C. H. Keitel. Photon photon sattering in ollisions of intense laser pulses. New Journal of Physis, 4, 202. [9] G. Sarpetta. Letter Nuovo Cimento,, 5, 984. [0] D. F. Falla and P. T. Landsberg. Blak holes and limits on some physial quantities. European Journal of Physis, 5, 994. [] I. Newton. Philosophiae Naturalis Prinipia Mathematis. London,686. [2] M. E. MCulloh. Gravity from the unertainty priniple. Astrophysis and Spae Siene, 349(2), 204. [3] M. E. MCulloh. Quantised inertia from relativity and the unertainty priniple. EPL (Europhysis Letters), 5(6), 206. [4] W. Heisenberg. Über den anshaulihen inhalt der quantentheoretishen kinematik und mehanik. Zeitshrift für Physik, (43):72 98, 927. [5] M. Plank. Naturlishe Maasseinheiten. Der Königlih Preussishen Akademie Der Wissenshaften, p. 479., 899.

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