Quantum Gravity via Newton

Transcription

1 4 Pearson: Quantum Gravity via Newton Vol. 9 Quantum Gravity via Newton Ron Pearson UK pearson98@googl .om Sine relativity theories are unsatisfatory and annot provide quantum gravity an alternative approah is presented based on Newtonian onepts. Sine the latter is only suitable for use at low values of v/ and in weak gravity it an only be regarded as an approximate theory in need of revision. xtension to an exat mehanis is provided to extend appliation to extreme onditions. Instead of time dilating as in relativity theory loks lose time as speeds inrease for eletromehanial reasons. Instead of gravitational time dilation it is the speed of light that redues as a massive objet is approahed and yields the idential equation but with a different interpretation. The effets instein puts down to urved spaetime are aused by the non-uniformity of the density of the bakground medium that extends to the very edge of spae. In this way the end equations are found to math all the ahievements for whih instein is famous. There is no need to question the validity of the experiments involved. Then sine the assumptions are fully quantum ompatible it is an easy matter to provide a satisfatory theory of quantum gravitation.. Introdution It is impossible to ahieve a theory of quantum gravity from instein's theories of relativity owing to the inompatibility of assumptions. Sine all relativity theories ontain internal ontraditions as soon as mass inrease due to speed is onsidered it is lear that it is not quantum theory that has to be modified. Unfortunately it is not possible to return to an unrevised Newtonian mehanis as soon as it is aepted that = m. The reason is that the aeleration of a mass needs to be onsidered and energy has to be supplied to aelerate that mass. Then sine this energy annot be lost it follows that, sine its energy is inreased, then its mass must also rise. This proves the mass inrease is real and not the illusion some people imagine it to be and also shows a replaement theory has to be based on absolute speeds. If as in speial relativity speeds have to be onsidered relative to the observer then two observers in relative motion will not agree on the mass inrease of any other objet both simultaneously observe. Hene it is neessary to aept the existene of a bakground medium suh as the quantum vauum said to onsist of a seething mass of virtual partiles. Then all speeds, for the purpose of assessing mass inrease and eletromagneti fields, have to be measured from the vauum. Then sine the vauum has fluid properties and is in a state of aelerating expansion it follows that the measurement has to be taken where the objet, whose speed is being measured, is loated. This will be alled loal spae or the loal frame. It requires a modifiation to the onept of absolute speed sine a different situation exists from the original based on rigid nonexpanding ether. The alternative presented in this paper is based on ulidean geometry and universal time and so is fully ompatible with the quantum field theories of Rihard Feynman []. The theory of quantum gravity that is offered is at least ompatible with that formulation whih satisfatorily desribes the other three fores of nature. The theory now to be presented was first published by the Russian Aademy of Sienes by Pearson [] in 99 and Pearson [3] in 994. A omplete book is available Pearson [5] see erivation Showing Objets Cannot xeed the Speed of Light.. Photon Mehanis For appliation at the marosopi level the behaviour of light had to be onsidered in the new approah. The partile nature rather than the wave nature of light had to be adopted for the methodology seleted for derivation. Photons arry momentum, m as otherwise they ould not produe radiation pressure when they hit a surfae. The photon mass m has therefore to be onsidered real, but photons have no rest mass. So they have a kineti mass (instead of an indeterminate effetive mass ). They are therefore onsidered as if made from kineti energy. Although = m has been derived from both Newtonian mehanis and speial relativity, these derivations annot be utilized sine the former is inexat and the latter starts from a different set of assumptions. It is therefore neessary to obtain another derivation that avoids the ritiism that would otherwise be invited. The simplest way of ahieving this end is to start from experimental observations, even though these may have been predited from earlier theories. Then Newton s three laws of motion form the starting point, though with a more sophistiated seond law. xperiments have shown radiation pressure to be diretly proportional to the produt of the rate at whih photons arrive on unit area of surfae and the frequeny of their assoiated eletromagneti waves. Now from Plank s study, that initiated quantum theory, photon energy is diretly proportional to their wave frequeny. This ombination means that the kineti mass of photons is proportional to their kineti energy. It follows that the kineti mass of any objet must be proportional to its kineti energy for onsisteny with photons... The Horizontal Aeleration of an Objet The energy supplied to produe aeleration of a massive objet inreases the objet s kineti energy.

2 Albuquerque, NM PROCINGS of the NPA 4 Fig.. The horizontal aeleration and defletion of an objet Then sine the kineti mass of an objet is proportional to its kineti energy the mass m of the objet must also inrease from its rest mass m. It an be inferred that the energy inrease must be added to an initial rest energy sine it breaks the rules of logi to add things having different units. The term sum energy of a massive objet will be defined as: = + K, where is the rest energy of the objet and K its kineti energy. Correspondingly inertial mass m = m + m K. Then an be related to the objet s inertial mass m by the relation: m B m = B (.) where B is a onstant to be determined. Both v and m, as well as are variables and so Newton s seond law has to be used in form: fore = rate of hange of momentum: dmv F (.) dt Using (.) this an be written as: dv F B (.3) dt ifferentiating by parts and multiplying both sides by dx we have: dv d Fdx B dx Bv dx dt dt (.4) Now Fdx d and dx dt v, and so (.4) an be written: Re-arranging (.5) we have: This an be arranged for integration as: d Bvdv Bv d (.5) d Bv Bvdv (.6) d B (.7) Bv vdv Put z = - Bv and then dz = -Bvdv. Substituting in (.7) and integrating we obtain: This an be re-arranged to read: Bv loge loge Bv (.8) (.9) Now as v inreases / inreases aording to (.9) until Bv = when beomes / whih is infinite. This indiates that an ultimate speed exists that will be defined as v =, the speed of light, sine it is known that photons travel at this speed. Hene from (.9), B = /v, whih now beomes: B = / (.) Substituting (.) in (.) and (.9) yields: v m (.) In this way two important equations have appeared simultaneously. Next when v = then = and m = m. These are the rest energy and rest mass respetively and it follows that = m. The equations of (.) look like two of those derived by speial relativity but there are subtle differenes:. The frame of referene is the loal bakground medium: not the observer as in instein s relativity theories. Consequently the inertial mass m and the sum energy are both higher than rest values by real amounts due to a real inrease in kineti energy, whilst in relativity this kineti energy has no definite value sine it an be different for eah observer. In speial relativity theory two observers in relative motion see the other as having greater mass and this is logially impossible. Furthermore observers in motion relative to one another aredit different values of kineti energy to any objet they both observe. Consequently kineti energy is regarded as having no speifi value and neither an the mass inrease be given a definite value. In the new approah both kineti energy and kineti mass have definite values showing both are fully real.. Furthermore in relativity these equations appear with time dilation and a Lorentz ontration of both matter and spae as fundamentally inorporated. These are absent in the present derivation. However, this is of no onsequene sine the equations involved are very similar to the first term in (.) exept that they are the inverse of eah other. They anel out! This does not mean that relative effets do not exist. Two objets having different absolute veloities will have relative veloities and relative momenta with respet to one another but the mass of eah objet must remain as given by equation (.) with v measured from the loal frame. 3. Furthermore kineti energy beomes exatly speified regardless of observer motion as K = - or using (.) beomes: K v (.) In onsequene the dilemma of whether mass inrease is real or not is fully resolved: the mass inrease is fully real. Fig.. Gravitational light bending ( d = g dr)

3 4 Pearson: Quantum Gravity via Newton Vol. 9 If the aelerated objet then makes an ideal boune from an inlined plate, as shown in Fig. the same speed is retained. This shows the theory to be appliable to motion in any diretion in a gravitational field - provided that aeleration due to that field is subsequently taken into aount. 3. Gravitational Bending of a Beam of Light What will be presented here is a simple approah but yields the idential result to that provided using total rigor by Pearson (5). Bending is assumed aused by the fore of gravity ating on the kineti mass of the photon with propagation of light always maintained perpendiular to the wave front. It follows that the photon will fall and aelerate to speed gt in the field diretion. Clearly from Fig. there are two similar triangles of sides d.t:dr and gt: quating these yields: d = gdr (3.) The assumption of a wave always remaining perpendiular to its diretion of propagation is shown to be fully onsistent in the omplete text provided in Pearson [5]. 4. nergy is the Universal Substane, not Mass quation (3.) shows that the speed of light must inrease with distane from the Sun. So the speed of light has to be onsidered as another variable and = m no longer means that energy and mass are equivalent sine if inreases then m must redue if remains onstant. Fig. 3. Lowering an objet on a able To deide whether rest energy or rest mass remains onstant in a gravitational field a thought experiment needs to be explored. First it is important to understand that in this exat theory the onept of gravitational potential energy GP has to be abandoned and regarded as a useful mathematial stratagem limited to the inexat Newtonian. Instead as an objet rises freely it is losing kineti energy by pushing against the field to transfer energy to the quantum vauum. When it falls bak energy from the vauum flows bak to the objet. This means that when the objet is onsidered in isolation its energy is not onserved. In other words GP is to be ignored altogether. A buket is to be imagined lowered on a able from a winding drum. The winding drum is fitted with a brake to stop aeleration so that lowering takes plae at a negligible onstant and very slow speed. Then all the energy released by the fall is absorbed by the brake and released as heat: non remains to hange the energy of the lowered objet. Alternatively the fore F on the able is seen to arry out negative work during lowering and this exatly anels the positive work done by gravity. Whatever test is applied it is seen that it is the rest energy of any objet that remains onstant as it hanges level in a gravitational field and so is a universal onstant for a given objet. It follows that the rest mass, m with suffix representing the primary datum taken arbitrarily at average arth orbital radius, varies aording to the relation: m m (4.) Sine redues as an objet is lowered so rest mass inreases, even though no energy has been added. Inertia, being a funtion of mass, has also inreased. A mass hanging on a spring vibrates at lower frequeny when the mass is inreased. Analysis gives an equation idential to one instein attributes to gravitational time dilation. xperiment has therefore already provided powerful support to the present methodology. This also shows that mass annot be regarded as a measure of substane but instead governs the dynamis of energy: the true substane. The relation between inertial mass m and sum energy is: K m m mk (4.) If a mediator flux of some kind, suh as a stream of gravitons, ats on an objet to produe a gravitational fore, then the absorption ross setion of the objet has to be proportional to its sum energy, sine this is the only substane from whih it is made and learly the energy of motion is just as important as rest energy when defining substane. This rule then applies equally to massive objets and partiles having no rest energy (like photons and possibly neutrinos) A new law of physis an now be stated: Sum energy is the building substane of the universe and so ouples with any kind of gravitational flux to produe a fore, but inertial mass governs the aeleration of that sum energy when any kind fore is applied, inlusive of the gravitational. 5. General Relation of Light to Sum nergy It will be assumed at this stage that some kind of mediator flux ouples with sum energy whose absorption ross setion is proportional to that sum energy, to produe a fore F, following the onlusion reahed in 4. Then: F = - (5.) This is -ve sine F is opposite the flux diretion. 5.. Change of Sum nergy with istane lement dr If sum energy moves a distane dr further from the Sun, then the energy gain will be d = Fdr. Substituting from (5.) yields:

4 Albuquerque, NM PROCINGS of the NPA 43 Then: d = - dr (5.) Now for the general ase the quantity is to be defined as: dr (5.3) d dr (5.4) 5.. Change of Light Speed with istane lement dr variation, for an objet or partile in free fall, inlusive of photons, is always: (5.) This is a very important relationship applying in any kind of gravitational field provided the objet is in free rise or fall i.e. without involving frition of any kind. 6. The quation for Gravitational Fore Fig. 4. A stak of light beams eah of small thikness Now the problem addressed in the previous ase of 3, in whih a single element dr was onsidered is to be extended to a multipliity of elements. The work done by the fore F of gravity ats on an element of thikness dr of a very wide beam of light as illustrated in Fig.4 showing a stak of suh elements of light beams. The problem now is to summate the effet of an infinite stak of suh elements in hanging the speed of light over a finite radial distane. Now sine =m : F F g m (5.5) And substituting for F from (5.) this gives: g (5.6) In (3.) g was onsidered positive measured downwards and so needs a negative sign before equating with (5.6). So we an write: d g (5.7) dr Rearranging and putting in integral form, the result is: d dr (5.8) Both sides of equation (5.4) need to be multiplied by to make positive. Then equation (5.4) an be equated with (5.8) above. xpressed in integral form the result beomes: d d log log e e (5.9) It follows that for any kind of gravitational field, inluding those arising from multiple attrating objets (sine the variation of with r was not speified) the relation between the and Fig. 5. The inverse square law of gravitational flux Sine energy has been shown to represent the substane from whih the universe is made, but not mass, it follows that objets will ouple with energy: not mass. And sine rest mass varies with altitude, but rest energy does not, this makes a differene. Furthermore the kineti energy of photons has been shown to ouple with gravity in order to ause the bending of light. It follows that the kineti energy of any objet must ouple with gravity just as does its rest energy. Consequently it is the sum energy of all objets that ouple with the gravitational flux: not their rest mass. The onverse follows that suh a flux must be proportional to the sum energy S of the attrating objet ausing the emission of that attrating flux. Suh a flux will have an intensity varying with an inverse square law sine it spreads out radially over a spherial surfae whose area is proportional to the square of its radius measured from the entral massive attrating objet. The distane between the enters of mass of the two objets will be denoted d. Then if / S is not negligible the objets will orbit about their baryenter that lies between the mass enters of the two. The orbital radius of the smaller objet will be denoted r. (In Fig.5 the differene between d and r is ignored.) Hene we an dedue that the fore F of gravity will be given by: G F (6.) d d S KS K S G 4 C The negative sign means the fore ats opposite the radial distane. Here G C is the new gravitational onstant. Its value an be dedued from the Newtonian onstant G by putting S / = m S and / = m in the Newtonian equation. Then: G GC Nm J (6.)

5 44 Pearson: Quantum Gravity via Newton Vol. 9 The datum speed of light is to be measured at arth orbit where G and are measured. It is important to note that G has to vary as gravitational potential hanges if G C = G/ 4 sine hanges. Hene G C is the true gravitational onstant but G has now beome another variable. quation (6.) unifies light and matter sine it is equally appliable to both. Indeed it applies to neutrinos and everything else. (So the entire energy of neutrinos ouples: not its minute mass.) quation (6.) is also highly signifiant sine sum energies S and have replaed the rest masses m S and m of the original Newtonian equation. Then sine inreases as r redues, it follows that the law of gravitational fore will have, in effet, a higher index than two in the denominator. This will produe part of the preession of elliptial planetary orbits. xtension to onsider ases where r = d is inaurate are overed in the omplete text see Pearson [5]. To summarize the mass energy of the Sun S is hanged to an equivalent energy Sm plaed at the baryenter that would produe the same fore on the orbiting planet. 7. The nergy quation for Planets Sine the baryenter remains stationary and the hypothetial objet Sm is entred there, to replae the massive objet of sum energy S, that hypothetial objet an be onsidered to maintain its energy value onstant. The only variables then redue to and r for the motions of even very large planets. The energy inrement d of the planet is d = Fdr and so (6.) an be substituted with r replaing d so that d = G C Sm dr/r and an be rearranged in the integral form: d dr GCSm (7.) r And this integrates to give: log e GCSm r r (7.) Now the gravitational radius r is onventionally defined as GmS/ but now is equivalent to: r = G C Sm (7.3) has already been defined in (5.4) and so an be written as: r r r r (7.4) Then substituting from (5.) and ombining (7.), (7.3) and (7.4) and also re-arranging in a more onvenient form the result is: e (7.5) Note that e is often written exp(). 8. Non-Uniformity of the Quantum Vauum ensity Novikov [4] (983) showed that the virtual partiles of the quantum vauum existed, on average eah within a ube of side L as illustrated in Fig. 6. Hene L is also the average distane of separation of these virtual partiles and is equal to something like the Compton wavelength given by: L (8.) m Fig. 6. Novikov s spae ubes ( L m ) Plank s onstant is h but this onstant is of no interest to our derivation. So equation (8.) an be divided by datum equivalents to yield: L m (8.) L m The ratio of m to m of the virtual partiles will be onstant anywhere in spae and so it is of no onsequene whether either of these is used in (8.). Now their average sum energy will be uniform over the entirety of spae and so an be regarded as a onstant. Then sine = m = m : m m Substituting in (8.) for m /m yields: L (8.3) L This shows the average separating distane of primaries in the quantum vauum to inrease with inrease of altitude in the same proportion as the speed of light inreases. It also follows that the density of spae must redue with distane from any objet. 9. The Observed Speed of Light T The L variation will affet the speed of light that is observed and so this is denoted T. However, the speed still needs to appear in the final basi equation sine (5.) remains unhanged. In order to proeed further it is neessary to onsider the motion of the photon as if it made a series of instantaneous jumps of distane S at datum position with a time delay t between eah jump. Then at greater distane r from the Sun the dwell hanges to t. Hene: = S /t and = S /t (9.) In flat spae i.e. without an L variation, then there is no S variation. To aount for the inrease in as r inreases t must

6 Albuquerque, NM PROCINGS of the NPA 45 fall in inverse proportion to the inrease in and so the two parts of (9.) an be divided one into the other to yield: t (9.) t Fig. 7. Strutured eletromagneti wave This time variation will not hange if L hanges but now the observed speed of light is to be defined as T where: T = S/t (9.3) The equation (9.3) an now be divided by (9.) to yield: T S t t S Now S/S = L/L by definition and from (9.) t /t = /. So substituting in the above and also using (7.3) and LHS of (7.5) a omposite equation results for the ase of orbiting planets or satellites yielding: T L e L (9.4). The First Composite Summary quation of CM Theory.. Modify the Ultimate Speed of q. (.)] Fig. 8. Veloity v from horizontal aeleration deflets to w The veloity transverse to the field diretion was alled the horizontal diretion v for the derivation of equation (.). In general there will be both transverse and radial omponents of veloity, the latter being denoted by symbol u. The vetor sum of u and v will be given the symbol w so that w u v. Now after aelerating from rest to speed v and with u = as imagined for deriving equation (.) it is now reasonable to imagine the objet making a perfet boune from a plate inlined to the horizontal. Then, as illustrated in Fig. and in Fig.8, it will have the veloity w but at the same speed as the original v. Hene equation (.) is still appliable to w in that kineti energy and kineti mass have the same relation to one another regardless of diretion of motion in a gravitational field. The ultimate speed, however, an no longer be regarded as as in (.). At datum level this is but at any other level it is T. So for the general ase where non-datum levels are onsidered T must replae in (.) Hene the equation represented by (.) an now be presented as: w T (.) w This ompletes all fators needed for presenting the omplete omposite basi equation of the CM theory and an be written by ombining, (.), (5.), (7.3), (7.4), (7.5), (8.3), (9.4) and (.) to yield:.. The First Composite CM Summary quations Nomenlature: L Average separating distane of virtual partiles of the quantum vauum (QV) Speed of light in flat spae i.e. QV of uniform density T Observed speed of light inlusive of non-uniform vauum density Suffix meaning datum, eg. q. (9.4) Rest energy of the objet (onstant) K Kineti energy (variable) Sum energy K of objets (variable) m Inertial mass (variable) m Rest mass (variable) w Veloity of objet relative to the sun s baryenter when at radius r from that baryenter Note: w has a radial omponent u and tangential omponent v (all variables). Note that w is the veloity at the datum and when w > meaning the apogee of the orbit has to be at least as high as the datum radius r. When this is not the ase, as for the orbit of Merury, then a seondary datum is hosen, suffix, at the perihelion and the relation between the two datums found from the general equation that follows this list. G C = N m J - is the new gravitational onstant. G C = G/ 4 Where G is the old Newtonian gravitational onstant - now seen as a variable sine is a variable. However G C an be inonveniently small but an be replaed by the gravitational radius r defined as r = G C S = Gm S /. The following equation is appliable to all ases of free fall/rise of both light and matter. Note that light is always in a state of free fall/rise but for matter resistive fores like frition or air resistane an exist and then the following will not apply. The ideal ase is: L T w m f L m w r where exp r f r r T (.)

7 46 Pearson: Quantum Gravity via Newton Vol. 9 The only universal onstants are G C, r and. quation (.) is derived from the gravitational fore equation (6.) where the new gravitational onstant G C is derived. All measurements of distane, suh as r are based on ulidean geometry with universal time t as a separate dimension. Sine L the average separating distane between virtual partiles of the quantum vauum QV are also measured by the same ulidean geometry, the energy density of the QV varies as the inverse ube of L so that: L (.3) L Note that for weak fields an expansion has to be used for e sine the value is very lose to unity. This is: exp e... (.4)!! 3! 4! Has the reader notied that in one plae the derivation was not suffiiently rigorous? The derivation turned out to be not quite exat so that inonsistenies arose when investigating extreme fields suh as those at the surfae of a neutron star. However, (.) is aurate to 8 deimals for fields as strong as those at the surfae of a white dwarf star. Spae does not permit omparison with experimental heks but the full treatment given by Pearson [5] shows equations idential with those of general relativity arise for the doubling of the defletion of starlight grazing the Sun, the gravitational red shift and the anomalous perihelion advane of Merury. In the latter ase ¾ of the shift is due to the mass inrease as the planet falls from apogee to perigee whilst the remainder is aused by the density inrease of the QV. The differene in the equation for the Shapiro time delay as given by GR is negligibly small but both fall far short of the measured values. Fortunately I disovered in 7 (Pearson [6]) that if the measured speed of light in vauum on arth was in error by only 6 m/s an exat math was obtained. This is only parts in 8 so it appears that the effets being explored experimentally are lose to resolution of the apparatus. Later full rigor was ahieved but the only effet was to inrease the perihelion shift of Merury in the ratio 4/3. Now, however, it beame possible to more aurately investigate neutron stars and blak holes.. Conlusion An alternative to both speial and general relativity has been presented that yields most of the same end equations and so satisfies the experimental heks equally well. However, starting from totally different and quantum ompatible assumptions all end equations had different interpretations. It has advantages of being free from internal ontradition and inompatibility with quantum theory so that, in fat, a satisfatory theory of quantum gravity had emerged by 989. Unfortunately all sientifi journals refused publiation on grounds, to quote a typial rejetion saying that Relativity has withstood the test of time so no alternative is required. So the first presentation had to appear in Russia in 99. This is a revision of Newton s approximate mehanis to make it exat. It shows mass inrease due to speed is fully real and so diffuses the unertainty arising from relativity theory. Instead of gravitational time dilation a gravitational mass inrease appears that has the same effet in ausing loks to lose time. This mass inrease is not neessarily aompanied by an energy inrease. Mass inrease due to speed has a similar effet. So there is no need to onsider time as a variable. Instead it is the speed of light that beomes the variable and inreases with altitude. Sum energies, being the sum of rest and kineti energies, replae the masses used in Newton s equation for the gravitational fore and a new gravitational onstant makes the old one obsolete sine it is shown to be a variable hanging with altitude. The effet instein onsiders to be due to urved spaetime is repliated by non-uniform density of the quantum vauum. The onept of gravitational potential energy has to be disarded when formulating exat mehanis so objets onsidered in isolation lose energy in rising and gain it when falling. Conservation is restored by interation with the quantum vauum. Some theorists are questioning the validity of all the experiments onsidered to support relativity theory. The reason is to justify a return to an unrevised Newtonian mehanis. From the perspetive of this paper that approah is totally invalid. Referenes [ ] R. P. Feynman, Q, The Strange Theory of Light and Matter, 57p (Prineton University Press, New Jersey 985). [ ] R.. Pearson, Alternative to Relativity inluding Quantum Gravitation, pp (Seond International Conferene on Problems in Spae and Time: St. Petersburg, Sep 99). [ 3 ] R.. Pearson, Quantum Gravitation and the Strutured ther, pp (Sir Isaa Newton Conferene. St. Petersburg, Marh 993). [ 4 ] I.. Novikov, volution of the Universe (Cambridge University Press, 983). [ 5 ] R.. Pearson, Quantum Gravity via An xat Classial Mehanis (CM Theory), p ( 9). [ 6 ] R.. Pearson, Creation Solved, 36p (7), spae.om.