Gravitation is a Gradient in the Velocity of Light ABSTRACT

Transcription

1 1 Gravitation is a Gradient in the Veloity of Light D.T. Froedge Formerly Auburn University Phys-dtfroedge@glasgow-ky.om ABSTRACT It has long been known that a photon entering a gravitational potential follows a path idential to that of a photon in a variable speed of light defined by the Shapiro veloity for Minkowski flat spae [1]. A spatially variable speed of light is impliitly present in General Relativity, and in fat has a long history starting in the pre GR efforts of Einstein and others. The differene in the approah presented in this paper is not that gravitation hanges the speed of light, but that gravitation is a hange in the speed of light. There are lots of impliations in this inluding whether or not gravitons and gravitational waves exist. It will be demonstrated that a pair of opposite going photons or massless bosoms, instantaneously loated at a point have the mass and inertial properties of a massive partile, and transform under a hange in the veloity of light exatly as a massive partile responds to a gravitational field. The mehanism for holding massless bosons together as a partile will not be addressed in this paper, but it is not neessary to illustrate onfinement to demonstrate the transformation properties on photon momentum in a variable speed of light. At least some massive partiles are in reality a onfinement of massless bosons, and if it an be shown that the olletive momentum is aelerated by a gradient in, then the premise vindiated. It is shown here that the enter of mass of a partile defined as a pair of opposite going null vetor photons, in a gradient light speed, is aelerated exatly as a massive partile is aelerated in a gravitational potential. It is thus asserted that gravitation an be defined as a gradient in produed by the presene of mass, and thus, Newton s apple falls not beause of an inrease in energy, but beause the speed of light at the branh is higher than the speed of light at the ground.

2 Introdution There are hundreds of papers on gravitation and a variable speed of light (VLS) some of whih preexist General Relativity and many thereafter[3]. All papers so far reviewed by this author assert that gravitation alters the veloity of light as well as providing an attrative potential for mass partiles. If however mass as a onfinement of bosons in a volume of spae is aelerated by a gradient in equivalent to a gravitational potential, then asserting that gravitation is anything but a gradient in is not neessary. There has been a historial aversion to defining mass for speed of light partiles, mass being only assoiated with a proper rest mass. This is the standard onvention although urrently used in some Bose ondensate papers [9], the onvention of using m E / as photon mass seems rare. This paper is not going to atually deviate from the norm. For a pair of non ollinear photons there is a less than light speed veloity of the enter of mass, and thus a rest mass an be defined for a pair or group. If photons are onfined suh as in a refletive avity or as a standing wave between two refletors, then there is a rest mass rest equivalent. Reently it has been noted by this author that loally ontained speed of light massless bosons, have an aeleration of the enter of mass by a gradient in equivalent to the aeleration of a massive partile in a gravitational field. By theorizing a Proto-partile model onsisting of two ollinear opposite phase photons, it an be shown that the relativisti dynamis are the same as a massive partile []. It will be the assertion then that some real partiles onsist of bosons that are similarly aelerated by a gradient in whih effetuates gravitation. The assertion in this development is then, not that gravitation alters the speed of light, but that gravitation is an alteration in the speed of light, and the gradient aelerates massive partiles. Appendix II will disuss the prospet that if gravitation is a gradient in, how it ould originate from the effets of Quantum Field Theory, thus making gravitation ompletely an eletromagneti phenomena. Relativist Lagrangian

3 3 Earlier papers by the author have developed an alternate theory of gravitation in Minkowski spae based on a loally onserved energy theory, in whih a partile infalling a gravitational potential maintains onstant energy. As the partile infalls, rest mass energy is onverted to kineti energy, but the total remains onstant. A partile stopped at a position in the potential has a mass defet [4]. This theory will be the basis for the work here From [4], the Lagrangian relation between the relativist, kineti and rest mass inluding gravitation is based on the summation over point partile masses is: v m 1 m 1 r (1) The right of this relation is the rest mass as a funtion of the gravitational potential: mr m 1 r, () and the left side is the kineti energy, and relativisti mass. m, is the rest mass in gravitational free spae. The onnetion to the standard relativisti Lagrangian an be demonstrated by taking the square root and dividing by the right side of this expression, and ignoring a third order terms. 1 L m m m v m r (3) The distint differene is that the right side of Eq.(1), is the loal rest mass and a funtion of r. The dynamis for the Proto-partile shown it this development will illustrate a onversion of rest energy to kineti as a result of a hange in exatly as proposed by Eq. (1). Variable Speed of light in flat Minkowski spae

4 4 Blandford et.al [1], and others [4],[5], have shown that photons operating aording to Fermat s priniple, in a medium having a speed of light with an index of refration defined by: 1 r, (4) follows a trajetory idential to that of a photon in a gravitation field. This veloity relation is arrived at by a projetion of the photon path defined in the GR Shwarzshild metri tensor, onto flat time Minkowski spae. Fermat s priniple, with an index of refration defined by Eq.(4), yields the proper trajetories for photons in the presene of gravitational fields and is ommonly used for making galati lensing alulations. A slightly different, but more symmetri relation is an be dedued from Eq.(1), [6] in a loally onserved gravitational potential paper, and is: For the purposes of this paper, either form is suitable. 1 r. (5) With the speed of light in a gravitational potential defined by Eq.(5), putting this into Eq.(1), the Lagrangian for the partile in a VLS is: v m 1 m (6) This is the relationship between the Lorentz relativisti mass and the veloity of light resulting from the relation between gravitation and the speed of light. It is expeted that this relation is valid for any vauum hange in the speed of light, indued by gravitation or otherwise. Two Photon Proto-partile

5 5 Premise A partile, designated as a Proto-partile will be defined in terms of a photon pair, shown to be a Lorentz invariant, idential, in inertial properties, to a massive partile, and aelerated loally by a gradient of. It is not illogial that onfining a photon to a volume of spae results in the reation of rest mass. If the photon is absorbed by a partile, mass in inreased, if a photon enters and is trapped in an apparatus the energy has inreased; the inertial properties of the box should reflet the inrease in rest mass. The inrease in the mass of an apparatus trapping a photon in a box inreases by, hν, and thus inertial rest mass inreases by hν /. If a massless speed of light partile is onfined to a volume of spae there is little doubt that the inertial properties of that volume of spae should reflet the energy as mass. For two non-ollinear onfined photons, there is always a fame in whih the momentum and energy of both are equal, and in opposite diretions. If the photons are not ollinear however there is also an angular momentum ouple assoiated with the motion. For defining a simple mass, ompliations inluding onfinement mehanisms, an be avoided, that do not ontribute to understanding the issue. Photons will be assumed sine the inertial dynamis are well understood, but any massless partile with energy and frequeny related by Plank s onstant would do as well. Definition We will define two ollinear; opposite phase veloity photons loated at the same point at an instant in time as a Proto-partile. From momentum onsiderations the motion of the enter of mass for this Protopartile for two opposite going photons is, as any other mass, defined by: v ν ν ν ν (7) 1 1 For a photon the frequeny is presumed to represent the energy and the Plank onstant will be set to one. So long as the relations are understood, the terms energy, and mass for photons an be used somewhat interhangeably.

6 6 Proto-partile Four Momentum Photons moving along ollinear vetor paths in the opposite diretion an be desribed by the null four-momentum in geometri algebra matrix as: hν1 k P1 k and the same with an opposite phase veloity:, (8) hν k P k (9) (The Weyl representation of the Dira gamma matrix is assumed.) Presuming these momentum vetors are the null vetors assoiated with the ation of a photon funtion, then they represent photons with the phase moving in opposite diretions at the speed of light, and instantaneously oloated. The square of the differene of the two null vetors is not zero, but is independent of time and spae and is an invariant under a Lorentz transformation of the vetor sum. [7] or: ν1 ν P P ν ν , (1) 1 1 4ν1ν ν ν ν ν ν (11)

7 Thus ν / is the invariant fixed momentum assoiated with the pair of opposite going photons, independent of a Lorentz transformation jointly applied to the null vetors. (It will be shown later that this expression requires a slight modifiation in regard to having invariane in. (See Eq.(33)) Fatoring the sum of the two partile momentum from Eq.(11), gives: 7 1 ν ν ν ν ν ν ν (1) Noting that from Eq.(7), for two opposite going photons: v ν ν ν 1 ν 1 (13) Where v is the veloity of the enter of mass, or energy. Putting this into, Eq.(1), it is found that the relativisti Lagrangian for the Proto-partile is exatly equivalent to the relativisti veloity relation for massive partiles: v p 1 p (14) With mass defined in the loal frame as the total energy and 1: ν ν ν ν m / (15) 1 1 Loal Photon Energy Veloity Differentials The loal energy, veloity differentials for single photons under a transformation in v, and an be evaluated. Lorentz- single photon v

8 8 For the photon transform, the relativisti Doppler relation, along the diretion of motion, for frequeny is, [1]: 1/ vt vt ν = ν 1 / 1 (16) vt is the transform veloity. Differentiating gives dν / dv for the single photon energy: dv dv T ν (17) In this expression vt is the transformation veloity and is opposite for opposite for opposite going photons. Speed of Light - single photon The loal total differential for the veloity of light in wavelength and frequeny for an individual photon is: or: dν d dν ν, (18) d d d d d d dν ν ν (19) Observing the individual differentials: So that from Eq(): dν d / d d / ν ()

9 9 d dν ν d, (1) Putting this into Eq.(19), gives dν / d for the single photon dν / d : dν ν d () This relation is independent of the diretion of the photon. Inserting Loal Single Photon differentials in Proto-partile Lorentz veloity transform For the Lorentz veloity transformation, inserting the differentials into the mass for the Proto-partile Eq.(15), gives: dvt dvt dvt dm dν1 ν ν1 ν 1 ν ν, (3) or: ν ν dv v dv dm ν1 ν m ν ν 1 T T 1 (4) The hange in the mass as a funtion of the Lorentz transforms veloity for the Proto-partile is then: dm v m dv T (5) Now the same relation an be found to ompare with the hange in the relativisti mass of a real partile as a funtion of veloity. or: v d m 1 dm, (6)

10 1 or: dv v mv 1 dm (7) dm dv v m (8) Comparing, Eq.(5), and Eq.(8), shows that the total energy hange for the two photons in the Proto-partile is the same as the energy hange of the relativisti mass. The Proto-partile mass energy thus transforms properly under a Lorentz veloity transformation. This math of Lorentz transform properties for the Proto-partile and a massive partile onstitutes a proof of onept for defining a rest mass for onfined photons. Speed of Light transform For observing the hange in the mass of the Proto-partile, as a result of a hange in the loald ν / d, from Eq,(), an be put into the mass. Thus the loal photon differential for a two photon energy Proto-partile as a funtion of a hange in from Eq.(). is: or: d d d ν ν ν ν ν ν ν ν, (9) dm d dm d 1 ν ν (3) This is a lear indiation that the Proto-partile mass as stated is not invariant under a hange in the veloity of light. More importantly, neither is the momentum defined in Eq.(11),

11 11 p 1 ν ν (31) This is not an altogether unexpeted. It would be true for a fixed veloity of light, but sine is variable this is no longer true. It is easily shown however that: p 1 ν ν, (3) is an invariant under as well as v, and thus Eq.(11), should properly be stated as: and Eq.(1), as: 1 1 4ν1ν ν ν ν ν ν 1 ν ν ν ν ν ν ν (33) (34) This relation is derived from the magnitude of two opposite going photons with an invariant momentum under a loal instantaneous Lorentz veloity or VSL transformation, and is independent of any referene to gravitation. It is lear that the right and left side of Eq.(34), are invariant under a Lorentz veloity or veloity hange. (The invariane of v/ in regard to VSL is presented in Appendix I). Rearranging Eq.(34): ν ν ν ν 1 ν ν1 ν 1 1 (35) This is then the relativisti mass or energy relation for the motion of the enter of mass of two opposite phase veloity photons as a result a transformation in v or. having no referene to gravitation. With identifiation of mass Eq.(15), and veloity Eq.(13), this is idential to Eq.(6),

12 1 v m 1 m (36) Solving this for v then gives: m v 1 m, (37) the veloity as a funtion of. Replaing with its value in gravitation gives the standard value of veloity for a partile falling from infinity: v (38) r Putting in the Shapiro dependene of the speed of light of gravitation gives bak the original, Eq.(1), gravitational Lagrangian. Now partile mass and veloity is a funtion of as defined by the Shapiro veloity: v m 1 m 1 r (39) If gravitation is only a hange in the veloity of light, the Proto-partile motion will be same as a massive partile in a gravitational field, thus if the onstituents of real partiles transform the same under a hange in, there is no need for any other postulated mehanism. Newton s apple falls not beause of an inrease in energy, but beause the speed of light at the branh is higher than the speed of light at the ground. Results Confined photons are aelerated in a gradient in just as a massive partile with the same mass.

13 Blak holes ould not exist in this theory sine photon energy is not diminished by a gravitational field. The presentation of the alternative neutron star, to a blak hole is developed in [11], and the alternative image is simulated in [1]. 13 If gravitation is a gradient in then there would not be gravitational radiation. To aount for the energy loss in a pulsar binary, a time varying gradient in must ouple into the eletromagneti spetrum. If this is true gravitational waves would not be found, but eletromagneti radiation signatures of gravitational energy loss should. Conlusion It has been shown that the effets of gravitation attration an be effetuated on a mass by a gradient in the speed of light without need of any other mehanism. Referenes: 1. Roger Blandford, Kip S. Thorne, in Appliations of Classial Physis, (in preparation, 4), Chapter 6 DT Froedge, The Conept of Mass as Interfering Photons, V Jan Broekaert,, 8 A spatially-vsl gravity model with 1-PN limit of February 5mhttp://arxiv.org/pdf/gr-q/4515v4.pdf 4 DT Froedge, The Gravitational Theory with Loal onservation of Energy, V914, 5. F. Karimi, S. Khorasani, Ray-traing and Interferometry in Shwarzshild Geometry, arxiv: [gr-q] arxiv: v1 [gr-q] 9 Jun 1 6. DT Froedge, The Veloity of Light in a Loally Conserved Gravitational Field,,V914, 7. Chris Doran, 3 Geometri Algebra for Physiists, Cambridge University Press, pp Feynman, Hibbs, 1965, Quantum Mehanis and Path Integrals MGraw-Hill. 3. D. Kharzeeva, K. Tuhinb, Vauum Self Fousing of Very Intense Laser Beams, arxiv:hep-ph/611133v 9. Alex Kruhkov Bose-Einstein ondensation of light in a avity

14 14 1. John 1 David Jakson, Classial Eletrodynamis Seond Edition pp DT Froedge, Blak Hole vs. Variable Rest Mass Neutron Star Field,, V4191, 1. DT Froedge, Image Comparisons of Blak Hole vs. Neutron Dark Star by Ray Traing,, V111514, Appendix I Ratio v/ The v/ ratio for the Proto-partile, an be shown to be independent of a gradient in. From Eq.(13), this ratio is: v ν ν dν dν ν ν dν dν d d ν ν ν ν ν ν ν ν ) Making substitutions from Eq(), into this gives: Thus: QED. d d d d ν ν ν ν d 1 1 v v d v d ν1 ν ν1 ν ν1 ν d v d 1.) 1.3)

15 15 Appendix II QFT Origin of Gravitation? This appendix is a bit of speulation, but indiated by the state of the art. From the above disussion: Gravitation ould be defined as the presene of mass altering the loal veloity of light. Sine there are well-known proesses defined in QFT and path integral formulations of QM, that alter the veloity of light in the proximity of moving partiles,[8] it is speulated that the proesses of QFT ould be the progenitor of gravitation. Consider an apparatus having a avity with opposing mirrors and having photons trapped between the mirrors. From onservation of energy the apparatus has more mass and generates more gravitational attration than the avity without the photons. There is not speulated to be interation between the photons, so the photons that are bouning bak and forth must be generating gravitation. Fig1 Photons trapped between mirrors of an apparatus inrease the mass and thus the gravitational attration of the apparatus. The inrease in energy of the system is hν so the mass of the apparatus inrease as a result of a trapped photon is:

16 16 m (.1) The gravitational potential due to a photon is: r G (.) 4 r Putting this into Eq.(5), then gives: or: G 1 4 r G 3 r (.3) (.4) 3 Noting that the square of the Plank radius is G / this an be stated as: r P, (.5) r whih if the motion of the photons generates gravitation, has to be the hange in at a distane r indued by a photon The fat that the Plank radius is the onstant in the equation is quite urious. By the methods of path integrals noted by Feynman the probability for the partile moving from point a to point b, exist throughout spaes, it has already been shown by the methods of Quantum Eletrodynamis that a photon beam indues a hange in the veloity of light in the viinity of the beam.

17 17 Fig..This illustration shows the path ations indued by a pair of osillating photons. From the work of D. Kharzeeva, et.al, [9] it is shown that for an intense laser beam the QFT effets related to eletron positron loops indue vauum self-fousing whih is a vauum alteration of the index of refration in the speed of light in the viinity of the beam A partile model being a reiproating bosons in a massless box, as asserted here, onstitutes an intense, highly energeti bak and forth reiproal motion, orders of magnitude greater than a laser. It is suggested here that the multiple path integration of the photon ation over all spae would alter the veloity of light near the path as a funtion of r, and if Eq.(.5), is realized the onnetion between gravitation and QFT would be established.