Gravitational Theory with Local conservation of Energy ABSTRACT

Transcription

1 Gravitational Theory with Loal onservation of Energy D.T. Froege Formerly Auburn University Phys-tfroege@glasgow-ky.om ABSTRACT The presentation here is base on the presumption that the total energy of a partile an photon in a gravitational fiel is loalize an onserve. A mass partile thus entering a stati gravitational fiel has an inreasing veloity, but a ereasing rest mass, or a mass efet. The total energy is onserve. This also means that as a photon rises in a gravitational fiel there is no loss of energy, an therefore a photon esapes the most intense fiel, preluing the formation of a blak hole. Sine there is no energy hange in an aelerating partile tehnially gravitation is not a fore. It will be shown that suh a theory of gravitation an be evelope, that properly preits known ynami, has proper ovariant transformations, the proper Shapiro veloity, an oes not require formulation in urve spae. Noether s theorem efinitively shows that ontrary to all other fores, energy annot be onserve nor loalize in a Riemannian gauge fiel representation. It is presume here that this is a flaw in GR, an it is asserte here that Noether s theorem is not an iniator of a physial reality, but an iniator of the approximate nature of GR. This an best be teste in the observation of the properties of objets ite to be blak holes. There are points of this evelopment that are testable, an provable or isprovable in experiments on Blak Holes an, Event 1

2 INTRODUCTION As is well known, but presume unimportant, are aspets of the Rii tensor representation that illustrate the theory is an approximation to the orret representation, but is not omplete or aurate representation. The obvious shortfalls are that: 1. Mass is represente as a ontinuous ensity funtion, when reality requires isrete point funtions, an there is insuffiient omplexity in the tensor expressions to represent mass as a olletion of point partiles.. GR oes not sale own, nor properly funtion at mirosopi level, though the theory reognizes no saling limits. 3.It is a gauge fiel, with an infinite number of infinitesimal generators. Beause of this, Noether s theorem illustrates that energy tensor is not ovariant uner general oorinate transformations, an there an be no loal onservation of energy, meaning the flow of energy in an out of a spaetime volume is not onserve [9]. This leas iretly the onept of blak holes, sine the soure of the kineti energy gaine by a partile entering a gravitational fiel omes from the fiel an not the rest mass. There is no rest mass efet. 4. It is not ovariant uner general oorinate transformations. Loal energy balane is epenent on the oorinates use for the alulation; onsequently ifferent results are obtaine for ifferent oorinate frames [4]. 5. The istane between any two points in the efine urve spae is ambiguous, an epenent on the path [4]. Researhers who o not view GR as an approximation o not onsier these issues shortomings, but the reality of physis. The ynami partile interations presente here are formulate in ovariant ifferential an algebrai relations. It is shown that the phenomenology of GR an be reproue without resorting to Riemannian spae urvature an

3 oes not result in unphysial singularities. This evelopment will ahere to a flat Minkowski spae ( t 0) an a variable spee of light. The total energy of a partile is presume loalize in the volume of the partile, an as the partile aelerates in the fiel the kineti energy inreases at expense of the rest energy. There is thus no work one on the partile an tehnially the fiel is not a fore, as work one on the partile woul inrease energy. The tehnial issues involve in the aeleration is more fully in evelope in The Conept of Mass as Interfering Photons, an the Originating Mehanism of Gravitation [1]. Current views of photon energy in GR are ontrary to Einstein s original view that the photon energy is onstant an the shift is ue slower lok assoiate with the emission. [1]. Current views of GR are that the photon loses energy to the fiel on rising. This evelopment is ast in Minkowski spae, an asribes the frequeny of a photon rising in a gravitational fiel to have a lower measure frequeny at the elevate reeptor ue to the lower rest mass of the emitter not a hange in the time sale or a loss of energy. After taking aount of the rest mass hange at ifferent elevations, the results of the Poun-Rebka-Snier experiment shows that the energy of the photon is onserve. [6] GENERAL DEVELOPMENT For a massive partile in a gravitational potential, the total mass of a partile at rest relative to an observer external to the fiel is efine to be: M M0 1 r, (1.1) where M0 is the rest mass external to the gravitational potential. The relativisti mass is then: 3

4 v M 1 M 0 1 r (1.) This relation Eq.(1.), is presume to be the funamental relation between mass partiles gravity an veloity or the primary postulate of the theory. With this an loal onservation of energy, all the other ynamial relations an be erive inluing the Shapiro veloity of light. GR presumes the funamental postulate to be the Rii tensor urvature of spaetime whih is a far more expansive assumption, an an only eal with aggregates of matter having no mirosopi saling for point partiles Though similar to stanar Lagrangian expressions Eq.(1.), is a n orer eparture but is easy to show that it an be ast in the form of the well known Lagrangian within measurable auray. i.e. GMm 1 M M Mv r 0 (1.3) Orbital Mehanis In the following it will be shown that the orbital preession preite by GR an alulate with Eq.(1.), without resorting to urve spae, only inluing known Speial Relativisti onsierations. Most alulations of orbital motion ten to neglet issues relate to the makeup of the potential term. GMm, (1.4) r First is that the mass terms have to be the relativisti mass. This is obvious from the fat that, if the partiles happen to be spinning the kineti energy must be inlue, meaning the mass is relativisti mass. In aition eah mass experiene the other as if it is moving with their relative veloity. 4

5 From our knowlege of the Thomas preession, it is known that the istane a partile traveling the irumferene of a irle aroun an attrating potential is shortene by the relativisti ontration. We woul assert that if the irumferene of a irle is ontrate as the result of the relativisti veloity, the raius must also be ontrate. With those onsierations the gravitation term in Eq.(1.3), shoul be: GMm G M0 m0 r r 1 v / 1 v / 1 v / 0 (1.5) or: GMm GMm lo 3 v 3 M 1 r r r 1 v / (1.6) This is a ifferential expression relating, the veloity, an the istane to the loal gravitating mass. This an now be solve for the orbital motion, without nee to make assumptions about the fore mass relation. In the following it will be shown that the equations of motion proues orbital relations, equivalent to the weak fiel GR relations, with the same perihelion avane: Noting. Eq.(1.6), an inserting into Eq. (1.), yiels: 4 lo lo 3 v 1 1 v 3 v M0 1 1 M r r r r 8 (1.7) Noting that there is only one signifiant ross term this beomes: 5

6 4 3 v 1 v 1 v 3 v M0 1 M 4, (1.8) r r r r 8 whih separates into: M M 1 1 v 3 v M0 r r r (1.9) Setting the left term in this to, an note that in a onservative system, this term is onstant. That is beause M0 is a efine onstant an the total energy is onstant. Using the proeures for fining as outline in Robertson & Noonan,[4] the perihelion preession, in agreement with GR is: 1 3 u 0 u 0 u 0 u u 0 u 3 (1.10) p The etaile alulations for this are inlue in Appenix I. q.e.. It has been shown that the proper orbital equations an be erive without resorting to Riemannian spaetime. Photon Energy From the efining relation of this theory Eq.(1.), the view of the Poun- Rebka-Snier[6], Mossbauer effet experiment ( )[6] hanges. Instea of the photon losing energy as the photon rises in the tower, the emission of the photon at the bottom of the tower is from a less massive generator, an at a lower frequeny. The generate frequeny plus the ae Doppler frequeny provie by the veloity of the soure in the experiment equals the frequeny at the top, thus the photon loses no energy in the flight up the tower. 6

7 1 r B D T. (1.11) This is a eparture from General Relativity. GR requires a photon esaping from a gravitational fiel to lose energy to the fiel, an in the ase of a blak hole the entirety of the energy is lost before esapement. Sine the energy in isussion here is loalize an not lost to the gravitational fiel, the Shwarzshil raius is no barrier. Proper Defletion an Veloity of Light The well known Shapiro veloity of light is arrive at by Blanfor, an others [7],[14],[15], by solving the GR fiel equations for a onstant time sale, thus giving the apparent veloity of light in three spae an the utility of Fermat's priniple to projet ray traes. 0 1, (1.1) r Using the onept of loally onserve energy, an knowing that a photon etete at a lower position seems to gain energy from the potential energy suh that: E 0 E ' 1 r, (1.13) then the loal spee of light an be eue. In the loally onserve system it is the measurement system that has a lower rest mass an thus the photon only appears to have gaine energy. As a photon enters a gravitational fiel it is presume here that the energy is not hange. The relation expete to be true is: 7

8 E h h (1.14) From the exterior observation point the frequeny oes not hange. This an be easily unerstoo by observing the phase of a raio wave transmitte from spae to the surfae an reflete bak. The number of waves an the frequeny of the returning signal is the same as the transmitte signal. Thus in Minkowski spae, the interior an the exterior frequeny are the same an onstant. From observations it is known that in the loal frames as a photon esens there is an apparent frequeny inrease ue to an apparent inrease in energy, thus: h ' E 1 r (1.15) As the partile esens, there is a hange in the loally observe value of the frequeny, where the prime iniates a loal value. GR woul asribe this to the effet of time ilation. In the loal frame; ' ' (1.16) ' In aition, the wavelength of a photon arriving at a loal loation in a gravitational fiel is erease by: ' 1 r (1.17) Putting Eq.(1.17), into Eq.(1.16), then that into Eq.(1.15), Eq.(1.14), gives: ' E h (1.18) 1 1 r r 8

9 Comparing Eq.(1.18), to Eq.(1.14), shows are equivalene if the loal spee of light at the loal loation is: ' 1 r (1.19) Comparing Eq.(1.19), with the GR equivalent, Eq.(1.1), shows them to be nearly the same aept that Eq.(1.19), has a seon orer term. The ifferene will only be etetable for motion of photons very near the gravitational raius, an thus observation of the photons in that proximity will istinguish the proper theory. In simplisti terms when the photon enters a gravitational fiel, a loal reeiver, along the path fins the frequeny goes up an the wavelength goes own, to aommoate both these effets from an external perspetive in whih the frequeny stays onstant, the spee of light must erease aoring to Eq.(1.19), or the inex of refration is: 0 / 1 / r (1.0) Whereas the GR inex woul be establishe by the Shapiro Veloity with inex of refration: 0 / 1 / r (1.1) The seon orer ifferenes in these two expressions shoul be soon testable by the Event Horizon Telesope [16]. CONCLUSION With simple assumptions regaring the relation between rest mass, an relativisti mass, proper gravitational ynamis an stellar efletion phenomena an be preite. The propose theory yiels the proper orbital equations, with the proper perihelion avane, efletion of light an gravitational re shift. The gravitational potential exhanges no energy with 9

10 photons, thus photons are not boun in a gravitational fiel, an there are no blak holes, a belief often expresse by Einstein [13]. Preision light efletion experiments near large masses, or isoveries of neutron star masses larger than GR allows, will valiate or invaliate this theory. Referenes: 1 DT Froege, The Conept of Mass as Interfering Photons, an the Originating Mehanism of Gravitation,V00914, 13 Noether's Disovery of the Deep Connetion Between Symmetries an Conservation Laws arxiv:physis/ H. P. Robertson, T.W. Noonan, in Relativity an Cosmology, (W. D. Sauners, Philaelphia, 1968), pp Conley Powell, General Relativity Unpublishe (005) 6 R.V. Poun an J.L. Snier, Effet of gravity on gamma raiation, Phys. Rev. B 140: (1965). 7 Roger Blanfor, Kip S.Thorne, in Appliations of Classial Physis, (in preparation, 004), Chapter TP. Purue, The gauge invariane of general relativisti tial heating arxiv:gr-q/ v 15 Jun E. Noether, "Invariante Varlationsprobleme", Nahr.. König. Gesellsh.. Wiss. zu Göttingen, Mathphys.E. Noether, "Invariante Varlationsprobleme", Nahr.. König. Gesellsh.. Wiss. zu Göttingen, Math-phys. Klasse (1918), 35-57; English translation M. A. Travel, Transport Theory an Statistial Physis 1(3) 1971, J. M. Bareen. Rapily rotating stars, isks an blak holes. In C. DeWitt an B.S. DeWitt, eitors, Blak Holes, Les astres olus, pages Goron an Breah Siene Publishers, New York, J. A. Ruea, V. S. Manko, E. Ruiz, an J. D. Sanabria-G omez. The ouble-kerr equilibrium onfigurations involving one extreme objet. Class. Quantum Grav., : , November Okun, L. B.; Selivanov, K. G.; Telegi, V. L.On the Interpretation of the Reshift in a Stati Gravitational Fiel Am.J.Phys. 68 (000) A Einstein, On a Stationary System with Spherial Symmetry Consisting of Many Gravitating Masses Journal Annals of Mathematis 40: F. Karimi, S. Khorasani, Ray-traing an Interferometry in Shwarzshil Geometry, arxiv: [gr-q] 10

11 15 Khorasani, Defetions of Light an Shapiro Delay: An Equivalent Meium Theory Approah, arxiv: v1 [gr-q] 9 Jun DT Froege, The Image Comparisons of Blak Hole vs. Neutron Dark Star by Ray Traing,V111514, Appenix I Details of Perihelion Avane The general rest mass veloity relation propose is: M 1 v M 1 0 r Where the veloity invariant potential is: 1 v M 0 1 M 1 3 r 1 v / Taking square root: 1/ 1/ 1 v M M 3 r 1 v / (.1) (.) (.3) Binomial expansions: 3 1/ x x x 1 x / x 3 1 x 1 x 8 x x 1 x x 8 16 The simple expansion woul be: 1/ 3 (.4) 1/ lo lo 1 lo lo 1 1 r r r r lo lo lo lo (.5) 11

12 Expaning all the terms in Eq. (.3). 4 3/ lo 1 lo 1 v 3 v M0 1 1 v / 1 M, (.6) rlo r r lo an: 4 lo lo 3 v 1 lo 1 v 3 v M M rlo rlo r rlo 8 (.7) There is only one ross term of signifiant value. v 4 3 v 1 1 v 1 v 3 v M0 1 M r r r r r 8 (.8) Simplifying an separating the mass terms: v 3 v 4 v M0 1 M r r r 8 r 1 1 v 3 v M M r r r 4 r (.9) 1 1 v 3 v M0 M0 1 4 M r r r 4 r M M v 3 v M0 r r r 4 r multiplying by, & noting that in a onservative system where the total energy is onstant, the mass term is onstant. M M0 (.10) M0 Thus: r r r 4 r 1 3 v v 1 4 (.11) The orresponing GR term per Robertson & Noonan. h v (.1) 3 r r 1

13 Some onventional oorinate transformations: u 1/ r u r u h h t r u v u r h u h t r u (.13) making the substitutions, we have: 1 3 v 1 u u v 1 4u v 4 Now taking the erivative with respet to the angular oorinate: (.14) 1 r 3 v 1 u u u r 1 4u v t 4 or: 1 1 u u r u h 1 4 u t 3 v v 4 Making some substitutions.,(.15) (.16) 1 1 u u r u h u t 0 h u 1 4 u u v h u h 4 3 h u h u 4 (.17) 13

14 14 Differentiating the three terms, esignating eah as A,B, & C: 1 u u 1 u 1 u h 1 u A h Parts of the B term: u h u u u u h u u u h u u 1 4 u 4 (.18) So the B term is: u h u 1 4 u u u h u 4 u u h u 1 4 u u u u h u u B 1 4 u (.19)

15 An the C term: h u h u 3 1 u u h h u u u u u u h h u u 4 h u 3 1 u u h u u u 3 1 u u h h u u 0 4 C (.0) Colleting an fatoring a ommon term gives: 0 h 1 u h u u u u u 1 4u u 3 h u u (.1) olleting ommon terms reues the number of terms: 15

16 1 u h u 3 h u u 1 4 u u 0 u (.) Diviing by the oeffiient of the seon orer term gives: 3 h u 1 u / 1 4 u h u 3 h u 0 u 1 4 u / (.3) u u or: 3 h u 1 u 4 u h u u 3 h u u 0 u 1 4 u 0 (.4) The equation for a irle is: u u u 0 +f (.5) Where f is a perturbation of the orbit. 16

17 The preession, per the proeure of Robertson & Noonan is 1 f. u Where in this ase f is: f 3 h u u0 u 4 u h u u 3 8 u 4 3 (.6) So: Where: h p 1 e 1 u 0. 3 h u u0 5u u 1 u 4 h u u (.7) Where is a ratio of the perihelion avane to the orbit irumferene h 5 u0 3 u 0u 4 u h 1 u0 3 h u u u 0 (.8) 1 u 4 u 0 Then we have for the preession: 1 3 u 0 u 0 u 0 u u 0 u 3 (.9) p The units are the ratio of the avane to the orbital irumferene. Comparing with the GR value from Robertson & Noonan: 17

18 1 r u uu u u h 3 h 3u u t u0 (.30) Thus our proeure yiels the proper perihelion preession. 18