Normal Coordinates Describing Coupled Oscillations in the Gravitational Field

Transcription

1 Normal Coordnates Descrbng Coupled Oscllatons n the Gravtatonal Feld Walter James Chrstensen Jr. Department of Physcs Cal State Unversty, Fullerton 8 N. State College Blvd. Fullerton, CA 983 And Department of Physcs Cal Poly Pomona Unversty 38 W. emple Blvd. Pomona, CA 9768 Publshed n General Relatvty and Gravtaton January 7 DOI:.7/s Abstract he moton of a local source nducng small oscllatons n the gravtatonal feld s nvestgated and shown to exhbt pure rotatonal knetc energy. Should the net affect of these slow, revolvng oscllatons cause large-scale rotatons n spacetme t would certanly result n anomalous celestal acceleratons. When ths angular rotatonal frequency of spacetme s appled to the anomalous acceleraton of the Poneer / spacecrafts, the correlaton s promsng. I. Introducton Fundamental to the theory of general relatvty s the couplng exstng between the gravtatonal feld and the energy-momentum source ; f one changes, so too, wll the other. In partcular, f the gravtatonal feld undergoes oscllatons then there must be a causal source nducng these oscllatons. If so, ths suggests the gravtatonal system can be treated lke a coupled sprng and drver. hough coupled moton can be qute

2 complex, not even perodc, t can always be descrbed n terms of a set of normal coordnates havng the property that each coordnate oscllates wth a sngle, well-defned frequency wth no couplng among them. he goal then wll be to descrbe the moton of the energy-momentum-source, smply by knowng that the gravtatonal feld s oscllatng. hs can be accomplshed analogously through the classcal approach of analyzng small dsplacements about a pont of equlbrum and then solvng for the normal coordnates, a procedure that s also well known n gravtatonal lterature. Once these coordnates are dentfed they can be brought nto the language of general relatvty. he metrc tensor s then constructed and energy-momentum tensor calculated from the Ensten ensor G. he resultng dagonal tensor wll be shown to have components of pure rotatonal knetc energy densty. In classcal physcs ths dagonal-knetc-energy result s a necessary condton mposed by normal coordnates. herefore the method presented here of extendng normal coordnates nto general relatvty s promsng. Furthermore, though the dagonal rank-two tensor s shown to be constant and real, the contravarant tensor necessarly turns out to be complex. However, becomes real and equal to every one-fourth the perod of the fundamental mode of oscllaton of the normal coordnates. Countable rotatonal symmetry, together wth Noether s theorem 3, suggests the energy-momentum tensor s conserved. II. Normal Coordnates

3 Ensten s gravtatonal feld equatons express a causal lnk between the energymomentum-source, and spacetme curvature assocated wth the tensor G. he purpose of ths paper wll be to determne the causal moton of the source nducng oscllatons n the gravtatonal feld. he problem s remnscent of a classcal oscllator and drver and wll be our startng pont. We begn wth a Lagrangan representng small oscllatons about a pont of equlbrum. L j V j j j (.) he s represent small devatons from the generalzed coordnates q, such that q '. Classcally the ' s subsequently become the generalzed coordnates for q the equatons of moton, wheren the knetc energy has dagonal components only. V (no sum over ) (.) j j he soluton 4 to (.) has the normal coordnate form of C e t (.3) Assumng these coordnates quas-descrbe oscllatons n the gravtatonal feld, by the prncple of equvalence let a general relatvstc coordnate bass e experence the acceleratons expressed by (.3). Furthermore let the coeffcents of (.3) be set equal to

4 one and let negatve one-half be ntroduced n front of the angular velocty. hese small changes allow for the moton of the energy-momentum source to become more apparent. Raylegh s prncple 5, 6 s appled, and the coordnate frequences reduce to the fundamental mode of oscllaton,, havng the greatest ntensty. he average knetc energy s then equal to the average potental energy U. Wth the precedng adjustments made, the bass for the general relatvstc coordnate system s constructed from the modfed normal coordnates 7, 8 : t e e (.4) By defnton the nner product of any two such bass elements e, e yelds the metrc tensor. t t g e e e e (.5) As wth the mechancal oscllaton problems t s understood that only the real part of ths complex metrc corresponds to physcal measurement. III. Energy-Momentum ensor

5 he constructed weak feld metrc g s appled to Ensten tensor G. A straghtforward calculaton produces the energy-momentum tensor 9, and together they form a lnearzed theory of gravtaton: 3 G 6 c (3.) s separated out to show ts rotatonal knetc energy densty form. 3 G 8 c (3.) he moment of nerta and angular frequency matrces are defned to be ; 3 G 8 c I ~ (3.3) In compact tensor notaton the energy-momentum tensor becomes

6 I ~ (3.4) If the angular velocty s replaced by v r, then equaton (3.) resembles an energymomentum tensor for a radaton domnated perfect flud --n partcular a perfect flud of gravtons. It s mportant to realze was derved from Ensten s gravtatonal wave equaton based on a varatonal prncple, and not upon the prejudce of defnton. Furthermore, t s nterestng to observe that, although, s completely real, ts contravarant counterpart s necessarly complex. g g e t (3.5) hs result shows the Ensten tensor G and ts metrc constructed from normal coordnates, are able to separate the energy momentum-tensor nto real and magnary parts through a tme rotaton. Although tme-wse there are uncountable many complex energy-momentum tensors, becomes completely real and equal to n one-fourth the perod of the fundamental mode of oscllaton; that s whenever t. 4 Countable symmetry together wth Noether s heorem suggests the energy momentum tensor s conserved under tme rotaton. every

7 IV. Estmate of the Fundamental Mode of Oscllaton he precedng developments for an oscllatng gravtatonal system are now appled to a regon of spacetme below the mcro-level. In ths ultra small regon both the source and feld must be comprsed of nearly the same partcle, otherwse the source would overdrve the gravtatonal system and no longer would small oscllatons occur about the pont of equlbrum. In the world of partcle physcs the gravtatonal feld s made up of self-nteractng gravtons, whch also nteract wth every other partcle n the unverse. Snce the gravton has an extremely small rest mass of less than x 65 Kg, and that the regon of spacetme beng consdered s so very tny, the proposed source and gravtatonal feld must be comprsed of coupled gravtons that oscllate wth rotatonal knetc energy. hese gravtons must therefore have an assocated spn I ~. he dscrete energy components are notably ndependent of Planck s constant. he model envsoned n ths tny regon of spacetme s a gyroscopc gravton creatng a pont of equlbrum from whch coupled gravtons not only oscllate at the fundamental frequency, they rotate at ths frequency as well. Furthermore, snce the regon s so small, accordngly ths frequency must be the de Brogle wave-frequency for a gravton. If such a coupled system s prevalent throughout the unverse, the net affect could add to cause large-scale rotatons n spacetme. Dependng on where an observer resdes relatve to the axs of rotaton, one mght see dstant bodes exhbtng anomalous acceleratons. hs assumpton s supported by recent radometrc data receved from

8 Poneer / spacecrafts, wheren an anomalous nbound acceleraton toward the sun s observed. Presently there s no conclusve explanaton for ths phenomenon. As a physcal check to determne f the anomalous acceleraton s related to the rotatonal frequency of coupled gravtons, the smple calculaton a r R s made for the large-scale rotaton frequency, of spacetme. he observed nward solar acceleraton s a r 8.74x m s,3. he dstance of AU s chosen for the radal dstance R from the axs of rotaton to the spacecraft because t s the approxmate dstance when the Poneer anomaly was frst dscovered. he angular velocty of the gravtatonal feld, and hence the de Brogle gravton wave frequency, s computed to be.7x sec (4.) he de Brogle gravton wave-frequency computed from the gravton mass s g mc.7x 4 sec (4.) hough these frequences are three orders of magntude apart, as a frst approxmaton the result s promsng, especally realzng other values for the gravton mass 4 are nearly sx orders of magntude heaver than the calculated Goldhaber and Neto result.

9 V. Concluson In ths paper normal coordnates were appled to an oscllatng gravtatonal feld as a method for determnng the moton of the energy-momentum-source. hs moton was calculated to be pure rotatonal knetc energy. By applyng ths result to a regon of spacetme below the sub-mcro level, the partcle s rotaton becomes gravton spn. he net affect of ths gyroscopc moton has a cumulatve affect causng spacetme to rotate at the gravton frequency. Calculatons based on the poneer spacecraft data support ths noton. Acknowledgements My warmest thanks to Alfonso Agnew for hs enthusastc support, gudance, and for verfyng my energy-momentum result s correct. My frendshp goes to John Fang, my early mentor on gravtatonal partcle physcs. I also wsh to extend my frendly thanks (for many reasons) to Ka Lam, Mary Mogge and Hed Fearn. Lastly, I would lke to thank Kp horne for spendng tme wth me at the nd Pacfc Coast Gravtatonal Meetng to answer questons related to ths paper.

10 he German mathematcan Karl Weerstrass (85-897) showed n 858 that the moton of a dynamcal system could always be expressed n terms of normal coordnates. W. Msner, K. S. horne and J. Wheeler. Gravtaton. W. H. Freeman and Company Pages N. Byers. Israel Mathematcal Conference Proceedngs Vol., (999) 4 H. Goldsten, C. Poole, J. Safko. Classcal Mechancs, Addson Wesley. 5 () 5 J. B. Maron and S.. hornton. Classcal Dynamcs of Partcles and Systems, HBJ. 458 (988, 3 rd edton) 6 R. M. Quck and H. G. Mller. Phys. Rev. D 3, 68 (985) 7 A. Majd, A. Allezy, R. Dufour. Journal of Vbratons and Acoustcs. 8, 5 (6) 8 H. Collns, B. Holdom. Phys. Rev. D 65, 44- () 9 Prvate Communcaton wth Prof. Alfonso Agnew, of Calforna State Unversty Fullerton, Mathematcs Department ndependently confrmed the energy-momentum tensor result of equaton (3.) W. Msner, K. S. horne and J. Wheeler. Gravtaton. W. H. Freeman and Company A. S. Goldhaber and M. M. Neto. Phys. Rev. D 9, 9 (974). J. D. Anderson, P. A. Lang, E. L. Lau, A. S. Lu, M. M. Neto and S. G. uryshev, Phys. Rev. D 65, 84 () 3 O. Bertolam and J. Parámos. Phys. Rev. D 7, 35-3 (5). 4 A. Cooray and N. Seto. Phys. Rev. D 69, 35 (4)