A Constructive Model of Gravitation

Transcription

1 A Constuctive Model of Gavitation agubans P. Sing Abstact Tis pape poposes a pysical model in wic gavitational inteaction between masses is mediated by tei mass-momentum fields. A mass in te massmomentum field of anote mass expeiences two gavitational foces: epulsion due to tei sepaation and attaction due to tei motions in te univese. Te pape ten fomulates gavitational inteaction between matte and matte and between matte and enegy quanta, calculates gavity s effect on spectal lines and clock time peiods, and estimates te speed of gavitational wave. 1. Intoduction In Geneal elativity 1 gavitation is due to te cuvatue wic matte ceates in te field of space-time geomety. Te cuved space-time is te gavitational field, wic, as enegy quanta, would be te so-called gavitons. Astonomic collisions and inteactions among celestial bodies notwitstanding, so fa tee is no evidence of suc gavitons (paticles) o distubances (waves) in te field of space-time geomety. Te pysicist s unsakable fait in te undelying simplicity of natue is leading te quest fo a unified teoy of te fundamental foces. Te electoweak teoy unifies te weak and electomagnetic foces. Te stong foce could be next; oweve, gavitational foce defies being unified wit te est.. A. Milne olds tat geomety can be selected pimaily by te natue of undelying penomenon and te convenience of epesenting and analyzing tat penomenon; and tansfomations of coodinates alone ae but tanslations of language and ave not necessaily muc to do wit penomena. How matte waps (o ceates) space is left unexplained. Space and time neite act no ae acted upon in te stong, te weak, and electomagnetic inteactions. Tose tee fundamental inteactions ae mediated by espective fields (bosons), wic ae ineently attaced to tei inteacting matte. So, it would be natual to ave gavitational inteaction be mediated similaly by gavitationally petinent fields of inteacting matte witout coodinates and obseves being a pat of te law of gavitation (natue). Gavitational inteaction ee will be efomulated as mentioned above at te macoscopic level in te nonelativistic famewok. At te macoscopic level, fields ae effectively continuous. (Continuous means a value at eac space-time point.) Micoscopic foces and quantum mecanics will be ignoed.. Assumptions: We will base te model on two assumptions: (a) Matte as an envelope of intinsic mass field. (b) Motion ceates an envelope of momentum field. Mass is a popety of matte. Te ange of mass field is infinity. (Te oigin of matte o mass is not petinent ee.) Momentum is a popety of mass-in-motion. Momentum field is effective witin a momentum field ange, wic vaies wit te momentum.. Caacteistics of mass, cage, and te fields Fom mecanics, electodynamics, and te Assumptions, Table I lists te petinent caacteistics of cage, mass, and te fields. nties in ow, columns 5-10 ae deduced; k 1, k, k, and k 4 ae constants; and extends fom te axis of moment of momentum o cuent to te mass o cage. Table I. Caacteistics of cage, mass, and fields Cage Mass (lectic cage) (Gavitational cage) Cage: e Mass: m Cuent: i eu Momentum: p mu Moment of cuent (Angula cuent): i L e Cage (electic) field: k1 e / Cuent (magnetic) field: k Le / epulsive foce between cages due to sepaation: k1 e1 e / Attactive foce between cages due to paallel motions: k i1 i / Speed of electomagnetic wave: c (k 1 /k ) Foce between cages due to sepaation and paallel motions: k ( 1 u u / c ) e e / Moment of momentum (Angula momentum): p L m Mass field: k m / Momentum field: k4 Lm / epulsive foce between masses due to sepaation: k m1 m / Attactive foce between masses due to paallel motions: k4 p1 p / Speed of gavitational wave: b (k /k 4 ) Foce between masses due to sepaation and paallel motions: k ( 1 u u / b ) m m 1 1 /

2 4. Te gavitational model Figue 1, secto I, sows masses m 1 and m at 1 and fom te Pimodial Point P (at te Big Bang). Te angle between 1 and at P is α. Figue sows te masses wit sepaation distance, velocity vectos u 1 and u elative to P, and mass fields (ligt sades) and momentum fields (dak sades). Te anges of te momentum fields ae S 1 and S. I Figue 1. Masses m 1 and m at 1 and fom P P 1 α II I P Figue. Masses wit mass and momentum fields We define fo a mass m its effective momentum field ange S m, wic is popotional to its momentum: S m σ m u, (1) wee σ is momentum field ange coefficient, a fundamental constant. A 1 kg mass wit a speed of 1 m/s as an effective momentum field ange of σ m. Te inte-momentum field ange between m 1 and m is: Fom (1) and (), we ave: S 1 S 1 + S () S i S i j m i / (m i + m j ) ; i j () Fom Assumption (a) and Table I, te epulsive foce between m 1 and m, due to tei sepaation in space, is mediated by tei mass fields and is expessed in (4): m1 m Fs Gs, (4) wee G s is te static gavitational constant. Fom Assumption (b) and Table I, te attactive foce on m 1 by m, due to tei momenta p 1 and p elative to te Pimodial Point, is mediated by tei momentum fields and is expessed in (5). Hee extends fom m 1 to m. p1 ( p) Fd Gd, (5) wee G d is te dynamic gavitational constant. 1 III IV α m 1 m S 1 u 1 m 1 S m u We will estimate momentum field ange coefficient late to be quite small ( 10 4 s/kg). Te age of te univese is close to 14 BY. So, at 1,, and S 1, α 0. Fom (5), te attactive foce is expessed in (6): p1 p Fd Gd (6) Te dimension of G s /G d is of te squae of speed. Denoting tis speed by b, wic would be te speed of mass-momentum (gavitational) wave, we ave: G / G b (7) s d Witin S 1 momentum fields ae effective. Fom (6), te foce between m 1 and m is attactive and given in (8). We will set u 1 u u as needed fo simplicity. m1m F1 Gd u ; S 1 (8) Beyond > S 1 mass fields ae pedominant. Fom (4), (6), and (7), te esultant foce between m 1 and m is expessed in (9), wose sign depends on u /b: u m1m F1 G 1 s ; > S 1 b (9) qs. (8) and (9) ae of te fom of Newton s Law of gavitation. In (8), as S 1, te Cavendis expeiment yields a value fo G d u G, te classical gavitational constant, wic now vaies wit u. Bot G and G s (1 u /b ) depend on u ; oweve, on te uman-time scale, as u is constant, tey ae constant. Table II as te signs of gavitational inteaction, wic is attactive in inne egions ( S 1 ) egadless of te value of u, and epulsive, attactive, o zeo in oute egions ( > S 1 ) depending on u/b. Table II. Inteaction signs wit espect to b and S 1 S 1 > S 1 u < b attaction epulsion u b attaction zeo u > b attaction attaction 5. Mass-enegy gavitational inteaction An enegy quantum (, c) is at distance fom a mass (m, u). Te enegy quantum as no mass field but as momentum field by its momentum p /c. Te attactive foce on te quantum is due to te inteaction between te momentum fields, and, fom (6), is given in (10): m F me κ ;, (10) wee κ is mass-enegy gavitational constant: κ G d u / c G /( uc) (11) q. (10) olds as well fo te gavitational inteaction between a mass and a poton.

3 Te angle of gavitational deflection θ of electomagnetic wave wit impact paamete d is expessed in (1): 1 m θ tan κ (1) d To escape a mass m of adius, an electomagnetic wave must be outside a citical adius e : tan 1 m 1 κ cos 0 (1) e e Te adius e fo θ 90 0 is given by: e κ m (14) We note tat, at u c/, θ in (1) and e in (14) agee wit Geneal elativity. 6. Pysical data Te following data ae petinent ee: (a) Speed of ligt (c):.0 x 10 8 m/s; (b) Deflection of ligt at te sun (θ):. ac secs () ; (c) G d u G (pesent-day): 6.67 x N kg m ; (d) Sun s mass: x 10 0 kg; (e) Sun s adius: 6.96 x 10 8 m; (f) at s mass: x 10 4 kg; (g) at s adius: 6.78 x 10 6 m; () Mecuy s mass:. x 10 kg; (i) Mecuy s sideeal peiod: days; (j) Mecuy s peielion pecession: 575 ac secs/centuy; (k) Fatest Kuipe-belt body fom te sun: ~ 10 AU; (l) Diamete of te Milky Way galaxy: ~ 10 5 l.y. It is not clea wete te obseved deflections of ligt at te sun wee coected fo efaction and ote effects toug te sun s and te eat s atmospees. 4 We will estimate u and b fo lack of obsevations. 6.1 stimates of G d, u, and κ Wit efeence to te sun, q. (1) and data 6(a-e) yield: u x 10 8 m/s (15) G d x 10 7 N kg s (16) κ x 10 7 N kg s (17) 6. stimate of σ Fo lack of obsevations, datum 6(l) would be consideed as te sun s appoximate momentum field ange. Fom (1), (), (15), and data 6(d, k), we ave: S sun 1.5 x m (18) σ 6. x 10 5 s/kg (19) A 1 kg mass wit a speed of 1 m/s as an effective momentum field ange of te ode 10 4 m. Fom (1), (), (14), (15), (19), and datum 6(l), te Black Hole at te cente of te Milky Way galaxy as a mass of about 6. x 10 6 kg and a escape adius of about 1. x m fo electomagnetic waves. 6. stimates of G s and b We estimate te magnitude of G s and te speed of gavitational wave (b) based on ef. [5] and data 6(-j). Planet Mecuy is unde two sets of gavitational foces: one due te sun and te ote due to te oute planets. q. (8) is applicable to te fome, q. (9) to te latte. Pice and us 5 deive Mecuy s apsidal angle ψ to be: ( 1 F / F f / F ) ψ π, (0) p wee foce F s (by te sun), foce F p (by planets Venus toug Satun), and foce f (adial oscillations) ae: s F s 1.18 x 10 N; (1) F p π Γ m N; () f π Γ m N; () wee m is Mecuy s mass, and instead of G we consideed Γ in F p and f in above: Γ G s (1 u /b ) (4) Te pecession ate of Mecuy s peielion is 575 ac secs/centuy ( x 10 6 ad/sideeal peiod). So, ψ π x 10 6 ad (5) Caying out te calculations wit (0) - (5), we get: Γ 7.41 x N kg m (6) G s 1.88 x N kg m (7) G s /G d.95 x m s (8) b 1.7 x 10 8 m/s (9) u/b 0.69 (pesent-day) (0) b/c (1) Te speed of gavitational wave would be appoximately 57.4% of te speed of ligt. 7. Vibating paticle in gavitational field We deive te cange in fequency ν of vibation of a paticle as its position elative to a mass m canges. Te mass, a pefect spee of adius and density ρ, is at 0. Te momentum of te paticle is given by p /c, and its enegy is popotional to ν. Te attactive foce F between te mass and te paticle is mediated by tei momentum fields and given by (10). 7.1 Vibating paticle outside te mass As te paticle is moved fom to, te cange in its enegy is given by: F d s m κ d ()

4 Caying out te integation, we ave: ν m + κ ν 1 () Te expession in te backet > 1, tus ν < ν. Te paticle vibates at lowe fequency close to te mass. If te fequency is tat of emitted ligt, its wavelengts, wit efeence to te sun ( sun ), fom (), (17), and data 6(d, e), ae given by: λ ( x 10 6 ) λ (4) Spectal lines poduced on te sun s suface ae edsifted by about 5. x 10 6 of tei wavelengts coesponding to tose poduced at infinity. If te vibating paticle seves as an atomic clock, its time peiods, wit efeence to te eat ( eat ), fom (), (17), and data 6(f, g), ae given by: τ ( x 10 9 ) τ (5) A 1.0 second peiod at infinity is dilated to seconds at te eat. 7. Vibating paticle inside te mass As te paticle is moved fom 0 to, te cange in its enegy is given by: F d 0 κ 0 0 m d, (6) wee m (4/ π ρ ) is te mass witin. Caying out te integation, we ave: ν m 0 1 κ ν (7) Te expession in te backet < 1 but > 0, tus ν 0 < ν. Te paticle vibates at lowe fequency close to te cente. Wit efeence to te sun ( sun ), electomagnetic wavelengts, fom (7), (17), and data 6(d, e), ae given by: λ (1.67 x 10 6 ) λ 0 (8) Spectal lines poduced at te sun s cente ae edsifted by about.67 x 10 6 of tei wavelengts coesponding to tose poduced at its suface. Wit efeence to te eat ( eat ), time peiods, fom (7), (17), and data 6(f, g), ae given by: τ ( x ) τ 0 (9) A 1.0 second peiod at te suface is dilated to seconds at te cente. 7. Vibating paticle nea a point-dense mass An infinitely ig point-dense mass may be indicated by m as 0, o m/. Fom (), as m/, τ /τ. Time peiod nea te suface tends to infinity; time exists but vitually stops. Fom (), as m/, te wavelengt of ligt nea te suface tends to infinity (λ, ν 0, ν λ c). Ligt vitually ceases to exist in wavefom but still popagates. Fom (1), as m/d, θ Ligt passing nea te mass migt go aound it and etun towad its souce. An example of a mass of infinitely ig point-density is a black ole. 7.4 Vibating paticle and no mass A no-mass may be indicated by m 0 as 0. Fom (), as m/ 0/0, τ /τ 0/0. Time is indeteminate in te absence of mass. 7.5 Fundamental inteactions and time Outside a mass, te un of time is slowe as te intensity of gavitational field inceases. Inside a mass, time uns slowe as te intensity of gavitational field deceases. Te un of time in ote fundamental fields is not known. 8. Poton falling in te gavitational field of a mass We calculate te cange in te enegy of a poton (, ν, p /c) as it falls fom a eigt in te gavitational field of a mass m of adius. Tei momentum fields mediate tei gavitational attactions. As te poton falls fom + to, te cange in its enegy is given by: + F d + Caying out te integation, we ave: m κ d (40) κ m κ m ν / ν 1 1 (41) + As (+) >, ν > ν. Te falling poton is bluesifted. Te Pound-ebka expeiment 6 sows factional cange in te enegy of a poton as it falls fom eigt.5 m to te eat to be δ/.5 x Fom (41), we get δ/ (ν ν ) / ν 6.17 x Poton-poton gavitational inteaction A poton as no mass field but as momentum field. Fom (6), absolute gavitational foce between potons is: Gd ν1ν Fν ν ; S1, (4) c wee is Planck s constant and (G d /c ) 0. Tee is no gavitational foce between potons. 10. Antimatte-antimatte gavitational inteaction Te model applies to antimatte-antimatte gavitational inteaction as well. 11. Matte-antimatte gavitational inteaction Te question about matte-antimatte gavitational inteaction is wide open. To esolve tis question expeiments ae needed to eveal te sign of antimatte gavitational mass and te sign of matte-antimatte gavitational inteaction. 4

5 1. emaks Gavity exists as epulsion and as attaction. Gavitational epulsion is ineent. Gavitational attaction is acquied, exists at close sepaations ( S 1 ), and as been evolving wit te speeds of te masses afte te Big Bang. Attaction is weake tan epulsion by a facto of (u/b). Te classical gavitational constant G is not a constant but vaies wit u ; oweve, it is constant on te uman-time scale. We note fom (0) tat u < b at pesent. We may ten infe fom Table II tat: te univese as bound systems due to attactions between masses in inne egions ( S 1 ); and te univese is expanding due patly to epulsions between masses in oute egions ( > S 1 ). Te univese migt undego one o moe cycles of expansion, steady, and contaction states. Te calculations and infeences ee acutely depend on te accuacy of te value of te gavitational deflection of ligt by a mass and of momentum field ange coefficient of a mass in motion. Tose values ougt to come fom delicate obsevations. 1. Addendum Faaday intoduced te concept and utility of field to pysics. Classical pysics ad gavitational field and electomagnetic field; moden pysics intoduced te stong field and te weak field. Geneal elativity intoduced te field of space-time geomety to explain gavity. Tis model intoduces mass-momentum field to explain gavity. Tus, te stong field, te weak field, electomagnetic field, and gavitational (mass-momentum) field now belong to te same class of fields tat is, fields wic ae ineently associated wit and depend on tei souces. Te ate o pobability of emission o absoption of a quantum is elated to te stengt of te undelying inteaction. Te elative stengts of te fundamental inteactions ae: (g : w : e : s) (1 : 10 0 : : 10 4 ). A nucleus takes t e 10 1 sec to emit a poton. So, elatively, a nucleus would emit a gaviton in ougly t g ( e / g ) t e secs 10 1 yeas! Tis is seveal odes of magnitude ige tan te known age of te univese ( 14 x 10 9 yeas)! Te following questions ae fundamentally significant to undestanding gavity and, vey possibly, te fundamental inteactions fute: (1) Is mass, as gavitational cage, a popety of matte (o antimatte), as ae te colo, te weak, and electical cages? () If so, wat endows matte (o antimatte) wit tese fundamental cages? () Do gavitational masses of matte and antimatte ave opposite signs? (4) Do matte and antimatte ave te fundamental cages of opposite signs? 14. efeences [1] Albet instein, Te Meaning of elativity, Pinceton Univesity Pess, Pinceton, NJ, []. A. Milne, elativity Gavitation and Wold- Stuctue, Oxfod Univesity Pess, Oxfod, 195. []. Feundlic, H. v. Klübe, A. v. Bun; Zs. f. Astopys.,, 171, 191. [4] Cales Lane Poo, Te deflection of ligt as obseved at Total Sola clipses, JOSA, v0n4, 190. [5] Micael P. Pice and William F. us, Nonelativistic contibution to Mecuy s peielion pecession, Am. J. Pys., v47n6, [6]. V. Pound and G. A. ebka, J., Gavitational ed- Sift in Nuclea esonance, Pys. ev. Letts., (9), Hendon, Viginia, USA Mac 1, 011 5