Inertial Field Energy

Transcription

1 Adv. Studes Theor. Phys., Vol. 3, 009, no. 3, Inertal Feld Energy C. Johan Masrelez 309 W Lk Sammamsh Pkwy NE Redmond, WA 9805, USA jmasrelez@estfound.org Abstract The phenomenon of Inerta may be explaned f the four-dmensonal scale of spacetme were to change durng acceleraton. Although a dynamcally changng scale cannot be modeled by coordnate transformaton t s compatble wth General Relatvty, and t s observatonally ndstngushable from Specal Relatvty, except that t admts cosmologcal smultanety. The new theory explans the orgn of the nertal force as beng a curved spacetme phenomenon, wth the mplcaton that acceleratng matter mght nfluence the metrcal scale of space and tme and consequently also by General Relatvty the energy-densty of spacetme. A close correspondence s found between an nertal lne-element and a gravtatonal lne-element, both expressng energy densty by Posson s equaton. Under certan condtons the feld energy nduced by acceleraton could become negatve. PAC s: p, q, 04.0.Cv, Kd, e Keywords: Inerta, Inertal Feld Energy, Negatve Feld Energy 1. Introducton In two recent papers the author proposes that Inerta, lke Gravtaton, could be a curved spacetme phenomenon caused by acceleratng moton [Masrelez, 007a and Masrelez, 008]. If the four spacetme metrcs of the Mnkowskan lne element change dynamcally durng acceleraton, t could explan Inerta wthout volatng the two postulates upon whch Ensten based hs paper on Specal Relatvty (SR) [Ensten, 1905]:

2 13 C. Johan Masrelez 1. All nertal frames are physcally equvalent.. The velocty of lght n these nertal frames s the same. The dynamc scale-factor, whch multples all four metrcal components n the Mnkowskan lne-element and explans Inerta, s 1-(v/c), where v s the relatve velocty and c the speed of lght. When Ensten derved the Lorentz transformaton n ths 1905 paper he mplctly made an addtonal assumpton, whch perhaps should be seen as a thrd postulate; he assumed that coordnate locatons n nertal frames are related va the Lorentz transformaton. However, f nertal frames were to dffer by metrcal scale-factors that depend on ther relatve veloctes they would not be related va the Lorentz transformaton, but va Vogt s transformaton [Vogt, 1887] that was derved pror to the Lorentz transformaton. The resultng dynamc scale theory would be observatonally ndstngushable from SR, but would allow smultanety regardless of moton. And, t would allow us to treat Inerta as a gravtaton-type phenomenon [Masrelez, 008]. The present paper shows that there s a close correspondence between an nertal lneelement and a gravtatonal lne-element, both expressng energy densty by Posson s equaton.. Inerta and energy Energy s ntmately related to moton and to Inerta; knetc energy would not exst wthout Inerta, snce a force would not be requred to accelerate or decelerate an object. The recent dscovery that moton mght change the metrcs of spacetme, ncludng the temporal metrc, opens up the possblty that the energy densty of spacetme mght be nfluenced by acceleratng moton. The proposton that acceleratng moton mght cause spacetme curvature s consstent wth the general understandng that Inerta s a phenomenon smlar to Gravtaton. Accordngly acceleratng partcles mght cause local changes of ther spacetme metrc, and we wll assume that an acceleratng stream of partcles correspond to an nertal feld smlar to a gravtatonal feld descrbng how the nduced spacetme curvature changes wth spatal locaton. By ths approach the spacetme curvature becomes a functon solely of a velocty that depends on poston. Ths pont of vew s allowed, f we assume that the nertal propertes of spacetme do not change wth tme, snce n ths case the temporal dependence s mplct when the velocty profle as a functon of locaton s known [Masrelez, 007a]. Accordng to ths vew, a flow of acceleratng partcles creates an nertal feld wth scalar feld potental v /, whch accordng to GR mght generate spacetme energy. Ths possblty s nvestgated n ths paper.

3 Inertal feld energy Dervng the nertal energy densty Consder the nertal lne element found n [Masrelez, 007a]: μ ν ds = g ( μν dx dx = (1 v ) dx0 dx1 dx dx3 ) (3.1) v = v( x, y, z) = v( x1, x, x3) In ths expresson the speed of lght s c=1 and x 0 =t, x 1 =x, x =y, x 3 =z. Also, repeated ndces ndcate summaton accordng to Ensten s conventon unless otherwse noted. Wth ths lne element all trajectores are geodescs of general relatvty, whch would explan Inerta. Ensten s GR equatons are: 1 Rμν gμνr= 8π GTμν (3.) R μν s the Rcc tensor, T μν the energy-momentum tensor and G the gravtatonal constant. The components of the Rcc tensor are gven by: α α α β α β R =Γ Γ +Γ Γ Γ Γ (3.3) μν μν, α μα, ν μν αβ μβ αν The Chrstoffel symbols are: α 1 (,,, ) g αβ Γ g g g μν = βμ ν + βν μ μν β (3.4) The Rcc scalar s formed by contractng the Rcc tensor: μν R = g R (3.5) μν In GR the energy densty s not defned n general, snce t depends on the chosen reference frame. I wll assume that the energy s measured n relaton to an nertal reference frame, whch could be arbtrary. I wll also assume that the total (net) Inertal Energy Densty (IED) relatve to ths reference frame s gven by: IED = T0 T1 T T3 (3.6) Ths defnton s used by Rchard Tolman [Tolman, 1930]. Makng use of the relatons (3.) and (3.5) we have: 1 0 IED = R0 (3.7) 4πG The expresson for the Rcc tensor components (3.3) contans dervatves of the metrcs wth respect to the four spacetme coordnates. All dervatves wth respect to tme (x 0 ) dsappear because the velocty s modeled as a functon solely of the three spatal coordnates. Let us evaluate the contrbuton to R 00 from dfferentaton of one of the spatal coordnates denoted by the ndex. In the expressons below the ndex s kept fxed. Thus, here repeated ndces do not mply summaton.

4 134 C. Johan Masrelez ( vv ) vv v Γ 00, = ( g )( g00, ) = ( vv) = r r (1 v ) (1 v ) (1 v ) (3.8) 0 Γ 00 = 0 (3.9) 1 vv Γ 00 = ( g )( g00. ) = (1 v ) (3.10) vv Γ 0 =Γ 1 =Γ =Γ 3 = ( g )( g00, ) = (1 v ) (3.11) Usng (3.3) ths gves a contrbuton to R 00 from each of the three spatal components =1,, and 3: R () = Γ +Γ Γ +Γ +Γ +Γ Γ Γ Γ Γ = 00 ( 00, 00 ( ) ) 0 ( 00, 00 0 ) = Γ + Γ Γ = ( vv ) ( vv ) vv v = + = (1 v ) (1 v ) (1 v ) vv v 1 ( v /) = = (1 v ) (1 v ) dx (3.1) Addng all three contrbutons the nertal energy densty becomes after rasng one ndex and usng (3.7): 1 ( v /) ( v /) ( v /) IED = + + = 4 πg(1 v ) dx1 dx dx3 (3.13) 1 v = Δ 4 πg(1 v ) Where Δ s the Laplace operator. The gravtatonal feld potental satsfes a smlar relaton, the Posson gravtatonal equaton: 1 ρ = matter ( ) 4π G Δ Φ (3.14) Φ s the gravtatonal potental, whch outsde a sphercally symmetrc mass M s MG/r. 4. The Posson equaton for gravtatonal matter energy densty We can derve a geometrc expresson for gravtatonal matter energy densty based on Tolman s energy densty relaton (3.6) usng the same approach as n secton 3.

5 Inertal feld energy 135 Consder the lne-element: N ( r ) N ( r ds = e dt e ) dr r dθ + sn θ dϕ (4.1) ( ( ) ) Ths lne-element has been nvestgated extensvely wth varous metrcs exp(n), the most well known beng Schwarzschld s exteror soluton wth exp(n)=1/(1-mg/r). It s also used n the Ensten-Born-Infeld soluton for an electromagnetc sphercally symmetrc spacetme feld [Born-Infeld, 1934], and n Hoffman s monopole soluton [Hoffman, 1935]. The same form of the lne-element was used by Ressner (1916) and Nordstrom (1918) for an electrcally charged mass accumulaton. More recent examples are Bekensten s (1975) lneelement and the geon of Demansk (1986), whch also under certan condtons reduces to the lne-element (4.1). Therefore, ths gravtatonal lne-element may be used to model gravtatonal energy densty. The components of Ensten s tensor for ths lne element may be found n many books on GR. Usng these results Ensten s tensor components for the lne element (4.1) become, lettng the lower ndex r denote dfferentaton wth respect to r: Temporal T 0 0 component: N ( 1 e ) 0 1 -L N r 0 R0 R = e + = 8π T 0 (4.) r r Radal T 1 1 component: R1 R= 8π T1 = 8π T0 (4.3) Angular T and T 3 3 components: α 1 N Nrr Nr N r α Rα R= e + + = 8π Tα (4.4) r Formng the Tolman energy densty expresson (3.6) we get: N Nr e Nrr + + N = 8π T r T T T Makng the substtuton: e N( r) ( ) r (4.5) = 1 ϕ( r) (4.6) We fnd that φ(r) satsfes: d ϕ dϕ ( + ) = 8π ( T0 T1 T T3 ) dr r dr (4.7) 1 ϕ Δ ( ) = T0 T1 T T3 4π Wth φ(r)=gm/r ths s Posson s equaton but wth opposte sgn.

6 136 C. Johan Masrelez The sgn conventon n the feld equatons of the present paper agrees wth Tolman [Tolman, 1930], wth the one used by Mser, Thorne and Wheeler [Mser, Thorne and Wheeler, 1973] and by Peeble [Peeble, 1993]. It yelds a Posson-type equaton wth negatve matter-energy densty. A possble explanaton to ths sgn reversal may be found n the Appendx. 5. Inertal matter energy densty Assumng that Inerta and Gravtaton are closely related, we mght speculate that nertal energy densty correspondng to matter energy densty also mght be generated by the nertal feld. Ths new knd of energy densty denoted ρ nertal would be the negatve of the nertal spacetme energy densty IED defned by (3.7). From (3.13) we would then have: 1 v ρnertal = Δ (5.1) 4 πg(1 v ) Rentroducng c ths relaton becomes: 1 v ρnertal = Δ (5.) 4π G 1 ( v/ c) And the correspondng Inertal Matter Energy Densty (IMED) s: c v IMED = Δ 4πG 1 ( v/ c) (5.3) 6. Inertal matter energy The Inertal Matter Energy (IME) wthn a spatal volume may be derved by ntegratng (5.3): c v c v IME = Δ g dv = Δ dv 4 1 ( / ) 4 V πg v c πg V Therefore the effectve nertal mass densty s: 1 v 1 ρnertal = Δ = dv( a) (6.) 4πG 4πG The acceleraton vector a(x 1, x, x 3 ) satsfes: (6.1)

7 Inertal feld energy 137 v a grad = (6.3) Ths mples: c IME = dv( a) dv 4πG (6.4) V Furthermore, usng the Dvergence Theorem: c v c IME = grad ds = a ds 4πG 4πG S S (6.5) Accordng to relaton (6.1) the nertal spacetme energy could be huge; n fact t appears to be unrealstcally large. In dervng t we assumed that the nertal spacetme curvature apples to all acceleratng volume elements. However, Inerta s a property of partcles, and t s lkely that the volume affected by the nertal scale factor s related to the volumes of partcles rather than to the total spatal volume of the flow. Ths could very sgnfcantly reduce the energy that mght be generated. However, snce the constant c /G s very large, even a small effectve volume mght correspond to a large amount of energy. 7. Negatve feld energy The expresson for (6.4) for nertal feld energy suggests that t mght be possble to generate both postve and negatve spacetme matter energy from acceleratng matter. If the acceleratng flow s such that the dvergence for the acceleraton s negatve, the nduced feld energy could also become negatve. Although ths s somethng new and totally unexpected, t should theoretcally be possble f acceleraton curves spacetme by the nertal scale factor. However, f the acceleraton s proportonal to -1/r as n a gravtatonal feld around a central mass accumulaton or n an electrostatc feld of a central charge, the nertal feld energy dsappears, snce the Laplace operator actng on 1/r dsappears. But, negatve feld energy mght be nduced wth other nertal feld potentals. Consder for example a statc sphercally symmetrc nertal feld potental v / proportonal to r a where a s a constant. The Laplace operator n sphercal coordnates s: Δ= + (7.1) r r r We fnd from (6.) that the nertal mass energy densty has the same sgn as a(1+a); t s negatve n the range -1<a<0 and s mnmum for a=-1/. However, wth cylndrcal coordnates the same nertal feld potental r a always yelds postve nertal energy, snce the sgn s the same as for a.

8 138 C. Johan Masrelez Furthermore, t appears that negatve energy also mght be nduced by charged partcles accelerated by tme-varyng voltages and by certan knds of rotatng moton. 8. Concludng comments At frst t mght seem untenable, or perhaps even objectonable, that negatve spacetme energy may be artfcally generated. Nevertheless, f Inerta and Gravtaton have the same orgn, t s not unreasonable that nertal energy correspondng to gravtatonal energy exsts. Two prevous papers suggest that the phenomenon of Inerta s caused by changng spacetme metrcs durng acceleraton [Masrelez, 007a and Masrelez, 008]. The present paper follows up on ths proposton by dervng the energy-momentum tensor that would be nduced by acceleraton f the proposed theory of nerta were vald. A close correspondence s found between an nertal lne-element and a gravtatonal lneelement, both expressng energy densty by Posson s equaton. Ths suggests that t mght be possble to generate nertal spacetme energy from acceleratng moton. Summarzng: If the spacetme metrcs durng acceleraton change n a relatve sense t would accordng to the proposed theory mply that an acceleratng partcle experences spacetme curvature, whch would explan Inerta. As a consequence of ths curvature a correspondng energy-momentum tensor could be nduced, wth feld energy densty that could be postve or negatve. If the moton s such that the mathematcal dvergence of the acceleraton s negatve, the nduced nertal feld energy densty could become negatve. The possblty of artfcally generatng negatve spacetme energy would be of great theoretcal as well as practcal nterest. References 1. Bekensten J. D., Ann.Phys. (N.Y.), 91, 75 (1975). Born M. and Infeld L., Proc. R. Soc. (London) A144, 45 (1934) 3. Demansk M., Found. Phys. 16, No., 187 (1986) 4. Ensten A., Annalen der Physk 17, (1905) 5. Ensten A., Out of my later years, Carol Publshng Group Edton, (1995). Orgnally publshed (1919) 6. Hoffmann B., Phys. Rev. 47, 887 (1935). 7. Masrelez C.J., The Scale Expandng Cosmos Theory, Ap&SS 66, Issue 3, , (1999) 8. Masrelez C. J., Scale Expandng Cosmos Theory I An Introducton, Aperon Aprl, (004a)

9 Inertal feld energy Masrelez C. J, The Scale-Expandng Cosmos theory, Nexus Magazne, June- July (006b) 10. Masrelez C. J, Does cosmologcal scale expanson explan the unverse? Physcs Essays, March (006c) 11. Masrelez C. J., Moton, Inerta and Specal Relatvty a Novel Perspectve, Physca Scrpta, 75, , (007a) 1. Masrelez C. J., Dynamc Incremental Scale Transton wth Applcaton to Physcs and Cosmology, Physca Scrpta, (007b) 13. Masrelez C. J., Specal Relatvty and Inerta n Curved Spacetme, Advanced Studes n Theoretcal Physcs,, no. 17, , (008) 14. Mser C. W., Thorne K. S. and Wheeler J. A., W. H. Freeman and Company, (1973) 15. Lev-Cvta T., Rend. Acc. Lnce. (5), 6, 381, (1917) 16. Longer A.,: On black holes and gravtatonal waves, La Golardca Pavese (00) 17. Lorentz H. A., Amst. Versl., 5, 468, (1916) 18. Nordström G., Verhandl. Konnkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam 6: (1918) 19. Peebles P. J. E., Prncples of physcal cosmology, Prnceton Unversty Press, (1993) 0. Ressner H., Annalen der Physk 50: (1916) 1. Tolman, R. C., Phys. Rev. 35, 875 (1930). Vogt, W., Göttnger Nachrchten 7: (1887) Appendx: A comment on matter-energy and the energy-momentum tensor In hs book Out of my later years (1919) Ensten makes the followng statement regardng hs feld equaton: But, t s smlar to a buldng, one wng of whch s made from fne marble (left part of the equaton), but the other wng of whch s made from low grade wood (rght sde of the equaton). The energy-momentum tensor, whch usually s found on the rght hand sde of the feld equaton, has to be put n by hand ; t must be postulated, gvng the mpresson that t contans matter or radaton energy, possbly of some other knd than what s gven by the spacetme geometry on the left hand sde. General relatvty does not tell us how ths matter energy s generated and sustaned, whch mght be why Ensten calls t low grade wood. It would be much more satsfyng f the rght hand also were made from the fne marble of spacetme geometry. Ths dea s not new. Tullo Lev-Cvta [Lev-Cvta, 1917] and Hendrk Lorentz [Lorentz, 1916] ndependently proposed that Ensten s tensor mplctly defnes gravtatonal

10 140 C. Johan Masrelez feld energy densty. Wth ths nterpretaton Ensten s equatons s an dentty that smply says that the gravtatonal feld energy densty s such that t always matches the source energy feld, but wth opposte sgn. Quotng Lev-Cvta [Longer, 00]: The nature of ds s always such as to balance all mechancal actons; n fact the sum of the energy tensor and the nertal (spacetme) one dentcally vanshes. We mght therefore speculate that snce the nertal spacetme feld energy derved above appears to be of the same knd as the gravtatonal matter energy, t seems possble that the tensor T μν on the rght hand sde of Ensten s equatons (3.) actually mght be generated by htherto unknown processes that nvolve metrcs as expressed by the geometry on the left hand sde. Processes that modulate these metrcs n tme and/or space could accordng to GR generate energy, whch mght be the essence of matter. And, f the geometry on the left hand sde models ths matter ts energy densty should appear wth negatve sgn. In ths case the energy-momentum tensor on the rght hand sde of Ensten s equatons could correspond to a spacetme tensor on the left hand sde wth opposte sgn. The geometry on the left hand sde would then consst of two parts; a frst part correspondng to the energy-momentum tensor wth negatve sgn, -8πGT μν, and a second part matchng the tradtonal gravtatonal spacetme curvature tensor (Ensten s tensor G μν ). In ths case both sdes of Ensten s equatons would be spacetme geometry. In Ensten s words, both sdes of hs equatons would be made from fne marble and the essence matter would be nothng but spacetme geometry. Receved: September, 008