Physics of Elemental Space-Time and Cosmology

Transcription

1 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology Physis of Elementl Spe-Time nd Cosmology Bin B.K. in Abstt We postlte tht o spe is filled by the mm elements hving enegy nd mss with the size ppoximtely given by the wvelength of the highest enegy gmm ys nd tht time nd distne e both disetized by the poess of light popgtion fom one mm element to the next. These postltes povide s with theoetil gond to explin why the speed of light,, shold emin onstnt to ll obseves egdless of thei inetil fmes of efeene. In the osmologil sle, the enegy of the mm elements filling spe hs been eqted to the enegy epesented by the osmologil onstnt, i.e., the dk enegy. When pplied to the expnding nivese, the EST model bings dditionl 5% of the totl mss simply s the eltivisti oetion to the non-eltivisti eslts; hene the dk mtte is losely identified s meely the eltivisti oetion to the dk enegy pedited by the Fiedmn eqtion. The geement with the obseved mgnitdes onviningly sppots this intepettion of dk enegy nd dk mtte. Keywods Spe Time Element vity Cosmologil Constnt Dk Enegy Dk tte ELEENTAL SPACE-TIE O spe is postlted to be filled with mm elements hving extemely smll mss, enegy, nd dimensions. It is fthe postlted tht light popgtes thogh the mm elements by enegizing them, sy by mens of some poess of eltivisti boost of intenl enegy. Light nnot popgte withot the medim of mm elements. The light popgtion thogh mm elements defines the eltivisti spe nd time. We fthe postlte tht eh mm element opies bile of spe with line dimension l p nd volme l p. Then the distne between two neighboing mm elements is lso l p. Fthemoe, the time eqied fo light enegy to popgte fom one mm element to the next is the elementl time intevl, whih we will denote t p. Ths t p nd l p e the elementl nits of time nd length, espetively, nd l p /t p defines the speed of light popgtion. Bin B.K. in 890 Los Robles Avene, Plo Alto, CA 906, U.S.A. e-mil: bmin@nbon.om

2 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology The piniple of eltivity nd tht is onstnt in ll inetil fmes leds to the theoy of the speil eltivity. These two onditions of speil eltivity e both pted by the ssmption tht o spe-time is disetized nd tht l p /t p by definition in ll inetil fmes. We will ll the ltte the piniple of elementl spe-time (EST), o simply the EST ondition. This lso ssets tht the elementl length nd time e the smllest nits of length nd time nd tht we meely ont the nmbe of l p nd t p to peeive the distne nd time, espetively. The mgnitdes of l p nd t p my hnge by the eltivisti effet bt thei onts don t, hene the onstny of the speed of light. We n bild bidge between the ontinm physis nd the elementl spe-time s following. Now let n s be the nmbe of the mm elements enegized pe seond by light, i.e., seond n s t p. The distne tveled by light in one seond then mst be n s l p. The speed of the light popgtion then is llted to be nsl p l p () n t t s p p whih is onstnt in ll inetil fmes. Unfotntely we don t find wy to dietly mese the vles l p o t p. We note, howeve, l p mst be te low limit fo the wvelength of eletomgneti wves. We find the smllest eletomgneti wvelength tht hs been expeimentlly obseved o pojeted omes fom the lthigh enegy gmm ys [,,, ] in the nge, λ -y x 0-9 m x 0-5 m. We then dede l p x 0-9 m x 0-5 m hene t p l p /. x 0-8. x 0 - s. We note tht these elementl nits, l p nd t p, hve the oigins nelted to the onventionl Plnk nits. TE COSOLOICAL CONSTANT The stess enegy tenso of the genel eltivity ssoited with the osmologil onstnt my be expessed s [5, 6]. ( v ) Λ Tµν µν 8 g (μ, ν 0,,, ) () whee T (v) μν is the stess enegy tenso of the vm, g μν the meti tenso, the gvittionl onstnt, nd Λ the osmologil onstnt. The time-time omponent of the bove is given by Λ v () 8 whee v is the vm enegy density nd hve the sme qlity s the enegy density of mm elements,, with the mss density of the mm elements. ene by eqting v /, we get 8 Λ ()

3 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology whee Λ hs dimension of [/m ]. In this wy, we now hve estblished the eqivlene of the enegy of mm elements nd the enegy of the osmologil onstnt. In othe wods, the existene of mm elements mens the existene of the osmologil onstnt nd vie ves. RELATIVISTIC EXTENSION FOR TE NEWTON S LAW OF RAVITY We begin by witing the Newton s gvittionl lw fo two bodies s following: F (5) R whee nd e the msses of the two bodies nd R is the distne between them. If they move with the veloity υ nd υ, espetively, withot losing genelity we ssme υ -υ υ. We n then ewite the bove to inlde the speil eltivisti effet, whee the foe tnsfoms s following: F (6) R with, the Loentz fto, given by υ. (7) The need fo tem is lso seen by the length onttion of R, speil eltivisti effet when υ is not too smll omped to. If υ <<, tking the fist ode ppoximtion we hve υ F ( + ). (8) R Fo the plnety motion ond the Sn, let s eple by 0, the mss of the Sn, by m, the mss of ey, nd R by R C, the vege distne (i.e., of il motion) between the Sn nd ey. The veloity of ey ond the Sn my be ppoximted by ignoing the eentiity of the obit, υ 0, (9) nd we then hve F 0 0 f ( + ) (0) m RC RC whee f is the speifi foe (i.e., pe nit mss) fo ey. We ell 0 R C, whee is the speifi ngl momentm of ey fo the obitl motion nd nely onstnt, the bove then beomes R C 0 0 C R C f. () R The seond tem is eognized, exept fo nmeil fto thee, s the genel eltivisti oetion, 0 /( R C ), fo the plnet motion tht fmosly leds to the oetion to the peihelion motion of ey [7, 8]. This lends edene to Eq. (6) being vlid eltivisti extension to the Newton s lw of gvittion fo moving bodies.

4 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology TENSILE PRESSURE IN TE AA ELEENT SPACE. Non-eltivisti Deivtion Let s tke ple in the mm element spe whee the gvity of ptil st o plnet does not dominte. ee mm element is inflened by the gvity of ll msses of the nivese t ptil time o epoh. (See Fig..) We sset tht the mm element spe is isotopi nd homogeneos nd we fthe ssme tht it behves s pefet flid, i.e., its behvio n be hteized by the mss density nd isotopi pesse p. ( p is tensile pesse.) Fo simpliity, insted of single mm element we now onside smll spheil volme of dis Δ of mm elements loted in the ente of the Ctesin oodinte system. Fo the gvity foe pe the nit volme of mm elements, f i, exeted by ll othe msses in the nivese, the stess eqilibim eqies σ ji, j + f i 0 (i, j,,; smmed ove epeted () indies). Bt σij p δ ij (i, j,,) () whee δ ij is the Koneke delt. ene we get fi i p (i,,). () Applying the spheil symmety, the only non-tivil omponent is tht in the dil dietion,, f p. (5) If we define the gvittionl potentil Φ ( ) (6) whee is the gvity onstnt, the mss density of the mm elements, the mss of mss element of the nivese, nd the distne between the mm element nd the mss element being onsideed. We hve f Φ Φ. (7) Fom Eqs. (5) nd (7), we then get ( p + Φ) 0, (8) hene p + Φ onstnt. (9) By ombining Eqs. (6) nd (9), we get p + onstnt. (0) We note tht while the potentil fntion Eq. (6) is veto in the -dietion, the pesse p is sl. To obtin the totl pesse exeted by the gvity of the nivese, we will onside, in ptil, the pesse p n exeted by the mss n t distne n,

5 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology n pn + onstnt. () n Now sine p is sl, we hve the totl pesse p pn () whee the smmtion is fo ll the msses n in the Univese (othe thn the smll volme of the mm elements itself being onsideed). The smmtion my then be pefomed s following: n n n n d whee is the mss density nd the dis of the Univese. UNIVERSE (Totl ss, Density, ) Δ A SALL VOLUE OF AA ELEENT SPACE Density, Pesse, p AA ELEENT SPACE Density, ss Element, n FI. An Elementl Spe-Time odel fo the Univese (Not to sle) The lowe bond of the integtion, Δ, my be epled with zeo withot losing y s long s Δ is sffiiently smll omped to the ppe bond,. ene we get p n C pn d () 0 n n n whee C is onstnt. is the vege density of the Univese (fom the dis Δ to ) tht is the sm of the density of the mm elements,, whih is onstnt nd independent of, nd the density of ll othe mttes tht is popotionl to -. Intepeting the bove s botopi eqtion of stte, the bove eqtion then pesents lol (inside the smll volme onsideed) sond wve speed g [9, 0], 5

6 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology g p, () whih my be eognized s the speed of the gvittionl tensile pesse wves. One n, theefoe, expess the dis of the Univese, g ( NR ) (5) whee (NR) indites this is non-eltivisti lltion. Let, g q, whee q is dimensionless onstnt, we then hve z (6) q whee we define z, onstnt. One n obtin simil eltionship by simply qntizing, Comping the two, we get m 0 m 0 kg s 0 kg. (7) q 0 m z. (8) 0 kg Aoding to the bble s lw, &, (9) the veloity of the expnsion wold begin to exeed the speed of light, &, t the itil dis,. (0) With the ent vle of the bble onstnt epoted by NASA [, ], 0.5 x 0-8 s -, we get. x 0 6 m (o. x 0 0 light-yes.) If g, the speed of light, then Eqs. () nd () with C 0 yields p /. () Wilzek [] lls the bove (exept fo the sign) the well-tempeed eqtion. Also if g, Eq. (5) is the sme s the Einstein s stti Univese obtined by the se of the Fiedmn eqtion bt with fto of diffeene. NASA [, ] epots the ent density of the Univese, 0 (NASA epoted) 9.90 x 0-7 kg/m. If we se this vle, Eq. (5) yields the ent dis of the Univese 0 (NR, llted).7 x 0 6 m (o.56 x 0 0 light yes) nd Eq. () with C 0 the tensile pesse p 6. x 0-0 kg m s - m -. 6

7 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology. Reltivisti (R) Coetion Sine this dis is gete thn the itil vle, ths oding to this non-eltivisti lltion, the ote edge the Univese is ently expnding t the speed gete thn the speed of light. This is nstisftoy sine oding to the speil eltivity (nd EST), nothing will tvel t the veloity gete thn the speed of light. To inlde the speil eltivisti effet, we dd the Loentz fto to ont fo the length onttion of in the gvity potentil to get Φ ( ) () whee &, () ( ) & denoting the time deivtive of o the dil veloity. Then Eq. () beomes n p C pn n d. n n n 0 & () ( ) Now we my pply the bble s empiil lw & to the bove eqtion ssming is onstnt thoghot the Univese t ptil time o epoh, nd integte ove the Univese (Δ to ) to get p C. (5) The bove eqtion Eq. (5) pesents gvittionl tensile pesse wve hteized by the sond wve speed p g. (6) If & <<, Eqs. (5) nd (6) e ppoximted to Eqs. () nd (), espetively, s expeted. We n lso expess the dis of the Univese, If g, this edes to g g 8. (7) ( R ) (8) 8 whee (R) indites this is eltivisti lltion. If g, the speed of light, then Eqs. (5) nd (6) with C 0 gin yields Eq. (), the well-tempeed eqtion. The well-tempeed eqtion is, theefoe, eltivisti. Agin, by the se of NASA [, ] - epoted ent vles, 0 nd 0, we get 7

8 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology 0 (R, llted). x 0 6 m (.0 x 0 0 light yes,) signifint edtion fom the non-eltivisti vle of 0 (NR). The tensile pesse p 6. x 0-0 kg m s - m -, emins the sme s the non-eltivisti vle. Ths, the dis of the Univese llted with the eltivisti effet is smlle thn the itil dis, Eq. (0) nd the ote edge of the Univese is ently expnding t the speed slightly lowe thn the speed of light. In this wy, we hve shown tht the mm element spe is onsistent with the pesene of negtive vm pesse nd pesene of the gvittionl wves with sond wve speed, g. Thee is, howeve, ently no fim theoetil gond to sset tht g is eql to the speed of light,. 5 TE EXPANDIN UNIVERSE We se Newton s shell theoem to llte the totl enegy, the sm of the kineti enegy (K.E.) nd the potentil enegy (P.E.), of the n th ptile hving mss, n, t the dis fom the ente. Note tht Newton s shell theoem ignoes the gvittionl pesse deived in the pevios setion. It is jstified sine sh pesse is in eqilibim nd does not ontibte to the motion of the ptile. ene we wite n eqtion fo the totl enegy, whee K.E. n + P.E. n n onst ± nκ (9) & (0) ( ) nd is the totl mss within the sphee of the dis (t), fntion of time. We hve set the onstnt bitily to be ± n κ whee κ is onstnt withot losing genelity. If κ > 0, the kineti enegy povides moe thn the espe veloity; if κ 0, the kineti enegy povides jst enogh n espe veloity; nd if κ < 0, the kineti enegy is insffiient fo n to oveome the gvity of nd espe. This leds to & κ ( ± + κ ). + () ( ± κ ) Fo sffiiently smll κ, i.e., κ <<, the bove ppoximtes to & κ ± The non-eltivisti vesion of the bove, i.e., withot the seond tem, is in the sme fom s the Fiedmn eqtion. A tem whih is popotionl to - is intepeted in the litete s dition dependent se of the fist tem s opposed to mtte dependent fist tem. ee the seond tem, whih is lso popotionl to -, is simply eltivisti oetion to the fist tem. We note +. () whee is the mss of ll mttes othe thn, the mss of the mm elements, within the sphee of the dis. Note tht is popotionl to ( ) while is 8

9 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology 9 onstnt oding to the model shown in Fig.. Also note tht is onstnt while is popotionl to - ( /.) Some sefl eltionships e listed below whee the sbsipt 0 denotes the vle t the pesent epoh: Constnt 0 0, Constnt 0 0, 0 0, , nd. Now we get fom Eq. (), κ ± &. () Eq. () my be lso be witten s ) ( 9 6 ) ( 8 κ ± + + & () o s κ δ δ δ ± + + Λ + Λ & (5) whee δ 0 0 / nd Λ 8 /. If κ 0, the bove yields some sefl eslts s following. 5. Reltivisti (R) Tetment The eltivisti (R) se is obtined fom Eq. () (with κ 0,) &. (6) This, togethe with the eltivisti eqtion of stte Eq. (6) nd g, e solvble fo nd to yield, (R) (7) nd

10 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology ( R) (8 ). (8) With 0.5 x 0-8 s -, we get 0 (R, llted).0 x 0-7 kg/m, 0 (R, llted).6 x 0 6 m (o. x 0 0 light yes) nd 0 <, being given by Eq. (0). Fom Eq. (5) we get (with κ 0,) 6 & Λ δ Λ δ δ (9) Eq. (9) my be witten in n integl fom to llte the ge of the Univese d Λ dt 6. (50) + δ ( + δ ) Λ Unfotntely, losed fom soltion is not esily vilble fo the bove eqtion. Some soltions e vilble fo limiting ses; we will not, howeve, pse those hee. 5. Non-Reltivisti (NR) Limit A non-eltivisti (NR) se is obtined by ignoing the seond tem of Eq. (). If lso κ 0, the sme eqtion beomes & 8. (5) This, togethe with the non-eltivisti ent eqtion of stte Eq. () nd with g, e solvble fo nd to yield nd (NR) 8 (5) ( NR). (5) With 0.5 x 0-8 s -, we get 0 (NR, llted) 9.0 x 0-7 kg/m, 0 (NR, llted).5 x 0 6 m o.6 x 0 0 light yes, nd 0 (NR) >, being given by Eq. (0). Eq. (5) my lso be witten s, & (5) The bove eqtion is identified s the Fiedmn eqtion fo the flt Univese, κ 0, nd my be integted fo time to yield (f. []), 0

11 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology We n invet Eq. (55) to get t (NR) sinh 6. (55). (56) [ sinh( 6 t) ] If we se the NASA - epoted vles, 0 7. x 0-7 kg/m nd t 0.7 x 0 0 yes, then we get 0 (llted).5 x 0-7 kg/m. Sine we mese only 0.6 x 0-7 kg/m fo the obsevble mtte density, the est of the mtte density, (.5-0.6) x 0-7 kg/m.05 x 0-7 kg/m, mst be ttibted to something nknown, i.e., dk mtte. This is in good geement with the NASA estimted dk mtte density,. x 0-7 kg/m. Clely, the dk mtte postlte is neessy to ont fo the mesed vles of the density nd the ge of the Univese so long s we tke the non-eltivisti Fiedmn eqtion to model o Univese s in the ΛCD model..5 DARK ENERY AND DARK ATTER An inventoy of the osmi mss densities is shown in Fig., divided into the obsevble mtte, dk enegy, nd dk mtte. Those obtined by the pesent EST model e omped with those obtined by the se of the stndd ΛCD model s epoted fom both WAP [, ] nd Plnk pojets [5, 6]. WAP epots the densities with el SI nits while Plnk only with the tios ove totl. The mss densities of the Univese oding to NASA mesements [, ] (s we sed some of them in the bove) e smmized below: 0 (Univese) 9.90 x 0-7 kg/m 0 (obsevble mtte-byoni nd ll othes) 0.6 x 0-7 kg/m (.6%) dm (dk mtte). x 0-7 kg/m (.%) de (dk enegy) 7. x 0-7 kg/m (7.%) Obsevble mtte inldes byoni mtte nd dition inlding the osmi miowve bkgond (CB) dition [7]. The key nmbes fom the EST lltion e: the eltivisti totl density 0 (R, llted).0 x 0-7 kg/m, Eq. (7), vs. non-eltivisti totl density 0 (NR, llted) 9.0 x 0-7 kg/m, Eq. (5). The diffeene is.0 x 0-7 kg/m nd shows tht the oetion de to the eltivisti effet monts to 5% of the eltivisti totl density. The non-eltivistilly llted density, 9.0 x 0-7 kg/m, is esonbly lose to the NASA-estimted density of the Univese, 9.90 x 0-7 kg/m.

12 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology ΛCD WAP 0-7 kg/m ΛCD Plnk (% of ) EST 0-7 kg/m 0 esed (NR) o Cllted Dk Enegy dm Cllted. 6.8 N/A Dk tte 0 (NR) Cllted R Cllted N/A N/A.0 Reltivisti Coetion (R) o Cllted N/A N/A.58 Dk Enegy 0 (R) Cllted N/A N/A.0 dm / 0 (NR).% 6.8% N/A R / 0 (R) N/A N/A 5.0% FI. Cosmi Enegy Inventoy (Unit: 0-7 kg/m o % of 0 ) ( sles with - nd is the enegy density of ll mttes exlding. The ltte, the enegy density of mm elements, is identified s the vm enegy density oesponding to the osmologil onstnt.) A diet mesement of the densities is possible only fo the obsevble mtte whih fom WAP mesement is 0.6 x 0-7 kg/m. With the eltivisti EST eslt, theefoe, we wold simply intepet the Univese being omposed of 0.6 x 0-7 kg/m obsevble mtte nd the emining (.0-0.6) x0-7 kg/m.58 x0-7 kg/m the enegy of mm elements o the dk enegy. If we sed the non-eltivisti limit of the EST lltion, this wold be 0.6 x0-7 kg/m obsevble mtte nd ( ) x 0-7 kg/m 8.57 x0-7 kg/m the enegy of mm elements o the dk enegy. We hve.0 kg/m simply s the eltivisti oetion. The Fiedmn eqtion n be deived fom the non-eltivisti Newtonin gvity eqtion hene is non-eltivisti speil soltion of the genel eltivity despite the ft tht it oigintes fom the genel eltivity. This mens tht Fiedmn eqtion wold miss the mont oesponding to the eltivisti oetion. Indeed, the eslts of the ΛCD model epots the nexplined dk mtte whih is ppoximtely.% of the totl by WAP nd 6.8% by the Plnk. The eltivisti EST model bings dditionl 5% of the totl mss simply s the eltivisti oetion to the non-eltivisti eslts llted by the Fiedmn eqtion. The geement mong the mgnitdes is exeedingly onvining tht this intepettion of the dk mtte is oet. In this wy, the eqivlene of the mm elements, the

13 Bin B.K. in Physis of Elementl Spe-Time nd Cosmology Einstein s osmologil onstnt, nd the dk enegy hs lso been onviningly demonstted. SUARY AND CONCLUDIN REARKS The EST model my be viewed s binging the ethe bk, only this time the ethe is not bsolte, bt ompised of mteil elements lled mm elements hving enegy nd oesponding mss. The hypothesis of spe-filling mteils is shown to be in line with the existene of the osmologil onstnt nd the oesponding enegy, i.e., the dk enegy. The spe filled with mm elements is theoetilly shown to be sbjet to negtive gvittionl pesse nd the gvittionl pesse wves. By this osmologil model ombined with eltivisti extension to the Newton s lw of gvity hene to the Fiedmn eqtion, we show the pedited dk enegy to be in lose geement with the obseved mgnitdes bt lso the dk mtte most likely to be mee eltivisti oetion to the non-eltivisti estimte of the dk enegy. Refeenes. Ask n Astophysiist: Is thee n ppe limit to the Eletomgneti Spetm?. Wikipedi, the fee enylopedi, (0).. Ahonin, F.: et l, The time veged TeV enegy spetm of kn 50 of the extodiny 997 otbst s mesed with the steeosopi Cheenkov telesope system of ERA, Xiv:stoph/99086v (Jl 999)...E.S.S. Collbotion, A. Abmowski, et l., ESS J n exeptionlly lminos TeV gmm-y spenov emnnt, Xiv:0.88v [sto-ph.e] (Feb 0). 5. Wikipedi, the fee enylopedi, (05) 6. Einstein, Albet (96). "The Fondtion of the enel Theoy of Reltivity" (PDF). Annlen de Physik 5 (7): 769. Bibode:96AnP E. doi:0.00/ndp Fitzptik, R.: Peihelion Peession of ey, (0). 8. Vnkov, A. A.: Einstein s Ppe: Explntion of the Peihelion otion of ey fom enel Reltivity Theoy, 9. Kolsky,.: Stess Wves in Solids, Dove Pblitions (96). 0. lven, L. E.: Intodtion to the ehnis of Continos edim, Pentie ll, In. (969). NASA, Wilkinson iowve Anisotopy Pobe (WAP) (0).. Bennett, C. L. et l.: Nine-Ye Wilkinson iowve Anisotopy Pobe (WAP) Obsevtions: Finl ps nd Reslts, Xiv:.55v (0). Wilzek, F.: The Lightness of Being, Chpte 8, Bsi Books (008).. Steine, F.: Soltion of Fiedmn Eqtion Detemining the Time Evoltion, Aeletion nd the Age of the Univese, 5. Wikipedi, the fee enylopedi, (0) 6. Plnk Collbotion, Plnk 0 eslts. XVI. Cosmologil pmetes, Xiv: [stoph.co] (0). 7. Fkgim,. & Peebles, P. J. E.: The Cosmi Enegy Inventoy, Xiv:sto-ph/006095v, (00)