Relativistic Non-local Physics in the Theory of Gravitational Field Interaction with a Laser Beam

Transcription

1 Ameran Journal of Modern Phss and Alaton 15; (6): Publshed onlne November 15 (htt:// Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam Bors V. Aleeev Phss Deartment Mosow Lomonosov State Unverst of Fne Chemal ehnologes Prosekt Vernadskogo Mosow Russa Emal address Bors.Vlad.Aleeev@gmal.om o te ths artle Bors V. Aleeev. Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam. Ameran Journal of Modern Phss and Alaton. Vol. No Abstrat Relatvst non-loal hss s aled to the roblem of a laser beam nteraton wth gravtatonal feld nludng gravtatonal waves. We ntend to answer the followng questons: a) Is t ossble to seak about nteraton of a laser beam (for eamle) wth gravtatonal feld? b) Can we admt the estene of gravtatonal waves from oston of non-loal relatvst hss? It s shown that both questons have the ostve answers. In other words - the non-loal treatment does not lead to the mathematal ontradtons n the theor. It s shown that the roblem of gravtatonal feld nteraton wth a laser beam annot be solved n the frame of loal theoretal hss n rnal. An analog of Newton' law for the hoton movement s obtaned. Man onrete results of alulatons are delvered. Kewords Relatvst Non-loal Phss Newton' Law for the Photon Movement Gravtatonal Feld Laser Beam Interaton 1. Introduton In loal hss gravtatonal waves are onsdered as rles n the urvature of sae-tme whh roagate as a wave travellng outward from the soure. Predted to est b Albert Ensten n 1916 on the bass of hs henomenologal theor of general relatvt gravtatonal waves theoretall transort energ as gravtatonal radaton. Soures of detetable gravtatonal waves ould ossbl nlude bnar star sstems omosed of whte dwarfs neutron stars or blak holes. Gravtatonal waves annot est n the Newtonan theor of gravtaton sne n t hsal nteratons roagate at nfnte seed. Although gravtatonal radaton has not been dretl deteted there s ndret evdene for ts estene. For eamle the 199 Nobel Pre n Phss was awarded for measurements of the Hulse-alor bnar sstem whh suggests gravtatonal waves are more than mathematal anomales. Varous gravtatonal wave detetors est [1 ]. Reentl (on Marh 14) astronomers at the Harvard Smthsonan Center for Astrohss lamed that the had deteted and rodued "the frst dret mage of gravtatonal waves aross the rmordal sk" wthn the osm mrowave bakground [-6]. But these new fndngs are ver far from real sentf onsensus. Pratall all attemts reman unsuessful n detetng suh henomena. Moreover t means that the effet of an nteraton between radaton and gravtatonal feld needs n addtonal theoretal nvestgaton and justfaton. Let us onsder ths roblem from oston of relatvst non-loal hss. We ntend to answer the followng questons: 1. Is t ossble to seak about nteraton of a laser beam (for eamle) wth gravtatonal feld?. Can we admt the estene of gravtatonal waves from oston of non-loal relatvst hss? It wll be shown that both questons have the ostve answers. In other words the non-loal treatment does not lead to the mathematal ontradtons n the non-loal theor.. Bas Equatons of Relatvst Non-loal Phss Havng Regard to Eternal Fores heor and man alatons of non-loal relatvst and non-relatvst hss an be found n [7 1]. he artle [7] ontans the treatment of Shannon-Nqust-Kotelnkov theorem from oston of relatvst non-loal hss for a hsal stuaton when the nfluene of eternal fores ould be omtted. he mentoned aer ontans all mathematal detals for ths ase and n the followng we ntend to dsuss onl other detals onneted wth ntroduton of the eternal fores n relatvst non-loal hss (see also [8-1]).

2 116 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam Let us ntrodue = ( 1 ) = ( t 1 ) as 4 radus vetor of a artle; = ( m 1 mv mv mv ) γ γ γ γ - 4- vetor of artle mulse; m - the rest mass of artle (salar nvarant); v γ = 1 1/ v - velot module of a artle n the 1 statonar oordnate sstem K of vewer; v v v - velot omonents. We ntrodue 4 fore vetor K 1 F F F F = (.1) m m m m = f β β d he followng tensor desgnatons are also used: 4 dmensonal tensor of the thrd rank = f βγ β γ d. (.6) (.7) 4 dmensonal vetor of the averaged fore atng on the unt mass K = K f d ; (.8) where and F = m F qe v j B k (.) jk = F F (.) s a fore of non-eletromagnet orgn atng on the unt mass of a artle k k B = B ( j k ) s magnet nduton qejkv B = q ( v B) - Lorent fore j k= 1 ; e jk = f = j = k or j = k e1 = e1 = e 1 = 1 e1 = e1 = e 1 = 1. q artle harge. he maroso desrton of the relatvst gas s based on the moments of the DF and defned b tensors =... f β... γδ β γ δ d. (.4) he frst moment transform (.4) nto the artle fourflow defned b N = f d. =1. (.5) K l K K l К l m = 4 dmensonal tensor of the seond rank K d β = K β f ; (.9) 4 dmensonal tensor of the frst rank K K d β = f ; (.1) β 4 dmensonal tensor of the frst rank K K K β d = K f. (.11) β he dervaton of the generaled relatvst Aleeev- Boltmann knet equaton an be found n [9 11]. l Multl resetvel ths equaton b m = m d he seond moment s the energ-momentum tensor moton equaton l lβ K l τ K l K l m ( τ ) ( τ ) β m energ equaton τ τ τ and ntegrate over. he desrbed roedure leads to the generaled non loal relatvst hdrodnam equatons of Enskog te. Namel: ontnut equaton β N K m τ m ( τ ) = (.1) β (.1) β K τ K K m ( τ ) ( τ ) β m K K K К m = τ τ τ (.14) where τ s non-loalt arameter.

3 Ameran Journal of Modern Phss and Alaton 15; (6): B gong to the three dmensonal oordnates ontnut equaton moton equaton energ equaton and tme t we fnd j K K mn τ m mn τ m = (.15) j t t t l l l τ K l K l m m t m t τ l lj l K l K l K l m m m j m t K l K l K K τ l К l m = t (.16) j τ K τ K K m m ( ) j t m t m t K K K K К K m τ m = t. (.17) Hdrodnam equatons (.15) (.17) an be used for ratal alulatons after obtanng the elt eressons for tensor moments.. Calulaton of the ensor Moments for the Laser Beam Evoluton Let us onsder the dstrbuton funton n the form f = nδ ɶ δ δ (.1) = m =. (.) hen for artles movng along х-as h ɶ = ɶ = ɶ ν (.) tal n the theor of a laser beam transort. herefore we where hν ɶ s energ of the orresondng quantum ɶ ν - onsder artles movng along the - as wth the mulse frequen. ɶ Defne all tensor omonents resentng n equatons = onst. Numeral artle denst п an deend on (.15) (.17). oordnates and tme. For artles wth the rest mass whh s equal to ero we have d d N = f = nδ ( ɶ ) δ ( ) δ ( ) = n (.4) Analogall we fnd hen d d N = f = nδ ( ɶ ) δ ( ) δ ( ) = n (.5) d d N = f = nδ δ δ =. (.6) ( ɶ ) N =. (.7)

4 118 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam n n N = (.8) For the tensor of the seond rank we have d d = f = nδ ( ɶ ) δ ( ) δ ( ) = nɶ (.9) d d = = f = nδ ( ɶ ) δ ( ) δ ( ) = nɶ (.1) d d = f = nδ ( ɶ ) δ ( ) δ ( ) = nɶ (.11) other omonents for tensor β are equal to ero. But ɶ = ɶ = ɶ = ɶ then β o nɶ nɶ nɶ nɶ = =. (.1) For tensor of thrd rank d р d р Т = ср р р f = с р р р nδ ( ɶ ) δ ( ) δ ( ) = n ( ɶ ) = nɶ (.1) р р and analogall ( ɶ ) = = = = = = Т Т Т Т Т Т Т = Т = n = n (.14) other omonents of Т βδ are equal to ero. Let us alulate now the tensor omonents ontanng the eternal fores. ake nto aount that the hoton harge s equal to ero ( q = ) we needn t use the Lorent fore. K 1 F F F F F F F F = = m m m m (.15) K d K f f F d = = = d ( F F F ) nδ ( ɶ ) δ ( ) δ ( ) = d ( F ) n ( ɶ ) ( ) ( ) = nf δ δ δ (.16) ɶ K1 d F d = = = ( ) = ɶ K f f F nδ δ δ d nf (.17) K d F d = = = ( ) = ɶ K f f F nδ δ δ d nf (.18) ɶ

5 Ameran Journal of Modern Phss and Alaton 15; (6): Relatons (.16) (.19) an be wrtten n the matres form Now we obtan all omonents K β we have K d = K f = nf K. (.19) nf nf =. (.) nf nf K d F ( = K f = fd = 1 ) 1 1 = 1 F F F nδ ɶ δ δ d nf ɶ (.1) K1 δ ɶ δ δ ( ɶ ) δ δ δ = F n d nf ɶ d F d d = K f = f = F F F n = K d F d = K f = f = ( F F F ) nδ ( ɶ ) δ ( ) δ ( ) d = (.) (.) K = K d f = (.4) K1 d F = K f = fd = F nδ ( ɶ ) δ ( ) δ ( ) d = nf ɶ K11 d F d = K f = f = F nδ ( ɶ ) δ ( ) δ ( ) d = nf ɶ K1 d F d = K f = f = δ ( ɶ ) δ δ 1 F n d = (.5) (.6) (.7) K1 d = K f = (.8) K d F = K f = fd = F nδ ( ɶ ) δ ( ) δ ( ) d = nf ɶ (.9) = = = ɶ (.) K d F K f fd nf

6 1 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam K1 d F d = K f = f = F nδ ( ɶ ) δ ( ) δ ( ) d = nf ɶ (.1) K1 d F d = = = ɶ (.) K f f nf K d F d = K f = f = δ ( ɶ ) δ δ 1 F n d = (.) K d F d = = = K f f (.4) K d F d = K f = f = δ ( ɶ ) δ δ 1 F n d = (.5) hen n the matres form K β K d F d = = =. (.6) K f f K K K K nf ɶ nf ɶ K K K K nf ɶ nf ɶ = K K K K =. (.7) nf ɶ nf ɶ K K K K ( 1 ) nf ɶ nf ɶ K K Let us fnd now the omonents of tensor K β d defned b relaton (.1) = f β. We have K K K K K d = = f F F F F d f = t nδ ( ɶ ) δ ( ) δ ( ) F F F F d = t F. 1 1 F d F F nδ ( ɶ ) δ ( ) δ ( ) = nɶ nɶ t t (.8) K 1 K K K K d = = f

7 Ameran Journal of Modern Phss and Alaton 15; (6): F F F F d f = t nδ ( ɶ ) δ ( ) δ ( ) F F F F d = t F 1 1 F d F F nδ ( ɶ ) δ ( ) δ ( ) = nɶ nɶ t t (.9) K K K K K d = = f F F F F d f = t nδ ( ɶ ) δ ( ) δ ( ) F F F F d = t F. 1 1 F d F F nδ ( ɶ ) δ ( ) δ ( ) = nɶ nɶ t t (.4) Analogall we reah K F F = nɶ nɶ t. (.41) and the orresondng matres form K K K β d Now we all attenton to tensor (.11) = K f. We resent alulatons β K F F nɶ t F F nɶ t =. (.4) ( 1 ) F F nɶ t ( 1 ) F F nɶ t K K K K K K d = K K K K f = F F F F d K K K K = f d F K F K F K f = (.4) δ ( ɶ ) δ δ F F F d F F F n f = n (( F ) ( F ) ( F ) ).

8 1 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam K K1 K K K K d = K K K K f = F F F F d K K K K = f d F d F K f = F f = F 1 F 1 F d n ( ɶ ) = F F F nδ δ δ f ( F ). (.44) K K K K K K d = K K K K = f () 1 () 1 () 1 () 1 F F F F d K K K K f = () 1 () 1 d () 1F d F K f = F f = () 1 () 1 () 1 () 1 () 1 () 1 () 1 () 1 F F F d n F F F nδ( ɶ ) δ δ f = F F (.45) Analogall one obtans K K n = F F (.46) and the matres form K K (( F ) ( F ) ( F ) ) n ( F ) n =. (.47) n F F ( 1 n ) F F 4. Non-loal Relatvst Equatons n the Elt Form Some relmnar remarks of rnal sgnfane: 1. Self-aton of the hoton eletro-magnet feld s out of onsderaton; eternal eletro-magnet feld has no affet on hoton. Interestng to note that the Comton effet (also alled Comton satterng) s the result of a hgh-energ hoton olldng wth a target whh an be onsdered as an effetve gravtatonal feld. We have a mture of an ndent radaton λ and the sattered radaton λ whh eerenes a wavelength shft λ that annot be elaned n terms of lassal wave theor; ( 1 os ) λ = λ λ = λ θ (4.1) С where the Comton wavelength λ С = h/ m whh s onneted (aart of the usual onstants) onl wth the rest mass m of the target. he Comton wavelength gves the sale of the wavelength hange of the ndent hoton. From Eq. (4.1) we note that the greatest wavelength hange ossble orresonds to one dmensonal ase wth θ =18 o when the wavelength hange wll be twe the Comton wavelength λ С whh an be wrtten also as a fore F mс E F = mсν С = = λ λ. (4.) Beause С λ =.4 nm for an eletron and even less for other artles owng to ther larger rest masses the mamum wavelength hange n the Comton effet s.485nm. С С

9 Ameran Journal of Modern Phss and Alaton 15; (6): he law of gravtatonal feld nfluene on hoton remans unknown. Let us suose that the menton nteraton (gravtatonal feld radaton GFR) ests. In ths ase the nfluene GFR nteraton s reresented b the term mf whh orresonds to a fore atng on a artle (n the ase under onsderaton on hoton). Wth the am to avod msunderstandngs hereafter we use F = mf F = mf F = mf as desgnaton for a fore atng on hoton b GFR-nteraton.. he hothetal transforms nto F 1 φ ( = ) nfluene on the laser beam an be dsovered after nvestgaton of the laser beam transort along and aganst the dreton of the gravtatonal feld. 4. Of ourse on a stage of our researh we should ntrodue some assumtons onernng the F form. After substtuton of all tensor omonents n equatons (.15) (.17) we reah the sstem of hdrodnam equatons n the elt form. Contnut equaton m N m m N m t t t j K K τ τ = j ( nɶ ) ( nɶ ) τ ( nɶ ) ( nɶ ) τ n nf n nf t m t m t τ τ nf nf =. m m (4.) Moton equaton along - as (.16) transforms nto ( nɶ ) ( nɶ ) τ nɶ t m t ɶ nf ( nɶ ) ( nɶ ) τ τ nɶ nf ɶ nf ɶ m t m τ ( 1 ) nf ɶ nf m (4.4) ( ɶ 1 1 ) ( ɶ ) τ nf nf F F n nɶ ( F ) =. m t t Moton equaton along - as (see also (.16)) τ K K m m t m t τ j K K K m m m j m t τ K K K K К m = transforms nto τ τ ( 1 ) nf ɶ ɶ nf nf t m m t ( ɶ 1 1 ) ( ɶ ) τ nf nf F F n nɶ F F =. (4.5) m t t

10 14 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam Analogall an be wrtten the moton equaton along as usng the ehange. Energ equaton takes the form j τ K τ K K m m ( ) j t m t m t K K K K К K m τ m = t ( nɶ ) ( nɶ ) τ nɶ t m t ( nɶ ) ( nɶ ) τ nɶ m t nf ɶ nf ɶ ( ɶ 1 1 ) ( ɶ ) τ τ ( 1 ) nf ɶ ɶ nf nf m m (( ) ( ) ( ) ) τ nf nf F F n nɶ F F F =. (4.6) m t t Now we an use the lmt transformaton usng m vald for hoton gas. We also ntend to onsder the 1D ase then F = F =. We fnd from (4.)(4.4)(4.6) ontnut equaton ( nɶ ) ( nɶ ) ( nɶ ) ( nɶ ) nf nf = (4.7) t t t moton equaton ( nɶ ) ( nɶ ) t t ( nɶ ) ( nɶ ) t ɶ nf t nf ɶ ( nɶ ) ( nɶ ) n = F F. (4.8) t energ equaton ( nɶ ) ( nɶ ) ( nɶ ) ( nɶ ) nf ɶ ɶ nf = t t t 1 ɶ ɶ n n n F ( F ). t (4.9) Imortant remarks: 1. Loal arts of all relatvst hdrodnam equatons for hoton gas are dsaeared. In artular t means that loal hss s not alable for ths ase.. Moton equaton (4.8) ondes wth the energ equaton (4.9).. he sstem of equatons does not deend on non-loal arameter τ. We have for the ase onl two ndeendent equatons defnng SYSEM 1: ( nɶ ) ( nɶ ) ( nɶ ) ( nɶ ) nf nf = (4.1) t t t

11 Ameran Journal of Modern Phss and Alaton 15; (6): ( nɶ ) ( nɶ ) nf ɶ t t ( nɶ ) ( nɶ ) ( nɶ ) ( nɶ ) n nf ɶ = F F. (4.11) t t Sstem of non-statonar 1D equatons (4.1) (4.11) n t - beam ontan two deendent varables namel numeral denst and ɶ ( t ) - hoton mulse. he mentoned sstem an be solved usng usual numeral methods. 5. Wave Solutons for GFR-nteraton Now we an onsder the roblem of rnle sgnfane s t ossble to fnd wave solutons of equatons (4.1) (4.11)? Wth ths am we ntrodue a new ndeendent varable ξ as ξ = t. hen the substtuton s used or From (4.1) follows = t ξ = ξ. (5.1) ( nɶ ) с nf = (5.) ξ ξ ( nɶ ) с nf = onst = С. (5.) ξ Equaton (5.) an be onsdered as an analog of the Newton seond law of hoton moton. Reall for better understandng let us onsder the relaton ξ = t n Newtonan ase where << t. In ths ase ξ ~ сt and ( nɶ ) nf =. (5.4) t t From (5.) follows the seond Newton s law for hoton gas f << t: ( nɶ ) = nf. (5.5) t Analogal transformaton s alable to equaton (4.11) one obtans nɶ 4 с ξ ξ Usng (5.) we fnd nɶ n nf ɶ = F ξ ( F ). (5.6) n 1 С ɶ ɶ СF =. (5.7) ξ ξ 4с We have for the ase onl two ndeendent dmensonless equatons defnng SYSEM : ( nɶ ) с nf = C (5.8) ξ n 1 С ɶ ɶ СF =. (5.9) ξ ξ 4с Let us transform SYSEM to the dmensonless form. We ntend to use the followng sales: = = = h ɶ ɶ ɶ ν - sale for mulse; n - sale for the beam numeral denst n ; F - sale for a fore F atng on hoton; - length sale. Usng the mentoned sales we fnd from (5.8) ( ˆˆ ɶ ) n nf ˆˆ F = ˆ ξ с сn CС ˆ (5.1) where uer hat orresonds to dmensonless values. he fore sale s reasonable to hoose n form с E F. (5.11) = = he sale (5.11) s wrtten also as atttude of ntal hoton energ E = hɶ ν to the length sale. hen ( n ˆˆɶ ) ˆ ξ where dmensonless onstant C ˆ nf ˆ = Cˆ (5.1) сn = = n E. (5.1) From equaton (5.9) follows dmensonless form nˆ 1 ˆ ˆ СF ξ ξ 4 As a result we obtan SYSEM ˆ ˆ ˆ ˆˆ ( 1 ) С ɶ ɶ = ( n ˆˆɶ ) ˆ ξ ˆ nf ˆ = Cˆ nˆ 1 ˆ ˆ СF ξ ξ 4 ˆ ˆ ˆ ˆˆ ( 1 ) С ɶ ɶ =.. (5.14)

12 16 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam Some estmaton an be done for hosen sales. he rub laser ulse wth a duraton of nanoseonds has energ radaton E = 1 Joule. he emsson wavelength s equal l λ = m. Determne the number of hotons emtted er laser ulse. In ths ase the energ of an ndvdual hoton E = h = h/ = ν λ Joules. he number of emtted hotons N = E / E = l. he length of the reated laser hannel L~ 1м. If the ross setonal area of the hannel s 1 mm the volume of the hannel s equal 4 ~1 m Consequentl the ulse ower s equal 1 P = E/ t =.5 1 watts. Power ulsed lasers b several orders of magntude hgher. For omarson hdroeletr 9 ower s P ~1 watts. In rnle t s ossble to rovde a ower of monohromat radaton n a ontnuous mode. Neodmum laser generates a ulse of energ 75 Joules the 1 duraton of whh 1 s. For the rub laser ulse 4 4 En = EN / Vl ~1 J / m. If = 1 m then С ~1 J 4 m. Some relmnar remarks before alulatons: 1. he followng onsderaton s vald f funton of ξ or onstant.. he elt form of funton ˆ ˆ F s a F an be obtaned from eermental data wth a laser beam dreted along and aganst dreton of gravtatonal feld.. Let the enter of attraton las a fore dreted along the negatve values. hen there must est a lmt value ostve values orresondng to the area of ossble enetraton of the radaton. he observer s n the ostve area. 4. Let us assume that ˆ F = onst ˆ ξ. hs s the smulaton of the laser beam nteraton wth the gravtatonal wave. 5. All followng alulatons are realed for SYSEM ɶ ˆ = 1 under ntal ondtons for Cauh roblem dnˆ n ˆ = 1 =. he followng fgures reflet the dξˆ results of alulatons realed wth the hel of the Male rogram. Male notatons are used ˆ nɶ =ɶ D(n)(t) = ɶ ξ n =ɶn ˆ t = ξ C= ˆС ; for the ase notaton s Fˆ = F ξ. ˆ ˆ = onst ˆ F ξ Male Is t ossble to obtan the wave te soluton for beam transort under these ondtons? Let us show that the SYSEM admts suh knd of solutons. he results of alulatons refleted on fgures 1 14 whh are organed b the followng wa. Fgures 1 orresond to the ase when ˆ F = onst > fgures 4 6 ontan results when ˆ Fˆ = onstξ onst >. Fgures 7 9 orresond to the ase when ˆ F = onst <. Fgures 1 14 orresond to the ase when Fˆ = onstξ onst <. he atons ontan ˆ numeral lmtatons of the wave desrton whh est n some ases. Fg. 1. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the (1) F =.1 С ˆ = 1 ˆ( ˆ ξ ) - dotted lne lm= Fg.. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1 C = 1 1 ˆ ˆ( ˆ ξ ) - dotted lne lm= Fg.. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1 C = 1 1 ˆ ˆ( ˆ ξ ) - dotted lne lm=

13 Ameran Journal of Modern Phss and Alaton 15; (6): Fg. 4. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F =.1 ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne lm= Fg. 7. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1 C = 1 1 ˆ ˆ( ˆ ξ ) - dotted lne lm= Fg. 5. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne. Fg. 8. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) ˆ for the F = 1 C = 1 1 ˆ ˆ( ˆ ξ ) - dotted lne lm= Fg. 6. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1 ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne. Fg. 9. Deendenes of numeral denst ˆ F =.1 ˆ C = 1 ˆ( ˆ ξ ) - dotted lne lm= nˆ( ˆ ξ ) for the ase

14 18 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam Fg. 1. Deendenes of numeral denst ˆ( ˆ ξ ) for the ase ˆ F =.1 ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne lm= Fg. 1. Deendenes of numeral denst ˆ F =.1 ˆ ˆ ξ C = 1 lm= nˆ( ˆ ξ ) for the ase Fg. 11. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1 ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne lm1= lm= Fg. 14. Deendenes of mulse ˆ( ˆ ξ ) for the ase lm1= lm= ˆ F =.1 ˆ Cˆ = 1 ξ Obvousl SYSEM reveals tremendous ossbltes for mathematal modelng. Choosng dfferent Cauh ondtons and values ˆ F Cˆ we an dsath the ondene the eermental and theoretal data. he hange n frequen an be fed n eerment. For eamle Fg. 1. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne lm1=-.884 lm= ν ν ˆ ν = = ==. (5.15) ξ h ξ ˆ ξ Fgures 15 and 16 reflet the alulatons of these dervatves.

15 Ameran Journal of Modern Phss and Alaton 15; (6): Fg. 15. Deendenes of mulse dervatve ˆ F =.1 ˆ ˆ ξ C = 1 lm=1.14. dˆ( ˆ ξ)/ dˆ ξ for the ase Fg. 18. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1/ ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne. 6. Conluson Fg. 16. Deendenes of mulse dervatve dˆ( ˆ ξ)/ dˆ ξ for the ase ˆ F =.1 ˆ ˆ ξ C = 1 lm1= lm= he deendene (4.) an lead to an aromaton of the te ˆ F = F/ ˆ ξ. Fgures 17 and 18 reflet the orresondng ase. Fg. 17. Deendenes of numeral denst nˆ( ˆ ξ ) and mulse ˆ( ˆ ξ ) for the F = 1/ ˆ ˆ ξ C = 1 ˆ( ˆ ξ ) - dotted lne. In Newtonan gravt gravtatonal effets are assumed to roagate at nfnte seed. But we stll don't know a entur after Ensten found the feld equatons whether gravtatonal rles travel at. Nevertheless we do have strong emral evdene that suh rles est. he tal eamle s the Hulse-alor sstem ontanng two neutron stars orbtng around ther ommon enter of mass and the erod of the orbt s observed to be dereasng graduall over tme. hs s usuall nterreted as evdene that the stars are losng energ to radaton of gravtatonal waves. wo questons were formulated: 1. Is t ossble to seak about nteraton of a laser beam (for eamle) wth gravtatonal feld?. Can we admt the estene of gravtatonal waves from oston of non-loal relatvst hss? It s shown that both questons have the ostve answers. In other words the non-loal treatment does not lead to the mathematal ontradtons n the theor. In onluson we note the followng mortant nformaton: 1. here s an area of hss n whh non-loal members of the relatvst hdrodnam equatons are determnant and moreover loal members dsaear. hs s theor of radaton nludng radaton n the gravtatonal feld. Moreover the roblem of gravtatonal feld nteraton wth a laser beam annot be solved n the frame of loal theoretal hss n rnal.. Whle studng the moton of massless artles (hotons) the arameter of non-loalt n the relevant relatvst hdrodnam equatons vanshes. It means that the soluton does not deend on the arameter of nonloalt.. Newton' law analog for the hoton movement s obtaned. he develoed theor an dsath the vast mathematal modelng of a laser beam nteraton wth gravtatonal feld nludng gravtatonal waves roagatng wth a seed of gravt.

16 1 Bors V. Aleeev: Relatvst Non-loal Phss n the heor of Gravtatonal Feld Interaton wth a Laser Beam Referenes [1] Weber J. Deteton and Generaton of Gravtatonal Waves Phss Revew Volume 117 Number (196). [] Ju L. Blar D. G. Zhao C. Deteton of Gravtatonal Waves Re. Prog. Phs () Prnted n the UK PII: S4-4885() [] Staff. BICEP 14 Results Release. Natonal Sene Foundaton. Retreved 18 Marh 14. [4] htt:// Harvard-Smthsonan Center for Astrohss. Retreved 17 Marh 14. [5] Clavn W. NASA ehnolog Vews Brth of the Unverse. NASA. Retreved 17 Marh 14. [6] Overbe D. Deteton of Waves n Sae Buttresses Landmark heor of Bg Bang. New York mes. Retreved 17 Marh 14. Frame of Non-loal Relatvst Phss and Shannon- Nqust-Kotelnkov heorem Ameran Journal of Modern Phss and Alaton 15; (): 4-57 Publshed onlne June (15). (htt:// [8] Aleeev B. V Generaled Boltmann Phsal Knets Elsever Amsterdam he Netherlands (4) 68. [9] Aleeev B. V Unfed Non-loal heor of ransort Proesses Elsever Amsterdam he Netherlands (15) 644. [1] Aleeev B. V. Ovhnnkova I. V. J. Nanoeletron. Otoeletron. Part (1). [11] Aleeev B. V. Ovhnnkova I. V. J. Nanoeletron. Otoeletron. Part (1). [1] Aleeev B. V. he Generaled Boltmann Equaton Generaled Hdrodnam Equatons and ther Alatons Phl. rans. Ro. So. Lond (1994). [7] Aleeev B. V. Conneton Between me Quantaton n the