gr-qc/ v2 28 Jul 1997

Transcription

1 This page is a plaehle insete by the aministats s that the system ul pue PDF f this pape. The eal pape is Walke.ps, whih fllws this page if yu e lking at the PDF. g-q/ v 8 Jul 997

2 Ppagatin Spee f Lngituinally Osillating Gavitatinal an Eletial Fiels William D. Walke an J. Dual (Geneal elativisti analysis pefme in llabatin with T. Chen) Institute f Mehanis, Swiss Feeal Institute f Tehnlgy, 809 Zuih, Switzelan walke@ifm.mavt.ethz.h, ual@ifm.mavt.ethz.h g-q/ v 8 Jul 997 F seveal yeas, the auths have been investigating the pssibility f evelping a labaty expeiment apable f measuing the spee f gavitatinal inteatin. Duing the 950s - 960s, seveal eseahes ppse that it might be pssible t lngituinally vibate a mass nea anthe mass an t etet the esultant gavitatinally inue lngituinal vibatin. The ase spee ul then be etemine fm the sillatin fequeny, the sepaatin istane between the masses, an the measuement f the ase iffeene f the vibatin between the tw masses. These ealy eseahes ha assume that the ase spee f gavity was equal t the spee f light. The ase shift expete f a typial expeimental set-up was n the e f miegee. Beause f the limite tehnlgy at the time, n gavitatinal expeiments wee pefme. In 96 R. P. Feynman publishe a geneal ysis bk in whih he analyse the eleti fiel f an sillating hage. Feynman s nlusin was that the sillating fiel ppagates nealy instantaneusly alng the axis f vibatin, muh faste than the spee f light. Beause f the similaity f the analgus sillating mass pblem, the ysis mmunity has sine nlue that the ase spee f bth a lngituinally sillating gavitatinal fiel an a lngituinally sillating eletial fiel ae t fast t measue with a nea-fiel labaty expeiment. Feynman s analysis is nt vali in the nea fiel an is theefe innlusive f a nea-fiel labaty expeiment. An analysis f the eletial fiel pue alng the axis f vibatin f an sillating hage, vali in the nea fiel, is pesente. The slutin iniates that the ase spee f the lngituinally sillating eleti fiel next t the vibating hage ( ) is nealy infinite, an apily eays t the fm /( ) in the fa fiel ( >> ), in whih is the vibatin amplitue f the hage, is the spee f

3 light, is the fequeny f vibatin, an is the istane fm the ente f the sillating hage. A peliminay alulatin f the gavitatinal fiel pue by the analgus sillating mass pblem is als pesente. The slutin iniates that the ase spee f a lngituinally sillating gavitatinal fiel is equal t lage than e /( ). Bth f these esults iniate that the ase spees f a lngituinally sillating eletial an gavitatinal fiel ae t lage t be measuable with a labaty expeiment. The pssibility f measuing the gup spee f a lngituinally sillating gavitatinal fiel, whih is mmnly thught t be equal t the spee f light, is nw being nsiee. The basi iea is t amplitue-mulate the lngituinal vibatin f a mass an t measue the esultant lngituinal vibatin f a neaby mass ue t gavitatinal inteatin. The mulatin signal an be extate using a ie etet an the gup spee an then be etemine fm the sillatin fequeny, the sepaatin istane between the masses, an the measuement f the ase shift f the mulatin signal. If the gup spee is equal t the spee f light, ase shifts n the e f miegee ul be geneate with a typial expeimental set-up. It is pesently unknwn if the ase stability f an expeimental system an be ntlle t this auay ve the measuement time. A bening gavitatinally inteating system that is apable f geneating a nanmete gavitatinally inue vibatin amplitue, whih is 4 es f magnitue lage than peviusly ahieve with the systems, has been evelpe an teste. The esultant gavitatinally inue vibatin is in g ageement with Newtnian alulatins. In aitin, a ase measuement system that is apable f measuing a 00 nanegee ase shift an a 5 nanvlt hange in amplitue, whih is 5 es f magnitue me sensitive than mmeial lk-in amplifies, has als been evelpe an teste.

4 Theetial Analysis f the Phase Spee f a Lngituinally Osillating Eletial Fiel Sin(t) q l in whih: p q q Chage Spee f light Distane t sue Chage vibatin amplitue Vibatin fequeny ε Dieleti nstant Figue : Vibating hage mel use t alulate the ase spee f an sillating eleti fiel alng the axis f vibatin. R. P. Feynman Slutin a : (Use multiple analysis. Vali nly in fa fiel i.e. >> ) E axisac pt pt + πε >> Althugh nt alulate by Feynman, the fllwing nlusins an be eue fm this esult: E Axis AC [ Sin( t ) ] q + + πε >> in whih: Tan >> Tayl expaning the esult f Cs Sin Cs + Sin < yiels the fllwing seies: << << << 0% auay a R. P. Feynman, Feynman letues in ysis, Aisn - Wesley Pub. C., Vl., Ch., (964).

5 4 The ase spee ( ) f an sillating fiel f the fm: Sin( t k), in whih k k(, ), an be etemine by setting the ase pat f the fiel t ze, iffeentiating the esultant equatin, an slving f : ( t k) 0 k k 0 Diffeentiating k with espet t yiels: k k k + k Cmbining these esults yiels: in whih, is the ase shift The ase spee f the lngituinally sillating eletial fiel is alulate t be: + >> lim > ~ << 0% auay 0% auay ~ λ π Analysis using the Liéna-Wiehet ptentials b : (Vali in nea fiel, ef. Figue ) A E ϕ in whih: t K ϑ v n l et K v A v n l et E Eleti fiel ϕ Sala ptential A Vet ptential v Velity f the mving hage Vibatin fequeny n Unit vet fm sue t bsevatin pint et ( ') Quantity in bakets t be evaluate at the etae time (T) q K 4πε b J. Jaksn, Classial Eletynamis, Jhn Wiley &Sns, (975).

6 5 E K v v n v n l et + fa-fiel tansvese aiatin tem E K Axis v v ( l ) in whih: x v x Sin( t) v Cs( t) β Cs( t) β t v l β Cs( T ) l Sin( t) l Sin( T) T t Fuie tansfming the esult yiels: E Axis Tx q ( ) Sin t O πε + hh.. in whih: + O The ase spee is alulate t be: 4 h.h. Highe hamnis O + 4 ~ New nea-fiel tem iffeent fm Feynman slutin 0% auay Cnlusin The ase spee f a lngituinally sillating eletial fiel is muh faste than the spee f light an nt measuable using men tehnlgy whih is nly pesently apable f measuing ase spees less than equal t the spee f light.

7 6 Theetial Analysis f the Phase Spee f a Lngituinally Osillating Gavitatinal Fiel (Analysis pefme in llabatin with T. Chen, Institute f Mehanis, ETH, Züih ) Sin(t) m l In whih: G µν Einstein tens T µν Enegy mmentum tens G Gavitatinal nstant Spee f light ϑ Gavitatinal ptential ρ Mass ensity Vibatin fequeny Figue : Vibating mass mel use t alulate the ase spee f an sillating gavitatinal fiel alng the axis f vibatin. Using the Einstein elatin: G πg 8 4 T µν µν Alng the axis f vibatin, f small masses an lw velities, the Einstein equatin eues t: 4πG ϑ t T The nly nnvanishing tem in the enegy mmentum tens t e β is: [ ] T ρ δ[ Sin( t) ] + O( β ) in whih: β Slving the patial iffeential equatin yiels: ϑ K l l ξsin t βcs t [ + O( β )] in whih: K -m G, ξ l, l Sin t T. Chen, W. D. Walke, J. Dual, Gavitatinal nea-fiel inteatin, Cuently une eview by Physial Review

8 7 The gavitatinal fiel (g) an then be alulate using the elatin g ϑ : mg g Sin t O O hh.. The ati f the ase spee f the gavitatinal fiel t the spee f light is alulate t be: O Cnlusin The ase spee f a lngituinally sillating gavitatinal fiel is muh faste than the spee f light an nt measuable using men tehnlgy whih is nly pesently apable f measuing ase spees less than equal t the spee f light. As in the ase f the lngituinally sillating eleti fiel, alulatin f the gavitatinal fiel t highe es in β may enable the ase spee t be etemine. Ppse Expeiment t Measue the Gup Spee f a Lngituinally Osillating Gavitatinal Fiel The pssibility f measuing the gup spee f a lngituinally sillating gavitatinal fiel, whih is mmnly thught t be equal t the spee f light, is nw being nsiee. The basi iea is t amplitue-mulate the lngituinal vibatin f a mass an t measue the esultant lngituinal vibatin f a neaby mass ue t gavitatinal inteatin. The mulatin signal an be extate using a ie etet an the gup spee an then be etemine fm the sillatin fequeny, the sepaatin istane between the masses, an the measuement f the ase shift f the mulatin signal. If the gup spee is equal t the spee f light, ase shifts n the e f miegee ul be geneate with a typial expeimental set-up. It is pesently unknwn if the ase stability f an expeimental system an be ntlle t this auay ve the measuement time. Nte that the gup spee f light has been measue using this type f set-up. R. Ba, T. Amstng, An inexpensive appaatus f measuement f the gup velity f light in tanspaent meia using a mifie Helium-Nen lase, Am J. Phys., Vl. 58, N., Nv. (990).

9 8 m + m m + m AM Detet x Ban Pass Filte 0 AM Signal Geneat m m Phase Detet m 0 m g Figue : Ppse expeimental set-up t measue the gup spee f a lngituinally sillating gavitatinal fiel. Gup Spee Analysis f a Lngituinally Osillating Eletial Fiel If ase spee f an sillating fiel is a funtin f fequeny spae, then the gup spee will iffe fm the ase spee. The gup spee f an sillating fiel f the fm: Sin( t k), in whih k k(, ) mpnents f a wave gup:, an be etemine by nsieing Fuie Sin( t k ) + Sin( t k ) Sin( t k) Sin( t k) in whih:, k k k, +, k k + k The gup spee ( g ) an then be etemine by setting the ase pat f the mulatin mpnent f the fiel t ze, iffeentiating the esultant equatin, an slving f : ( t k) 0 k k 0 g k k + k Diffeentiating k with espet t yiels: k Cmbining these esults yiels: g lim small

10 The gup spee f a lngituinally sillating eletial fiel, using the Feynman fa-fiel slutin, is alulate t be: 9 g >> ( + ) lim > g ~ - g << 4 g 0% auay 0% auay 4 ~ - λ π Using the Liéna-Wiehet ptentials, the gup spee f a lngituinally sillating eletial fiel in the nea fiel, is alulate t be: g 0 New nea-fiel tem iffeent fm Feynman slutin + O 0 4 g ~ g 0% auay Nte that these slutins iniate that gup spee in the semi-nea-fiel is faste than the spee f light, whih appeas t vilate ausality an shul nt be pssible. An altenate appah t alulating the gup spee is pesente whih uses amplitue mulatin (AM) an emulatin (ef. Figue ): The signal ut f the AM signal geneat is: [ ( m )] ( ) AM Sig + Sin t Sin t Afte the eleti fiel has ppagate a istane, the signal ut f the multiplie in the AM etet is: AM Sig Det Cs ( m) t + + Sin t + Cs ( + m) t + [ ] [ ] [ ]

11 0 Inseting the ase slutin btaine fm the Feynman slutin ( << ): in whih: ( ) m ( ) Simplifying the esult, the ase f the esultant m mpnent yiels: m << ( ) + m m ( + ) m The gup spee f the mulate signal an be alulate, yieling the same esult as btaine using the lassial efinitin ( << ): g m m << >> m 0 ( + ) m Cnlusin The analysis f the gup spee f a lngituinally sillating eletial fiel is uently innlusive. The gup spee is mmnly thught t be equal t the spee f light, but peliminay analysis iniates that the gup spee is muh faste than light whih is nt thught pssible ue t ausality vilatin. Expeimental Gavitatinally Vibating System Simple Mass Sping Gavitatinally Inteating System u m Tx m Rx κ Rx b Rx Figue 4: Mathematial mel f a mass sping gavitatinally inteating system.

12 Inseting a sillating gavitatinal fe int a ampe mass sping patial iffeential equatin yiels the fllwing eeive mass vibatin amplitue (at esnane): u AC amp GQRxmTx Tx This esult lealy iniates that the eeive s gavitatinally inue vibatin (u AC ) is inease by maximising the eeive mass quality fat (Q Rx ), the vibating mass (m Tx ), an the amplitue f the tansmitting mass vibatin ( ). In aitin, minimising the vibatin fequeny ( Tx ) an the istane between the tw masses () is espeially effetive. New Bening Gavitatinally Inteating System A new bening beam gavitatinally inteating system has been built an teste e. The system is apable f geneating nanmete gavitatinally inue vibatins whih is 4 es f magnitue lage than pevius systems. The system nsists f a bass beam (Tx) whih is eletmagnetially vibate at its fist me 40 Hz bening fequeny. An ajaent quatz glass beam late.5 m away has been bseve t vibate at the same fequeny ue t gavitatinal inteatin. The bseve vibatin is nt affete by hanging the austial, mehanial, eletmagneti upling between the tw beams. When the spaing between the beams is inease fm.0 m t 7.0 m, a fat f 7.4 eutin in the Rx beam vibatin is bseve, as peite fm Newtnian they. Speifiatins: Tx beam - Bass, 95.9 m x.0 mm x.0 mm.0 m beam en, 40 Hz fist me vibatin Rx beam - Quatz glass, m x 7.6 mm ia., Q ~ 0,000.8 nm beam en, 40 Hz fist me vibatin expeimentally bseve.8 nm beam en vibatin peite fm Newtnian they e W. D. Walke, J. Dual, Expeiment t measue the ppagatin spee f gavitatinal inteatin, Vig gavity wave nfeene peeings, (996).

13 Steel vauum hambe 9.0 m inne ia..4 m lng Expeiment Tp View Glass winws Quatz glass beam (Rx) 7.6 mm ia m lng Bass beam (Tx).0 mm wie,.0 mm high, m lng Vetial ubbe bans suppting glass >.5m Cbalt magnet Eletmagnet Mehanial pin f nal suppt Vetial sping suppting tansmitte tay Vetial ubbe bans suppting glass Quatz glass beam (Rx) Lase Intefemete Expeiment fl Steel vauum hambe Expeiment En View Invetible austi shiel.5m Vetial sping suppting tansmitte tay Bass beam (Tx) Cbalt magnet Eletmagnet Ne pin suppts Glass winws Optial Islatin Table Figue 5: Expeimental set-up f bening beam gavitatinally inteating system.

14 00 Nanegee Phase an 5 Nanvlt Amplitue Measuement System A new iffeential ase an amplitue measuement system using a mmeial lk-in amplifie has been evelpe an teste f. The system is apable f measuing 00 nanegee ase shifts an 5 nanvlt hanges in amplitue, whih is 5 es f magnitue me sensitive than mmeial lk-in amplifies. A B D X Y ny nd Tan ny nb + nx small ny nb Amp A - B B + nx n + ny n B small nx n nx a N f a-b a N f n(a-b) na nn f n(a-b) nn 0 LPF n(a-b) na f 0 if feebak i.e. a b Expeiment (ase elay) A B Subtat D Amp nd nx x gain(n) LPF nx LPF n(a-b) nnn 0 f n(a-b) na f Nte: b 0 f N Signal an Lk-in white nise n π M ( TC )( LPF BW in [ + ( πftc LPF) ] # Ple LPF PLL +90eg f h Lk-in In whih: M.57 f Ple LPF. f Ple LPF.05 f Ple LPF.05 f 4 Ple LPF x ny n(a-b) f LPF na nn 0 f ny LPF LPF n(a-b) na na nnn 0 f n(a-b) na Figue 6: Signal pessing iagam f iffeential ase an amplitue measuement. The maximum sensitivity f this tehnique is limite by the intenal lk-in amplifie nise (X nise an Y nise typially: 5nV / Hz ) an the banwith (BW): min 80 ( Ynise / Hz) BW π A in whih: BW Ampmin ( X Nise / Hz ) BW M TC Time nstant f lk-in amplifie π TC f W. D. Walke, J. Dual, Expeiment t measue the ppagatin spee f gavitatinal inteatin, Vig gavity wave nfeene peeings, (996).