THE NEW TRANSFORMATION EQUATION AND TRANSVERSE PHYSICAL EFFECT K.VAITHIYANATHAN

Transcription

1 THE NEW TRANSFORMATION EQUATION AND TRANSVERSE PHYSICAL EFFECT BY K.VAITHIYANATHAN E.Mal : vathyanathan_k@yahoo.co.n Andkkadu Taml Nadu. South Inda. 1

2 Introducton PART 1 Any one can see the new transformaton equaton n my frst paper A new concept for classcal mechancs and relatvstc mechancs whch s publshed n General scence journal In that paper we already dscussed some conflcts n Newton s classcal mechancs and Ensten s relatvstc mechancs. In ths paper we dscuss some specal nature of the new euaton why we need that new transformaton equaton? What s the conceptual dfference between the new and the old? The two fathers of our basc scence the Galleo and the Newton ntroduced a transformaton equaton to descrbe relatvty prncple n mechancs. X-vt yy zz tt. The basc quanttes of physcs space tme and matter are nvarant wth respect to the above equaton. Later eperments well proved that the three quanttes are not nvarant and they are affected by the second order factor ( ( v / c ) and also the above transformaton equatons are nconsstent for electro dynamcs and optcs. To overcome these dffcultes Ensten etended the relatvty prncples from mechancs to electrodynamcs by Lorentz transformaton equaton (-vt) / 1 v / c yy zz t(t-v/c ) / 1 v / c. These equatons unfy the mechancs electro dynamcs and optcs. But t cannot help us to etend the relatvty prncple for all states of moton and eneral co ordnate transformaton (.e. ravtatonal acceleraton). So Ensten seeks another route to etend the relatvty prncple for all state of moton. He used two concepts Equvalence prncple and the General covarance prncple. Here he does a pvotal mstake. Why he selects the conceptual framework? Why he does not take any step to construct a compact transformaton equaton for all state of moton? All physcsts have these questons but no one succeeds. The journey contnue to fnd out the new one. Ensten spent eleven years and lot of conceptual confuson to arrve hs ravtatonal feld equaton. At one stae he abandoned hs covarance prncple also. Then he reaned t Even thouh he met lot of dffcultes he could not unfy the specal and eneral theory of relatvty prncple by a snle transformaton equatons. The two pllars of modern physcs quantum mechancs electrodynamcs feld equatons are untouched by ths concept. In ths way hs conceptual framework arses a lot of problems. From Ensten s tme to tll ths date no one takes any step to ntroduce a new transformaton equaton whch s sutable for all physcal phenomena and all states of moton n ths way the new transformaton equaton satsfes the small amount from our lare demand.

3 Basc dea All physcal process n our unverse consst of one or more events each event has specfc space co-ordnates (yz) and a tme co ordnate (t). The rato between space and tme co ordnate are called event velocty ( y z )/ t C. Ths modern noton of space-tme s enouh to descrbe any physcal process. To transfer the space-tme co ordnates of same types of two events n (k k 1 ) two dfferent states of moton. We use that event velocty c whch takes an mportant role n our equaton. F:1 K ds dd j K 1 ds d d j V Same type of two events s s They may be electrodynamcs optcs or sound or mechancal event. One occurs n K other n k yzt and 1 y 1 z 1 t 1 are space and tme co ordnates of s and s 1 wth respect to ther local observers the event velocty c s consant n all drectons whch does not affect the frame velocty v we use that constant event velocty c to transfer the space-tme coordnate from k to k 1 so c ( y z )/ t ( y z )/ t Ths nature s common for all types of events The event velocty C s constant n all drectons wth respect to ther local observers measurement of a partcular state of moton that state of moton does not affect the event s velocty (c) Ths nature s common for all types of event s velocty. (Electrodynamcs optcs sound and mechancs) we use above prncple n our transformaton equaton (constant events velocty) to transfer the same type of two event s space tme co-ordnate n dfferent state of moton. So we can use our new transformaton equaton for all types of event s the factor (v/c) s common for all events (where v s the relatve velocty c s the events velocty nstead of Ensten s lht velocty c). For electro dynamcs and optcs the event velocty s m/s for sound waves C s 340 m/s all other mechancal event s the event velocty s wth respect to ther numercal value. (We can see ts dervaton n our frst paper) Fnally the new transformaton equaton s as follow. vt y y 1 v / c z z 1 v / c t tv/ c Newtransformaton 3

4 6RPHVSHFLDOFDVHV &DVH.1 *DOLOHR VWUDQVIRUPDWLRQ put the Galleo s condton c n the new equaton vt y z y 1 (v/ ) z 1 (v/ ) /( )( ) t t v We have vt y z t y z t *DOLOHR VWUDQVIRUPDWLRQ &DVH /RUHQW]WUDQVIRUPDWLRQ equate the transverse drecton y y z z form the new equaton ( y z t c ) (-vt) y (1 v / c ) z (1 v / c ) c ( tv/ c ) ( y z ) ( y z ) 0 t c c ( tv/ c ) /(1 v / c ) -(-vt) /(1 v / c ) t c c ( t v/ c ) /(1 v / c ) 0 -(-vt) /(1 v / c ) 0 We have ( vt)/ 1 v / c y z y z t ( t v / c )/ 1 v / c /RUHQW]WUDQVIRUPDWLRQ Galleo s and Newton s classcal mechancs Enstens and Lorentz relatvty mechancs are specal cases of the new equaton n other words they are lmted valdty. We can use t n nertal frame only. 4

5 Case.3 Prncple of General Covarance 1. Wrte physcal law as tensor equaton. Show that t s true n some coordnate system 3. The equaton holds n all coordnate systems and therefore s covarant From the New Transformaton Equaton we have physcal law for nertal and non-nertal system. S y z ct ( y z ct ) ds d d dd It s Tensor form. j j For Inertal co-ordnate system (CCS) (GCS) 1 For Non nertal system (1 v / c ) and t s frst and second dervatve. The value of the metrc tensor only chaned wth respected to selected co-ordnate system. The tensor form of the physcal law s unchaned. So the new equaton satsfy the eneral covarance prncple. Case.4 Prncple of Equvalence 1. Gravtatonal mass s same as nertal mass. Ths s an emprcal fact reconzed by Newton e.. and whch use the same mass m.. Ths mples that all objects eperence the same acceleraton n a ven ravtatonal feld (e.. Galleo) 3. Ensten reconzed that the ravtatonal feld s undetectable n a freely falln (nertal) frame! Inertal and ravtatonal acceleraton have complete physcal equvalence Inertal acceleraton ravtatonal acceleraton ths prncple only holds ood for small scale In lare-scale ravtatonal acceleraton have a specal nature In a local frame f you have two partcles and wat too lon the partcles wll move towards each other whch s detectable from nertal acceleraton. To overcome ths conflct Ensten showed that the ravtatonal acceleraton have curved space-tme. Here Ensten unnotced a pont why nertal acceleraton lose ths property. Ensten s Lorentz transformaton cannot help us to remove the dffcultes but our new transformaton equaton help us to arrve same curved space-tme nertal acceleraton. See t n part II so our new equaton satsfy the Prncple of Equvalence 5

6 SPECIAL NATURE OF THE NEW TRANSFORMATION EQUATION In ths chapter we dscuss some specal nature of the new transformaton equaton whch are not present n Gallean and Lorenz transformaton equatons. Transverse Physcal Effect (1) We can arrve transverse (second order ( v / c ) ) physcal effect from our equaton for all types of events. () We can use t for (all types of moton) nertal and non nertal frames. In the new equaton we have Newton s frst order effect (v/c) for and t only the y and z have no frst order effect but they have relatvstc second order effect ( v / c ). We can arrve the same second order effect for and t from ts frst order effect. For t we compute the ve and -ve sdes of the frst order effect. To observe the effect n the ve and ve sdes put V -V n our new transformaton equatons we have two sets vt y y 1 v / c z z 1 v / c t tv/ c vt y y 1 v / c z z 1 v / c t t v/ c The (-V) and (V) The sn chane produce a chane n the frst order effect only but the second order remans unaltered. The space-tme co-ordnates are med n the equaton. To separate them ndvdually we use the follown relaton s constant n all drectons 0 0 ct y z ct a vt. The event velocty c whch 0 y 0 ct 0 0 z ct t s the event s tme whch s common for all space co-ordnates. So the space co-ordnates yzhave equal numercal values n ther drectons n the relaton avt V s the relatve velocty between the two frames. T s the tme of the event n one unt of tme nterval above velocty component s equal to space coordnate 6

7 vt (1 a/ ) vt (1 a / ) y y 1 v / c y 1 a / y y y 1 v / c y 1 a / y z z 1 v / c t tv/ c z 1 a / z t(1 a/ ) z z 1 v / c t t v/ c z 1 a / z t(1 a / ) (1 / ) t t v c t t 1 v / c t t 1 v / c (1 / ) t t v c (1 / ) t t v c t t 1 v / c t t 1 v / c (1 / ) t t v c The above equatons are Classfed as frst and second order. (1 / ) v c y y 1 v / c z z 1 v / c (1 / ) v c (1 / ) v c y y 1 v / c z z 1 v / c (1 / ) v c (1 / ) t t v c t t 1 v / c t t 1 v / c (1 / ) t t v c (1 / ) t t v c t t 1 v / c t t 1 v / c (1 / ) t t v c Multple the (-ve) (ve) sdes of the equaton and ts square root 1 v / c y y y 1 v / c z z z 1 v / c tt t 1 v / c tt t 1 v / c tt t 1 v / c Fnally we come to the concluson from the new equatons that the Newton s classcal frst order effect (v/c) only arses n the movn drectons of ve and -ve sdes only. There s no such effect n the y and z drectons the Gallean transformaton ndcates ths physcal phenomena. The relatvstc second order effect v / c arses n all drectons.e. yzt. Lorentz transformaton faled to epress ths nature so Lorentz transformaton has lmted valdty ( e.. nertal frame only). 7

8 Form the equaton 1 and yy zz y z y z v c ( ) ( ) [( )(1 / )] ctt t c [ ct (1 v / c )] 0] Equate the tme co-ordnate to zero ctt t c ct (1 v / c ) 0] We have second order physcal effect for space coordnate L L L L 1 v / c For the frst order y and z have no such effect v c (1 / ) v c (1 / ) Equate the space co-ordnate to zero [( y y z z ) ( y z )] [( y z )(1 v / c )] 0 Second order effect for tme coordnate t t t t 1 v / c There s no frst order effect for Y and Z t t t (1 v / c ) t t t v c (1 / ) t1unt of event tme t < t 1unt of event tme t > 1 1 t t / 1 v / c t t v c /(1 / ) t t v c /(1 / )

9 Part II New transformaton equaton & Space-tme eometry n non nertal frame From the equaton (1) and () we have the follown relaton vt y y 1 v / c z z 1 v / c t t v / c vt y y 1 v / c z z 1 v / c t t v / c vt y y 1 v / c z z 1 v / c t t v / c Square of the event For the equaton 1 S y z ct (1 v / c )( y z ct ) For the equaton S [( y y zz ctt (1 v / c )( y z ct ) S [( yy zz ctt ) ( y z ct )] [( y z ct )(1 v / c )]--17 Metrc relaton ds d d (1 v / c ) dd j (1 v / c ) η < > where v / c η 1 c v (1 v / c ) For Cartesan co-ordnate 1 The unform constant relatve velocty V n the new transformaton equaton affects the metrc value. ts not constant and ts value vares wth respect to the relatve velocty v. 9

10 The metrc transformaton for two states of moton whch have relatve velocty v. V 0 V C 1Cartesan co-ordnate n one state of moton j (1 v / c ) j 1 j j j 0 j j j The metrc value chanes from 0 to 1 for the value of V C to 0 1.e The eometrcal nature chanes wth respect to the relatve velocty v. Ths result s very mportant whch we arrve from the new transformaton equaton. The transformaton of the determnant mples j j j.. j From our new transformaton equaton we have v c (1 / ) (1 v / c ) 10

11 dv dd 1 d3d4 dv (1 v / c ) dd 1 d3d From our new transformaton equaton we have That the volume of the element chanes wth respect to relatve velocty v. ths s another mportant result from the new transformaton equaton. When vo The relatve velocty between these two frames k and k 1 s zero Apply ths condton n the new equatons. y yz z t t ) S ( y z ct )] [( y z ct ) dd dd j j Here both frames are n Cartesan co ordnates now these two frames are nertal frames. If we concde the orns of the two events n each frame. They always concde durn the complete events tme there s no chanes n orns between these two frames so the eometrcal nature of the spacetme s nvarant. The value of the metrc tensor s are equal. The element volumes (V V) have equal values. The orns of the events take mportant roles. K K F: o o The orns o and o concde When V>O The relatve velocty V between these two frames k and k s constant. Here one frame moves wth unform constant velocty wth respect to other F:3 K K V The local Cartesan co ordnate (LCCS) system n k s transformed n to eneral co ordnates (GCS) system due to the cause of the relatve velocty V. 11

12 Accordn to the transformaton law we have metrc relaton between LCCS and GCs j j Use the new transformaton equaton n the metrc transformaton law. We have (1 v / c ) v / c η η Here the movn frame s n non-nertal condton. The event orns of these two frames not concde durn the event tme. The chanes of orn ( o to o ) affect eometrc nature the chanes of eometrc nature metrc. When V>O η affect space-tme The relatve velocty v s unformly chanes t means unform acceleraton. In ths case the frame k 1 moves wth unform constant acceleraton wth respect k. The observer nsde k cannot aware ts d acceleraton dt 0. The local co-ordnate system LCS s Cartesan. Its metrc s 1. But the same tme the frame k moves wth unform acceleraton wth respect to k. The metrc of the LCCS unformly chanes to GCS so the metrc have a dervatve Unform Inertal acceleraton k The frst dervatve nduced Chrstaffel symbol l l ljγ k lγ kj Inter chane the ndces k and I Γ Γ Smlarly wth k J nto kj l l lj k kl

13 Work out (30)(31)-(3) to obtan Recall l kj k j Γ Γ l l lk j l jk l jk Γ Γ Recall lk kl So after canceln common term (30)(31)-(3) k ljγ kj l k j To solate the frame Γ we multply by / jn n l n lj δγ l k Γ k Unform nertal acceleraton nduced frst dervatve of metrc tensor k. k nduced chrstoffel symbol n Γ k The Chrstoffel symbol nduces eodesc equaton. A partcle moves n a straht lne n a local coordnate system k whch moves wth unform acceleraton wth respect to k the observer n LCS can t aware t s acceleraton. d So t s acceleraton s dt d Its velocty s constant dt Wth respect to k the local Cartesan co-ordnate system s converted nto eneral co-ordnate system. We can derve the equaton of moton d d ( ) 0 dt dt Usn the chan rule d d ( ) dt dt Usn the product rule d d d dt dt dt

14 Applyn the chan rule aan d d d dt dt dt j 0 j In equaton at moton look for terms nd order n tme multplyn throuh d d d dt dt dt k j k 0 j There fore d k ves k d d j k δ Γ dt dt dt d k k Γ d d dt dt dt k k j Γ Affne connecton From the above calculaton The unform chane n relatve velocty V unform acceleraton Frst dervatve of metrc n tensor k Chrstoffel symbol Γ k eodesc equaton d k k Γ d d dt dt dt Wth respect to our new equaton we have S y z c t (1 v / c )( y z c t ) It means the space-tme co-ordnate chanes wth respect relatve velocty. In the unform acceleraton the velocty unformly chanes ths effect affect the space tme co-ordnate unformly. The unform chane of space tme co-ordnate produce a chane t s volume(rch Tensor) and t s shape(weyl Tensor). Ths nature nduced devaton of eodesc equaton ts nothn but second order dervatve of metrc tensor l k so the unform nertal acceleraton nduced second order dervatve l k also. We can see t follown relaton 14

15 Accordn to Taylor seres k k l k kl 1/ To compare two eodesc devaton nter chane l & k so we have R l k l k kl Relatve nertal acceleraton relatve eodesc devaton Remannan Manfold curved space- tme Wth respect to the new transformaton equaton the unform nertal acceleraton nduced Remann s curved space tme. Eperment If the events orn chanes that chane affect space tme co ordnate and eometry. We can prove t follown some eperment. We can arrve above transverse Physcal effect by observn transverse Doppler effect for lht and sound waves. Generally Doppler effect are calculated by wave lenth and ts frequency. For wave lenth we have follown equaton from our results. Wave lenth observed Wave lenth observed Ve & movn drecton -Ve movn drecton ½ - Rest state wave lenth Rest state wavelenth Source velocty 1/ Snal velocty λλ λ λ(1 v/ c) λ(1 v/ c) λ 1v λ λ c 15

16 We can draw wave front daram wth systematc unt scale C1cm/s V5cm/s When V0 Vt 0 a0 F :4 16

17 When V>0 V>0 a> F:5 For lht snals transverse Doppler effect are already observed by Ives stll wll. We use the same eperment s data we can calculated classcal Doppler effect for Ve and Ve movn drecton s λ λ(1 v/ c) λ λ(1 v/ c) and λ λ λ λ A C 3105 km/s Source velocty λ λ λ Producton relatvty Observed value Above calculaton from Ives stll wll epermental data confrmed our new concept. 17

18 Eperment II For sound waves To observe transverse Doppler effect for sound waves any one can use classcal Doppler effect. Tll ths date physcsts cannot take step for t because they have stron confdence about Newton and Ensten s relatvstc Concept. Wth respect to ther concept we have no possblty to observe transverse Doppler effect for sound waves. But our new concept helps us to observe such effect. Here I descrbed a smple method to observe transverse Doppler effect for sound waves. A movn sound source (vc/170m/s) produce sound waves at varous space ponts wth respect to non-movn ar medum. Each wave front s produced n dfferent orn (AB) n the ar medum. The chane n orn affects ve and ve sde wave lenths (see the systematc daram) due to the eometrcal chane. To satsfed the above condton we set two source A and B n the non movn ar medum wth a dstance c/340/170m/s. Three observers P Q R are placed at a dstance at 340m from the source A P Q are placed n ve and ve sdes of the drecton R s placed n perpendcular drecton from the source B. The source A produces frst pulse of sound waves at a tme t0 after one second tme nterval the source B produced another sound waves these two waves are produced n one second tme nterval wth varous space ponts A B n the non movn ar medum. Ths nature s equal to movn sound sources physcal nature. These two waves are observed from the pont s P Q and R. The observed tme ntervals from the ponts PQ and R are as follow wth respect to our calculaton value. From the results anyone can verfy that the observed results n the perpendcular drecton s equal calculated value n ths way any one can arrve transverse Doppler effect for sound waves wth respect to our new concept. F:6 (ve) sde t t(1v/c) -ve) sde t t(1v/c) 18

19 t t 1 v / c Wth respect to our calculaton. tt t vc vc (1 / )(1 / ) t 1 v / c Any one can verfy that the observed result n the perpendcular drecton s equal to calculated value. In ths way any one can arrve transverse Doppler effect for sound waves. Eperment Select four rd stcks whch have equal lenth OAOBOCOD6cm and arrane them n y - -y drecton as shown n the fer F:7 B y 6Cm A O C - 6Cm 6Cm D -y In the arrane ment ve and -ve sdes of the and y drectons have equal lenths from the orn O and also they have Cartesan eometrcal nature. ( ) y (66)6cm The orn has no chane) the ve and ve sdes of drecton lenth also no chane Net to produce a chane n the ve and - ve sdes of drecton we cut the rd stcks acm from the ve sde and remove t from that place and mpose t n the opposte drecton. Now the orn O s shfted at a dstance of a cm the new orn s O In the process the -ve and ve sdes of rd strcks are chaned ther values. AOOO(a)68cm OC-OO(-a)6-4cm 19

20 The total value of AC s not chane AC1CM F:8 B A B A O a O C D D Above chane n ve -ve sdes of drecton affect the eometry of the y drecton and t devate from the ornal poston OBOD to new poston OBOD But the total value of BOD not chaned (OBOD)(66)1cm The eometrcal nature only Chaned. Now the four rd stcks AO8cm 6-4 CmOBOD6cm are arraned n the new eometrcal nature ths nature s ven n the f- The calculated value of y n the new eometrcal nature by our new trans formaton equaton result. Measurement values are (a)(6)8cm (-a)(6-)4cm Y y (1-a /y ) 6 (1- /6 ) 5.65 Cm The measurement value concdes wth the calculated value and also t reveals that the frst order chare n -ve ve sdes of drecton affects the y drecton wth second order ((-a) (a)) v ((6) (6-)) 5.65 Y y ((-a) (a)) 6 (1- / ) 5.65 Select another four stcks wth respect to the second order value 5.65cm and concde the f-1 The second order value of the stcks chanes from 6cm to 5.65cm ths value s equal n all drecton t s drectly professonal to the poston of the event orn. If a events orn chanes and ts chane produce equal second order effect n yz drecton. Due to the cause of the velocty chane V My sncere thanks to My frends Mr.P.Uthraju Mr.P.Mankandan and Mr.Dawood. Those who are work hard day and nht wth me to prepare ths paper. Ther best help and cooperaton encouran and nsprn to me aan my sncere thanks to all of them. 0

21 REFERENCE 1. The foundaton of the eneral theory of relatvty By A.Enten. The Relatvstc theory of Gravtaton By A.Lounv and Mestvrshvl Mr Publsher Mascow. Revsed from the 1986 Russan Edton. 3. Introducton to the theory of relatvty by Peter Gabrel Bermann Prentce Hall of Inda Prvate Lmted New Delh The Feynman Lectures on Physcs pae By Feynman Lehton Sands Narosa Publshn House New Delh