Optical coordinate system for a local observer in a weak gravitational field

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Optical coordinate system for a local observer in a weak gravitational field"

Πελαγία Δήλια Τομαραίοι
7 χρόνια πριν
Προβολές:

1 tcal coodnate sste fo a local obseve n a weak avtatonal feld A.N. Aleandov, E. Fedoova, V.I. Zhdanov Natonal Taas Shevchenko nvest of Kv, Astonocal bsevato kane We develo the ethod to constuct the efeence sste of a local obseve wthn the lnea aoaton on sace-te cuvatue. Tansfoaton to otcal o Fe coodnates s based on the solutons of the sotoc o sace-lke eodesc equaton and aallel tansot equaton. The an advantae of the otcal coodnates les n the dect lnk wth obsevable ostons of dstant objects on the celestal shee. We also aled ths ethod to constuct the otcal and Fe coodnates fo the case of acceleated and otatn obseve n avtatonal feld nduced b ovn bodes of the Sola sste. The tansfoaton foulae fo ntal coodnates to new ones wee obtaned and the etc tensos n otcal and Fe faes wee found n the cubc aoaton on the obseves veloct.

2 tcal coodnates C ae the ost aoate fo coason wth obsevatonal ostons of objects on celestal shee Can be easl constucted usn the av. lensn theo IA ecoendatons: ecoendatons about efeence faes, based on haonc coodnates HC We develo hee the eonental an to fnd the tansfoaton foula HC>C and HC>FC Fe coodnates, and to fnd a etc tenso n these coodnates Based on: local obseve conceton based on eodescs and aallel tansot equaton - > Sne G. Geneal elatvt; Пирагас К., Жданов В., Александров А., Кудря Ю., ПирагасЛ. Качественные и аналитические методы в релятивистской динамике., гл.3, MTW Gavt Qute close to the "obsevaton coodnates b.maatens

3 FC wee dscussed befoe qute actvel: Bahde T.// Fe Coodnates of an bseve Movn n a Ccle n Mnkowsk Sace: Aaent Behavo of Clocks D. Klen, P. Collas//Eact Fe coodnates fo a class of sace-tes Ashb N., Betott B. elatvstc Effects n Local Inetal Faes // Phs. ev. D Vol. 34, 8. - P Mazln K. Fe coodnates fo a weak avtatonal felds // Phs. ev. D vol.5, N, - Р Александров А., Жданов В., ПарновскийС. Релятивистская система отсчёта в околоземном пространстве и радиоинтерферомерическиенаблюдения // КФНТ Т.6,. -С.? Александров А., Жданов В., ПарновскийС. Релятивистские поправки в системах отсчёта локального наблюдателя при радиоинтерферометрии в космосе: Препр. / АН СССР, ин-т прикладной астрономии,. -Л.:99. Fukusha T. The Fe coodnates fo weak avtatonal felds // Cel. Mech Vol P Fukusha T., Fujoto M., Knoshta H., Aok S. Coodnate sstes n the eneal elatvstc faewok // "elatvt n celestal echancs and astoet: Hh ecson dnacal theoes and obsevatonal vefcatons". - Lennad SS Poceedns of the Sosu. - P ЖдановВ., Александров А. Координаты Ферми и радиоинтерферометрическиенаблюдения // Вестник КУ. Астрономия вып. 3. -Киев, Лыбидь. -С.4-8. C ae studed less ntensvel: Александров О., ЖдановВ. До теорії релятивістських систем, побудованих на основі оптичних координат // Вісник КУ, Фіз.-мат С.6-.

4 х с - obseve s woldlne fathe onefeence cuve, s hs oe te; х,s eodescs lne fo on to a souce wth natual aaete s х,х с, and v α - tanent vecto of ths eodescs; α set of efeence vectos of the ntal efeence sste vecto nube s n backets e α u α Then eannan noal v coodnates NC, adoted to the efeence set ae: у v s. e α е α efeence vectos set of s the obseve aallel tansoted wth h, zeo th one concdes wth the obseve s veloct: c d c d u e Aleande I Nesteov 999 Class. Quantu Gav eann noal coodnates, Fe efeence sste and the eodesc devaton equaton

5 efeence sets of obseve and HC ae connected wth each othe va Loentz tansfoaton at Λ α : Λ α е α. Ths set s tanslated aallel alon the obseve s woldlne: D e Hee the 4 otaton tenso Ω Ω α -а α u а u α -ε α u α ω, а α - 4- acceleaton, ω - anula 4-veloct of obseve s otaton. ε e ε

6 Tansfoaton foula can be found nteatn the equaton of the efeence eodescs: d ds Γ d ds d ds hee Γ α, Cstoffel s sbols, and d v ds s Addtonall, fo C: Geodescs s sotoc: v α v α, and fo FC Geodescs s sace-lke and othoonal to the obseves woldlne: u α v α. Then FC zα ae: z к e к, z, And ОК α ae: α e α,.

7 - etc tenso n the ntal coodnatesх αη αhα whee h ae sall n coason to. Then Γ ρ, ρ, ρ,, ρ h ρ, hρ, h, ρ η ρ γ γ γ Inteaton ve us the tansfoaton foula n lnea aoaton on etc c whee: σ σ f h α -h α c η I J, σ σ η I I J s s s f f s' s' ds ' s' ds'

8 FC : etc tenso Γ u ~ D, F F z j F j z z κ κ. σ α α α σ σ α α α σ σ σ σ σ η η η η,, I J I J I I u ε ε Ω D

9 F c γ u u c γ Ω u Σ c ρ γ Ω Ω Σ ρ, Σ α s s ds' v ds' s' α v s' ds" α s" s' v v s" ds" s" α Hee αχδ - eannan tenso σ ρ σ I α, I, α J ασ, J σ, α J σρ σρ, α Σα f α I α I { } α K J f s { } J α α α α Γ,, α, ~

10 K α s ds' s s' γ ς f s' F, v ' α α, ς c s Takn nto account that: z e F { γ α γ Σ u Σ Ω } e α c α α c F j δ α j Σαe e

11 C: etc tansfoaton: j j κ κ

12 α c α α Metc tenso n C: γ c Σ Ω e e u u j j δj e e j Σ j j j Σ u u Σ u e u e γ α Σ c γ u u c γ Ω u Σ c ρ γ Ω Ω Σ ρ

13 An eale: ovn ont asses Intal etc: k δ ϕ k ϕ 4π ϕ whee π і s a veloct, - ass And avtatonal otental s: ϕ

14 Tansfoaton foula: 4 4 D c π ϖ c π ln v d d D π ϖ π v v v d

15 C: n n u u oj j o j j j v v v v F Σ Σ Σ σ d F Σ D π π Σ π Σ 4 3

16 E D C B z Ae π γ γ 3 A π 3 A B π 3 D d C 3 D d E

Σχετικά έγγραφα

Curvilinear Systems of Coordinates

Curvilinear Systems of Coordinates A Cuvilinea Systems of Coodinates A.1 Geneal Fomulas Given a nonlinea tansfomation between Catesian coodinates x i, i 1,..., 3 and geneal cuvilinea coodinates u j, j 1,..., 3, x i x i (u j ), we intoduce