On the second order relativistic deviation equation and its applications
Kuupäev
2010-07-30
Autorid
Ajakirja pealkiri
Ajakirja ISSN
Köite pealkiri
Kirjastaja
Abstrakt
Maailmafunktsiooni mõistet kasutades on tuletatud geodeetilise hälbe täpne üldistatud võrrand, mis kirjeldab kahe kiirendatava punktmassi suhtelist liikumist kõveras aegruumis nende osakeste meelevaldsete kiirenduste, kiiruste ja maailmajoonte suvaliste parametrisatsioonide korral. Võrrandi teist järku lähend rittaarenduses hälbevektori komponentide suhtes on esitatud nii üldistes koordinaatides kui ka ühe punktmassiga kaasaliikuva vaatleja Fermi koordinaatides. Kiireneva vaatleja Fermi koordinaatides on võrrandit põhjalikumalt uuritud kahel erijuhul – radiaalsihis kiirenev vaatleja Schwarzschildi väljas ja Weberi ostsillaator nõrga monokromaatilise gravitatsioonilise tasalaine väljas. Schwarzschildi väljas on arvutatud prooviosakese vaba langemise kiirendus radiaalsihis kiireneva vaatlejaga seotud taustsüsteemis. Vaba langemise kiirenduse Taylori reaksarenduses sisaldavad teist järku liikmed ka vaatleja 3-kiirust tsentraalkeha kui gravitatsioonivälja allika suhtes ning vaatleja kiiruse lähenemine valguse kiirusele toob kaasa nende liimete piiramatu kasvu. Seega muutuvad vaatleja relativistlikul liikumisel teist järku liikmed võrreldavateks esimest järku liikmetega ja nende mõju ei saa jätta arvestamata, nii nagu mitterelativistliku suhtelise kiirusega liikuva vaatleja korral, kus tähtsust omavad vaid esimest järku liikmed, mis teatavasti ei sõltu vaatleja kiirusest tsentraalkeha suhtes. Tähelepanu ongi pööratud teist järku liikmetega kaasnevaile uut tüüpi relativistlikele efektidele, mida esimest järku võrrand ei kajasta. Parema ülevaate saamiseks uuritakse liikumisesuunalisi loodejõududest tingitud efekte ka kõrgemates lähendustes, kasutades Mouldi lõõtsvarda kontseptsiooni.
By means of the concept of the world function the exact generalized deviation equation is derived, describing the relative motion of two accelerated point masses in curved spacetimes. The obtained equation is valid in case of arbitrary velocities and accelerations of the point masses and also of arbitrary parametrization of their world lines. The Taylor expansion of the deviation equation with respect to the components of the deviation vector is presented in general coordinates and in the Fermi coordinates of an accelerated observer. The second approximation of the relativistic deviation equation is examined in two cases – the radially accelerated observer in the Schwarzschild spacetime and the Weber oscillator in the field of a weak monochromatic plane gravitational wave. Using the second order deviation equation in the Schwarzschild spacetime, the acceleration of free fall of a test particle is calculated in the Fermi coordinates of an observer, accelerated in the radial direction. The second order terms in the Taylor expansion of the acceleration of free fall contain also the observer’s 3-velocity relative to the source of the gravitational field. If the observer’s 3-velocity approaches to the velocity of light, it causes the unlimited growth of the abovementioned second order terms. Ergo, if the observer’s motion is relativistic, the second order terms become comparable with the first order ones and their contribution can not be neglected as in the case of an observer moving at the nonrelativistic 3-velocity. The main attention has been paid to the relativistic effects, “hidden” in the second order terms and non predictable by means of the first order equation. For better understanding the relativistic tidal effects, caused by the observer’s motion, the longitudinal effects in the wave field are also examined in the higher approximations, using the Mould concept of the accordion rod.
By means of the concept of the world function the exact generalized deviation equation is derived, describing the relative motion of two accelerated point masses in curved spacetimes. The obtained equation is valid in case of arbitrary velocities and accelerations of the point masses and also of arbitrary parametrization of their world lines. The Taylor expansion of the deviation equation with respect to the components of the deviation vector is presented in general coordinates and in the Fermi coordinates of an accelerated observer. The second approximation of the relativistic deviation equation is examined in two cases – the radially accelerated observer in the Schwarzschild spacetime and the Weber oscillator in the field of a weak monochromatic plane gravitational wave. Using the second order deviation equation in the Schwarzschild spacetime, the acceleration of free fall of a test particle is calculated in the Fermi coordinates of an observer, accelerated in the radial direction. The second order terms in the Taylor expansion of the acceleration of free fall contain also the observer’s 3-velocity relative to the source of the gravitational field. If the observer’s 3-velocity approaches to the velocity of light, it causes the unlimited growth of the abovementioned second order terms. Ergo, if the observer’s motion is relativistic, the second order terms become comparable with the first order ones and their contribution can not be neglected as in the case of an observer moving at the nonrelativistic 3-velocity. The main attention has been paid to the relativistic effects, “hidden” in the second order terms and non predictable by means of the first order equation. For better understanding the relativistic tidal effects, caused by the observer’s motion, the longitudinal effects in the wave field are also examined in the higher approximations, using the Mould concept of the accordion rod.
Kirjeldus
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Märksõnad
doktoritööd, üldrelatiivsusteooria, geodeetiline hälve