Summary
Bondi’s Newtonian inductive energy transfer by gravitation is rederived and new forms are presented to exhibit the lack of uniqueness of the Newtonian gravitational Poynting vector. Einstein’s power transfer theory is employed in a straightforward manner to deduce the energy transfer for weak fields in general relativity. The relativistic and classical expressions, developed in terms of the multipole moments of the matter distributions participating in the interaction, coincide in the limit asc→∞.
Riassunto
Si deduce nuovamente il trasferimento induttivo di energia newtoniana di Bondi e si presentano nuove forme per mettere in evidenza la mancanza di unicità del vettore di Poynting gravitazionale newtoniano. Si impiega in modo diretto la teoria del trasferimento di potenza di Einstein per dedurre il trasferimento di energie per campi deboli nella relatività generale. Le espressioni relativistiche e classiche, sviluppate in termini dei momenti di multipolo della distribuzione della materia che partecipa all'interazione, coincidono nel limitec→∞.
Пезюме
Заново выводится выражение Бонди для для ньютоновского переноса индуктивной энергии благодаря гравитации. Предлагаьтся новые формы, чтобы показать отсутствие единственности ньютоновского гравитационного вектора Пойнтинга. Теория переноса энергии Эйнштейна непосредственно применяется для получения переноса энергии из-за слабых полей в общей теории относительности. Релятивистское и классическое выражения, определенные в терминах мультипольных моментов раслределений вещества, участвующих во взаимодействии, совпадают в пределе, когдаc→∞.
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References
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ϱ is the mass density,v is the velocity and a dot denotes differentiation with respect to time.
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F. I. Cooperstock andD. J. Booth:Phys. Rev.,187, 1796 (1969).
T ikis the energy-momentum tensor,R is the source-point-to-field-point separation,\(\xi _1 ^a \) defines the source point. Latin indices range from 0 to 3, Greek indices range from 1 to 3 apart from ε which always assumes the values 1, 2. A subscript immediately to the right of a symbol indicates the system to which it refers. A comma denotes partial differentiation. Thus Ψ 1 ∝β,0≡ ∂ Ψ 1 ∝β,/∂(ct)and\(\Psi _1 ^{ \propto \beta } \) are the spatial components of the field generated by system 1.
They may be found in ref. (4).
C. Møller: {jtMat. Fys. Medd. Dan. Vid. Selsk.}, {vn35}, {snNo. 3} ({dy1966});A. Trautman: inGravitation: An Introduction to Current Research, edited byL. Witten (New York, 1962), p. 169.
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Supported, in part, by the National Research Council of Canada, Grant A5340.
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Cooperstock, F.I., Booth, D.J. General-relativistic and Newtonian gravitational energy transfer. Nuov Cim B 2, 139–147 (1971). https://doi.org/10.1007/BF02722238
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DOI: https://doi.org/10.1007/BF02722238