There exist different kinds of averaging of the differences of the energy–momentum and angular momentum in normal coordinates NC(P) which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [J. Garecki, Rep. Math. Phys. 33, 57 (1993); Int. J. Theor. Phys. 35, 2195 (1996); Rep. Math. Phys. 40, 485 (1997); J. Math. Phys. 40, 4035 (1999); Rep. Math. Phys. 43, 397 (1999); Rep. Math. Phys. 44, 95 (1999); Ann. Phys. (Leipzig) 11, 441 (2002); M.P. Dabrowski and J. Garecki, Class. Quantum. Grar. 19, 1 (2002)] giving the canonical superenergy and angular supermomentum tensors. In this paper we present another averaging of the differences of the energy–momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy–momentum and angular momentum densities. We have called these tensorial quantities “the averaged relative energy–momentum and angular momentum tensors”. These tensors are very closely related to the canonical superenergy and angular supermomentum tensors and they depend on some fundamental length L > 0. The averaged relative energy–momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to coordinate independent analysis (local and in special cases also global) of this field. Up to now we have applied the averaged relative energy–momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general, only homogeneous) universes. The obtained results are interesting, e.g., the averaged relative energy density is positive definite for the all Friedman and other universes which have been considered in this paper.
Similar content being viewed by others
References
Garecki J. (1993) . Rep. Math. Phys. 33, 57
Garecki J. (1996) Int. J. Theor. Phys. 35: 2195
Garecki J. (1997). Rep. Math. Phys. 40, 485
Garecki J. (1999) . J. Math. Phys. 40: 4035
Garecki J. (1999) . Rep. Math. Phys. 43, 397
Garecki J. (1999) . Rep. Math. Phys. 44, 95
Garecki J. (2002) . Ann. Phys. (Leipzig) 11, 441
Da̧browski M.P., Garecki J. (2002) . Class. Quantum. Grav. 19, 1
Mashhoon B. et al. (1997) . Phys. Lett. A 231, 47
Mashhoon B. et al. (1999) . Class. Quant. Grav. 16: 1137
B. Mashhoon, in Proc. Spanish Relativity Meeting, J.F. Pascual, S. L. Floría, A. San Miguel, and F. Vincente eds. (World Scientific, Singapore, 2001) (preprint: gr-qc/0011014).
B. Mashhoon, preprint: gr-qc/0311030.
Garecki J. (1978). Zeszyty Naukowe WSP Szczecin 26: 25 (in Polish, summary in English).
Bergmann P.G., Thomson R. (1953) . Phys. Rev. 89, 401
Garecki J. (2001) . Grav. Cosmol. 7, 1
Landau L.D., Lifschitz E.M. (1975) . The Classical Theory of Fields. Pergamon, Oxford
Szabados L.B. (1992) . Class. Quant. Grav. 9: 2521
Szabados L.B. (2004) . Living Rev. Rel. 7, 4
A. Ashtekar, preprint: gr-qc/9901023.
A. Ashtekar, preprint: gr-qc/0112038.
Muxin Han et al., preprint: gr-qc/0509064.
Rovelli C. (2004) . Quantum Gravity. Cambridge University Press, Cambridge
Thiemann T (2003) Lect. Notes Phys. 631, 41 (preprint: gr-qc/0210094).
A. Ashtekar, preprint: math-ph/0202008.
Bojowald M. (2005) Living Rev. Rel. 8, 11 (preprint: gr-qc/0601085).
Ashtekar A., Lewandowski J. (2004) Class. Quant. Grav. 21, R53 (preprint: gr-qc/0404018).
L. Smolin, preprint: hep-th/0303185.
Ashtekar A. (2005) New J. Phys. 7, 198 (preprint: gr-qc/0410054).
A. Perez, preprint: gr-qc/0409061.
Albrow M.G. (1973) . Nature 241, 56
Tryon E.P. (1973) . Nature 246, 396
Cooperstock F.I. (1992) . Found. Phys. 22: 1011
Cooperstock F.I. (2000) . Ann. Phys.(N.Y.) 282, 115
S. S. Xulu, preprint: hep-th/0308077.
Bringley T. (2002) Mod. Phys. Lett. A 17, 157 (preprint: gr-qc/0204006).
Gad R.M. (2005) Astrophys. Space Sci. 95, 451 (preprint: gr-qc/0307010).
Bonnor W.B. (2000) . Gen. Relat. Gravit. 32: 1627
Garecki J. (2005) Class. Quant. Grav. 22: 4051 (preprint: gr-qc/0410013).
Misner Ch.W., Thorne K.S., Wheeler J.A. (1973) . Gravitation. Freeman, San Francisco
Rosen N. (1994) . Gen. Relat. Gravit. 26, 319
Johri V.B. et al. (1995) . Gen. Relat. Gravit. 27, 313
Banerjee N., Sen S. (1997) . Pramana J. Phys. 49, 609
S. S. Xulu, preprint: hep-th/0308070.
Salti M., Ali Havare (2005) Int. J. Mod. Phys. A 20: 2169 (preprint: gr-qc/0502060).
Salti M. et al. (2005) Astrophys. Space Sci. 299, 227 (preprint: gr-qc/0505079).
Salti M. (2005) Mod. Phys. Lett. A 20: 2175 (preprint: gr-qc/0505078).
Salti M. (2005) Nuovo Cim. B 120, 53 (preprint: gr-qc/0506061).
Salti M. (2006) Czech. J. Phys. 56, 177 (preprint: gr-qc/0511095).
O. Aydogdu,“Gravitational energy–momentum Density in Bianchi Type II Spacetimes”, accepted for publication in Int. J. Mod. Phys. D (preprint: gr-qc/0509047).
Aydogdu O. (2006) Fortsch. Phys. 54, 246 (preprint: gr-qc/0602070).
A. Trautman, in Gravitation: An Introduction to Current Problems, L. Witten, ed. (Wiley New York-London, 1962).
J. Goldberg, in General Relativity and Gravitation, A. Held, ed. (Plenum Press, New York, 1980).
Komar A. (1959) . Phys. Rev. 113, 934
Komar A. (1962) . Phys. Rev. 127: 1411
Garecki J. (1995) . Gen Relat. Gravit. 27, 55
Katz J. et al. (1997) Phys. Rev. D 55: 5957 (preprint: gr-qc/0504041).
M. Salti et al., preprint: gr-qc/0502042.
P. Halpern, preprint: gr-qc/0606095.
Radinschi I. (2000) . Fizika B (Zagreb) 9, 203
O. Aydogdu et al., preprint: gr-qc/0601133.
M. S. Berman, preprint: gr-qc/0605063.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garecki, J. The Tensors of the Averaged Relative Energy–Momentum and Angular Momentum in General Relativity and Some of Their Applications. Found Phys 37, 341–365 (2007). https://doi.org/10.1007/s10701-007-9107-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-007-9107-y
Keywords
- energy
- energy–momentum complexes
- superenergy tensors
- averaged relative energy–momentum tensors
- friedman universes
- homogeneous universes