Surfaces of Revolution in the General Theory of Relativity

Taxiarchis Papakostas

Department of Electrical Engineering, Technological Educational Institute of Crete, Heraklion, Greece.

DOI: 10.4236/jmp.2015.614206                

Abstract

We present a class of axially symmetric and stationary spaces foliated by a congruence of surfaces of revolution. The class of solutions considered is that of Carter’s family [A] of spaces and we try to find a solution to Einstein’s equations in the presence of a perfect fluid with heat flux. This approach is an attempt to find an interior solution that could be matched to a corresponding exterior solution across a surface of zero hydrostatic pressure. The presence of a congruence of surfaces of revolution, described as the quotient space of the commoving observers, can be useful to the determination of the surface of zero pressure. Finally we present two formal solutions representing ellipsoids of revolutions.

Keywords

General Relativity, Exact Solutions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Papakostas, T. (2002) Anisotropic Fluids and Rotation in GR, in Recent Developments in Gravity. Proceedings of the Hellenic Relativity Conference, World Scientific, 128.
[2]	Papakostas, T. (2005) Journal of Physics, Conference Series, 8, 13. http://dx.doi.org/10.1088/1742-6596/8/1/002
[3]	Papakostas, T. (2009) Journal of Physics, Conference Series, 189, Article ID: 012027. http://dx.doi.org/10.1088/1742-6596/189/1/012027
[4]	Papakostas, T. (2005) Rotating Figures of Equilibrium in GR. arxiv.org/pdf/gr-qc/0511078.
[5]	Chandrasekhar, S. (1987) Ellipsoidal Figures of Equilibrium, Dover.
[6]	Geroch, R. and Lindblom, L. (1990) Physical Review D, 41, 1855. http://dx.doi.org/10.1103/PhysRevD.41.1855
[7]	Krasinski, A. (1978) Annals of Physics, 112, 22-40. http://dx.doi.org/10.1016/0003-4916(78)90079-9
[8]	Plebanski, J. and Krasinski, A. (2006) An Introduction to General Relativity and Cosmology. Cambridge University Press, Cambridge, 490. http://dx.doi.org/10.1017/cbo9780511617676
[9]	Racz, I. (1992) Classical and Quantum Gravity, 9, L93-L98. http://dx.doi.org/10.1088/0264-9381/9/8/004
[10]	Zsigrai, I. (2003) Classical and Quantum Gravity, 20, 2855. http://dx.doi.org/10.1088/0264-9381/20/13/330
[11]	Carter, B. (1968) Communications in Mathematical Physics, 10, 280-310.
[12]	Debever, R. (1971) Bulletin de la Société Mathématique de Belgique, 23, 360-371.
[13]	Carter, B. and McLenaghan, R. (1979) Physical Review D, 19, 1093-1097. http://dx.doi.org/10.1103/PhysRevD.19.1093
[14]	Papakostas, T. (1988) Journal of Mathematical Physics, 29, 1445. http://dx.doi.org/10.1063/1.527939
[15]	Papakostas, T. (1998) International Journal of Modern Physics D, 7, 927-941. http://dx.doi.org/10.1142/S0218271898000619
[16]	Hauser, H. and Malhiot, R. (1976) Journal of Mathematical Physics, 17, 1306. http://dx.doi.org/10.1063/1.523058
[17]	Debever, R. (1964) Cahiers de Physique, 18, 303.
[18]	Debever, R., McLenaghan, R.G. and Tariq, N. (1979) General Relativity and Gravitation, 10, 853-879. http://dx.doi.org/10.1007/BF00756664
[19]	Debever, R., et al. (1981) General Relativity and Gravitation, 22, 1711.
[20]	Wahlquist, H.D. (1968) Physical Review, 172, 1291.