Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as the unique solution, which essentially is the statement of the well known Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo-Riemanian space-times provide more isometries than expected from the original metric holonomy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the corresponding Differential Equation. Additionally, applying the Noether Point Symmetry method we have obtained the conserved quantities corresponding to the generators of the Schwarzschild Lagrangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.
References
[1]
Hawking, S.W. and Ellis, G.F.R. (1975) Large Scale Structures of Space-Time (Cambridge Monographs of Mathematical Physics). Cambridge University Press, Cambridge.
[2]
Maciel, A., Le Dellion, M. and Mimoso, J.P. (2018) Revisiting Birkhoff Theorem from a Dual Null Point of View. Physical Review D, 98, Article ID: 024016. https://doi.org/10.1103/PhysRevD.98.024016
[3]
Hobson, G.E.M.P. and Lasenby, A.N. (2006) General Relativity: An Introduction to Physicists.
[4]
Birkhoff, G.D. (1923) Relativity and Modern Physics. Harvard University Press, Cambridge, 255.
[5]
Goenner, H. (1970) Einstein’s Tensor and Generalization of Birkhoff Theorem. Communications in Mathematical Physics, 16, 34-47. https://doi.org/10.1007/BF01645493
[6]
Schmidt, H.J. (1997) A New Proof of Birkhoff Theorem. Gravitation and Cosmology, 3, 185-190.
[7]
Schmidt, H.J. (2013) The Tetralogy of Birkhoff Theorems. General Relativity and Gravitation, 45, 395-410. https://doi.org/10.1007/s10714-012-1478-5
[8]
Rindler, W. (1998) Birkhoff’s Theorem with Λ-Term and Bertrotti-Kasuer Space. Physics Letters A, 245, 363. https://doi.org/10.1016/S0375-9601(98)00428-9
[9]
Bojowald, M., Kartrup, H., Schramn, F. and Strobl, T. (2000) Group Theoretical Quantization of Phase Space and the Mass Spectrum of Schwarzschild Black Hole in D Dimension. Physical Review D, 62, Article ID: 044026. https://doi.org/10.1103/PhysRevD.62.044026
[10]
Havas, P. (1977) Theories of Gravitation with Higher Order Field Equations. General Relativity and Gravitation, 8, 631-645. https://doi.org/10.1007/BF00756315
[11]
Schmidt, H.J. (1991) Scale-Invariant Gravity in Two Dimensions. Journal of Mathematical Physics, 32, 1562. https://doi.org/10.1063/1.529267
[12]
Mignemi, S. and Schmidt, H.J. (1995) Two Dimensional Higher Derivative Gravity and Conformal Transformation. Classical and Quantum Gravity, 12, 849-858. https://doi.org/10.1088/0264-9381/12/3/021
[13]
Bona, C. (1988) A New Proof of the Generalized Birkhoff Theorem. Journal of Mathematical Physics, 29, 1440-1441. https://doi.org/10.1063/1.527936
[14]
Bronnikov, K. and Melnikov, V.N. (1995) The Birkhoff Theorem for Multidimensional Gravity. General Relativity and Gravitation, 27, 465-474. https://doi.org/10.1007/BF02105073
[15]
Ellis, G.F.R. and Goswami, R. (2012) Birkhoff Theorem and Matter. General Relativity and Gravitation, 44, 2037-2050. https://doi.org/10.1007/s10714-012-1376-x
[16]
Ellis, G.F.R. and Goswami, R. (2011) Almost Birkhoff Theorem in General Relativity. General Relativity and Gravitation, 43, 2157-2170. https://doi.org/10.1007/s10714-011-1172-z
[17]
Ovsyannikov, L.V. (1982) Group Analysis of Differential Equations. Academic Press, New York.
[18]
Stephani, H. and McCallum, M. (1989) Differential Equations: Their Solution Using Symmetries. Cambridge University Press, Cambridge.
[19]
Olver, P.J. (2000) Applications of Lie Groups to Differential Equations (Graduate Text in Mathematics). 2nd Edition, Springer, Berlin.
[20]
Noether, E. (1918) Invariant Variations Problems. Nachrichtea der Akademie der Wissenschaften in Göttingen, Mathematisch—Physikalische Klasse, 2, 235-257. (English Translation in Transport Theory and Statistical Physics, 1(3), 186-207 (1991)) https://doi.org/10.1080/00411457108231446
[21]
Carroll, S.M. (2003) An Introduction to General Relativity Spacetime and Geometry. Pearson Publication, London.
