The Mathematical Foundations of General Relativity Revisited

ABSTRACT

The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. **1) VESSIOT VERSUS CARTAN****:** The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature + torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). **2) JANET VERSUS SPENCER****:** The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. **3) ALGEBRA VERSUS GEOMETRY****:** Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of electromagnetism and gravitation must be revisited within this formal framework, though striking it may look like. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.

Cite this paper

J. Pommaret, "The Mathematical Foundations of General Relativity Revisited,"*Journal of Modern Physics*, Vol. 4 No. 8, 2013, pp. 223-239. doi: 10.4236/jmp.2013.48A022.

J. Pommaret, "The Mathematical Foundations of General Relativity Revisited,"

References

[1] J.-F. Pommaret, “Partial Differential Control Theory,” Kluwer, Dordrecht, 2001. doi:10.1007/978-94-010-0854-9

[2] E. Cartan, Annales Scientifiques de l’école Normale Supérieure, Vol. 21, 1904, pp. 153-206.

[3] E. Cartan, Annales Scientifiques de l’école Normale Supérieure, Vol. 40, 1923, pp. 325-412.

[4] H. Goldschmidt, Journal of Differential Geometry, Vol. 6, 1972, pp. 357-373,

[5] A. Kumpera and D. C. Spencer, “Lie Equations,” Annals of Mathematics Studies 73, Princeton University Press, Princeton, 1972.

[6] J.-F. Pommaret, “Differential Galois Theory,” Gordon and Breach, New York, 1983.

[7] J.-F. Pommaret, “Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics,” In: Y. Gan, Ed., Continuum Mechanics-Progress in Fundamentals and Engineering Applications, InTech, 2012. http://www.intechopen.com/books/continuum-mechanics-progress-in-fundamentals-and-engineering-applications/spencer-operator-and-applications-from-continuum-mechanics-to-mathematical-physics

[8] J.-F. Pommaret, “Partial Differential Equations and Group Theory,” Kluwer, 1994. doi:10.1007/978-94-017-2539-2

[9] E. Vessiot, Annales scientifiques de l’école Normale Supérieure, Vol. 20, 1903, pp. 411-451.

[10] V. Ougarov, “Théorie de la Relativité Restreinte,” MIR, Moscow, 1969.

[11] L. P. Eisenhart, “Riemannian Geometry,” Princeton University Press, Princeton, 1926.

[12] J.-F. Pommaret, “Lie Pseudogroups and Mechanics,” Gordon and Breach, New York, 1988.

[13] A. Lorenz, “Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method,” Acta Applicandae Mathematicae, Vol. 1, 2008, pp. 205-213. doi:10.1007/s10440-008-9193-7

[14] D. C. Spencer, Bulletin of the American Mathematical Society, Vol. 75, 1965, pp. 1-114.

[15] M. Janet, Journal de Mathématiques Pures et Appliquées, Vol. 8, 1920, pp. 65-151.

[16] H. Weyl, “Space, Time, Matter,” Berlin, 1918.

[17] E. Cosserat and F. Cosserat, “Théorie des Corps Déformables,” Hermann, Paris, 1909.

[18] J.-F. Pommaret, Acta Mechanica, Vol. 149, 2001, pp. 23-39. doi:10.1007/BF01261661

[19] J.-F. Pommaret, Acta Mechanica, Vol. 215, 2010, pp. 43-55. doi:10.1007/s00707-010-0292-y

[20] J. J. Rotman, “An Introduction to Homological Algebra,” Academic Press, Cambridge, 1979.

[21] R. E. Kalman, Y. C. Yo and K. S. Narenda, Journal of Differential Equations, Vol. 1, 1963, pp. 189-213.

[22] M. Kashiwara, Mémoires de la Société Mathématique de France, Vol. 63, 1995, pp. 1-72.

[23] V. P. Palamodov, “Linear Differential Operators with Constant Coefficients,” Grundlehren der Mathematischen Wissenschaften 168, Springer, 1970. doi:10.1007/978-3-642-46219-1

[24] J.-F. Pommaret, “Algebraic Analysis of Control Systems Defined by Partial Differential Equations,” Advanced Topics in Control Systems Theory, Springer, Lecture Notes in Control and Information Sciences 311, 2005, pp. 155-223.

[25] E. Kunz, “Introduction to Commutative Algebra and Algebraic Geometry,” Birkhauser, 1985.

[26] W. Pauli, “Theory of Relativity,” Pergamon Press, London, 1958.

[1] J.-F. Pommaret, “Partial Differential Control Theory,” Kluwer, Dordrecht, 2001. doi:10.1007/978-94-010-0854-9

[2] E. Cartan, Annales Scientifiques de l’école Normale Supérieure, Vol. 21, 1904, pp. 153-206.

[3] E. Cartan, Annales Scientifiques de l’école Normale Supérieure, Vol. 40, 1923, pp. 325-412.

[4] H. Goldschmidt, Journal of Differential Geometry, Vol. 6, 1972, pp. 357-373,

[5] A. Kumpera and D. C. Spencer, “Lie Equations,” Annals of Mathematics Studies 73, Princeton University Press, Princeton, 1972.

[6] J.-F. Pommaret, “Differential Galois Theory,” Gordon and Breach, New York, 1983.

[7] J.-F. Pommaret, “Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics,” In: Y. Gan, Ed., Continuum Mechanics-Progress in Fundamentals and Engineering Applications, InTech, 2012. http://www.intechopen.com/books/continuum-mechanics-progress-in-fundamentals-and-engineering-applications/spencer-operator-and-applications-from-continuum-mechanics-to-mathematical-physics

[8] J.-F. Pommaret, “Partial Differential Equations and Group Theory,” Kluwer, 1994. doi:10.1007/978-94-017-2539-2

[9] E. Vessiot, Annales scientifiques de l’école Normale Supérieure, Vol. 20, 1903, pp. 411-451.

[10] V. Ougarov, “Théorie de la Relativité Restreinte,” MIR, Moscow, 1969.

[11] L. P. Eisenhart, “Riemannian Geometry,” Princeton University Press, Princeton, 1926.

[12] J.-F. Pommaret, “Lie Pseudogroups and Mechanics,” Gordon and Breach, New York, 1988.

[13] A. Lorenz, “Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method,” Acta Applicandae Mathematicae, Vol. 1, 2008, pp. 205-213. doi:10.1007/s10440-008-9193-7

[14] D. C. Spencer, Bulletin of the American Mathematical Society, Vol. 75, 1965, pp. 1-114.

[15] M. Janet, Journal de Mathématiques Pures et Appliquées, Vol. 8, 1920, pp. 65-151.

[16] H. Weyl, “Space, Time, Matter,” Berlin, 1918.

[17] E. Cosserat and F. Cosserat, “Théorie des Corps Déformables,” Hermann, Paris, 1909.

[18] J.-F. Pommaret, Acta Mechanica, Vol. 149, 2001, pp. 23-39. doi:10.1007/BF01261661

[19] J.-F. Pommaret, Acta Mechanica, Vol. 215, 2010, pp. 43-55. doi:10.1007/s00707-010-0292-y

[20] J. J. Rotman, “An Introduction to Homological Algebra,” Academic Press, Cambridge, 1979.

[21] R. E. Kalman, Y. C. Yo and K. S. Narenda, Journal of Differential Equations, Vol. 1, 1963, pp. 189-213.

[22] M. Kashiwara, Mémoires de la Société Mathématique de France, Vol. 63, 1995, pp. 1-72.

[23] V. P. Palamodov, “Linear Differential Operators with Constant Coefficients,” Grundlehren der Mathematischen Wissenschaften 168, Springer, 1970. doi:10.1007/978-3-642-46219-1

[24] J.-F. Pommaret, “Algebraic Analysis of Control Systems Defined by Partial Differential Equations,” Advanced Topics in Control Systems Theory, Springer, Lecture Notes in Control and Information Sciences 311, 2005, pp. 155-223.

[25] E. Kunz, “Introduction to Commutative Algebra and Algebraic Geometry,” Birkhauser, 1985.

[26] W. Pauli, “Theory of Relativity,” Pergamon Press, London, 1958.