Theoretical Physics

Download PDF (1418.2 KB) PP. 9 - 17 Pub. Date: June 30, 2021

DOI: 10.22606/tp.2021.62001

• Miguel Socolovsky^*
  Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Cd. Universitaria, 04510, Ciudad de México, México

Using the Raychaudhuri equation, we show that a quantum probability amplitude (Feynman propagator) can be univocally associated to any timelike or null affinely parametrized geodesic congruence.

Raychaudhuri equation, geodesic congruence, Feynman propagator, Schwarzschild metric

[1] Raychaudhuri, A. Relativistic Cosmology. I, Physical Review 98, 1123-1126, (1955).

[2] Kar, S. and Sengupta, S. The Raychaudhuri equations: A brief review, Pramana 69, 49-76, (2007).

[3] Tipler, F.J. General Relativity and Conjugate Ordinary Differential Equations, Journal of Differential Equations 30, 165-174, (1978).

[4] Teschl, G. Ordinary Differential Equations and Dynamical Systems, Providence: American Mathematical Society, (2012).

[5] Feynman, R.P. and Hibbs, A.R. Quantum Mechanics and Path Integrals, McGraw-Hill, N.Y., (1965).

[6] Schulman, L.S. Techniques and Applications of Path Integration, Wiley, N.Y., (1981).

[7] Kruskal, M.D. Maximal extension of Schwarzschild metric, Physical Review 119, 1743-1745, (1960).

[8] Szekeres, G. On the singularities of a Riemannian manifold, Publ. Math. Debrecen 7, 285-301, (1960).

[9] Ehlers, J. The Nature and Structure of Spacetime, in The Physicist’s Conception of Nature, J. Mehra (ed.), D. Reidel Pub. Co., Dordrecht-Holland, 71-91, (1973); Ehlers, J., Pirani, F.A.E., and Schild, A., The geometry of free fall and light propagation, republication: General Relativity and Gravitation 44, 1587-1609, (2012).

[10] Rovelli, C. Quantum Gravity, Cambridge University Press, pbk. ed. (2008).

[11] álvarez, E. Windows on Quantum Gravity, Fortschritte der Physik 69, 2000080, 1-29, (2021).

[12] Baez, J.C. and Bunn, E.F. The meaning of Einstein equation, American Journal of Physics 73, 644-652, (2005).

[13] Gradshtein, I.S. and Ryzhik, I.M. Table of Integrals, Series, and Products, Academic Press, (1980).

[14] Arfken, G.B. and Weber, H.J. Mathematical Methods for Physicists, Academic Press, San Diego, (2001).

[15] Poisson, E. A Relativistic Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge University Press, (2004).

[16] Socolovsky, M. Eikonal Equations for Null Radial Geodesics in the Schwarzschild Metric, Theoretical Physics 5, 41-49, (2020).

[17] Modesto, L. Disappearance of the black hole singularity in loop quantum gravity, Physical Review D 70, 124009, 1-5, (2004).

[18] Senovilla, J.M.M. and Garfinkle, D. The 1965 Penrose singularity theorem, Classical and Quantum Gravity 32, 124008, 1-45, (2015).