Theoretical Physics

Download PDF (795.1 KB) PP. 109 - 114 Pub. Date: September 19, 2017

DOI: 10.22606/tp.2017.23001

• H. A. Camargo
  Facultad de Ciencias, Universidad Nacional Autónoma de México, México
• E. J. Gonzalez de Urreta
  
  Instituto de Ciencias, Universidad Nacional de General Sarmiento; Posdoctoral fellowship, CONICET, Argentina
• M. Socolovsky^*
  
  Instituto de Ciencias, Universidad Nacional de General Sarmiento, Argentina; With a leave of abscence from Instituto de Ciencias Nucleares, UNAM, México

Using Weyl geometry, we study some properties of the scalar Φ-field representing the conformally flat part of the Schwarzschild-Kruskal-Szekeres (S-K-S) metric in the black hole region. As expected, the classical field diverges (logarithmically) at the singularity, where it is infinitely heavy, but remains finite at the horizons, where it becomes massless. The quantum propagator in the radial direction is also computed, exhibiting rapid oscillations and a divergent absolute value at the singularity.

Schwarzschild black hole, scalar field, quantization.

[1] H. Weyl, Gravitation und Elektrizit¨at, Sitzungsber. d. Preuss-Akad. d. Wissensch. 465 (1918). English translation: L.O’Reifearteigh, The Dawning of Gauge Theory (Princeton Univ. Press, Princeton, 1997), 24-37.

[2] C. Romero, J.B. Fonseca-Neto, and M.L. Pucheu, Conformally flat spacetimes and Weyl frames, Found. Phys. 42, 224 (2012); arXiv: 1101.5333v1 [gr-qc].

[3] H.A. Camargo, E.J. Gonzalez de Urreta, and M. Socolovsky, Weyl geometry, anti-de Sitter space, and Φ⁴-theory, Eur. Phys. J. Plus 131, 172 (2016); arXiv: 1511.04403v1 [phys. gen-ph].

[4] K. Schwarzschild, Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzber. Deut. Akad. Wiss. Berlin, Kl. Math.-Phys. Tech. 189-196 (1916). English translation: On the Gravitational Field of a Mass Point According to Einstein Theory, Gen. Rel. Grav. 35, 951-959 (2003).

[5] M.D. Kruskal, Maximal Extension of Schwarzschild Metric, Phys. Rev. 119, 1743-1745 (1960).

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

[7] R.P. Feynman, and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

[8] L.S. Schulman, Techniques and Applications of Path Integration (J. Wiley, New York, 1981), pp. 31-41.

[9] A. Fabbri, and J. Navarro-Salas, Modeling Black Hole Evaporation (Imperial College Press, London, 2005), pp. 32-39.

[10] A.E. Sikkema, and R.B. Mann, Gravitation and cosmology in (1+1) dimensions, Class. Quantum Grav. 8, 219-235 (1991).