Global Solutions in Gravity. Lorentzian Signature

M. O. Katanaev

Informatics and Automation, Problems of the modern mathematical physics, Tome 228 (2000), pp. 168-195.

Résumé

A constructive method of conformal blocks is developed for the construction of global solutions for two-dimensional metrics having one Killing vector. The method is proved to yield a smooth universal covering space with a smooth pseudo-Riemannian metric. The Schwarzschild, Reisner–Nordström solutions, extremal black hole, dilaton black hole, and constant curvature surfaces are considered as examples.

Bibliographie

[1] Kruskal M. D., “Maximal extension of Schwarzschild metric”, Phys. Rev., 119:5 (1960), 1743–1745 | DOI | MR | Zbl

[2] Szekeres G., “On the singularities of a Riemannian manifold”, Publ. Math. Debrecen, 7:1–4 (1960), 285–301 | MR | Zbl

[3] Schwarzschild K., “Über das Gravitationssfeldeines Massenpunktes nach der Einsteinschen Theorie”, Sitzungsber. Akad. Wiss. Berlin, 1916, 189–196 | Zbl

[4] Bogolyubov N. N., “Pro deyaki ergodichni vlastivosti sutsilnikh grup peretvoren”, Nauk. zap. KDU im. T. G. Shevchenka. Fiz.-mat. zb., 4:5 (1939), 45–52

[5] Carter B., “Black hole equilibrium states”, Black holes, eds. C. DeWitt, B. C. DeWitt, Gordon and Breach, N. Y., 1973, 58–214 | MR

[6] Katanaev M. O., “All universal coverings of two-dimensional gravity with torsion”, J. Math. Phys., 34:2 (1993), 700–736 | DOI | MR | Zbl

[7] Katanaev M. O., Volovich I. V., “String model with dynamical geometry and torsion”, Phys. Lett. B, 175:4 (1986), 413–416 | DOI | MR

[8] Klösch T., Strobl T., “Classical and quantum gravity in $1+1$ dimensions. II: The universal coverings”, Class. Quant. Gravity, 13 (1996), 2395–2421 | DOI | MR | Zbl

[9] Katanaev M. O., Kummer W., Liebl H., “Geometric interpretation and classification of global solutions in generalized dilaton gravity”, Phys. Rev. D, 53:10 (1996), 5609–5618 | DOI | MR

[10] Katanaev M. O., Kummer W., Liebl H., “On the completeness of the black hole singularity in $2$d dilaton theories”, Nucl. Phys. B, 486 (1997), 353–370 | DOI | MR | Zbl

[11] Katanaev M. O., Klösch T., Kummer W., “Global properties of warped solutions in general relativity”, Ann. Phys., 276 (1999), 191–222 | DOI | MR | Zbl

[12] Klösch T., Strobl T., “Classical and quantum gravity in $1+1$ dimensions. III: Solutions of arbitrary topology”, Class. Quant. Gravity, 14 (1997), 1689–1723 | DOI | MR | Zbl

[13] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya. Metody i prilozheniya, 4-e izd., Nauka, M., 1998 | MR

[14] Landau L. D., Lifshits E. M., Teoriya polya, 7-e izd., Nauka, M., 1988 | MR

[15] Katanaev M. O., “Novaya integriruemaya model – dvumernaya gravitatsiya s dinamicheskim krucheniem”, DAN SSSR, 309:3 (1989), 591–593 | MR

[16] Katanaev M. O., “Complete integrability of two-dimensional gravity with dynamical torsion”, J. Math. Phys., 31:4 (1990), 882–891 | DOI | MR | Zbl

[17] Witten E., “String theory and black holes”, Phys. Rev. D, 44:2 (1991), 314–324 | DOI | MR | Zbl

[18] Mandal G., Sengupta A. M., Wadia S. R., “Classical solutions of two-dimensional string theory”, Mod. Phys. Lett., 6 (1991), 1685–1692 | DOI | MR | Zbl

[19] Eddington A. S., “A comparison of Whitehead's and Einstein's formulae”, Nature, 113 (1924), 192 | DOI

[20] Finkelstein D., “Past-future asymmetry of the gravitational field of a point particle”, Phys. Rev., 110:4 (1958), 965–967 | DOI | MR | Zbl