Abstract
We solve a class of boundary value problems for the stationary axisymmetric Einstein equations involving a disk rotating around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.
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References
Abramowicz, A., et al: Theory of black hole accretion discs. Edited by M. A. Abramowicz, G. Björnsson, J. E. Pringle. Cambridge: Cambridge University Press, 1999
Bardeen J.M., Wagoner R.V.: Uniformly rotating disks in general relativity. Astrophys. J. 158, L65–L69 (1969)
Bardeen J.M., Wagoner R.V.: Relativistic disks. I. Uniform rotation. Astrophys. J. 167, 359–423 (1971)
Bičák, J.: Selected solutions of Einstein’s field equations: their role in general relativity and astrophysics. In: Einstein’s field equations and their physical implications, Edited by B. G. Schmidt, Lecture Notes in Physics, Vol. 540, Berlin: Springer-Verlag, 2000, pp. 1–126
Chandrasekhar, S.: The mathematical theory of black holes. Reprint of the 1992 edition, Oxford Classic Texts in the Physical Sciences, New york: The Clarendon Press Oxford University Press, 1998
Farkas, H. M., Kra, I.: Riemann surfaces. 2nd edition, Graduate Texts in Mathematics 71, New York: Springer-Verlag, 1992
Fay J.D.: Theta functions on Riemann surfaces Lecture Notes in Mathematics 352. Berlin-New York, Springer-Verlag (1973)
Fokas A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. Roy. Soc. Lond. A 453, 1411–1443 (1997)
Fokas A.S.: Integrable nonlinear evolution equations on the half-line. Com. Math. Phys. 230, 1–39 (2002)
Fokas, A. S.: A unified approach to boundary value problems. CBMS-NSF regional conference series in applied mathematics, Philadelphia: SIAM, 2008
Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett 19, 1095–1097 (1967)
Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978
Kerr R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963)
Klein C.: Counter-rotating dust rings around a static black hole. Class. Quantum Grav 14, 2267–2280 (1997)
Klein C., Korotkin D., Shramchenko V.: Ernst equation, Fay identities and variational formulas on hyperelliptic curves. Math. Res. Lett. 9, 27–45 (2002)
Klein C., Richter O.: Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces. Phys. Rev. D 58, 124018 (1998)
Klein C., Richter O.: Ernst equation and Riemann surfaces. Analytical and numerical methods. Lecture Notes in Physics, 685. Springer-Verlag, Berlin (2005)
Korotkin D. A.: Finite-gap solutions of stationary axisymmetric Einstein equations in vacuum, Theoret. and Math. Phys. 77, 1018–1031 (1989)
Korotkin, D. A., Matveev, V. B.: On theta-function solutions of the Schlesinger system and the Ernst equation. (Russian) Funkt. Anal. i Pril. 34(4), 18–34, 96 (2000); translation in Funct. Anal. Appl. 34(4), 252–264 (2000)
Lenells J., Fokas A.S.: Boundary value problems for the stationary axisymmetric Einstein equations: a rotating disk. Nonlinearity 24, 177 (2011)
Neugebauer G., Meinel R.: The Einsteinian gravitational field of the rigidly rotating disk of dust. Astroph. J 414, L97–L99 (1993)
Neugebauer G., Meinel R.: General relativistic gravitational field of a rigidly rotating disk of dust: Axis potential, disk metric, and surface mass density. Phys. Rev. Lett. 73, 2166–2168 (1994)
Neugebauer G., Meinel R.: General relativistic gravitational field of a rigidly rotating disk of dust: Solution in terms of ultraelliptic functions. Phys. Rev. Lett. 75, 3046–3047 (1995)
Meinel R., Ansorg M., Kleinwächter A., Neugebauer G., Petroff D.: Relativistic figures of equilibrium. Cambridge University Press, Cambridge (2008)
Pringle J.E.: Accretion discs in astrophysics. Ann. Rev. Astron. Astrophys. 19, 137–162 (1981)
Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E.: Exact solutions of Einstein’s field equations Second Edition. Cambridge University Press, Cambridge (2003)
Zakharov V.E., Shabat A.B.: Exact theory of two-dimensional self-focussing and one- dimensional self-modulation in nonlinear media. Soviet Physics-JETP 34, 62–69 (1972)
Yamada A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 114–143 (1980)
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Communicated by P.T. Chruściel
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Lenells, J. Boundary Value Problems for the Stationary Axisymmetric Einstein Equations: A Disk Rotating Around a Black Hole. Commun. Math. Phys. 304, 585–635 (2011). https://doi.org/10.1007/s00220-011-1243-8
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DOI: https://doi.org/10.1007/s00220-011-1243-8