Abstract
In this note I discuss the conformal rescaling of spacetimes, with focus on the constraint equations. I give conditions on the Cauchy data at the conformal boundary which are sufficient for regularity, and show how to construct solutions to the conformal constraint equations. Finally, I discuss gauge conditions in the conformal setting and indicate the first steps in analyzing the conformally rescaled version of the Einstein vacuum evolution equations.
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References
M. Alcubierre. Appearance of coordinate shocks in hyperbolic formalisms of general relativity. Phys. Rev. D (3), 55, 5981–5991, 1997.
M. Alcubierre and J. Massó. Pathologies of hyperbolic gauges in general relativity and other field theories. Phys. Rev. D (3), 57, R4511–R4515, 1998.
L. Andersson and P. T. Chruściel. Hyperboloidal Cauchy data for vacuum Einstein equations and obstructions to smoothness of null infinity. Phys. Rev. Lett., 70, 2829–2832, 1993.
L. Andersson and P. T. Chruściel. On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of scri. Comm. Math. Phys., 161, 533–568, 1994.
L. Andersson and P. T. Chruściel. Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”. Dissertationes Math. (Rozprawy Mat.), 355, 100, 1996.
L. Andersson, P. T. Chruściel, and H. Friedrich. On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Comm. Math. Phys., 149, 587–612, 1992.
L. Andersson and M. S. Iriondo. Existence of constant mean curvature hypersurfaces in asymptotically flat spacetimes. Ann. Global Anal. Geom., 17, 503–538, 1999.
L. Andersson and V. Moncrief. Elliptic-hyperbolic systems and the Einstein equations. gr-qc/0110111, 2001.
J. Frauendiener. Conformal infinity. Living Rev. Relativ., 3, 2000–4, 92 pp. (electronic), 2000.
S. Frittelli and O. Reula. Well-posed forms of the 3 + 1 conformally-decomposed Einstein equations. J. Math. Phys., 40, 5143–5156, 1999.
H. Friedrich. Cauchy problems for the conformal vacuum field equations in general relativity. Comm. Math. Phys., 91, 445–472, 1983.
P. Hübner. How to avoid artificial boundaries in the numerical calculation of black hole spacetimes. Class. Quant. Grav., 16, 2145–2164, 1999.
P. Hübner. A scheme to numerically evolve data for the conformal Einstein equation. Class. Quant. Grav., 16, 2823–2843, 1999.
P. Hübner. From now to timelike infinity on a finite grid. Class. Quant. Grav., 18, 1871–1884, 2001.
P. Hübner. Numerical calculation of conformally smooth hyperboloidal data. Class. Quant. Grav., 18, 1421–1440, 2001.
R. Penrose and W. Rindler. Spinors and space-time. Vol. 2 (Cambridge University Press, Cambridge, 1988), second edn. Spinor and twistor methods in space-time geometry.
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Andersson, L. (2002). Construction of Hyperboloidal Initial Data. In: Frauendiener, J., Friedrich, H. (eds) The Conformal Structure of Space-Time. Lecture Notes in Physics, vol 604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45818-2_9
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DOI: https://doi.org/10.1007/3-540-45818-2_9
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