Simple numerical solutions to the Einstein constraints on various three-manifolds. (English) Zbl 1515.83038
Summary: Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 70H45 | Constrained dynamics, Dirac’s theory of constraints |
| 54E45 | Compact (locally compact) metric spaces |
| 83C80 | Analogues of general relativity in lower dimensions |
| 35A35 | Theoretical approximation in context of PDEs |
Software:
ReginaReferences:
| [1] | Cook, G.B.: Living reviews in relativity 3(5), 53(2000). doi:10.12942/lrr-2000-5 · Zbl 1024.83001 |
| [2] | Pfeiffer, HP; Kidder, LE; Scheel, MA; Teukolsky, SA, Comput. Phys. Commun., 152, 253-273 (2003) · Zbl 1196.65179 · doi:10.1016/S0010-4655(02)00847-0 |
| [3] | Okawa, H., Int. J. Mod. Phys. A, 28, 1340016 (2013) · Zbl 1277.83053 · doi:10.1142/S0217751X13400162 |
| [4] | Ossokine, S.; Foucart, F.; Pfeiffer, HP; Boyle, M.; Szilágyi, B., Class. Quant. Gravity, 32, 245010 (2015) · Zbl 1331.83019 · doi:10.1088/0264-9381/32/24/245010 |
| [5] | Tichy, W.; Rashti, A.; Dietrich, T.; Dudi, R.; Brügmann, B., Phys. Rev. D, 100, 124046 (2019) · doi:10.1103/PhysRevD.100.124046 |
| [6] | Lindblom, L., Rinne, O., Taylor, N.W.P: J. Comp. Math. (in press) (2022) |
| [7] | Burton, B A., Budney, R., Pettersson, W. et al.: Regina: software for low-dimensional topology, (1999-2021). http://regina-normal.github.io/ |
| [8] | Yamabe, H., Osake J. Math., 12, 21-37 (1960) · Zbl 0096.37201 |
| [9] | Bartnik, R., Isenberg, J.: The constraint equations the Einstein equations and the large scale behavior of gravitational fields. In: Crusciel, P., Friedrich, H., (eds.)The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity, p1. Birkhäuser, Basel, Switzerland (2004) (ISBN 3-7643-7130-7) · Zbl 1073.83009 |
| [10] | Three-dimensional geometry and topology. Princeton University Press (1997) · Zbl 0873.57001 |
| [11] | Scott, P., Bull. Lond. Math. Soc., 15, 401-487 (1983) · Zbl 0561.57001 · doi:10.1112/blms/15.5.401 |
| [12] | Isenberg, J., Class. Quant. Gravity, 12, 2249-2274 (1995) · Zbl 0840.53056 · doi:10.1088/0264-9381/12/9/013 |
| [13] | Lindblom, L.; Szilágyi, B., J. Comput. Phys., 243, 151-175 (2013) · Zbl 1349.65651 · doi:10.1016/j.jcp.2013.02.031 |
| [14] | Balay, S., Abhyankar, S., Adams, M.F., Benson, S., Brown, J., Brune, P., Buschelman, K., Constantinescu, E., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W.D., Hapla, V., Isaac, T., Jolivet, P., Karpeev, D., Kaushik, D., Knepley, M.G., Kong, F., Kruger, S., May, D.A., McInnes, L.C., Mills, R.T., Mitchell, L., Munson, T., Roman, J.E., Rupp, K., Sanan, P., Sarich, J., Smith, B.F., Zampini, S., Zhang, H., Zhang, H., Zhang, J.: PETSc/TAO users manual Tech. Rep. ANL-21/39 - Revision 3.16 Argonne National Laboratory, (2021) |
| [15] | Lindblom, L.; Taylor, NW; Rinne, O., J. Comput. Phys., 313, 31-56 (2016) · Zbl 1349.65626 · doi:10.1016/j.jcp.2016.02.029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
