Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Summary:
We investigate which three dimensional near-horizon metrics $g_{NH}$ admit a compatible 1-form $X$ such that $(X, [g_{NH}])$ defines an Einstein-Weyl structure. We find explicit examples and see that some of the solutions give rise to Einstein-Weyl structures of dispersionless KP type and dispersionless Hirota (aka hyperCR) type.
References:
[1] Calderbank, D.M.J., Kruglikov, B.: Integrability via geometry: dispersionless differential equations in three and four dimensions. arXiv:1612.02753.
[2] Cartan, E.: Sur une classe d’espaces de Weyl. Ann. Sci. École Norm. Sup. (3) 60 (1943), 1–16. DOI 10.24033/asens.901 | MR 0014292 | Zbl 0028.30802
[3] Doubrov, B., Ferapontov, E.V., Kruglikov, B., Novikov, V.: On the integrability in Grassmann geometries: integrable systems associated with fourfolds $Gr(3,5)$. arXiv:1503.02274.
[4] Dunajski, M., Ferapontov, E.V., Kruglikov, B.: On the Einstein-Weyl and conformal self-duality equations. J. Math. Phys. 56 (8) (2015), 10pp., 083501. DOI 10.1063/1.4927251 | MR 3455337 | Zbl 1325.53058
[5] Dunajski, M., Gutowski, J., Sabra, W.: Einstein-Weyl spaces and near-horizon geometry. Classical Quantum Gravity 34 (2017), 21pp., 045009. DOI 10.1088/1361-6382/aa5992 | MR 3605882 | Zbl 1358.83081
[6] Dunajski, M., Kryński, W.: Einstein-Weyl geometry, dispersionless Hirota equation and Veronese webs. Math. Proc. Cambridge Philos. Soc. 157 (1) (2014), 139–150. DOI 10.1017/S0305004114000164 | MR 3211812 | Zbl 1296.53036
[7] Dunajski, M., Kryński, W.: Point invariants of third-order ODEs and hyper-CR Einstein-Weyl structures. J. Geom. Phys. 86 (2014), 296–302. DOI 10.1016/j.geomphys.2014.08.012 | MR 3282331 | Zbl 1316.34036
[8] Dunajski, M., Mason, L.J., Tod, K.P.: Einstein-Weyl geometry, the dKP equation and twistor theory. J. Geom. Phys. 37 (2001), 63–93. DOI 10.1016/S0393-0440(00)00033-4 | MR 1807082 | Zbl 0990.53052
[9] Eastwood, M.G.: Notes on conformal differential geometry. Rend. Circ. Mat. Palermo (2) Suppl. 43 (1996), 57–96. MR 1463509 | Zbl 0911.53020
[10] Eastwood, M.G., Tod, K.P.: Local constraints on Einstein-Weyl geometries. J. Reine Angew. Math. 491 (1997), 183–198. MR 1476092 | Zbl 0876.53029
[11] Ferapontov, E.V., Kruglikov, B.: Dispersionless integrable systems in 3D and Einstein-Weyl geometry. J. Differential Geom. 97 (2014), 215–254. DOI 10.4310/jdg/1405447805 | MR 3263506 | Zbl 1306.37084
[12] Hitchin, N.J.: Complex manifolds and Einstein’s equations. Lecture Notes in Math., Springer, Berlin-New York, 1982, Twistor geometry and nonlinear systems (Primorsko, 1980), 73–99. DOI 10.1007/BFb0066025 | MR 0699802 | Zbl 0507.53025
[13] Kunduri, H.K., Lucietti, J.: A classification of near-horizon geometries of extremal vacuum black holes. J. Math. Phys. 50 (2009), 41pp., 082502. DOI 10.1063/1.3190480 | MR 2554413 | Zbl 1223.83032
[14] Kunduri, H.K., Lucietti, J.: Classification of near-horizon geometries of extremal black holes. Living Rev. Relativity 16 (2013), 8pp., arXiv:1306.2517v2. DOI 10.12942/lrr-2013-8 | MR 2554413 | Zbl 1320.83005
[15] LeBrun, C., Mason, L.J.: The Einstein-Weyl equations, scattering maps, and holomorphic disks. Math. Res. Lett. 16 (2009), 291–301. DOI 10.4310/MRL.2009.v16.n2.a7 | MR 2496745 | Zbl 1176.53071
[16] Lewandowski, J., Pawlowski, T.: Extremal isolated horizons: A local uniqueness theorem. Classical Quantum Gravity 20 (2003), 587–606, arXiv:gr-qc/0208032. DOI 10.1088/0264-9381/20/4/303 | MR 1959394 | Zbl 1028.83025
[17] Lewandowski, J., Racz, I., Szereszeweski, A.: Near horizon geometries and black hole holograph. Phys. Rev. D 96 (2017), 044001, arXiv:1701.01704. DOI 10.1103/PhysRevD.96.044001
[18] Lewandowski, J., Szereszeweski, A., Waluk, P.: When isolated horizons met near horizon geometries. 2nd LeCosPA Symposium Proceedings, “Everything about Gravity" celebrating the centenary of Einstein's General Relativity, December 14-18, Taipei, 2015, arXiv:1602.01158. MR 1744050
[19] Li, C., Lucietti, J.: Three-dimensional black holes and descendants. Phys. Lett. B 738 (2014), 48–54. DOI 10.1016/j.physletb.2014.09.012 | MR 3272448 | Zbl 1360.83037
[20] Nurowski, P.: Differential equations and conformal structures. J. Geom. Phys. 55 (1) (2005), 19–49. DOI 10.1016/j.geomphys.2004.11.006 | MR 2157414 | Zbl 1082.53024
[21] Tod, K.P.: Einstein-Weyl spaces and third-order differential equations. J. Math. Phys. 41 (2000), 5572–5581. DOI 10.1063/1.533426 | MR 1770973 | Zbl 0979.53050
Search
Partner of
