{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T11:03:52Z","timestamp":1761563032105,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,14]],"date-time":"2024-02-14T00:00:00Z","timestamp":1707868800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper presents a brief study of (2-dimensional, spacelike) wave surfaces to a null direction l on a space-time (M,g) and studies how certain imposed symmetries on the set of such wave surfaces can be used to describe other geometrical features of l and (M,g). It is mainly a review of known material but contains some novelties. For example, the brief discussion of the nature of wave surfaces (when viewed geometrically as wave fronts to a null ray direction) in Wave Surfaces Section is new in the sense that although it appeared in the author\u2019s work by the present author, it has not, to the best of his knowledge, appeared in this form anywhere else. Further, the work on conical symmetry and plane waves are, to the best of the author\u2019s knowledge, original with him from earlier papers and are reviewed here while the work on complete wave surface (sectional curvature-) symmetry is believed to be entirely new. Geometrical use of the sectional curvature function is employed in many places. The consequences of the various symmetry conditions imposed on the collection of all wave surfaces to a null direction spanned by a null vector l are described in terms of l spanning a principal null direction of the Weyl tensor (if non-zero) at the point concerned (in the sense of Petrov and Bel).<\/jats:p>","DOI":"10.3390\/sym16020230","type":"journal-article","created":{"date-parts":[[2024,2,14]],"date-time":"2024-02-14T06:59:26Z","timestamp":1707893966000},"page":"230","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Wave Surface Symmetry and Petrov Types in General Relativity"],"prefix":"10.3390","volume":"16","author":[{"given":"Graham","family":"Hall","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hall, G.S. (2004). Symmetries and Curvature Structure in General Relativity, World Scientific.","DOI":"10.1142\/1729"},{"key":"ref_2","unstructured":"Witten, L. (1962). An Introduction to Current Research, Wiley."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1088\/0305-4470\/6\/5\/008","article-title":"On the Petrov classification of gravitational fields","volume":"6","author":"Hall","year":"1973","journal-title":"J. Phys. A."},{"key":"ref_4","unstructured":"Hall, G.S. (2015, January 10\u201314). Celebrating 100 Years of Einstein\u2019s Theory of General Relativity. Proceedings of the International Conference on Relativistic Astrophysics, Lahore, Pakistan."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Petrov, A.Z. (1969). Einstein Spaces, Pergamon Press. (see also Petrov, A.Z. The classification of spaces defining gravitational fields. Gen. Rel. Grav. 2000, 32, 1665\u20131685.).","DOI":"10.1023\/A:1001910908054"},{"key":"ref_6","first-page":"59","article-title":"Radiation states and the problem of energy in general relativity","volume":"16","author":"Bel","year":"1962","journal-title":"Cah. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Stephani, H., Kramer, D., MacCallum, M.A.H., Hoenselaers, C., and Herlt, E. (2003). Exact Solutions to Einstein\u2019s Field Equations, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511535185"},{"key":"ref_8","first-page":"309","article-title":"Gravitational waves in general relativity. VI. The outgoing radiation condition","volume":"264","author":"Sachs","year":"1961","journal-title":"Proc. Roy. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s40065-018-0235-3","article-title":"Some remarks on wave solutions in general relativity theory","volume":"8","author":"Hall","year":"2019","journal-title":"Arab Journ. Maths."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hall, G.S. (2023). Four-Dimensional Manifolds and Projective Structure, Chapman and Hall.","DOI":"10.1201\/9781003023166"},{"key":"ref_11","unstructured":"Defrise, L. (1969). Groupes d\u2019Isotropie et Groupes de Stabilite Conforme dans les Espaces Lorentziens. [Ph.D Thesis, Universite Libre de Bruxelles]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/230\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:59:38Z","timestamp":1760104778000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/2\/230"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,14]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["sym16020230"],"URL":"https:\/\/doi.org\/10.3390\/sym16020230","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,2,14]]}}}