Submission declined on 8 January 2026 by SafariScribe (talk). This draft's references do not show that the subject meets Wikipedia's criteria for inclusion. The draft requires multiple published secondary sources that: provide significant coverage: discuss the subject in detail, not just brief mentions or routine announcements; are reliable: from reputable outlets with editorial oversight; are independent: not connected to the subject, such as interviews, press releases, the subject's own website, or sponsored content. Please add references that meet all three of these criteria. If none exist, the subject is not yet suitable for Wikipedia. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by SafariScribe 4 months ago. Last edited by SafariScribe 4 months ago. Reviewer: Inform author. Resubmit Please note that if the issues are not fixed, the draft will be declined again.

• Comment: there is only one independent source here, can you find others? Theroadislong (talk) 17:41, 18 December 2025 (UTC)

• Comment: Please rewrite the lead. As an example see Folk theorem (physics). Ldm1954 (talk) 16:01, 20 October 2025 (UTC)

This is a draft article. It is a work in progress open to editing by anyone. Please ensure core content policies are met before publishing it as a live Wikipedia article. Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL Easy tools: Citation bot (help) | Advanced: Fix bare URLs · Article logs · Draft logs. Last edited by SafariScribe (talk | contribs) 4 months ago. (Update) Finished drafting? Submit for review

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (September 2025) (Learn how and when to remove this message)

In general relativity, Chandrasekhar waves are a class of standing cylindrical gravitational-wave solutions of Einstein’s field equations. They describe vacuum space-times with cylindrical symmetry in which gravitational waves oscillate in time without net energy transport along the radial direction. Chandrasekhar waves form one of the best-studied examples of exact standing-wave solutions in general relativity and are closely related to, but distinct from, Einstein–Rosen waves.

The solution was first derived by Subrahmanyan Chandrasekhar in 1986, who showed that nontrivial standing cylindrical waves can exist without violating regularity conditions on the symmetry axis.^[1] Subsequent work clarified the physical interpretation of the solution, its relation to other cylindrical wave models, and its behavior under linear perturbations and backreaction effects.^[2][3][4][5]

A key distinguishing feature of Chandrasekhar waves is that the associated C-energy—a quasi-local measure of gravitational energy adapted to cylindrical symmetry—remains constant in time, in contrast to Einstein–Rosen waves, for which the C-energy oscillates periodically.^[6]

In units where ${\displaystyle c=1}$, the space-time metric of Chandrasekhar waves can be written as^[1]

${\displaystyle ds^{2}=e^{2\nu }[(dt)^{2}-(d\rho )^{2}]-e^{-2\mu }(\rho d\varphi )^{2}-e^{2\mu }(dz-qd\varphi )^{2}}$

where the metric functions depend on the time coordinate ${\displaystyle t}$ and the radial coordinate ${\displaystyle \rho }$; both ${\displaystyle t}$ and ${\displaystyle \rho }$ are measured in units of ${\displaystyle 1/\sigma }$, with ${\displaystyle \sigma }$ denoting the wave frequency. The functions ${\displaystyle \mu }$, ${\displaystyle q}$, and ${\displaystyle \nu }$ are given by

${\displaystyle \mu ={\frac {1}{2}}\ln {\frac {1-F^{2}}{1+F^{2}-2F\cos t}},}$

${\displaystyle q\sigma =-{\frac {2\rho F_{\rho }}{1-F^{2}}}\cos t-4\int _{0}^{\rho }{\frac {\rho F^{2}}{(1-F^{2})^{2}}}d\rho ,}$

${\displaystyle (\nu +\mu )_{t}=0,\quad (\nu +\mu )_{\rho }={\frac {\rho }{(1-F^{2})^{2}}}(F^{2}+F_{\rho }^{2}),}$

where ${\displaystyle F=F(\rho )}$ satisfies the nonlinear ordinary differential equation

${\displaystyle F(1+F^{2})+{\frac {1-F^{2}}{\rho }}(\rho F_{\rho })_{\rho }+2FF_{\rho }^{2}=0.}$

Regularity on the symmetry axis requires the boundary conditions

${\displaystyle F=F_{0}\,\,(>0\,\,{\text{and}}\,\,<1)\quad {\text{and}}\quad F_{\rho }=0\quad {\text{for}}\quad \rho =0.}$

Because ${\displaystyle (\nu +\mu )_{t}=0}$, the associated C-energy,

${\displaystyle C=\nu +\mu ,}$

is conserved in time, reflecting the standing-wave character of the solution.

For large radial distances, the function ${\displaystyle F(\rho )}$ exhibits the asymptotic behavior^[1]

${\displaystyle F\sim {\frac {\text{const.}}{\sqrt {\rho }}}\cos \left(\rho +{\frac {1}{16}}\ln \rho +b\right)+O\left(\rho ^{-{\frac {3}{2}}}\right)\quad {\text{as}}\quad \rho \to \infty .}$

Near the axis and at infinity, the metric functions and the C-energy behave as

${\displaystyle e^{2\mu }\to {\frac {1-F_{0}^{2}}{1+F_{0}^{2}-2F_{0}\cos t}}\sim O(1),\quad q=O(\rho ^{2}),\quad C=O(\rho ^{2})\quad {\text{as}}\quad \rho \to 0,}$

${\displaystyle e^{2\mu }\to 1,\quad q\to -{\text{const.}}{\sqrt {\rho }}\cos t-{\text{const.}}\rho ,\quad C=O(\rho )\quad {\text{as}}\quad \rho \to \infty .}$