I've edited (ten days ago) a question on Physics Stack Exchange, this [*Mathematical characterization of gravitational geons*](https://physics.stackexchange.com/questions/726281/mathematical-characterization-of-gravitational-geons), post with identifier **726281** the users of the site were kind adding in the comment thread links about papers related to gravitational geons. And a question on Meta of MathOverflow, identifier **5432** of this [post.](https://meta.mathoverflow.net/questions/5432/asking-if-a-post-about-a-mathematical-characterization-of-gravitational-geons-ca) I think that I should to delete both posts if it is edited an answer for this post.



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> **Question (Updated after the interesting comments of professor in comments).** Please, can you identify an article and the corresponding equations/formulation (equation and page from the article) characterizing a gravitational geon? Secondly the post has more focus on electromagnetic fields and classical gravity, but I'm interested in the discussion of possible singularities in the solution. Can you tell us, from your knowleges or as a reference request, the properties of these solutions (the gravitational geons)?**Many thanks.**
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The companion question is from an advice that was given from kind professors on Meta: (in my words) summarize the more important properties of gravitational geons from your knowledges or from the literature answering my post as a reference request.


I'm not able to read the article but I'm motivated to study your answer.


In Wikipedia is edited the article [*Geon (physics)*](https://en.wikipedia.org/wiki/Geon_(physics)) of an hypothetical object in theoretical physics, that studied the physicist John Archibald Wheeler. In the post of Physics Stack Exchange were added from kind users comments about references for the literature.



Mathematical characterization of gravitational geons as reference request, and their properties as main question
A gravitational geon is a space-time configuration that is bounded (asymptotically flat at spatial infinity) and stable (held together for all times by its own gravitational attraction). No such object is believed to exist, a geon should be radiating gravitational waves, lose energy, and decay for long times. No-go theorems exist, see for example [On Derrick's theorem in curved spacetime](https://arxiv.org/abs/1906.00702).


A simple explanation why the non-linear Klein-Gordon equation in Minkowski 3+1 dimensional spacetime does not support stable, time-independent solutions of finite energy is given in [lecture notes](https://indico.cern.ch/event/951466/contributions/4009710/attachments/2109820/3548883/Aveiro_Set_2020_Lecture_4_final_print.pdf) by Herdeiro (page 7-10).


There are ways to avoid the no-go theorem. Periodic (rather than time-independent) solutions are one possibility ([Q-balls](https://en.wikipedia.org/wiki/Q-ball)). The obstruction does not apply to 1+1 dimensions, see [Gravitational geons in 1+1 dimensions.](https://arxiv.org/abs/0807.0611)


