On regular geons in general relativity

We present the exact regular solution of vacuum Einstein equations, which may be interpreted as "mass without mass'' (geon – in terminology of J.A. Wheeler), moving along a straight line with constant velocity. The feature of this classical object, localized in three-dimensional space...

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Дата:	2012
Автор:	Olyeynik, V.P.
Формат:	Стаття
Мова:	English
Опубліковано:	Odessa National University 2012
Назва видання:	Вопросы атомной науки и техники
Теми:	Section C. Theory of Elementary Particles. Cosmology
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/107062
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	On regular geons in general relativity / V.P. Olyeynik // Вопросы атомной науки и техники. — 2012. — № 1. — С. 171-172. — Бібліогр.: 7 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1070622016-10-13T03:02:16Z On regular geons in general relativity Olyeynik, V.P. Section C. Theory of Elementary Particles. Cosmology We present the exact regular solution of vacuum Einstein equations, which may be interpreted as "mass without mass'' (geon – in terminology of J.A. Wheeler), moving along a straight line with constant velocity. The feature of this classical object, localized in three-dimensional space, is that its scalar invariant constructed of two Riemann curvature tensors does not impose any restrictions on the velocity of the object. Представлено точное регулярное решение вакуумных уравнений Эйнштейна, которое можно интерпретировать как «массу без массы» (геон – по терминологии Дж.А. Уилера), движущуюся вдоль прямой с постоянной скоростью. Особенностью этого классического объекта, локализованного в трехмерном пространстве, является то, что его скалярный инвариант, составленный из двух тензоров Римана, не содержит каких-либо ограничений на скорость объекта. Представлено точне регулярне розв'язання вакуумних рівнянь Ейнштейна, яке можна інтерпретувати як «масу без маси» (геон – по термінології Дж.А. Уілера), що рухається вздовж прямої з постійною швидкістю. Особливістю цього класичного об'єкта, локалізованого у тривимірному просторі, є те, що його скалярний інваріант, складений з двох тензорів Рімана, не містить будь-яких обмежень на швидкість об'єкта. 2012 Article On regular geons in general relativity / V.P. Olyeynik // Вопросы атомной науки и техники. — 2012. — № 1. — С. 171-172. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 04.20.Ib, 11.27.+d http://dspace.nbuv.gov.ua/handle/123456789/107062 en Вопросы атомной науки и техники Odessa National University
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology
spellingShingle	Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology Olyeynik, V.P. On regular geons in general relativity Вопросы атомной науки и техники
description	We present the exact regular solution of vacuum Einstein equations, which may be interpreted as "mass without mass'' (geon – in terminology of J.A. Wheeler), moving along a straight line with constant velocity. The feature of this classical object, localized in three-dimensional space, is that its scalar invariant constructed of two Riemann curvature tensors does not impose any restrictions on the velocity of the object.
format	Article
author	Olyeynik, V.P.
author_facet	Olyeynik, V.P.
author_sort	Olyeynik, V.P.
title	On regular geons in general relativity
title_short	On regular geons in general relativity
title_full	On regular geons in general relativity
title_fullStr	On regular geons in general relativity
title_full_unstemmed	On regular geons in general relativity
title_sort	on regular geons in general relativity
publisher	Odessa National University
publishDate	2012
topic_facet	Section C. Theory of Elementary Particles. Cosmology
url	http://dspace.nbuv.gov.ua/handle/123456789/107062
citation_txt	On regular geons in general relativity / V.P. Olyeynik // Вопросы атомной науки и техники. — 2012. — № 1. — С. 171-172. — Бібліогр.: 7 назв. — англ.
series	Вопросы атомной науки и техники
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last_indexed	2025-07-07T19:26:42Z
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fulltext	ON REGULAR GEONS IN GENERAL RELATIVITY V.P. Olyeynik ∗ Odessa National University, 65026, Odessa, Ukraine (Received November 1, 2011) We present the exact regular solution of vacuum Einstein equations, which may be interpreted as “mass without mass” (geon – in terminology of J.A. Wheeler), moving along a straight line with constant velocity. The feature of this classical object, localized in three-dimensional space, is that its scalar invariant constructed of two Riemann curvature tensors does not impose any restrictions on the velocity of the object. PACS: 04.20.Ib, 11.27.+d 1. INTRODUCTION Wheeler [1] was the first who suggested, that vac- uum Einstein equations allow the existence of regu- lar, localized in curved space-time classical objects – geons, such that the distant observer sees the curva- ture concentrated in the central region with persisting large-scale structure. However, various attempts to construct such objects as static or stationary curved space-time, possibly coupled to other zero-mass fields such as massless neutrinos or the electromagnetic field [2, 3], so far not been successful. In particu- lar, these geons are however believed to be unstable, owing to the tendency of massless fields either to dis- perse to infinity or to collapse into a black hole [4]. In 1985, Sorkin [5] generalized Wheeler’s geon into a topological geon by allowing that its central re- gion may have complicated topology. This geon has a regular Euclidean-signature section, but the topolog- ically nontrivial central region may evolve into black hole [6]. So, recent researches in this area are study- ing the properties of families of geon-like black holes in D ≥ 4 space-time dimensions [7]. 2. THE REGULAR VAQUUM GEON All the above mentioned objects, described by non- linear differential equations, are static. This means that there exists a system of reference, in which parts of the structure under consideration are in rest rel- ative to each other. It is interesting to consider the localized nonlinear object, in which separate parts move relative to one another. In the simplest case it can be assumed, that a three-region of a curved four- dimensional space-time is moving relative to another with a constant 3-velocity �υ (υ, 0, 0) In such a space-time the spatial direction x as well as the plane (yz), perpendicular to it, are selected, so the space-time metric can be chosen in the form: ds2 = uc2dt2−fdx2−χ ( dy2 + dz2 ) +2ψcdtdx, (1) where u, f, χ, ψ are functions of the parameter: ξ = √ (x− υt)2 + y2 + z2. Determinant of this metric tensor is g = −(uf + ψ2)χ2. (2) Vacuum Einstein equations: Rik = 0 (3) have the following exact solution: u = (1 − υ c a4)2 a2 3 a2 (a1 − ξ)2 (a1 + ξ)2 − υ2 c2 a2 ξ4 (a1 + ξ)4, f = −a2 4 a2 3 a2 (a1 − ξ)2 (a1 + ξ)2 + a2 ξ4 (a1 + ξ)4, χ = a2 (a1ξ) 4 (a1 + ξ)4, (4) ψ = ( 1 − υ c a4 ) a4 a2 3 a2 (a1 − ξ)2 (a1 + ξ)2 + υ c a2 ξ4 (a1 + ξ)4, where a1, . . . , a4 are the integration constants. The scalar invariant, composed of two curvature tensors, for this space-time has the form: I = 1 48 RiklmR iklm = ( 2ξ3 a1a2 )2 ( 1 + ξ a1 )−12 . (5) In contrast to the Schwarzschild solution, for which this invariant is of the form: ISchw = ( rg 2r3 )2 , (6) where r is the distance to the center of the object, rg is Schwarzschild radius, the invariant (5) is regu- lar everywhere in the space-time: it goes to zero as r6, when the center of the geon is approached, and behaves like r−6 on large distances from it. ∗E-mail address: olyeyvp@onu.edu.ua PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 171-172. 171 3. CONCLUSIONS We obtain exact solution for vacuum Einstein equa- tions, which is asymptotically flat and regular every- where in the space-time. The scalar invariant (5) as well as the determinant of the metric tensor (2) con- tains the velocity of the geon only through the para- meter ξ. The integration constant a4 is not included in (2) and (5) at all. As is seen from (5), this invari- ant does not include any limitations on the velocity of the geon. The question of stability of this object requires further study. References 1. J.A. Wheeler. Geons // Phys. Rev. 1955, v. 97, p. 511-536. 2. J.A. Wheeler. Geometrodynamics. New York: Academic Press, 1962, 332 p. 3. D.R. Brill, J.B. Hartle. Method of the self- consistent field in general relativity and its ap- plication to the gravitational geon // Phys. Rev. 1964, v. B135, p. 271-278. 4. G. Gundlach. Critical phenomena in gravita- tional collapse // Arxiv : gr-qc/0210101, 2003, 65 p. 5. R.D. Sorkin. Introduction to topological geons // Topological properties and global structure of space-time. Proceedings of the NATO Ad- vanced Study Institute. Erice, Italy, May 12- 22, 1985, ed. P.G. Bergmann and V.De Sabbata. New York: Plenum, 1986, p. 249-270. 6. D. Gannon. On the topology of space-like hyper- surfaces, singularities, and black holes // Gen. Rel. Grav. 1976, v. 7, p. 219-232. 7. J. Louko, R.B. Mann, D. Marolf. Geons with spin and charge // Arxiv : gr-qc/0412012, 2005, 26 p. � �� ! �� " �� #�$%$ &� ��'� ��(�)�� )$ �� ) +�� ,�� -�� ,��$ �� .�� /� � � 0� �� / �� ! �� " �� / � � /1 #�$%$ &/ ��'� (� ��2�� 1 � ��/� �) ��/��)$ �� /��) 3�� .2�� /�� /� �� /� 2 �� (� �� / ��/� �� /� -/�� /�� /�� .2��$ 456

On regular geons in general relativity

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