https://doi.org/10.1140/epjp/s13360-025-06239-7
Regular Article
Revisiting of spherically symmetric exterior solutions and the validity of Jebsen-Birkhoff theorem in scalar field
1
Department of Mathematics and Statistics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur, India
2
Government Girls Inter College, 274509, Salempur, Deoria, India
Received:
10
March
2025
Accepted:
15
March
2025
Published online:
11
April
2025
Scalar fields are likely to play a significant role in astrophysics, as they are already crucial in particle physics and quantum field theory. In the early universe, they may have been responsible for inflation or linked to the spontaneous breaking of symmetries that give particles their mass. Studying spherically symmetric solutions is vital for understanding the weak-field limit of scalar field theory and comparing it with solar system observations. Additionally, investigating strong field instabilities, which could potentially undermine the theory, is conducted within spherical symmetry. It is therefore essential to understand how the Jebsen-Birkhoff theorem (well-known result in general relativity) satisfy in scalar field theory. The present paper studies the validity, or lack thereof, of the Jebsen-Birkhoff theorem in scalar field by generalizing it and considering quintessence and phantom fields as effective dark energy candidates. We focus on discussing the exact solution of the Einstein field equation exterior to an isolated, spherically symmetric, non-rotating spherical object. We showed that the gravitational field of external empty space-time for a spherically symmetric distribution of source matter, within the framework of scalar fields, must be static, but not asymptotically flat; that is, the theorem is partially valid within the framework of scalar field theory. We have also discussed the existence of time-like Killing vector field for the exterior solution in scalar field.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
