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Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes
Open accessAbstract
A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented. These spacetimes represent exact gravitational waves of arbitrary profiles which propagate in background universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson and type D (anti-)Nariai or Plebański–Hacyan spaces, and their generalizations. All possibilities are discussed and interpreted using a unifying simple metric form. Sandwich and impulsive waves propagating in the above background spaces with different geometries and matter content can easily be constructed. New solutions are identified, e.g. type D pure radiation or explicit type II electrovacuum waves in the (anti-)Nariai universe. It is also shown that, in general, there are no conformally flat Einstein–Maxwell fields with a non-vanishing cosmological constant.
Overview
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TL;DR
The paper presents a unified explicit family of Kundt-class non-diverging spacetimes that contains Petrov types O, D, N, and II and can represent exact gravitational waves propagating in various conformally flat and type D backgrounds. It identifies new solutions, including type D pure radiation in the (anti-)Nariai universe and type II electrovacuum waves in the (anti-)Nariai background, and proves that there are no conformally flat Einstein–Maxwell fields with non-vanishing cosmological constant.
Key findings
- A single metric family (1)–(2), with constants α, β, ε and functions C(u), D(u), H(ζ, ζ̄, u), yields Kundt spacetimes of Petrov type O, D, N, or II depending on whether D = D₀ and/or H = H₀.
- The backgrounds include Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson, and type D (anti-)Nariai or Plebański–Hacyan spaces; adding H_w = H − H₀ introduces exact gravitational waves of arbitrary profile.
- New solutions reported include a type D pure radiation solution in the (anti-)Nariai universe, with Ψ₂ = −Λ/3 and R = 4Λ, and an explicit type II electrovacuum solution for electromagnetic plus gravitational waves in the (anti-)Nariai background.
- The appendix proves that there are no conformally flat Einstein–Maxwell solutions with Λ ≠ 0; for Λ = 0 the only possibilities are the Bertotti–Robinson universe for a non-null Maxwell field and the special pp-wave for a null field.
Methods
- Theoretical derivation within the Kundt class: the authors adopt an explicit metric ansatz, compute the Newman–Penrose Weyl and Ricci scalars, and classify Petrov types and matter content by conditions on D and H.
- They analyze special subclasses corresponding to conformally flat and type D backgrounds, impose Einstein–Maxwell or pure radiation conditions, and construct radiative type N/II solutions by adding a generalised Kerr–Schild term −(Q/P)H_w du².
- Physical interpretation uses the geodesic deviation equation in an orthonormal frame, decomposing effects of R, Φ₁₁, Φ₂₂, Ψ₂, and Ψ₄ into isotropic, matter, Coulomb, and gravitational-wave contributions.
Limitations
- Stated: the general solution describing different Petrov types and matter fields explicitly is still impossible; even with symmetries, solutions can usually only be given implicitly.
- Stated: the Einstein–Maxwell restrictions rule out a purely electromagnetic field in general spacetimes of the form (1)–(2); only special choices of α, β, ε, C, D, and H satisfy the field equations.
- Evident from the text: the work is a theoretical exact-solution study with no observational data or numerical analysis, and its results are restricted to the non-diverging Kundt class and the specific metric ansatz considered.
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In the paper
“Such an explicit solution for electromagnetic (plus gravitational) waves in the (anti-)Nariai universe seems to have remained unnoticed in the literature so far (whereas pure gravitational waves were already considered [36,30]).”
§ Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes > 4.3 Electrovacuum solutions
What the sources say
- Partly supportedcites [36] Khlebnikov V I, 1986, Class. Quantum Grav. sourceExperimental — accuracy not yet measured
The Additional information section says the solutions describe strong gravitational waves propagating in generalized electromagnetic universes, and the Abstract specifies that the electromagnetic field is non-radiative. This supports the cited aside about gravitational waves without electromagnetic radiation, but our copy does not identify the background specifically as the (anti-)Nariai universe. (LLM, conf=0.81)
- Supportedcites [30] source
The citation is attached to the parenthetical claim that pure gravitational waves had already been considered, not to the claim about an unnoticed electromagnetic-wave solution. The Abstract describes a subclass of pure gravitational waves in Nariai and a generalization to anti-Nariai; the Concluding Remarks likewise describe impulsive gravitational waves in Nariai. (LLM, conf=0.92)
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In the paper
“It is also shown that, in general, there are no conformally flat Einstein-Maxwell fields with a non-vanishing cosmological constant.”
§ Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes > Abstract
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In the paper
“A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented.”
§ Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes > Abstract
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“There is no type D vacuum solution of the form (1) for Λ = 0 since (38) reduces to Minkowski space in such a case.”
§ Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes > 3.2 Type D spacetimes
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None of the sources cited for this claim have been checked yet.
In the paper
“In fact, introducing the additional term -(Q/P ) H w du 2 in the metric corresponds to a generalised Kerr-Schild transformation (see e.g.[31,32]) of the background geometries”
§ Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes > 4 Exact gravitational waves of arbitrary profile on the above backgrounds
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“From these diagrams I exclude the conformally flat case, as it implies 36 Λ = 0 with the only non-null member being therefore 40 the Bertotti-Robinson 2,28,37 metric and the null members being given by a special class of plane waves 1, 30 .”
From Algebraically special Einstein–Maxwell fields (2017) · cites this paper as [36]
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Metadata
- Authors
- Podolsk, J., & Ortaggio, M.
- Affiliations
- University of Trento (IT)
- ORCID
- Ortaggio, M
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2003-04-10
- Volume
- 20
- Issue
- 9
- Pages
- 1685-1701
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2036184054
- Cited by
- 61
- References
- 42
- Citations by year
- 2024: 2 · 2023: 2 · 2022: 2 · 2021: 2 · 2020: 6 · 2019: 3 · 2018: 6 · 2017: 6 · 2016: 2 · 2015: 1 · 2014: 2 · 2013: 4 · 2012: 1
- FWCI
- 1.90
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Univerzita Karlova v Praze
- Grantová Agentura České Republiky
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Keywords
- Kundt spacetimes, type D spacetimes, anti-de Sitter space, de Sitter space, Bertotti-Robinson spacetime, cosmological constant, Einstein-Maxwell fields, electrovacuum, impulse waves, conformally flat spacetimes, exact solutions
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