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Nariai, Bertotti-Robinson, and anti-Nariai solutions in higher dimensions
Open accessAbstract
We find all higher dimensional solutions of Einstein-Maxwell theory that are the topological product of two manifolds of constant curvature. These solutions include the higher dimensional Nariai, Bertotti-Robinson and anti-Nariai solutions and the anti--de Sitter Bertotti-Robinson solutions with toroidal and hyperbolic topology (Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski-Hacyan solutions). We give explicit results for any dimension $D>~4.$ These solutions are generated from the appropriate extremal limits of the higher dimensional near-extreme black holes in de Sitter and anti--de Sitter backgrounds. Thus, we also find the mass and charge parameters of higher dimensional extreme black holes as a function of the radius of the degenerate horizon.
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TL;DR
The paper constructs all higher-dimensional Einstein-Maxwell solutions that are topological products of two constant-curvature manifolds, including Nariai, Bertotti-Robinson, anti-Nariai, and Plebański-Hacyan solutions for any \(D \ge 4\). These are generated as extremal limits of near-extreme higher-dimensional de Sitter and anti-de Sitter black holes.
Key findings
- The higher-dimensional Nariai solution is the product \(dS_2 \times S^{D-2}\), with fixed \((D-2)\)-sphere radius \(B^{-1/2}\); the Bertotti-Robinson solution is \(AdS_2 \times S^{D-2}\), and the flat Bertotti-Robinson is its \(\Lambda=0\) limit.
- The Nariai-Bertotti-Robinson solution is \(M^{1,1} \times S^{D-2}\), with \(M^{1,1}\) two-dimensional Minkowski spacetime in Rindler coordinates, and is identified as a Plebański-Hacyan solution.
- In the AdS case, the solutions include spherical \(AdS_2 \times S^{D-2}\), toroidal/cylindrical/planar \(AdS_2 \times E^{D-2}\), and anti-Nariai \(AdS_2 \times H^{D-2}\); only the \(k=-1\) anti-Nariai case admits a neutral \(Q=0\) solution.
- For extreme dS black holes, the mass and charge parameters as functions of the degenerate horizon \(\rho\) are \(M = 2\rho^{D-3}\left(1 - \frac{D-2}{D-3}\frac{\Lambda}{3}\rho^2\right)\) and \(Q^2 = \rho^{2(D-3)}\left(1 - \frac{D-1}{D-3}\frac{\Lambda}{3}\rho^2\right)\), with \(\rho_{\max} = \sqrt{\frac{D-3}{D-1}\frac{3}{\Lambda}}\). For extreme AdS black holes, \(M = 2\rho^{D-3}\left(k - \frac{D-2}{D-3}\frac{\Lambda}{3}\rho^2\right)\) and \(Q^2 = \rho^{2(D-3)}\left(k - \frac{D-1}{D-3}\frac{\Lambda}{3}\rho^2\right)\).
Methods
- Theoretical study of the \(D\)-dimensional Einstein-Maxwell action with cosmological constant \(\Lambda\), using static higher-dimensional Tangherlini black-hole metrics in asymptotically dS and AdS backgrounds.
- The authors apply the Ginsparg-Perry near-extremal limiting procedure: set horizons to differ by a small parameter \(\varepsilon\), introduce new time and radial coordinates, and take \(\varepsilon \to 0\) to generate Nariai-like, Bertotti-Robinson-like, and anti-Nariai-like solutions.
- They derive explicit mass and charge parameters for extreme black holes as functions of the degenerate horizon \(\rho\), then identify the resulting product spacetimes and their topology for \(k=1,0,-1\).
Limitations
- The paper states that for \(D \ge 6\) it was not able to write \(f(r)\) as a function of the degenerate horizon \(\rho\), because one must deal with polynomial functions of degree higher than four.
- The text makes evident that stability analyses, including quasinormal modes and quantum stability, are left for future work; the paper focuses on constructing the exact solutions and their extremal black-hole parameters.
- The construction is limited to solutions generated from extremal limits of near-extreme static black holes; the paper does not present empirical data or observational tests.
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In the paper
“Thus, besides the degenerate horizon r = ρ, there is another horizon at σ = 3/Λ -2ρ 2 .”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > A. Higher dimensional extreme black holes in an asymptotically dS background
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In the paper
“There is no neutral (Q = 0) Bertotti-Robinson solution.”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > 2. Higher dimensional dS Bertotti-Robinson solution
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In the paper
“F = - Q ADM ρ D-2 u dt ∧ dx , (29”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > 3. Higher dimensional Nariai-Bertotti-Robinson solution
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“The condition Q 2 ≥ 0 implies that ρ ≤ 3/(2Λ).”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > A. Higher dimensional extreme black holes in an asymptotically dS background
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In the paper
“Now, the above construction can be extended to Ddimensional extreme dS black holes.”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > A. Higher dimensional extreme black holes in an asymptotically dS background
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In the paper
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > 3. Higher dimensional Nariai-Bertotti-Robinson solution
“F = - Q ADM ρ D-2 u dt ∧ dx , (29”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > 3. Higher dimensional Nariai-Bertotti-Robinson solution
“the Maxwell field (9) of the higher dimensional Nariai-Bertotti-Robinson solution is F = Q ADM ρ -D+2 u χ dT ∧ dχ.Now, the spacetime factor -χ 2 dT 2 + dχ 2 is just M 1,1 (2-dimensional Minkowski spacetime) in Rindler coordinates.Therefore, under the usual coordinate transformation χ = √ x 2 -t 2 and T = arctanh(t/x)”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > 3. Higher dimensional AdS black holes with hyperbolic topology
“One has mmin = -4 D-1 ( D-3 D-2 ) (D-3)/2 , and q0 = ( D-3 D-2 ) (D-3)/2 / √ D -2.”
§ Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions > 3. Higher dimensional AdS black holes with hyperbolic topology
“ρ min = - D -3 D -1 3 Λ .(35) For ρ = ρ min , the extreme black hole has no electric charge (Q = 0) and its mass is negative, M = -4ρ D-3 min /(D -1).”
Citations
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- H. Nariai, 1950, Sci. Rep. Tohoku Univ., Ser. 1, 34, 160
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Cited by
Quasinormal modes of black holes and black branes
Berti, E., Cardoso, V., & Starinets, A. O. (2009) · Classical and Quantum Gravity · doi:10.1088/0264-9381/26/16/163001 · Cited by 2,095
Self-citationGarraffo, C., & Giribet, G. E. (2008) · Modern Physics Letters A · doi:10.1142/s0217732308027497 · Cited by 213
Lovelock black holes with a nonlinear Maxwell field
Maeda, H., Hassaïne, M., & Martínez, C. (2009) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.79.044012 · Cited by 176
Static response and Love numbers of Schwarzschild black holes
Hui, L., Joyce, A., Penco, R., Santoni, L., & Solomon, A. R. (2021) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2021/04/052 · Cited by 165
Classification of the Weyl tensor in higher dimensions and applications
Coley, A. A. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/3/033001 · Cited by 129
Metadata
- Authors
- Cardoso, V., Dias, Ó. J. C., & Lemos, J. P. S.
- Affiliations
- Instituto Superior Técnico (PT)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2004-07-19
- Volume
- 70
- Issue
- 2
- ISSN
- 0556-2821 (ISSN-L)
- 1550-7998
- 1550-2368
- OpenAlex
- W1996802537
- Semantic Scholar
- a7883655b8fbb542977a5e8a20f7de99bd5b7895
- Cited by
- 107
- References
- 63
- Citations by year
- 2026: 3 · 2025: 2 · 2024: 8 · 2023: 4 · 2022: 3 · 2021: 2 · 2020: 4 · 2019: 2 · 2018: 5 · 2017: 2 · 2016: 4 · 2015: 2 · 2014: 2 · 2013: 4 · 2012: 6
- FWCI
- 3.85
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (acceptedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- ePrints Soton (University of Southampton) (submittedVersion)
- arXiv (Cornell University)
- Funders
- CERN
- Fundação para a Ciência e a Tecnologia
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- Einstein-Maxwell theory, higher-dimensional black holes, extremal black holes, de Sitter space, anti-de Sitter space, constant curvature manifolds, degenerate horizons
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