Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions
Abstract We study the free data in the Fefferman–Graham expansion of asymptotically Einstein $$(n+1)$$ ( n + 1 ) -dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at $${\mathscr {I}}$$ I , D, and the free data at $...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-10-01
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| Series: | European Physical Journal C: Particles and Fields |
| Online Access: | https://doi.org/10.1140/epjc/s10052-021-09704-6 |
| _version_ | 1819024223420547072 |
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| author | Marc Mars Carlos Peón-Nieto |
| author_facet | Marc Mars Carlos Peón-Nieto |
| author_sort | Marc Mars |
| collection | DOAJ |
| description | Abstract We study the free data in the Fefferman–Graham expansion of asymptotically Einstein $$(n+1)$$ ( n + 1 ) -dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at $${\mathscr {I}}$$ I , D, and the free data at $${\mathscr {I}}$$ I , namely a certain traceless and transverse part of the n-th order coefficient of the expansion $$\mathring{g}_{(n)}$$ g ˚ ( n ) . In the case $$\Lambda <0$$ Λ < 0 and Lorentzian signature, it was known [23] that conformal flatness at $${\mathscr {I}}$$ I is sufficient for D and $$\mathring{g}_{(n)}$$ g ˚ ( n ) to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero $$\Lambda $$ Λ . We then explore whether conformal flatness of $${\mathscr {I}}$$ I is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic $$\Lambda $$ Λ -vacuum spacetimes exist. In the case of $${\mathscr {I}}$$ I non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve $$\mathring{g}_{(n)}$$ g ˚ ( n ) from the (now singular) electric part of the Weyl tensor. We then concentrate in the $$\Lambda >0$$ Λ > 0 case where the Cauchy problem at $${\mathscr {I}}$$ I of the Einstein vacuum field equations is known to be well-posed when the data at $${\mathscr {I}}$$ I are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at $${\mathscr {I}}$$ I to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity. |
| first_indexed | 2024-12-21T04:51:22Z |
| format | Article |
| id | doaj.art-f7ba7624c20f414a803726d483bf0a6f |
| institution | Directory Open Access Journal |
| issn | 1434-6044 1434-6052 |
| language | English |
| last_indexed | 2024-12-21T04:51:22Z |
| publishDate | 2021-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | European Physical Journal C: Particles and Fields |
| spelling | doaj.art-f7ba7624c20f414a803726d483bf0a6f2022-12-21T19:15:26ZengSpringerOpenEuropean Physical Journal C: Particles and Fields1434-60441434-60522021-10-01811012210.1140/epjc/s10052-021-09704-6Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensionsMarc Mars0Carlos Peón-Nieto1Instituto de Física Fundamental y Matemáticas, Universidad de SalamancaInstituto de Física Fundamental y Matemáticas, Universidad de SalamancaAbstract We study the free data in the Fefferman–Graham expansion of asymptotically Einstein $$(n+1)$$ ( n + 1 ) -dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at $${\mathscr {I}}$$ I , D, and the free data at $${\mathscr {I}}$$ I , namely a certain traceless and transverse part of the n-th order coefficient of the expansion $$\mathring{g}_{(n)}$$ g ˚ ( n ) . In the case $$\Lambda <0$$ Λ < 0 and Lorentzian signature, it was known [23] that conformal flatness at $${\mathscr {I}}$$ I is sufficient for D and $$\mathring{g}_{(n)}$$ g ˚ ( n ) to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero $$\Lambda $$ Λ . We then explore whether conformal flatness of $${\mathscr {I}}$$ I is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic $$\Lambda $$ Λ -vacuum spacetimes exist. In the case of $${\mathscr {I}}$$ I non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve $$\mathring{g}_{(n)}$$ g ˚ ( n ) from the (now singular) electric part of the Weyl tensor. We then concentrate in the $$\Lambda >0$$ Λ > 0 case where the Cauchy problem at $${\mathscr {I}}$$ I of the Einstein vacuum field equations is known to be well-posed when the data at $${\mathscr {I}}$$ I are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at $${\mathscr {I}}$$ I to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.https://doi.org/10.1140/epjc/s10052-021-09704-6 |
| spellingShingle | Marc Mars Carlos Peón-Nieto Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions European Physical Journal C: Particles and Fields |
| title | Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions |
| title_full | Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions |
| title_fullStr | Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions |
| title_full_unstemmed | Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions |
| title_short | Free data at spacelike $${\mathscr {I}}$$ I and characterization of Kerr-de Sitter in all dimensions |
| title_sort | free data at spacelike mathscr i i and characterization of kerr de sitter in all dimensions |
| url | https://doi.org/10.1140/epjc/s10052-021-09704-6 |
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