Bianchi identities in higher dimensions

Abstract

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension $n$. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in $n-4$ spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

25 pages, v3: Corrections to Appendix B as given in Erratum-ibid.24:1691,2007 are now incorporated (A factor of 2 was missing in certain Bianchi equations.)

References in corpus (4)

• Classification of the Weyl Tensor in Higher Dimensions
• Particle and light motion in a space-time of a five-dimensional rotating black hole
• All spacetimes with vanishing curvature invariants
• Generalizations of pp-wave spacetimes in higher dimensions

Cited by in corpus (77)

• Black Holes in Higher Dimensions
• Black holes, hidden symmetries, and complete integrability
• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Alignment and algebraically special tensors in Lorentzian geometry
• Vanishing Scalar Invariant Spacetimes in Higher Dimensions
• On Spacetimes with Constant Scalar Invariants
• Kundt Spacetimes
• Robinson-Trautman spacetimes in higher dimensions
• Algebraic classification of higher dimensional spacetimes based on null alignment
• Generalization of the Geroch-Held-Penrose formalism to higher dimensions
• Type D Einstein spacetimes in higher dimensions
• Ultraspinning instability of rotating black holes
• Ricci identities in higher dimensions
• Higher dimensional VSI spacetimes
• General Kundt spacetimes in higher dimensions
• Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension
• Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction
• Higher dimensional Kerr-Schild spacetimes
• Lorentzian spacetimes with constant curvature invariants in three dimensions
• Kerr-Schild spacetimes with (A)dS background
• WANDs of the Black Ring
• Type III and N universal spacetimes
• Robinson-Trautman spacetimes with an electromagnetic field in higher dimensions
• Higher dimensional bivectors and classification of the Weyl operator
• Exact solutions to quadratic gravity
• Killing-Yano tensors and multi-hermitian structures
• Interpreting spacetimes of any dimension using geodesic deviation
• On a five-dimensional version of the Goldberg-Sachs theorem
• Type III and N solutions to quadratic gravity
• Perturbations of higher-dimensional spacetimes
• Separability of the massive Dirac's equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor
• Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
• Algebraically special axisymmetric solutions of the higher-dimensional vacuum Einstein equation
• Classification of Higher Dimensional Spacetimes
• Type III and N Einstein spacetimes in higher dimensions: general properties
• Spinor classification of the Weyl tensor in five dimensions
• The complex Goldberg-Sachs theorem in higher dimensions
• Electromagnetic fields with vanishing scalar invariants
• On asymptotically flat algebraically special spacetimes in higher dimensions
• A higher-dimensional generalization of the geodesic part of the Goldberg-Sachs theorem
• Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment
• Type II universal spacetimes
• On higher dimensional Einstein spacetimes with a warped extra dimension
• The perturbation theory of higher dimensional spacetimes a la Teukolsky
• Isolated horizons in higher-dimensional Einstein-Gauss-Bonnet gravity
• Explicit algebraic classification of Kundt geometries in any dimension
• Charging Kerr-Schild spacetimes in higher dimensions
• Asymptotic structure of radiation in higher dimensions
• Physical interpretation of Kundt spacetimes using geodesic deviation
• Extended Kerr-Schild spacetimes: General properties and some explicit examples
• Universal electromagnetic fields
• Refinements of the Weyl tensor classification in five dimensions
• Optical structures, algebraically special spacetimes, and the Goldberg-Sachs theorem in five dimensions
• Ultrarelativistic boost of the black ring
• Asymptotic behaviour of the Weyl tensor in higher dimensions
• On the Goldberg-Sachs theorem in higher dimensions in the non-twisting case
• Kerr-Schild double copy for Kundt spacetimes of any dimension
• On algebraically special vacuum spacetimes in five dimensions
• Supersymmetric isolated horizons
• Absence of gyratons in the Robinson-Trautman class
• On the uniqueness of the Myers-Perry spacetime as a type II(D) solution in six dimensions
• A new special class of Petrov type D vacuum space-times in dimension five
• On the algebraic classification of spacetimes
• Type II Einstein spacetimes in higher dimensions
• Spinor calculus on 5-dimensional spacetimes
• Gravitational wave solutions of quadratic curvature gravity using a null coframe formulation
• Weyl double copy in type D spacetime in four and five dimensions
• Five dimensional Weyl double copy
• On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction
• Spacetimes of Weyl and Ricci type N in higher dimensions
• All non-expanding gravitational waves in $D$-dimensional (anti-)de Sitter space
• On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions
• Killing vectors in higher dimensional spacetimes with constant scalar curvature invariants
• On well-posedness and algebraic type of the five-dimensional charged rotating black hole with two equal-magnitude angular momenta
• Asymptotic properties of gravitational and electromagnetic fields in higher dimensions
• A peeling theorem for the Weyl tensor in higher dimensions
• On type II(D) Einstein spacetimes in six dimensions