Type III and N Einstein spacetimes in higher dimensions: general properties

Abstract

The Sachs equations governing the evolution of the optical matrix of geodetic WANDs (Weyl aligned null directions) are explicitly solved in n-dimensions in several cases which are of interest in potential applications. This is then used to study Einstein spacetimes of type III and N in the higher dimensional Newman-Penrose formalism, considering both Kundt and expanding (possibly twisting) solutions. In particular, the general dependence of the metric and of the Weyl tensor on an affine parameter r is obtained in a closed form. This allows us to characterize the peeling behaviour of the Weyl "physical" components for large values of r, and thus to discuss, e.g., how the presence of twist affects polarization modes, and qualitative differences between four and higher dimensions. Further, the r-dependence of certain non-zero scalar curvature invariants of expanding spacetimes is used to demonstrate that curvature singularities may generically be present. As an illustration, several explicit type N/III spacetimes that solve Einstein's vacuum equations (with a possible cosmological constant) in higher dimensions are finally presented.

19 pages

References in corpus (11)

• Robinson-Trautman spacetimes in higher dimensions
• Generalization of the Geroch-Held-Penrose formalism to higher dimensions
• Type D Einstein spacetimes in higher dimensions
• Ricci identities in higher dimensions
• Higher dimensional VSI spacetimes
• General Kundt spacetimes in higher dimensions
• Higher dimensional Kerr-Schild spacetimes
• Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
• On asymptotically flat algebraically special spacetimes in higher dimensions
• Higher Dimensional Bondi Energy with a Globally Specified Background Structure
• Twisting type N vacuums with cosmological constant

Cited by in corpus (21)

• Algebraic classification of higher dimensional spacetimes based on null alignment
• Kerr-Schild spacetimes with (A)dS background
• Type III and N universal spacetimes
• Interpreting spacetimes of any dimension using geodesic deviation
• On a five-dimensional version of the Goldberg-Sachs theorem
• AdS-Wave Solutions of f(Riemann) Theories
• Type III and N solutions to quadratic gravity
• Spinor classification of the Weyl tensor in five dimensions
• On higher dimensional Einstein spacetimes with a warped extra dimension
• Charging Kerr-Schild spacetimes in higher dimensions
• Refinements of the Weyl tensor classification in five dimensions
• On the Goldberg-Sachs theorem in higher dimensions in the non-twisting case
• Asymptotic behaviour of the Weyl tensor in higher dimensions
• Exact Solutions of General Relativity and Quadratic Gravity in Arbitrary Dimension
• Spacetimes of Weyl and Ricci type N in higher dimensions
• Anisotropic Schrödinger black holes with hyperscaling-violation
• On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction
• Lovelock vacua with a recurrent null vector field
• On the uniqueness of the Kerr-(A)dS metric as a type II(D) solution in six dimensions
• All non-expanding gravitational waves in $D$-dimensional (anti-)de Sitter space
• On type II(D) Einstein spacetimes in six dimensions