Current Topics in Exact Solutions

8-11 April 2014

Department of Mathematical Analysis

S22, lecture room A, Galglaan 2, 9000 Gent, Belgium

Tuesday 8

9:00-9:30

coffee

9:30-10:30

Marcello Ortaggio: Asymptotic properties of gravitational and electromagnetic fields in higher dimensions

10:45-11:45

Alan Barnes: Einstein Spacetimes with Constant Weyl Eigenvalues

12:00-14:00

lunch

14:00-15:00

Vladimir Manko: A novel approach to extending the Kerr solution

15:15-16:15

Radu Slobodeanu: Shear-free fluids with linear EoS in general relativity

17:00-18:00

reception
 
Wednesday 9

9:00-9:30

coffee

9:30-10:30

Raül Vera: Perturbed matching theory to second order

10:45-11:45

Borja Reina: Revisiting Hartle's perturbative model for slowly rotating stars

12:00-12:30

lunch

12:30-21:30

excursion

 
Thursday 10

9:00-9:30

coffee

9:30-10:30

Timothy Clifton: The Initial Value Problem in Discrete Cosmological Models

10:45-11:45

Kjell Rosquist: The Evolution of Discrete Cosmological Models

12:00-14:00

lunch

14:00-15:00

José Natário: Gravito-electromagnetic analogies, physical significance of the curvature invariants and all that

15:15-16:15

Filipe Costa:  Dynamics of extended test bodies in General Relativity --- the problem of the representative worldline

 
Friday 11

9:00-9:30

coffee

9:30-10:30

José Senovilla: Spacetimes with linear differential conditions on the curvature

10:45-11:45

Michael Bradley: The quadrupole moment of rotating fluid balls

12:00-14:00

lunch

14:00-15:00

Lode Wylleman: Five dimensional Petrov type D spacetimes

15:15-16:15

t.b.a.

 

Abstracts and reading material

 

Alan Barnes Einstein Spacetimes with Constant Weyl Eigenvalues

Einstein spacetimes (that is vacuum spacetimes possibly with a non-zero cosmological constant Λ) with a constant non-zero Weyl eigenvalue are considered. For type II & D this assumption alone is sufficient to allow one to prove that the non-repeated eigenvalue necessarily has the value 2Λ/3 and it turns out that the only possible spacetimes are some Kundt-waves considered by Lewandowski which are type II and a Robinson-Bertotti solution of type D. For Petrov type I the only solution in which all three Weyl eigenvalues are constant turns out to be a homogeneous pure vacuum solution found long ago by Petrov using group theoretic methods. These results can be summarised by the statement that the only vacuum spacetimes with constant Weyl eigenvalues are either homogeneous or are Kundt spacetimes. This result is similar to that of Coley et al. who proved their result for general spacetimes under the assumption that all scalar invariants constructed from the curvature tensor and all its derivatives were constant. Some preliminary results are also presented for Petrov Type I vacua in which either only one of the Weyl eigenvalues is constant or in which the ratios of the Weyl eigenvalues are constants. In particular in each case there is a simple algebraic relation between the Newman-Penrose Weyl tensor component Ψ2 and the ’cross-ratio’ of the optical scalars κνσλ of the associated principal null tetrad of the Weyl tensor.

(a provisory manuscript can be obtained from the author or from the organizers)

Michael Bradley The quadrupole moment of rotating fluid balls

 

 

Timothy Clifton: The Initial Value Problem in Discrete Cosmological Models

 

In this talk I will present recent results on the application of geometrostatics to cosmological models. This technique allows one to find vacuum solutions to the initial value problem that describe an arbitrary number of black holes. We arrange these black holes in regular configurations on a 3-sphere, and compare the resulting geometry to that of a Friedmann-Lemaitre-Robertson-Walker solution with the same total mass. This process allows new cosmological models to be constructed that are (in some sense) approximately homogeneous and isotropic on large scales, while being highly inhomogeneous on small scales.

 

Suggested references:

Dynamics of a lattice universe by the Schwarzschild-cell method (http://adsabs.harvard.edu/abs/1957RvMP...29..432L)
The method of images in geometrostatics (http://adsabs.harvard.edu/abs/1963AnPhy..24..102M)
Interaction energy in geometrostatics (http://adsabs.harvard.edu/abs/1963PhRv..131..471B)
An exact quantification of backreaction in relativistic cosmology (http://arxiv.org/abs/1203.6478)
Backreaction and continuum limit in a closed universe filled with black holes (http://arxiv.org/abs/1312.0494)

 

Vladimir Manko A novel approach to extending the Kerr solution

 

Singularities in the Kerr–Newman and charged δ = 2 Tomimatsu–Sato spacetimes endowed with negative mass

 

Are known maximal extensions of the Kerr and Kerr-Newman spacetimes physically meaningful and analytic?

 

Filipe Costa Dynamics of extended test bodies in General Relativity --- the problem of the representative worldline

José Natário Gravito-electromagnetic analogies, physical significance of the curvature invariants and all that

Gravito-electromagnetic analogies

Marcello Ortaggio Asymptotic properties of gravitational and electromagnetic fields in higher dimensions

http://arxiv.org/abs/1403.7559

Borja Reina Revisiting Hartle's perturbative model for slowly rotating stars

We explore the perturbative stationary and axisymmetric matching around a spherically symmetric background static matched configuration to second order between a rotating perfect fluid interior and an asymptotically flat vacuum exterior. To do that we provide a consistent analysis using modern perturbative theory, and, in particular, the theory of perturbative matchings to second order [1]. The paticularisation to Hartle's setting, i.e. the explicit assumptions used in [1], thus provides a firm ground where to study the importance (or lack thereof) of the implicit assumptions made to construct the original model in [1] and further developments.

Previous related material can be found in the paper by Michael Bradley et al. "Slowly rotating fluid balls of Petrov type D" Phys.Rev.D75:024013,2007

Kjell Rosquist The Evolution of Discrete Cosmological Models

In this talk I will report on general relativistic cosmological models in which the sources are discrete. The distribution of the sources is assumed to be homogeneous and isotropic on large scales. Such models belong to the class of locally inhomogeneous models.  One motivation for considering this kind of models is to obtain a closer correspondence with the actual universe as compared to models in which the matter is a fluid. The relation between the evolution of exactly homogeneous vs locally inhomogeneous models is a nontrivial and difficult problem because of the nonlinear nature of the Einstein equations. In the talk I will give examples of discrete models in which the evolution is tracked by analytical and semi-analytical methods. Their relation to exactly homogeneous and isotropic (FLRW) models will also be discussed.

José Senovilla  Spacetimes with linear differential conditions on the curvature.

 

In Lorentzian geometry, the traditional locally symmetric spacetimes, and the recurent spacetimes, can be generalized to involve higher order derivatives of the Riemann tensor. In particular, we have recently studied and identified the "2nd-order symmetric" spacetimes ---those with a vanishing second covariant derivative of the Riemann tensor. This opens up a new entire world of interesting spacetimes, by assuming linear differential conditions on the Riemann tensor. Some properties of these spacetimes and their interest will be analyzed. In particular, their intimate relationship with plane-wave spacetimes and thereby with Penrose limits will be considered".

Radu Slobodeanu Shear-free fluids with linear EoS in general relativity

Shear-free perfect fluids with linear equation of state

Raul Vera Perturbed matching theory to second order

The aim of this talk is to present the tools needed to revisit, from a consistent and rigorous point of view, Hartle's perturbative model of slowly rotating stars in GR [1]. Hartle's model consists essentially of the perturbative matching of a perfect fluid interior with an asymptotically flat vacuum exterior, up to second order in a rotation parameter. The plan for this talk is to review the general theory of perturbative matchings to second order, constructed by Marc Mars in 2005 [2].

References
[1] James B. Hartle The Astrophysical Journal 150 (1967) 1005-1029
[2] Marc Mars "First- and second-order perturbations of hypersurfaces" CQG 22 (2005) 3325–3347

Lode Wylleman

 

Last update: 5.3.2014