Conformal regularization of Einstein's field equations

Röhr, Niklas; Uggla, Claes

doi:10.1088/0264-9381/22/17/026

Cited by 19 publications

(66 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This could have been avoided to a considerable extent by using other variables. Replacing s i with E i = g ii /H , i.e., the Hubble-normalized spatial frame variables of [7,13], and using y = m 2 H −2 instead of z, yields a single Kasner circle on the massless boundary instead of three. The latter variables, however, are not bounded; indeed, they blow up towards the future in the present case.…”

Section: Discussionmentioning

confidence: 99%

“…However, E i -variables, or 'E i -based' variables, would have yielded a more direct physical interpretation, and would have been more suitable to relate the present results to a larger context; but it is not difficult to translate our results to the E i -variables, used in e.g. [7,13], where the relationship between the dynamics of inhomogeneous and spatially homogeneous models was investigated and exploited.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamics of the spatially homogeneous Bianchi type I Einstein–Vlasov equations

Heinzle

Uggla

2006

Class. Quantum Grav.

Self Cite

View full text Add to dashboard Cite

We investigate the dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations with Bianchi type I symmetry by introducing a new formulation that allows an efficient use of dynamical systems methods. We find that all models are forever expanding and that they isotropize towards the future; towards the past there exists a singularity-we identify and describe all possible past asymptotic states. In this context, we establish the existence of a heteroclinic network, which is a new type of feature in general relativity. The past asymptotic structure illustrates that the dynamics of Vlasov matter models differs significantly from that of perfect fluid models.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Dynamics of the spatially homogeneous Bianchi type I Einstein–Vlasov equations

Heinzle

Uggla

2006

Class. Quantum Grav.

Self Cite

View full text Add to dashboard Cite

show abstract

“…the metric G. The object Σ αβ is the (trace-less) shear, R α is the Fermi-rotation, which describes how the frame rotates w.r.t. a Fermi propagated frame, associated with the vector ∂ ∂ ∂ 0 and the conformal metric G. In (3.4b), N αβ and A α are spatial commutator functions that describe the three-curvature of G, see below; we also refer to [12,24] where the analogous non-normalized objects are described.…”

Section: Conformal Hubble Normalizationmentioning

confidence: 99%

“…From the Hubble-normalized commutator equations, the Einstein field equations, and the Jacobi identities, see [24], we obtain a dimensionless system of coupled equations; we split the system into gauge equations, evolution equations, and constraint equations:…”

Section: Conformal Hubble Normalizationmentioning

confidence: 99%

The cosmological billiard attractor

Heinzle¹,

Uggla²,

Röhr³

2009

Advances in Theoretical and Mathematical Physics

Self Cite

231

View full text Add to dashboard Cite

This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike singularities, we derive in a step-by-step manner the cosmological billiard conjecture: we show that the generic asymptotic dynamics of solutions is represented by (randomized) sequences of heteroclinic orbits on the "billiard attractor". Our analysis rests on two pillars: (i) a dynamical systems formulation based on the conformal Hubblenormalized orthonormal frame approach expressed in an Iwasawa frame; (ii) stochastic methods and the interplay between genericity and stochasticity. Our work generalizes and improves the level of rigor of previous work by Belinskii, Khalatnikov, and Lifshitz; furthermore, we establish that our approach and the Hamiltonian approach to "cosmological billiards", as elaborated by Damour, Hennaux, and Nicolai, can be viewed as yielding "dual" representations of the asymptotic dynamics.e-print archive: http://lanl.arXiv.org/abs/gr-qc/0702141 J. MARK HEINZLE, CLAES UGGLA, AND NIKLAS RÖHR

show abstract

“…Writing the Einstein equations in terms of Hubble normalized scale invariant frame variables due to Uggla et al [1,47,52] or the approach of Damour et al [22], based on Iwasawa decomposition, each of which relies on a long history of previous formulations, gives a description of the asymptotics of the gravitational field in terms of billiard dynamical systems. In the case of the Hubble normalized formulation, one gets a billiard in the Kasner plane, cf [56], while for the Iwasawa formulation one gets for the case of 3 + 1 gravity, a billiard in a domain of hyperbolic space, which is analogous to the Misner-Chitre billiard.…”

Section: Stochastic Aspects Of Generic Singularitiesmentioning

confidence: 99%

Summary of session A3: mathematical studies of the field equations

Andersson¹

2008

Class. Quantum Grav.

View full text Add to dashboard Cite

In this report I will give a summary of some of the main topics covered in session A3, mathematical studies of the field equations, at GRG18, Sydney. Unfortunately, due to length constraints, some of the topics covered at the session will be very briefly mentioned or left out altogether. The summary is mainly based on extended abstracts submitted by the speakers and some of those who presented posters at the session. I would like to thank all participants for their contributions and help with this summary.PACS numbers: 04.20.−q, 04.25.−g, 04.40.−b The Buchdahl inequalityThe Buchdahl inequality, which is included in most textbooks on general relativity, states that for a static, self-gravitating body, 2M/R 8/9,where M is the ADM mass and R the area radius of the boundary of the static body. The proof by Buchdahl, cf [16] of (1) assumed that the energy density is non-increasing outwards and that the pressure is isotropic. A bound on 2M/R has an immediate observational consequence: if 2M/R < 8/9 then the gravitational red shift is less than 2 but if 2M/R approaches 1 the red shift is unbounded. The assumptions used to derive the inequality are very restrictive, and as, e.g. pointed out by Guven andÓ Murchhada [27] neither of them hold in a simple soap bubble and they do not approximate any known topologically stable field configuration. In addition to the restrictions implied by the hypotheses made by Buchdahl there are two other disadvantages associated with this inequality: it refers to the boundary of the body (i.e. the interior is excluded) and the solution which gives equality in the inequality is the Schwarzschild interior solution which has constant energy density and for which the pressure blows up at the center. In particular it violates the dominant energy condition. Andréasson explained recent work [2], where all four restrictions described above are eliminated. He considered matter models for which the energy density ρ and the radial pressure p is non-negative and which satisfies p + 2p T ρ, where is a non-negative constant and p T is the tangential pressure, and showed thatHere m is the quasi-local mass so that M = m(R). The case = 1 gives the bound 2m/r 8/9. Among the matter models which satisfy the conditions stated are Vlasov matter and matter which satisfies the dominant energy condition and has non-negative pressure. Inequality (2) is sharp in the sense of measures and there are examples which come arbitrarily close to saturating the inequality. Andréasson has recently generalized the inequality to the case of charged spheres [3]. Initial-boundary value problemsA long standing problem in the analytical and numerical study of the Einstein equations is the choice of boundary conditions for boundaries near 'infinity'. As is well known, the Einstein equations in harmonic coordinates reduces to a quasilinear system of wave equations. It was proved in previous work by Kreiss and Winicour cf [36] that the initial-boundary value problem is well posed for such systems with the Sommerfeld outgoin...

show abstract

Conformal regularization of Einstein's field equations

Cited by 19 publications

References 27 publications

Dynamics of the spatially homogeneous Bianchi type I Einstein–Vlasov equations

Dynamics of the spatially homogeneous Bianchi type I Einstein–Vlasov equations

The cosmological billiard attractor

Summary of session A3: mathematical studies of the field equations

Contact Info

Product

Resources

About