Einstein Euler System Research Articles

Naked Singularities in the Einstein-Euler System

In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of selfsimilar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a selfsimilar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to flatten the selfsimilar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.

Local Stability to the Einstein-Euler System in Schwarzschild Space-time

Local Stability to the Einstein-Euler System in Schwarzschild Space-time

The linear stability of the n + 1 dimensional FLRW spacetimes

Here we prove the linear stability of a family of ‘n + 1’-dimensional cosmological models of general relativity locally isometric to the Friedmann Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. We show that the solutions to the linearized Einstein–Euler field equations around a class of FLRW metrics with compact spatial topology (negative Einstein spaces and in particular hyperbolic for n = 3) arising from regular initial data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish the result, we express the Einstein–Euler system in constant mean extrinsic curvature spatial harmonic gauge and linearize about the chosen FLRW background. Utilizing a Hodge decomposition of the fluid’s n-velocity one-form, the linearized system becomes elliptic–hyperbolic (and non-autonomous) in the CMCSH gauge facilitating an application of an energy type argument. Utilizing the estimates derived from the associated elliptic equations, we first prove the uniform boundedness of an energy functional (controlling an appropriate norm of the data) in the expanding direction. Utilizing the uniform boundedness, we later obtain a sharp decay estimate which suggests accelerated expansion (for Λ > 0) of this particular Universe model may be sufficient to control the non-linearities (including possible shock formation) of the Einstein–Euler system in a potential future proof of the fully non-linear stability. In addition, the rotational and harmonic parts of the fluid’s n-velocity field only couple to the remaining degrees of freedom in higher orders, which once again indicates a straightforward extension of current analysis to the fully non-linear setting in the sufficiently small data limit. In addition, our results require a certain integrability condition on the expansion factor and a suitable range of the adiabatic index γ ad ( i.e. in the physically relevant ‘3 + 1’ Universe) if the barotropic equation of state p = (γ ad − 1)ρ is chosen.

Turning Point Principle for Relativistic Stars

Upon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960’s Zel’dovich (Voprosy Kosmogonii 9:157–170, 1963) and Harrison et al. (Gravitation Theory and Gravitational Collapse. The University of Chicago press, Chicago, 1965) formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity.

Uniqueness of static, isotropic low-pressure solutions of the Einstein\u2013Vlasov system

In Beig and Simon (Commun Math Phys 144:373–390, 1992) the authors prove a uniqueness theorem for static solutions of the Einstein–Euler system which applies to fluid models whose equation of state fulfills certain conditions. In this article it is shown that the result of Beig and Simon (1992) can be applied to isotropic Vlasov matter if the gravitational potential well is shallow. To this end we first show how isotropic Vlasov matter can be described as a perfect fluid giving rise to a barotropic equation of state. This Vlasov equation of state is investigated, and it is shown analytically that the requirements of the uniqueness theorem are met for shallow potential wells. Finally the regime of shallow gravitational potential is investigated by numerical means. An example for a unique static solution is constructed, and it is compared to astrophysical objects like globular clusters. Finally we find numerical indications that solutions with deep potential wells are not unique.

Self–gravitating fluid flows with Gowdy symmetry near cosmological singularities

ABSTRACTWe consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T3 and, in this class, we solve the singular initial value problem for the Einstein–Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein–Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.

Elliptic equations in weighted Besov spaces on asymptotically flat Riemannian manifolds

This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein’s constraints equations. We establish existence theorems for the Hamiltonian and the momentum constraints with constant mean curvature and with a background metric that satisfies very low regularity assumptions. These results extend the regularity results of Holst et al. (Commun Math Phys 288:547–613, 2009) about the constraint equations on compact manifolds in the Besov space \({B_{p,p}^{s}}\), to asymptotically flat manifolds. We also consider the Brill–Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds (Choquet-Bruhat et al. in Chin Ann Math Ser B 27(1), 31–52, 2006; Maxwell in Commun Math Phys 253(3):561–583, 2005), as well as they enable us to construct the initial data for the Einstein–Euler system.

Local Existence of Solutions of Self Gravitating Relativistic Perfect Fluids

This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein–Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density might vanish or tend to zero at infinity, and that the pressure is a fractional power of the energy density. In this setting we prove local in time existence, uniqueness and well-posedness of classical solutions. The zero order term of our system contains an expression which might not be a C ∞ function and therefore causes an additional technical difficulty. In order to achieve our goals we use a certain type of weighted Sobolev space of fractional order. In Brauer and Karp (J Diff Eqs 251:1428–1446, 2011) we constructed an initial data set for these of systems in the same type of weighted Sobolev spaces. We obtain the same lower bound for the regularity as Hughes et al. (Arch Ratl Mech Anal 63(3):273–294, 1977) got for the vacuum Einstein equations. However, due to the presence of an equation of state with fractional power, the regularity is bounded from above.

Beyond second-order convergence in simulations of binary neutron stars in full general relativity

Abstract Despite the recent rapid progress in numerical relativity, a convergence order less than the second has so far plagued codes solving the Einstein–Euler system of equations. We report simulations of the inspiral of binary neutron stars in quasi-circular orbits computed with a new code employing high-order, high-resolution shock-capturing, finite-differencing schemes that, for the first time, go beyond the second-order barrier. In particular, without any tuning or alignment, we measure a convergence order above three both in the phase and in the amplitude of the gravitational waves. Because the new code is already able to calculate waveforms with very small phase errors at modest resolutions, we are able to obtain accurate estimates of tidal effects in the inspiral that are essentially free from the large numerical viscosity typical of lower order methods, and even for the challenging large compactness and small-deformability binary considered here. We find a remarkable agreement between our Richardson-extrapolated waveform and the one from the tidally corrected post-Newtonian (PN) Taylor-T4 model, with a de-phasing smaller than 0.4 rad during the seven orbits of the inspiral and up to the contact point. Because our results can be used reliably to assess the validity of the PN or other approximations at frequencies significantly larger than those considered so far in the literature, at these compactnesses, they seem to exclude significant tidal amplifications from next to next-to-leading-order terms in the PN expansion.

Linear perturbations of self-gravitating spherically symmetric configurations

We present a new covariant, gauge-invariant formalism describing linear metric perturbation fields on any spherically symmetric background in general relativity. The advantage of this formalism relies in the fact that it does not require a decomposition of the perturbations into spherical tensor harmonics. Furthermore, it does not assume the background to be vacuum, nor does it require its staticity. In the particular case of vacuum perturbations, we derive two master equations describing the propagation of arbitrary linear gravitational waves on a Schwarzschild black hole. When decomposed into spherical harmonics, they reduce to covariant generalizations of the well-known Regge-Wheeler and Zerilli equations. Next, we discuss the general case where the metric perturbations are coupled to matter fields and derive a new constrained wave system describing the propagation of three gauge-invariant scalars from which the complete metric perturbations can be reconstructed. We apply our formalism to the Einstein-Euler system, dividing the fluid perturbations into two parts. The first part, which decouples from the metric perturbations, obeys simple advection equations along the background flow and describes the propagation of the entropy and the vorticity. The second part describes a perturbed potential flow, and together with the metric perturbations it forms a closed wave system.

Spherically Symmetric Equilibria for Self-Gravitating Kinetic or Fluid Models in the Nonrelativistic and Relativistic Case---A Simple Proof for Finite Extension

We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding growth condition on the equation of state p=P(\rho) at \rho=0. These purely local conditions are slight generalizations to known such conditions.

Well-posedness of the Einstein–Euler system in asymptotically flat spacetimes: The constraint equations

This paper deals with the construction of initial data for the coupled Einstein–Euler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order. The common Lichnerowicz–York scaling method (Choquet-Bruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state.

Nonconformally flat initial data for binary compact objects

A new method is described for constructing initial data for a binary neutron-star (BNS) system in quasi-equilibrium circular orbit. Two formulations for non-conformally flat data, waveless (WL) and near-zone helically symmetric (NHS), are introduced; in each formulation, the Einstein-Euler system, written in 3+1 form on an asymptotically flat spacelike hypersurface, is exactly solved for all metric components, including the spatially non-conformally flat potentials, and for irrotational flow. A numerical method applicable to both formulations is explained with an emphasis on the imposition of a spatial gauge condition. Results are shown for solution sequences of irrotational BNS with matter approximated by parametrized equations of state that use a few segments of polytropic equations of state. The binding energy and total angular momentum of solution sequences computed within the conformally flat -- Isenberg-Wilson-Mathews (IWM) -- formulation are closer to those of the third post-Newtonian (3PN) two point particles up to the closest orbits, for the more compact stars, whereas sequences resulting from the WL/NHS formulations deviate from the 3PN curve even more for the sequences with larger compactness. We think it likely that this correction reflects an overestimation in the IWM formulation as well as in the 3PN formula, by $\sim 1$ cycle in the gravitational wave phase during the last several orbits. The work suggests that imposing spatial conformal flatness results in an underestimate of the quadrupole deformation of the components of binary neutron-star systems in the last few orbits prior to merger.

Local existence of classical solutions for the Einstein–Euler system using weighted Sobolev spaces of fractional order

We prove the existence of a class of local in time solutions, including static solutions, of the Einstein–Euler system. This result is the relativistic generalisation of a similar result for the Euler–Poisson system obtained by Gamblin (1993). As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article. To cite this article: U. Brauer, L. Karp, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The initial value problem for a class of general relativistic fluid bodies

A body or collection of bodies made of perfect fluid can be described in general relativity by a solution of the Einstein–Euler system where the mass density has spatially compact support. It is shown that for certain equations of state there exists a wide class of solutions of this type corresponding to appropriate initial data given on a spacelike hypersurface. This class is not constrained by any symmetry requirements. The key element of the proof is to write the equations as a symmetric hyperbolic system which is regular both for nonvanishing density and in vacuum.