Einstein-scalar Field Research Articles

On static solutions of the Einstein-Scalar Field equations

In this article we study self-gravitating static solutions of the Einstein-Scalar Field system in arbitrary dimensions. We discuss the existence of geodesically complete solutions depending on the form of the scalar field potential \(V(\phi )\), and provide full global geometric estimates when the solutions exist. The most complete results are obtained for the physically important Klein–Gordon field and are summarised as follows. When \(V(\phi )=m^{2}|\phi |^{2}\), it is proved that geodesically complete solutions have Ricci-flat spatial metric, have constant lapse and are vacuum, (that is \(\phi \) is constant and equal to zero if \(m\ne 0\)). In particular, when the spatial dimension is three, the only such solutions are either Minkowski or a quotient thereof (no nontrivial solutions exist). When \(V(\phi )=m^{2}|\phi |^{2}+2\Lambda \), that is, when a vacuum energy or a cosmological constant is included, it is proved that no geodesically complete solution exists when \(\Lambda >0\), whereas when \(\Lambda <0\) it is proved that no non-vacuum geodesically complete solution exists unless \(m^{2}<-2\Lambda /(n-1)\), (n is the spatial dimension) and the spatial manifold is non-compact. The proofs are based on novel techniques in comparison geometry a la Bakry-Emery that have their own interest.

Solutions to the Einstein-scalar field constraint equations with a small TT-tensor

In this paper, we prove a far-from-CMC result similar to Holst et al. (Phys Rev Lett 100(16):161101, 4, 2008), Holst et al. (Commun Math Phys 288(2):547–613, 2009), Maxwell (Math Res Lett 16(4):627–645, 2009), Gicquaud and Ngo (Class Quantum Grav 31(19):195014 (20 pp), 2014) for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe invariant.

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

Determination of the spacetime from local time measurements

We consider an inverse problem for a Lorentzian spacetime (M, g), and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold $$\Sigma \subset M$$ determine the $$C^\infty $$ -jet of the metric in the Fermi coordinates associated to $$\Sigma $$ . We use this result to study the global determination of the spacetime (M, g) when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that (M, g) is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry—the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.

Geometrization conditions for perfect fluids, scalar fields, and electromagnetic fields

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from the metric are unified and extended to arbitrary dimensions, to include a cosmological constant, and to include any self-interaction potential. Necessary and sufficient conditions on a four-dimensional spacetime metric for it to be an electrovacuum and formulas for constructing the electromagnetic field from the metric are generalized to include a cosmological constant. Both null and non-null electromagnetic fields are treated. A number of examples and applications of these results are presented.

Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory

In the present paper we prove a uniqueness theorem for the static and asymptotically flat solutions to the Einstein-scalar field equations which possess a photon sphere. We show that such solutions are uniquely specified by their mass $M$ and scalar charge $q$ and that they are isometric to the Janis-Newman-Winicour solution with the same mass and scalar charge subject to the inequality $\frac{{q}^{2}}{{M}^{2}}l3$.

Self tuning scalar fields in spherically symmetric spacetimes

We search for self tuning solutions to the Einstein-scalar field equations for the simplest class of `Fab-Four' models with constant potentials. We first review the conditions under which self tuning occurs in a cosmological spacetime, and by introducing a small modification to the original theory—introducing the second and third Galileon terms—show how one can obtain de Sitter states where the expansion rate is independent of the vacuum energy. We then consider whether the same self tuning mechanism can persist in a spherically symmetric inhomogeneous spacetime. We show that there are no asymptotically flat solutions to the field equations in which the vacuum energy is screened, other than the trivial one (Minkowski space). We then consider the possibility of constructing Schwarzschild de Sitter spacetimes for the modified Fab Four plus Galileon theory. We argue that the only model that can successfully screen the vacuum energy in both an FLRW and Schwarzschild de Sitter spacetime is one containing `John' ∼ Gμν ∂μϕ∂νϕ and a canonical kinetic term ∼ ∂αϕ ∂αϕ. This behaviour was first observed in [1]. The screening mechanism, which requires redundancy of the scalar field equation in the `vacuum', fails for the `Paul' term in an inhomogeneous spacetime.

A scenario for critical scalar field collapse in AdS3

We present a family of exact solutions, depending on two parameters α and b (related to the scalar field strength), to the three-dimensional Einstein-scalar field equations with negative cosmological constant Λ. For b = 0 these solutions reduce to the static Bañados–Teitelboim–Zanelli (BTZ) family of vacuum solutions, with mass . For , the solutions become dynamical and develop a strong spacelike central singularity. The solutions are black-hole like, with a global structure topologically similar to that of the BTZ black holes, and a finite effective mass. We show that the near-singularity behavior of the solutions with agrees qualitatively with that observed in numerical simulations of sub-critical collapse, including the independence of the near-critical regime on the angle deficit of the spacetime. We analyze in the approximation the linear perturbations of the self-similar threshold solution, , and find that it has only one unstable growing mode, which qualifies it as a candidate critical solution for scalar field collapse.

Neutron star structure in the presence of conformally coupled scalar fields

Neutron star models are studied in the context of scalar–tensor theories of gravity in the presence of a conformally coupled scalar field, using two different numerical equations of state (EoS) representing different degrees of stiffness. In both cases we obtain a complete solution by matching the interior numerical solution of the coupled Einstein-scalar field hydrostatic equations, with an exact metric on the surface of the star. These are then used to find the effect of the scalar field and its coupling to geometry, on the neutron star structure, particularly the maximum neutron star mass and radius. We show that in the presence of a conformally coupled scalar field, neutron stars are less dense and have smaller masses and radii than their counterparts in the minimally coupled case, and the effect increases with the magnitude of the scalar field at the center of the star.

Existence Results for the Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds in the Null Case

This is the second in our series of papers concerning positive solutions of the Einstein-scalar field Lichnerowicz equations. Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension $${n \geqslant 3}$$ , f and $${a \geqslant 0}$$ are two smooth functions on M with $${\int_M f dv_g < 0}$$ , sup M f > 0, and $${\int_M a dv_g > 0}$$ . In this article, we prove two results involving the following equation arising from the Hamiltonian constraint equation for the Einstein-scalar field equation in general relativity $$\Delta _g u = f u^{2^\star - 1} + \frac{a}{u^{2^\star +1}},$$ where $${\Delta_g = -{\rm div}_g(\nabla_g)}$$ and $${2^{\star} = 2n/(n - 2)}$$ . First, we prove that if either sup M f and $${\int}$$ M a dv g or sup M a is sufficiently small, the equation admits one positive smooth solution. Second, we show that the equation always admits one and only one positive smooth solution provided $${{\rm sup}_M f \leqslant 0}$$ . We should emphasize that we allow a to vanish somewhere. Along with these two results, existence and non-existence for related equations are also considered.

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions \(n = 3, 4, 5\). As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique.

On the constraints problem for the Einstein–Yang–Mills–Higgs system

We provide proofs of some key propositions that were used in previous work by Dossa and Tadmon dealing with the characteristic initial value problem for the Einstein–Yang–Mills–Higgs (EYMH) system. The aforesaid proofs were missing, making the considered work difficult to understand. This work is presented with a view to have an almost self-contained paper. With this respect we completely recall the process of constructing initial data for the EYMH system on two intersecting smooth null hypersurfaces as done in the work of Dossa and Tadmon mentioned above. This is achieved by successfully adapting the hierarchical method set up by Rendall to solve the same problem for the Einstein equations in vacuum and with perfect fluid source. Many delicate calculations and expressions are given in details so as to address, in a forthcoming work, the issue of global resolution of the characteristic initial value problem for the EYMH system. The method obviously applies to the Einstein–Maxwell and the Einstein-scalar field models as well.

New class of cosmological solutions for a self-interacting scalar field

New cosmological solutions are found to the system of Einstein scalar field equations using the scalar field φ as the argument. For a homogeneous and isotropic Universe, the system of equations is reduced to two equations, one of which is an equation of Hamilton–Jacobi type. Using the hyperbolically parameterized representation of this equation together with the consistency condition, explicit dependences of the potential V of the scalar field and the Hubble parameter H on φ are obtained. The dependences of the scalar field and the scale factor a on cosmic time t have also been found. It is shown that this scenario corresponds to the evolution of the Universe with accelerated expansion out to times distant from the initial singularity.

The spherically symmetric Einstein-scalar field system with positive and vanishing cosmological constant: a comparison

We review recent results concerning the spherically symmetric Einstein-scalar field system with positive cosmological constant. We do so by comparing with the classical results of Christodoulou concerning the asymptotically flat case (vanishing cosmological constant) and by discussing some of the issues which have emerged since the publication of our main results. Concerning the positive cosmological constant case, we also sketch an alternative proof of global in (Bondi) time existence, based on energy estimates, which is presumably more flexible and, consequently, amenable to generalizations; other potential improvements and generalizations of our main results are also discussed.

A new variable in scalar cosmology with an exponential potential

We present a new way describing the solution of the Einstein-scalar field theory with exponential potential $V\propto e^{\sqrt{6}\beta \phi/M_{Pl}}$ in spatially flat Friedmann-Robertson-Walker space-time. We introduced a new time variable, $L$, which may vary in $[-1,1]$. The new time represents the state of the universe clearly because the equation of state at a given time takes the simple form, $w= -1+ 2L^2$. The universe will inflate when $|L|<1/\sqrt{3}$. For $\beta\leq 1$, the universe ends with its evolution at $L=\beta$. This implies that the equation of state at the end of the universe is nothing but $w=-1+2\beta^2$. For $\beta \geq 1$, the universe ends at L=1, where the equation of state of the universe is one. On the other hand, the universe always begins with $w=1$ at $L=\pm 1$.

INFLATION AS AN ATTRACTOR IN SCALAR COSMOLOGY

We study an inflation mechanism based on attractor properties in cosmological evolutions of a spatially flat Friedmann–Robertson–Walker spacetime based on the Einstein-scalar field theory. We find a new way to get the Hamilton–Jacobi equation solving the field equations. The equation relates a solution "generating function" with the scalar potential. We analyze its stability and find a later time attractor which describes a Universe approaching to an eternal-de Sitter inflation driven by the potential energy, V0&gt;0. The attractor exists when the potential is regular and does not have a linear and quadratic terms of the field. When the potential has a mass term, the attractor exists if the scalar field is in a symmetric phase and is weakly coupled, λ&lt;9V0/16. We also find that the attractor property is intact under small modifications of the potential. If the scalar field has a positive mass-squared or is strongly coupled, there exists a quasi-attractor. However, the quasi-attractor property disappears if the potential is modified. On the whole, the appearance of the eternal inflation is not rare in scalar cosmology in the presence of an attractor.

Power law inflation with electromagnetism

We generalize Ringström’s global future causal stability results (Ringström 2009) [11] for certain expanding cosmological solutions of the Einstein-scalar field equations to solutions of the Einstein–Maxwell-scalar field system. In particular, after noting that the power law inflationary spacetimes (Mn+1,gˆ,ϕˆ) considered by Ringström (2009) in [11] are solutions of the Einstein–Maxwell-scalar field system (with exponential potential) as well as of the Einstein-scalar field system (with the same exponential potential), we consider (nonlinear) perturbations of initial data sets of these spacetimes which include electromagnetic perturbations as well as gravitational and scalar perturbations. We show that if (as in Ringström (2009) [11]) we focus on pairs of relatively scaled open sets UR0⊂U4R0 on an initial slice of (Mn+1,gˆ), and if we choose a set of perturbed data which on U4R0 is sufficiently close to that of (Mn+1,gˆ,ϕˆ,Aˆ=0), then in the maximal globally hyperbolic spacetime development (Mn+1,g,ϕ,A) of this data via the Einstein–Maxwell-scalar field equations, all causal geodesics emanating from UR0 are future complete (just as in (Mn+1,gˆ)). We also verify that, in a certain sense, the future asymptotic behavior of the fields in the spacetime developments of the perturbed data sets does not differ significantly from the future asymptotic behavior of (Mn+1,gˆ,ϕˆ,Aˆ=0).

The Problem of a Self-Gravitating Scalar Field with Positive Cosmological Constant

We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data given on a truncated null cone. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves to a causally geodesically complete spacetime (with boundary), which approaches a region of de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration, as well as a realization of the cosmic no-hair conjecture.

Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds

In this paper, we study the stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on compact Riemannian manifolds. In particular, in dimension no more than 5, we can find a different way (comparing with the previous result of Hebey, Pacard and Pollack) by showing that there are at least two positive solutions or a unique positive solution according to the coercivity property of a quadratic form defined by the minimal solution obtained by the monotone method. When the coercive condition fails, we prove a uniqueness result. A positive solution of the Lichnerowicz equation is also found in a complete non-compact Riemannian manifold.

On the spherically symmetric Einstein–Yang–Mills–Higgs equations in Bondi coordinates

We revisit and generalize, to the Einstein–Yang–Mills–Higgs (EYMH) system, previous results of Christodoulou and Chae concerning global solutions for the Einstein-scalar field and the Einstein–Maxwell–Higgs (EMH) equations. The novelty of the present work is twofold. For one thing, the assumption on the self-interaction potential is improved. For another thing, explanation is furnished why the solutions obtained here and those proved by Chae for the EMH system decay more slowly than those established by Christodoulou in the case of self-gravitating scalar fields. Actually, this latter phenomenon stems from the non-vanishing local charge in EMH and EYMH models.