Einstein spaces as attractors for the Einstein flow
Open accessAbstract
In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of $n + 1$-dimensional, spatially compact spacetimes, which generalizes the $k = -1$ Friedmann-Lemaître-Robertson-Walker vacuum spacetime. This work extends the result from Future complete vacuum spacetimes. The background spacetimes we consider are Lorentz cones over negative Einstein spaces of dimension $n \ge 3$. We use a variant of the constant mean curvature, spatially harmonic (CMCSH) gauge introduced in Elliptic-hyperbolic systems and the Einstein equations. An important difference from the $3+1$ dimensional case is that one may have a nontrivial moduli space of negative Einstein geometries. This makes it necessary to introduce a time-dependent background metric, which is used to define the spatially harmonic coordinate system that goes into the gauge. Instead of the Bel-Robinson energy used in Future complete vacuum spacetimes, we here use an expression analogous to the wave equation type of energy introduced in Elliptic-hyperbolic systems and the Einstein equations for the Einstein equations in CMCSH gauge. In order to prove energy estimates, it turns out to be necessary to assume stability of the Einstein geometry. Further, for our analysis it is necessary to have a smooth moduli space. Fortunately, all known examples of negative Einstein geometries satisfy these conditions. We give examples of families of Einstein geometries which have nontrivial moduli spaces. A product construction allows one to generate new families of examples. Our results demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.
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[13] showed through studying the Hamilton-Jacobi equation that (g,K,N,X) solves the Einstein equations for all τ given initial data satisfying the constraints.
“This is exactly shown in [13] through studying the associated Hamilton Jacobi equation for the reduced dynamics.”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Section 5 (Local Existence) establishes a solution for small initial data and says that constraints satisfied initially remain satisfied during evolution. Theorem 8.1 (Section 8, Future Complete Spacetimes) extends this to the future only under additional small-data and geometric assumptions. The paper uses energy estimates and a continuation argument, not an analysis of the Hamilton–Jacobi equation; it also does not establish existence for all τ without those qualifications. (LLM, conf=0.91)
Experimental — accuracy not yet measuredThe phase space of the reduced 2+1 gravity dynamics is the cotangent bundle T*TΣ_p (≈ R^{12p-12}).
“where the phase space of the dynamics was identified with the co-tangent bundle of Teichmüller space ( ≈ R 12 p -12 ).”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Section 2.5.2 (page unavailable) identifies Teichmüller space as the deformation space for a genus-p surface and gives its dimension as 6p−6, consistent with the stated dimension of a cotangent bundle. But our copy does not identify the reduced 2+1 gravity phase space as that cotangent bundle or state the claimed global equivalence to R^(12p−12). (LLM, conf=0.83)
Experimental — accuracy not yet measured[13] explicitly showed that every non-trivial solution curve of the reduced dynamics in the configuration space TΣp approaches a point ρ lying in the interior of Teichmüller space as τ -> 0-.
“Analyzing the associated Monge-Ampere equation, [13] explicitly showed that every non-trivial solution curve of the reduced dynamics in the configuration space ( T Σ p ) approaches a point ( ρ ) lying in the interior of the Teichmüller space, that is,”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Corollary 8.3 states that solution curves covered by Theorem 8.1 converge to a limiting metric in the deformation space. But Theorem 8.1 is restricted to sufficiently small data near a suitable Einstein metric; the paper does not establish this for every non-trivial curve in TΣp, nor does it derive the result by analyzing a Monge–Ampère equation. Its discussion of convergence to a point in Teichmüller space (Additional information; page unavailable) describes analogous results from other works. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThere is a natural homeomorphism between the space of transverse traceless tensors on (Σp,γ) and the Teichmüller space from the Einstein flow.
“In addition, we have a natural homeomorphism between the space of transverse traceless tensors on (Σ p , γ ) and the Teichmüller space from the Einstein flow (for a detailed analysis see [13]).”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Section 1.4 (page unavailable) says that, for higher-genus surfaces in two dimensions, TT tensors describe the tangent space to the Teichmüller space of self-similar Einstein-flow solutions. Section 2.5.2 identifies Teichmüller space as the deformation space in that setting. These passages support a tangent-space correspondence, but do not establish the claimed natural homeomorphism between the full space of TT tensors and Teichmüller space. (LLM, conf=0.86)
Experimental — accuracy not yet measuredMoncrief's ray structure of Teichmüller space is defined by equation (3.39) together with an associated Hamilton-Jacobi equation.
“Now we will describe Moncrief's ray structure [13] of the Teichmüller space, which will be of crucial in obtaining the main result. The ray structure defined by Moncrief is the following equation”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Section 2.5.2 identifies Teichmüller space as a deformation space, but does not describe a ray structure. Searches for “ray structure,” “(3.39),” and “Hamilton-Jacobi” found no defining equation in our copy; “Hamilton-Jacobi” appears only in a reference title. (LLM, conf=0.75)
Experimental — accuracy not yet measured
S satisfies the Hamilton-Jacobi equation for all T in (0,∞); in other words, S is dynamically complete.
“that is, S satisfies the Hamilton-Jacobi equation”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Section 4.2 describes S as a harmonic slice in the constraint set, while Section 5 discusses continuation of the Einstein flow. Neither those passages nor searches for “Hamilton-Jacobi” and “dynamically complete” address whether S satisfies a Hamilton-Jacobi equation for all T > 0. (LLM, conf=0.76)
Experimental — accuracy not yet measured[13] constructed a fully non-linear elliptic equation of Monge-Ampere type and showed that a unique solution exists.
“Using this equation (which is effectively the same as the Gauss map equation), [13] constructed a fully non-linear elliptic equation of Monge-Ampere type and showed that a unique solution of such equation exists.”
From Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space (2023) · cites this paper as [13]
Section 5 (Local Existence) discusses local well-posedness of the Einstein evolution system and uniqueness for a shift equation, not a Monge–Ampère-type equation. Searches for “Monge-Ampère,” “Gauss map equation,” and a unique solution to a nonlinear elliptic equation found no passage addressing the attributed construction or uniqueness result. The indexed passages provide no page numbers. (LLM, conf=0.76)
Experimental — accuracy not yet measured
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Cites
Lorentz spacetimes of constant curvature
Mess, G. (2007) · Geometriae Dedicata · doi:10.1007/s10711-007-9155-7 · Cited by 278 · see citing works
On the global evolution problem in 2 + 1 gravity
Anderson, L., Moncrief, V., & Tromba, A. J. (1997) · Journal of Geometry and Physics · doi:10.1016/s0393-0440(97)87804-7 · Cited by 72 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 5
- Lars Andersson, Thierry Barbot, Riccardo Benedetti, Francesco Bonsante, William M. Goldman, François Labourie, Kevin P. Scannell, & Jean-Marc Schlenker, Notes on: "Lorentz spacetimes of constant curvature" [Geom. Ded- icata 126 (2007), 3-45]
- by G. Mess, Geom. Dedicata 126 (2007), 47-70.
- Future complete vacuum spacetimes, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 299-330.
- Arthur L. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987.
- Vincent Moncrief, Relativistic Teichmüller theory-a Hamilton-Jacobi approach to 2 + 1-dimensional Einstein gravity, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 203-249.
Cited by
Sorted by citations
Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space
Mondal, P. (2023) · Journal of Mathematical Physics · doi:10.1063/5.0136631 · Cited by 1 · see citing works
- Authors
- Andersson, L., & Moncrief, V.
- Affiliations
- Max Planck Society (DE)
- University of Miami (US)
- Corresponding author
- Moncrief, Vincent
- ORCID
- Andersson, Lars
- Venue
- Journal of Differential Geometry
- Publisher
- International Press of Boston
- Published
- 2011-09-01
- Volume
- 89
- Issue
- 1
- Pages
- 1-47
- ISSN
- 0022-040X (ISSN-L)
- OpenAlex
- W1498155539
- Cited by
- 63
- References
- 2
- Citations by year
- 2026: 3 · 2025: 3 · 2024: 6 · 2023: 6 · 2022: 3 · 2021: 5 · 2020: 6 · 2019: 4 · 2018: 2 · 2017: 4 · 2016: 5 · 2015: 4 · 2014: 3 · 2013: 4 · 2012: 1
- FWCI
- 4.00
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Differential Geometry (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
- Project Euclid (Cornell University) (submittedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Max Planck Digital Library (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0647331, 0407732, 0707306, 0354391)
- Yale University
- University of Miami
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- Einstein spaces, global existence, cosmic expansion, moduli spaces, Lorentz cone, energy estimates, product construction
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