On Local Characterization Results in Geometry and Gravitation

Mars, Marc

doi:10.1007/978-3-319-60039-0_18

Cited by 5 publications

(13 citation statements)

References 57 publications

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“…This notion has a natural generalization (T a [g, φ] = 0) to spacetimes equipped with scalar or tensor fields (φ), with equivalence still given by isometric diffeomorphisms that also transform the additional scalars or tensors into each other. A nice historical survey of this and other local characterization results can be found in [22].…”

Section: Introductionmentioning

confidence: 92%

IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

Canepa

Dappiaggi

Khavkine

2018

Class. Quantum Grav.

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In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric g0 consists of a set of tensorial equations T [g] = 0, constructed covariantly out of the metric g, its Riemann curvature and their derivatives, that are satisfied if and only if g is locally isometric to the reference spacetime metric g0. The same notion can be extended to also include scalar or tensor fields, where the equations T [g, φ] = 0 are allowed to also depend on the extra fields φ. We give the first IDEAL characterization of cosmological FLRW spacetimes, with and without a dynamical scalar (inflaton) field. We restrict our attention to what we call regular geometries, which uniformly satisfy certain identities or inequalities. They roughly split into the following natural special cases: constant curvature spacetime, Einstein static universe, and flat or curved spatial slices. We also briefly comment on how the solution of this problem has implications, in general relativity and inflation theory, for the construction of local gauge invariant observables for linear cosmological perturbations and for stability analysis.

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Section: Introductionmentioning

confidence: 92%

IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

Canepa

Dappiaggi

Khavkine

2018

Class. Quantum Grav.

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“…Remark 3.8. It is well-known that the Ricci curvature Ric g of a Lorentzian four-manifold admitting parallel spinors is of the form Ric g = f u ⊗ u for some function f ∈ C ∞ (M ) [20]. Nonetheless, and to the best of our knowledge, Equation (3.13) is the first precise characterization of such function f in the case of globally hyperbolic Lorentzian four-manifolds.…”

Section: 1mentioning

confidence: 99%

“…In order to illustrate the various uses of Proposition 2.3 and make contact with the existing literature, in this subsection we recover the well-known local characterization of a Lorentzian four-manifold (M, g) admitting a parallel spinor, obtaining along the way the global characterization of standard Brinkmann space-times that admit a parallel spinor, which seems to be new in the literature. Recall that by definition a Brinkmann space-time [6,20] is a Lorentzian four manifold equipped with a complete parallel null vector. Let (u, [l]) be a parallel parabolic pair on (M, g), which by Proposition 2.3 is equivalent to the existence of a parallel spinor.…”

Section: Parallel Spinors On Lorentzian Four-manifoldsmentioning

confidence: 99%

“…By uniformization, we conclude that X is diffeomorphic to either R 2 , R 2 \ {0} or T 2 . Appropriately choosing local coordinates the previous result directly implies that a four-dimensional space-time admitting parallel spinors is locally isometric to a pp-wave [15,20].…”

Section: Parallel Spinors On Lorentzian Four-manifoldsmentioning

confidence: 99%

“…This problem is well-posed by the results of Leistner and Lichewski, who proved the statement in arbitrary dimension [19,18]. Existence of a parallel spinor field is obstructed since it implies (M, g) to be a solution of Einstein equations with pure radiation type of energy momentum tensor [20]. This fact connects the study of parallel real spinors to the study of globally hyperbolic Lorentzian manifolds satisfying a given curvature condition, which has been a fundamental problem in global Lorentzian geometry since the seminal work of G. Mess [21], see [5] and references therein for more details.…”

Section: Introductionmentioning

confidence: 99%

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Parallel spinor flows on three-dimensional Cauchy hypersurfaces

Murcia¹,

Shahbazi²

2021

Preprint

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The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian metrics admitting parallel spinors are not necessarily Ricci flat, the parallel spinor flow preserves the vacuum momentum and Hamiltonian constraints and therefore the Einstein and parallel spinor flows coincide on common initial data. Using this result, we provide an initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds. Furthermore, we explicitly solve the left-invariant parallel spinor flow on simply connected Lie groups, obtaining along the way necessary and sufficient conditions for the flow to be immortal. These are, to the best of our knowledge, the first non-trivial examples of evolution flows of parallel spinors. Finally, we use some of these examples to construct families of η -Einstein cosymplectic structures and to produce solutions to the left-invariant Ricci flow in three dimensions. This suggests the intriguing possibility of using first-order hyperbolic spinorial flows to construct special solutions of curvature flows.

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From Riemannian to Relativistic Diffusions

Franchi

2017

From Riemann to Differential Geometry and Relativity

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On Local Characterization Results in Geometry and Gravitation

Cited by 5 publications

References 57 publications

IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

From Riemannian to Relativistic Diffusions

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