Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.     

Über Volumendefekte und Krümmung in Riemannschen Mannigfaltigkeiten mit Anwendungen in der Relativitätstheorie

On Defects of the Volume and Curvature in Riemannian Manifolds with Applications to General Theory of Relativity Due to J. Bertrand and V. Puiseux the Gaussian curvature K of a surface can be determined by geodesic measurements: construct the geodesic circle of radius r to some point P, measure the circumference L, then K can be calculated from the defect 2πr - L. There are similar relations for n-dimensional Riemannian manifolds with positive definite metric, H. Vermeil has proved that the curvature invariant can be determined from defects of the volume. In this paper we study 4-dimensional Riemannian manifolds with signature (? + + +). There are connections between defects of the volume, curvature invariant R and physical quantities of the general theory of relativity.     

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

Generalised G 2–Manifolds

We define new Riemannian structures on 7–manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G ₂, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type II supergravity theory with bosonic background fields. In particular, this geometry can be described by two linear metric connections with skew–symmetric torsion. Finally, we construct explicit examples by introducing the device of T–duality.On leave at: Centre de Mathématiques Ecole Polytechnique 91128 Palaiseau, France. E-mail: fwitt@math.polytechnique.fr     

The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.     

Proof of the Symmetry of the Off-Diagonal¶Hadamard/Seeley–deWitt's Coefficients in¶C ∞ Lorentzian Manifolds by a “Local Wick Rotation”

Completing the results achieved in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth D-dimensional Lorentzian manifolds. This result is relevant because it plays a central rôle in Physics, in particular in the theory of the stress-energy tensor renormalization procedure in quantum field theory in curved spacetime. To this end, it is shown that, in any Lorentzian manifold, a sort of "local Wick rotation" of the metric can be performed provided the metric is a (locally) analytic function of the coordinates and the coordinate are appropriate. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point (more generally, in a neighborhood of a space-like (Cauchy) hypersurface) into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or Kählerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to C^X non analytic Lorentzian manifolds by approximating Lorentzian C^X metrics by analytic metrics in common geodesically convex neighborhoods.     

Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature

Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent, or dual, metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles move on the shortest path. Thus one can visualize how acceleration in a gravitational field is explained by particles moving freely in a curved spacetime. Freedom in the dual metric allows us to display, with substantial curvature, even the weak gravity of our earth. This may provide a nice pedagogical tool for elementary lectures on general relativity. I also study extensions of the dual metric scheme to higher dimensions.     

Canonical connections of Finsler metrics and Finslerian connections on Riemannian metrics

A concept of canonical connection of a Finsler metric is developed. Connections that are compatible with Finsler metrics are compared with the canonical connection itself. They are also compared with the corresponding Cartan connection. A necessary and sufficient condition on metric Finsler connections is given for the metric to be Riemannian. This study unearths different ways in which Finsler geometry could be used to generalize the theory of general relativity.     

On the isotropy subalgebras of Lie algebras of conformal vector fields

The conformal isotropy algebra of a point m in an n-manifold with a metric of arbitrary signature is shown to be locally reducible, by a conformal change of the metric, to a homothetic algebra near m iff, by choice of a chart, its constituent vector fields are simultaneously linearisable at m and, for n≥3, a necessary and sufficient condition for this in terms of the first and second derivatives of these fields at m is given. The implications for the Riemannian case and the Lorentzian case are investigated. In contrast to the former, a Lorentzian manifold admitting a conformal vector field that is not linearisable at some point need not be conformally flat. Relevant four-dimensional examples are provided.     

Generalized Goldberg-Sachs theorems for pseudo-Riemannian four-manifolds

It has been recently observed that the generalized Goldberg-Sachs theorem in general relativity as well as some of its corollaries admit appropriate Riemannian versions. In this paper we use the formalism of spinors to give alternative proofs of these results clarifying the analogy between positive Hermitian structures of oriented Riemannian four-manifolds and shear-free congruences of oriented Lorentzian four-manifolds. We also prove similar results for oriented pseudo-Riemannian four-manifolds when the metric is of zero signature. This allows us to describe compact oriented four-manifolds possibly admitting a pseudo-Riemannian Einstein metric of zero signature whose positive Weyl tensor has two distinct eigenvalues corresponding to non-isotropic eigenspaces.     

Geometry from the spectral point of view

In this Letter, we develop geometry from a spectral point of view, the geometric data being encoded by a triple (A. H. D.) of an algebraA represented in a Hilbert spaceH with selfadjoint operatorD. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.

This paper is dedicated to the memory of J. Schwinger     

On a new formulation of the many-body problem in general relativity

A generalized Riemannian geometry is studied where the metric tensor is replaced by a matrix g of metrics. In this context new geometric quantities arise, which are trivial in ordinary Riemannian geometry. An application of this formalism to many-body alignments in general relativity is proposed, where the sub-constituents of the overall gravitational field are described by the components of g. The mutual gravitational interactions between the individual particles are encoded in specific tensors. In particular, very specific approximation schemes for Einstein’s field equations may be considered, which exclusively approximate those terms in the field equations which are due to interactions. The Newtonian limit as well as the first post-Newtonian approximation of the presented formalism is studied in order to display the interpretability of the presented formalism in terms of many-body alignments and in order to deduce a physical interpretation of the new geometric quantities.     

Variational equations on mixed Riemannian–Lorentzian metrics

A class of elliptic–hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fields on such a metric can be interpreted as the hodograph image of extremal surfaces in Minkowski 3-space. This suggests an approach to generalized Plateau problems in three-dimensional space-time via Hodge theory on the extended projective disc. Analogous variational problems arise on Riemannian–Lorentzian flow metrics in fiber bundles (twisted nonlinear Hodge equations), and on certain Riemannian–Lorentzian manifolds which occur in relativity and quantum cosmology. The examples surveyed come with natural gauge theories and Hodge dualities. This paper is mainly a review, but some technical extensions are proven.     

The L p Spectrum of Locally Symmetric Spaces with Small Fundamental Group

We determine the L ^p spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L ^p spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L ² case, the restriction on the injectivity radius is not needed.      

The Geometrical Meaning of Time

It is stated in many text books that the any metric appearing in general relativity should be locally Lorentzian i.e. of the type η _{μ ν} =diag (1,−1,−1,−1) this is usually presented as an independent axiom of the theory, which can not be deduced from other assumptions. The meaning of this assertion is that a specific coordinate (the temporal coordinate) is given a unique significance with respect to the other spatial coordinates. In this work it is shown that the above assertion is a consequence of requirement that the metric of empty space should be linearly stable and need not be assumed.     

The general four-dimensional spherically symmetric Finsler space

Finsler geometries give natural generalisations of Riemannian geometries, and hence possible natural extensions of general relativity. In this latter (gravitational) context, it is of particular interest to find the general spherically symmetric Finsler metric on ⁴. In this paper, we derive this metric. The general solution is given in two alternative forms, the second of which allows easy comparison with Riemannian-like metrics. We also derive the general axially symmetric Finsler metric on ⁴.     

Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z _ss =S(z,z _s ,z _t ,z _st ,s,t) and z _tt =T(z,z _s ,z _t ,z _st ,s,t), with z _s and z _t the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z _s ,z _t ,z _st ,s,t). When the S and T satisfy differential conditions analogous to those of the 3^rd order ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,z _s ,z _t ,z _st ) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. Received: 10 October 2000 / Accepted: 26 June 2001     

Mathematical Structures of Space-Time

At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables. Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christofel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychauduri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Namely, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with important previous papers. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy-momentum tensor, and we emphasize the role played by the full extrinsic curvature tensor and by the variation formulae.     

Basic principles of the Finsler theory of relativity. I

Physical principles are considered from which may be developed a generalization of relativity theory based on Finsler geometry, which is a metric generalization of Riemannian geometry.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 58–62, 1978.