DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 597–600

SECTIONAL CURVATURE AND EINSTEIN’S EQUATIONS

G. S. HALL

Abstract. Some results regarding the sectional curvature function and its relation to the metric tensor for a space-time are given. They are then used to suggest an alternative viewpoint for Einstein’s general theory of relativity.

1. Introduction

Let M be a smooth manifold of dimension n ≥ 2 admitting a smooth metric g of arbitrary signature and, if p ∈ M, let T_{p}M denote the tangent space to M at p and let G_{p} denote the 2(n−2)-dimensional Grassman manifold of all 2-dimensional subspaces (2-spaces) of T_{p}M. Consider the real valued function σ_{p} defined on a certain subset Ḡ_{p} of G_{p} and given, for F ∈ Ḡ_{p}, by

σ_{p}(F) = \frac{g(R(X, Y)X, Y)}{g(X, X)g(Y, Y) − g(X, Y)^{2}} (1)

where R is the curvature structure arising from the Levi-Civita connection of g and where F is spanned by X, Y ∈ T_{p}M. It is easily checked that σ_{p}(F) does not depend on the spanning set X and Y chosen and Ḡ_{p} is then taken as the subset of members of G_{p} for which the denominator in (1) does not vanish (this non-vanishing also being independent of the spanning set chosen for F). The function σ_{p} is called the sectional curvature function at p and σ_{p}(F) the sectional curvature of F. If G = ⋃_{p∈M} Ḡ_{p} the function σ : G → R defined for F ∈ G by σ(F) = σ_{p}(F) if F ∈ Ḡ_{p} is the sectional curvature function (of M).

If g is positive definite, Ḡ_{p} = G_{p} and σ_{p} : G_{p} → R is a bounded real valued function on the compact connected Hausdorff manifold G_{p}.

Now let M be a space-time so that M is a smooth, connected, Hausdorff, 4dimensional manifold and g is smooth with Lorentz signature (taken as (+ + +−)). For p ∈ M, Ḡ_{p} is now the family of non-null 2-spaces at p and is neither compact nor connected. This complication leads to a richer and more interesting structure for σ. Let S_{p}, T_{p} and N_{p} be, respectively, the subsets of spacelike, timelike and null members of G_{p} defined, respectively, by the conditions that the denominator in (1) is positive, negative or zero. Thus G_{p} = S_{p} ∪ T_{p} ∪ N_{p} and Ḡ_{p} = S_{p} ∪ T_{p}. Now G_{p} is 4-dimensional and S_{p} and T_{p} are diffeomorphic open (4-dimensional) submanifolds of G_{p} whilst N_{p} is a 3-dimensional submanifold of G_{p} which is the boundary of S_{p} and T_{p} [1].

1991 Mathematics Subject Classification. 83C05.

Key words and phrases. Sectional curvature, Einstein’s equations, Petrov Classification, Symmetries.

This paper is in final form and no version of it will be submitted for publication elsewhere.

2. Space-Time Sectional Curvature

Let M be a space-time manifold with Lorentz metric g and sectional curvature function σ. If p ∈ M, σ_{p} is defined on Ḡ_{p} and is a constant function if and only if R takes the familiar “constant curvature” form at p. Under these conditions σ_{p} can, of course, be continuously extended to the whole of G_{p}. In [2] it was shown that if σ_{p} can be extended to the whole of G_{p} then R took the constant curvature form at p and σ_{p} is a constant function on G_{p}. However, a much stronger result is available [3]-[5] which says that if σ_{p} is continuously extendible to a single member of N_{p} then it is a constant function on Ḡ_{p} and hence extendible to G_{p}. Now suppose that the sectional curvature function σ is given on M and one asks how many Lorentz metrics on M could give rise to this sectional curvature. If one imposes the restriction that the given σ is such that for no p ∈ M is σ_{p} a constant function, one sees from the above discussion that if g and g′ each have σ as their sectional curvature function then g and g′ have the same subset of G_{p} (= domain σ_{p}) as their set of null 2-spaces for each p ∈ M. As a consequence of this g and g′ have the same null cones at each p ∈ M and hence are conformally related on M so that g′ = ϕg for some smooth function ϕ : M → R. But more can be said. If one applies further algebraic analysis and imposes the differential constraints of the Bianchi identities one finds that, apart from some very special cases, ϕ ≡ 1 on M and so σ uniquely determines the metric that gave rise to it.

Theorem 1. [5],[6] Let M be a space-time admitting Lorentz metrics g and g′ which give rise to the same sectional curvature function σ on M which for no p ∈ M is a constant function. Then, generically, g′ = g.

Here, the word generic should be clarified. In fact if Γ^{∞}(L) denotes the topological space of all C^{∞} Lorentz metrics on M with Whitney C^{∞} topology then there exists an open dense subset of Γ^{∞}(L) every member of which is uniquely determined by its sectional curvature function [7],[4]. The special types of sectional curvature functions to which the conclusions of theorem 1 do not apply are essentially the conformally flat null Einstein-Maxwell (or null fluid) plane waves [5] and need not be considered further here. A very important class of space-times in general relativity are the non-flat vacuum space-times (for which the Ricci tensor vanishes everywhere on M and the curvature tensor does not vanish over a non-empty open subset of M). For these space-times a much tighter result follows from theorem 1 without clauses.

Theorem 2. [5] Let M be a space-time and g a non-flat vacuum metric on M with associated sectional curvature function σ. If g′ is any other Lorentz metric for M with the same sectional curvature function σ, then g′ ≡ g.

The analogue of theorem 1 for positive definite metrics was given in [8].

3. The Description of Gravitational Fields

In general relativity the gravitational field is described by the metric tensor of space-time. However, theorems 1 and 2 suggest that the sectional curvature function may be an alternative field variable for general relativity, especially theorem 2 in view of the extra strength of this result and the importance of vacuum space-times [9].

Two directions for further research present themselves. Firstly (and perhaps most importantly) it would be interesting to study the geometry of M from the point of view of the sectional curvature function σ (σ-based geometry). Secondly one might try to give an interpretation, in terms of σ, of the standard results and techniques that one already has for the geometry of M as described by the Lorentz metric g (g-based geometry). The first of these will probably be difficult since it is an unusual and unexplored avenue of research. Progress in the second direction may be partly misdirected given the general idea of this paper but at least it is easier and may suggest approaches to the first. In the remainder of this section some results in the second direction will be briefly described.

The Petrov classification of the Weyl tensor [10] and the consequent algebraic (Petrov) types for this tensor at a point of space-time have proved of immense value in both the practical and theoretical study of Einstein’s theory. If this classification, or something like it, is to be retained one needs a σ-based description of the Petrov types. Following work for the vacuum case in [2] such a scheme was given for general space-times in terms of the critical point structure of the function σ_{p} [4].

The energy-momentum tensor in general relativity is taken as representative of the sources of the gravitational field. If one wishes to continue to represent such sources in this way a method which is a mixture of σ-based and g-based theory

is to consider a function ϕ_{p} : Ḡ_{p} → R defined by ϕ_{p}(F) = σ_{p}(F) − σ_{p}(F^{∗}) where

F ∈ Ḡ_{p} and F^{∗} is its orthogonal complement. Then the algebraic (Segre) type of the energy-momentum tensor can be described in terms of the critical point structure of ϕ_{p} at each p ∈ M [3]. A special case of this result is known [11] and says that

σ_{p}(F) = σ_{p}(F^{∗}) (⇔ ϕ_{p}(F) = 0) for each F ∈ Ḡ_{p} and each p ∈ M if and only if the space-time is an Einstein space.

There has been much recent interest in various aspects of symmetries in general relativity (for a recent review see [12]). It is hence interesting to define σ-based symmetry concepts and to relate them to the usual g-based ones. For example let X be a smooth vector field on M with associated local flow ϕ_{t} for suitable t ∈ R. Then X will be called sectional curvature preserving if for each ϕ_{t} (and with p ∈ M and q = ϕ_{t}(p)) and each F ∈ G_{p} with F spanned by X, Y ∈ T_{p}M

σ_{p}(F) = σ_{q}(ϕ_{t∗}F) (2)

where ϕ_{t∗}F is the member of G_{q} spanned by ϕ_{t∗}X and ϕ_{t∗}Y and in the sense that either neither side of (2) is defined or both are and equality then holds. It turns out that such vector fields are Killing vector fields (the trivial case) or they are essentially a subalgebra of the Lie algebra of conformal vector fields on conformally flat plane wave space-times [13]. As another example let p ∈ M and let L be the Lorentz group regarded as a 6-dimensional Lie group of isomorphisms of T_{p}M. If H is any Lie subgroup of L one has an action H × T_{p}M → T_{p}M and hence an action ψ : H × Ḡ_{p} → Ḡ_{p} in an obvious way. If F ∈ Ḡ_{p} and h ∈ H let ψ_{h}(F) = ψ(h, F). Then σ_{p} will be called H-invariant if σ_{p}(F) = σ_{p}(ψ_{h}(F)) for each h ∈ H and F ∈ Ḡ_{p}. Using this idea one can show that SO(3)-invariance for σ_{p} for each p ∈ M essentially leads to the well known space-time cosmological models of Friedmann, Robertson and Walker [4].

It is remarked, in conclusion, that a convenient description of the material described here can be achieved using quadric surface theory [1],[4].

References

[1] Hall, G. S., General Relativity in terms of the Sectional Curvature Function, In Proceedings of the First Hungarian Relativity Workshop, Balatonszéplak (1985), 141.

[2] Thorpe, J. A., Curvature and the Petrov canonical forms, J. Math. Phys. 10 (1969), 1.

[3] Hall, G. S., Sectional Curvature and Lorentz Manifolds, (1978) unpublished - see Cormack, W. J. and Hall, G. S., Riemannian Curvature and the Classification of the Riemann and Ricci Tensors in Space-Time, Int. J. Theor. Phys. 18 (1979), 279.

[4] Hall, G. S., Rendall, A. D., Sectional Curvature and General Relativity, Gen. Rel. Grav. 19 (1987), 771.

[5] Hall, G. S., Sectional Curvature and the Determination of the Metric in Space-Time, Gen. Rel. Grav. 16 (1984), 79.

[6] Ruh, B., Kruemmungestreue Diffeomorphismem Riemannscher und pseudo-Riemannscher Mannigfaltigkeiten, PhD. Thesis, Zurich, (1982) - see also Math. Z. 184 (1985), 371.

[7] Rendall, A. D., Curvature of Generic Space-Times in General Relativity, J. Math. Phys. 29 (1988), 1569.

[8] Kulkarni, R. S., Curvature and Metric, Ann. Math. 91 (1970), 311.

[9] Hall, G. S., The significance of Curvature in General Relativity, Gen. Rel. Grav. 16 (1984), 495.

[10] Petrov, A. Z., Einstein Spaces, Pergamon, New York (1969).

[11] Ehlers, J., Kundt, W., In Gravitation: An Introduction to Current Research, ed. L. Witten (Wiley, New York) (1962).

[12] Hall, G. S., The Mathematical Study of Symmetry in General Relativity, Gravitation and Cosmology 2 (1996), 270.

[13] Hall, G. S., Hossack, A. D., Pulham, J. R., Sectional Curvature, Symmetries and Conformally Flat Plane Wawes, J. Math. Phys. 33 (1992), 1408.

Department of Mathematical Sciences, University of Aberdeen Aberdeen AB24 3UE, Scotland, U.K. E-mail: gsh@maths.aberdeen.ac.uk