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«Wave Fields in Weyl Spaces and Conditions for the Existence of a Preferred Pseudo-Riemannian Structure J. Audretsch 1, F. Gahler2, and N. Straumann3 ...»

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Communications in

Mathematical

Commun. Math. Phys. 95, 41-51 (1984)

Physics

© Springer-Verlag 1984

Wave Fields in Weyl Spaces and Conditions for the

Existence of a Preferred Pseudo-Riemannian Structure

J. Audretsch 1, F. Gahler2, and N. Straumann3

1 Fakultat fur Physik der Universitat Konstanz, D-7750 Konstanz, Federal Republic of Germany

2 Institute for Theoretical Physics, ETH-Hόnggerberg, CH-8093 Zurich, Switzerland

3 Institute for Theoretical Physics of the University of Zurich, Schonberggasse 9, CH-8001 Zurich, Switzerland Abstract. Previous axiomatic approaches to general relativity which led to a Weylian structure of space-time are supplemented by a physical condition which implies the existence of a preferred pseudo-Riemannian structure.

It is stipulated that the trajectories of the short wave limit of classical massive fields agree with the geodesies of the Weyl connection and it is shown that this is equivalent to the vanishing of the covariant derivative of a "mass function" of nontrivial Weyl type. This in turn is proven to be equivalent to the existence of a preferred metric of the conformal structure such that the Weyl connection is reducible to a connection of the bundle of orthonormal frames belonging to this distinguished metric.

1. Introduction In the past, there have been several attempts to deduce the pseudo-Riemannian structure of general relativity theory from a few axioms. An interesting approach in this direction has been worked out by Ehlers, Pirani, and Schild [1], which starts with a set of events, M, and two families of subsets of M which represent the collection of all (possible) light rays, and of all (possible) free fall world-lines of structureless test particles.

From a number of qualitative assumptions about light propagation and free fall which are mathematical idealizations of well-established facts and which appear to be minimal requirements of the local validity of special relativity, it has been shown by these authors that there is a unique Lorentzian conformal structure, (i.e. an equivalence class of pseudo-Riemannian metrics of Lorentzian signature) whose null geodesies are identical to the light rays. Moreover, the freely falling particles determine the affine geodesies (a geodesic path structure) of a class of projectively equivalent symmetric linear connections. The compatibility requirement that the null geodesies defined by the conformal structure 42 J. Audretsch, F. Gahler, and N. Straumann belong to the ("deparametrized") geodesies of the distinguished class of connections leads to a Weyl structure of space-time.

One of the present authors (J. A.) has recently proposed an additional assumption [2], which implies in a natural and convincing manner the space-time structure of general relativity theory: If one requires that the trajectories of massive wave fields in the short wave limit coincide with the geodesies of the Weyl space, then this space is already a Lorentzian manifold, i.e., there is a distinguished metric of the conformal structure with respect to which lengths and times at different points can be compared in a path independent manner.

In this paper, we try to improve the technical treatment of ref. [2] by using the language of fiber bundles. The concepts and methods of fiber bundles provide a natural and convenient framework for a (global) discussion of the reduction of a Weyl space to a pseudo-Riemannian manifold, which is along the line of other examples of bundle reductions which appear in gauge theories (e.g. spontaneous symmetry breaking). We hope that the structural aspects of the problem will thereby be clarified.

Section 2 contains a discussion of the geometry of Weyl spaces. In Sect. 3 we introduce spin structures on Weyl spaces and formulate field equations for massive spinor fields. The Klein-Gordon theory is generalized to Weyl spaces in Sect. 4.

In Sect. 5 we derive a necessary and sufficient condition that the trajectories of the WKB-limit agree with the geodesies of the Weyl connection. The geometrical significance of this condition will be elucidated in Sect. 6 and our conclusions will be summarized in Sect. 7.

2. Geometry of Weyl Manifolds Let M be a four-dimensional smooth manifold which is connected and paracompact.

A conformal structure on M is an equivalence class [0] of conformally equivalent Lorentz metrics with signature (H ). The pair (M, [g~\) is called a conformal manifold. In a conformal manifold we can introduce the bundle of conformal frames, which are the linear frames consisting of pairwise orthogonal tangent vectors of equal length (relative to any ge[g]\ We assume that M is space and time orientable and that an orientation is chosen. The set W(M) of oriented conformal frames on M can be regarded in an obvious manner as the total space of a principal fiber bundle π: W(M) -»M whose structure group G is the subgroup of the conformal group consisting of all nonzero multiples of homogeneous Lorentz transformations in L\. Clearly, G is isomorphic to Lτ+ x (R+.

The principle bundle of (oriented) conformal frames, which we call the conformal bundle, is a reduction of the bundle L(M) of linear frames. The canonical l-form θ on W(M) is just the restriction of the soldering form on L(M). A Weyl connection is a connection ω on W(M) with vanishing torsion form (i.e. Dωθ = 0, where Dω is the exterior covariant derivative belonging to ω). Since the connection form has values in the Lie algebra © of G, i.e., in so(l, 3)φlR, we can split ω uniquely ω = ώ -h φ 1, (2.1) Preferred Pseudo-Riemannian Structure 43

–  –  –





One sees immediately from (2.11) that, for a given g, τ (g ) is independent of the representative g, but depends in general on the path γ which connects p with q.

Thus the comparison of lengths of vectors at p and q is path dependent.

If τt denotes the transport of metrics from p to y(ί), one finds easily that and thus with (2.10)

–  –  –

This formula shows that τ t agrees with the parallel transport of g belonging to the given Weyl connection ω.

Since locally F(g) = 2σ*φ, we see from (2.12) and Stokes' theorem that the parallel transport of metrics is locally path independent iff dφ = 0.

The representations (p, V) of the group G have the form p: λΛ h-λ w p(Λ\ λ 0, ΛEL\, (2.14) where p is a representation of L\ and w is any real number, called the weight of the representation p. F-valued differential forms that transform with the representation (2.14) in Eq. (2.5) are said to have weight w. The space V will always be assumed to have a metric h (not necessarily positive definite) with respect to which p is orthogonal.

Consider two F-valued /c-forms α, β which transform with the same representation p relative to L\, but have in general different weights w(α) and w(/?). For each section σ we choose ge [0] such that it has the form (2.9) relative to σ. Then we can define a natural scalar product (v.,βy = g(v°,βb)h(va,vh), (2.15) Preferred Pseudo-Riemannian Structure 45

–  –  –

This is gauge invariant if the weight of the mass function m is — as expected from dimensional reasons — equal to — 1.

The Euler-Lagrange equation of (3.4) is Dirac's equation

–  –  –

4. Klein Gordon Fields in Weyl Spaces Scalar fields have dimension [mass] and thus we assign the weight — 1 to such fields in Weyl spaces.

The Lagrangian density for a minimally coupled complex scalar field φ is (4.1)

–  –  –

5. WKB-Limit for Wave Fields in Weyl-Spaces In this section we study the short wave limit for the unquantized Dirac and KleinGordon fields, and investigate under which conditions the currents belonging to these wave fields are in lowest order geodesic (relative to a Weyl connection).

(a) Dirac Fields In the short wave limit we can make the usual WKB-ansatz (5.1)

–  –  –

The first two terms on the right-hand side are by (5.7) proportional toj'g. In the last term we use which follows from the general formula, D2ψ = p^(Ω) Λ ψ, for tensorial forms of type p, and the vanishing of the torsion. Indeed, since S is a scalar of weight zero we have D2S = 0 = D(dμSθμ) = Dλ(dμS)θλ /\θμ + dμSDΘμ

–  –  –

6. Discussion of Dm = 0 We have shown that the WKB-limit leads to geodesic particle trajectories if and only if Dm = 0. This condition can be discussed in various ways.

Relative to a section σ we have D(σ*m) - d(σ*m) - σ*(φ)σ*(m) = 0, (6.1) because m is a Lorentz scalar and has weight - 1. Thus σ*(φ) = d ln(σ*w), and hence φ is closed, dφ = 0. (6.2) Consequently, the [R-component of the curvature in (2.4') vanishes, Ω = dώ + ώ Λ ώeso(l, 3). (6.3) Furthermore, dF(g) = 0 (6.4) for every g, since F(g) is locally the pull-back of φ by an appropriate section map.

From (2.12) we see with Stokes' theorem that the metric transport is locally pathindependent.

In bundle language we can argue as follows. Dm = 0 implies that m is constant along every horizontal curve. Thus the set of points W(p) in W(M) that can be joined with p with a horizontal curve consists oϊ frames with the same mass. By the reduction theorem, W(p) is a reduced sub-bundle of W(M) whose structure group is the holonomy group φ(p) with reference point p, and the connection ω in W(M) is reducible to a connection in W(p) (see, e.g., [3], Sect. II. 7). Clearly, φ(ρ) is contained in Lr+, because all frames in the fiber of W(p) through p have the same length relative to any ge[g\.

Furthermore, by a theorem of Ambrose and Singer ([3], Sect. II.8) the Lie algebra of φ(p) is equal to the sub-space of (5 (Lie algebra of G) spanned by all elements of the form Ωq(X, Y\ where qe W(p) and X and Y are arbitrary (horizontal) vectors at q. Since φ(p) c L\, we conclude that Ω takes values in so(l, 3), which shows again that dφ = 0.

Let U be a simply connected open neighborhood of any point xeM and consider the induced connection in W\U = π~*(U). From (6.4) we know that there exists in U a function λ such that F(g) = dλ in £7, where g is a given metric in [0]. Then by (2.1 1) F(eλg) = 0, and hence (2.10) implies D(eλg) = 0. Thus the connection is metrical relative to g = eλg. This means that relative to any orthonormal tetrad (with respect to §) the connection form ω is so(l, 3)-valued. Hence the Weyl connection ω in W U is reducible to the orthonormal bundle O s (t7).

50 J. Audretsch, F. Gahler, and N. Straumann We show now that the restriction to simply connected neighborhoods is not necessary and that there exists a ge[g~] such that ω is reducible globally to O~(M).

For proving this we introduce the principle bundle with total space Ji — { g χ ; ge[0], xeM}, projection π: Jί - M given by π(gχ) = x, and structure group R + operating as gχ^λgχ.

There is a natural bundle homomorphism/from W(M) onto Jt(M) which sends a frame at xeM to the metric at x with respect to which the frame is orthonormal.

Since m is a Lorentz scalar, but has nonvanishing weight, m projects to a function rri on Jί such that m =f*m'. Furthermore, to each connection ω in W(M) there is a unique connection ω' in Jί(M) such that horizontal subspaces are mapped into horizontal subspaces by/(see [3], Sect. II.6). The condition Dωm = 0 goes over to ω D 'm' = 0.

The crucial point is now that the bundle Jt(M) is trivial. In fact, any 0e[0] defines a trivialization Tg: Jί - M x [R+ by Tg(g'χ) = (x, λ\ where λ is determined Repeating the argument which led to (6.4) we see now that F(g) is (globally) exact for every g and thus there exists a #e[#] with Dg = 0 on M. Hence we conclude that the connection ω is reducible to O^(M).

By the construction above, we see that O~(M) = f ~ l ( σ ( M ) ) as a set, where σ:

M - Jί(M) is a global section with constant mass ra0 (and O^(M) consists of all points in W(M) with mass m 0 !). Clearly, we stay within O*(M) along horizontal curves, because m remains constant. This shows again that ω is reducible to O,(M).

7. Concluding Remarks The previous discussion can easily be generalized to arbitrary spin by studying the short wave limit of the massive Fierz-Pauli equations on Weyl manifolds.



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