Light cone and Weyl compatibility of conformal and projective structures

In the literature different concepts of compatibility between a projective structure and a conformal structure on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. An often cited paper [Ehlers-Pirani-Schild:1972] introduces still another criterion which is natural from the physical point of view: every light like geodesics of of the conformal structure is a geodesics of the projective structure. Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned in [Trautman:2012] and [Scholz:2019]. Here it is proved that the conjecture of EPS is correct.

💡 Research Summary

The paper addresses a long‑standing question in the foundations of spacetime geometry: under what conditions do a projective structure P (encoding the free‑fall trajectories of massive test particles) and a conformal structure C (encoding the causal light‑cone structure) together determine a Weylian metric on a differentiable manifold M? In the seminal 1972 work of Ehlers, Pirani, and Schild (EPS) it was claimed that the “light‑cone compatibility” condition—namely that every null geodesic of the conformal class C is also a geodesic of the projective class P—is sufficient to guarantee the existence of a Weylian metric whose associated affine connection belongs to P. This claim has been widely used but never rigorously proved; recent critiques (Trautman 2012, Scholz 2019) have even suggested that the EPS statement might be false when interpreted in the strict Riemannian sense.

The authors first clarify the hierarchy of compatibility notions. Light‑cone compatibility (Definition 2(i)) is the weakest: it requires only that null curves of any representative metric g∈C are unparameterised geodesics of some connection in P. Riemann compatibility (Definition 2(ii)) demands that the Levi‑Civita connection of a particular g lies in P. Weyl compatibility (Definition 2(iii)) is stronger still: there must exist a 1‑form ϕ such that the Weyl connection Γ(g,ϕ) (the unique torsion‑free connection satisfying ∇g+2ϕ⊗g=0) belongs to P. Clearly, Weyl ⇒ Riemann ⇒ light‑cone, but the converse implications are non‑trivial.

The central result (Theorem 1) proves that light‑cone compatibility already forces Weyl compatibility. The proof proceeds by considering any null geodesic γ of g. Because γ is a null curve for the conformal structure, we have ∇_g ·γ·γ=0. Since γ is also a geodesic of the projective class, there exists a scalar function β(γ,·γ) such that ∇_Γ ·γ·γ=β·γ. Subtracting yields a relation involving the difference tensor D:=Γ−z, where z is the Levi‑Civita connection of g:

 D_{ijk} ·γ^j ·γ^k = β ·γ_i .  (4)

Because this must hold for every null tangent vector v, the authors study the cubic polynomial

 P_i(v) := D_{ijk} v^j v^k v^s − D_{sjk} v^j v^k v^i .  (5)

P_i(v) vanishes whenever g(v,v)=0. Since the null cone {v | g(v,v)=0} is an irreducible quadric in ℝⁿ (n≥3), the polynomial must be divisible by g(v,v). Hence there exists a skew‑symmetric 2‑form ω such that

 P_i(v) = g(v,v) ω_i(v) .  (6)

Further algebraic analysis shows that ω must have rank two and can be written as ω_i(v)=ϕ_i g(v,v)−v_i ϕ·v for some 1‑form ϕ. Substituting back, the authors obtain the general solution for D:

 D_{ijk}=ϕ_i g_{jk}+δ_{ij} η_k+δ_{ik} η_j ,  (7)

with η another arbitrary 1‑form. This expression is precisely the difference between a Weyl connection Γ(g,ϕ) and the Levi‑Civita connection z, as given by the standard formula

 Γ(g,ϕ)^i_{jk}=z^i_{jk}+δ^i_j ϕ_k+δ^i_k ϕ_j−g_{jk} ϕ^i .  (1)

Thus Γ(g,ϕ) belongs to the projective class P, establishing Weyl compatibility. The construction also yields an explicit formula for ϕ in terms of D (see Remark 2), showing that the Weyl 1‑form is uniquely determined (up to gauge) by the given structures.

Consequently, any pair (P,C) satisfying the EPS light‑cone condition determines a unique Weylian metric