A paper by Ehlers, Pirani, and Schild axiomatizes the geometry of general relativity in what seems like a nice way. However, Jacobson criticizes one aspect of the system as not natural:

One deep question is why the causal cone is given by a quadric in the tangent space. After all, one can easily imagine a partial ordering relation that arises from an infinitesimal conical structure which is not a quadric. In the EPS paper, the quadratic nature of the light cone is derived from their axioms. This is not very satifying however, since one of the axioms [$\text{L}_1$] is not particularly physically natural.

Aside from any axioms, there is a special property of quadrics that might underlie the fact that the causal structure is given by one. Namely, quadrics have the largest possible symmetry group of any conical subset of the tangent space. This is the Lorentz group, together with the conformal rescalings, a group with 7 continuous parameters.

I should have a reference for this but I don’t know of one. Perhaps Herman Weyl proved it. Perhaps it is not even true (see Problem 11).

Is this actually an open problem?

In general, is it possible to characterize "proper" Finsler metrics (those not arising from a bilinear form) as being Finsler metrics that lack symmetry? It's not even obvious to me how to formalize this question, since normally I would talk about the symmetry of a metric in terms of its Killing vectors, but I don't think that machinery applies to a proper Finsler metric. Therefore I'm not sure how to describe the relevant symmetry in a way that doesn't depend on the choice of a basis.

*References*

Jürgen Ehlers, Felix A. E. Pirani, Alfred Schild, "The geometry of free fall and light propagation," republished in General Relativity and Gravitation, 2012, Volume 44, 1587, https://doi.org/10.1007/s10714-012-1353-4

T. A. Jacobson, "A spacetime primer," http://terpconnect.umd.edu/~jacobson/spacetimeprimer.pdf

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