# Are quadrics the cones of maximal symmetry?

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

• The tangent space is a real vector space, identify it with $\mathbb{R}^n$. Given any $S \subset \mathbb{R}^n$, you can ask what is the largest subgroup of $GL(n)$ that preserves $S$. If it is a cone, then automatically it is preserved by scaling, so you can reduce to looking at stuff in $SL(n)$. To make sense of the quote, probably you have to require that $S$ has codimension 1? Additionally, if you consider the degenerate case where $S = \mathbb{R}^3 \subset \mathbb{R}^4$, the set of $GL(4)$ elements that preserve $S$ is, if I am not mistaken, 13 dimensional. – Willie Wong Nov 1 '19 at 19:17
• Following up on Willie's comment, this short note by Winternitz goes through the details of how to identify maximal subalgebras of Lie algebras, with $sl(3,\mathbb{R})$ as an explicit example. Your question is essentially the same exercise for $sl(4,\mathbb{R})$. – Igor Khavkine Nov 1 '19 at 20:22
• Also, the highest dimensional subalgebras seem to be those fixing a linear subspace. For physical reasons, these should be discarded, by the criterion that there should not exist preferred spatial hyperplanes, or some generalized statement of that kind. – Igor Khavkine Nov 1 '19 at 20:25
• There's a theorem that for a smooth nondegenerate algebraic variety of degree greater than $2$ in $\mathbb{P}^n$, the group of projective linear transformations leaving the variety invariant is finite, mathscinet.ams.org/mathscinet-getitem?mr=476735 (item (2.1) in the second paper). – Zach Teitler Nov 2 '19 at 1:02
• @IgorKhavkine Thanks for pointing that out. In fact I would argue that "the largest group that fixes a salient conical subset" is a better realization of the intuition behind the statement than "the largest group that fixes a conical subset but doesn't fix a hyperplane", but perhaps only for aesthetic reasons. The point being that some groups fix both a hyperplane and another conical subset, and these groups are bad because they are too small, not because they fix the hyperplane. – Will Sawin Nov 2 '19 at 12:46

There is a whole theory of Lorentz-Finsler spaces, and their casuality has been considerably developed in recent years. This fact alone shows that it is not compelling to work with round light cones (i.e. cones that intersected by a hyperplane not passing through the origin of $$T_xM$$ give an ellipsoid. Notice that the notion of ellipsoid is affine, so the notion of round cone is well posed). The indicatrix (velocity/observer space) $$I_x\subset T_xM$$, is the locus $$\{y: 2L(x,y)=-1\}$$, where $$L(x,y)$$ is the Finsler Lagrangian whose vertical Hessian has Lorentzian signature $$(-, +, +, +)$$. It can be shown to be asymptotic to the boundary of a sharp convex cone with non-empty interior (the light cone). The sharpness of the cone expresses the finiteness of the speed of light in every direction. The non-roundness the fact that the speed of light is not isotropic.
What distinsuishes general relativity is the fact that the indicatrix (and hence the cone) is sent into itself by a group of endomorphisms of $$T_xM$$ which has maximal dimension. This is not the case in general. When this action is transitive the velocity space is said to be homogeneous. One can say that the relativity principle still holds true, in the sense that an observer cannot tell her position in $$I_x$$, i.e. her velocity (so velocity is only relative), by local measurements. However, the velocity space need not be isotropic (examples with affine sphere indicatrices can be constructed).
Suppose that any two spacetime points have isomorphic indicatrices, i.e. the velocity space does not change its geometry from point to point. The interesting fact is that the so called Berwald spaces, i.e.\ those Lorentz-Finsler spaces for which the connection does not depend on the velocity variable $$y$$, that are proper in your sense (i.e. the indicatrix has no symmetry) locally are necessarily translationally invariant (flat).