We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion.

中文翻译：

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Berwald Finsler时空的不可度量性

我们研究Szabo的可量化性定理是否可以扩展到不确定签名的Finsler空间。对于光滑的正定Finsler度量，这个重要定理指出，如果度量是Berwald类型（即，其Chern-Rund连接定义了基础流形上的仿射连接），则它仿佛等于Riemann空间，这意味着它的仿射联系是某种黎曼度量的Levi–Civita联系。我们首次表明，该结果不会扩展到一般的Finsler时空。更准确地说，我们发现仿射连接的Ricci张量不对称的一大类Berwald时空。与正定Finsler空间的根本差异（使这种不对称成为可能）是，通常，Finsler时空仅在切线切线束的适当圆锥子集上满足某些平滑性。确实，我们证明了当Finsler Lagrangian在整个狭缝切线束上是光滑的时，Ricci张量必须是对称的。但是，对于大类的Finsler时空，Berwald属性并不意味着仿射结构等同于伪黎曼度量的仿射结构。相反，仿射结构是具有消失的扭转的公制仿射几何结构。Berwald属性并不表示仿射结构等效于伪黎曼度量的仿射结构。相反，仿射结构是具有消失的扭转的公制仿射几何结构。Berwald属性并不表示仿射结构等效于伪黎曼度量的仿射结构。相反，仿射结构是具有消失的扭转的公制仿射几何结构。