  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 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 metric-affine geometry with
vanishing torsion.
