  We study the following problem: given an Einstein metric on a manifold,
characterize and study all Einstein metrics which are pointwise projective to
the given one. By definition, two metrics are said to be pointwise projectively
related if they have the same geodesics as point sets. This is closely related
to Hilbert's Fourth Problem. In this paper, we show that pointwise projectively
related Einstein metrics satisfy a simple equation along geodesics. In
particular, we prove the following rigidity theorem: if two pointwise
projectively related Einstein metrics are complete with negative Einstein
constants, then one is a multiple of another.
  A Riemann metric has two features: length and angle, while a Finsler metric
has only one feature: length. In projective metric geometry, the second feature
of Riemann metrics is generally not used. Thus, we do not restrict our
attention to Riemannian metrics. Our method is ``Finslerian''.
