  The operational approach to time is a cornerstone of relativistic theories,
as evidenced by the notion of proper time. In standard quantum mechanics,
however, time is an external parameter. Recently, many attempts have been made
to extend the notion of proper time to quantum mechanics within a relational
framework. Here, we use similar ideas combined with the relativistic
mass-energy equivalence to study an accelerating massive quantum particle with
an internal clock system. We show that the ensuing evolution from the
perspective of the particle's internal clock is non-Hermitian. This result does
not rely on specific implementations of the clock. As a particular consequence,
we prove that the effective Hamiltonian of two gravitationally interacting
particles is non-Hermitian from the perspective of the clock of either
particle.
