  The precision of optical atomic clocks is approaching a regime where they
resolve gravitational time dilation on smaller scales than their own
extensions. Hence, an accurate description of quantum clocks has to take their
spatial extension into account. In this article, as a first step towards a
fully relativistic description of extended quantum clocks, we investigate a
quantized version of Einstein's light clock fixed at a constant distance from a
large massive object like the Earth. The model consists of a quantum light
field in a one-dimensional cavity in Schwarzschild spacetime, where the
distance between the mirrors is fixed by a rigid rod. By comparing a vertical
and a horizontal clock, we propose an operational way to define the clock time
when the clock resolves gravitational time dilation on scales smaller than its
extension. In particular, we show that the time measured by the vertical light
clock is equivalent to the proper time defined at its center. We also derive
fundamental bounds on the precision of these clocks for measurements of proper
time and the Schwarzschild radius.
