  A generic prediction of quantum gravity is the vacuum dispersion of light,
and hence that a photon's speed depends on its energy. We present further
numerical evidence for a scale dependent speed of light in the causal dynamical
triangulation (CDT) approach to quantum gravity. We show that the observed
scale dependent speed of light in CDT can be accounted for by a scale dependent
transformation of geodesic distance, whose specific functional form implies a
discrete equidistant area spectrum. We make two non-trivial tests of the
proposed scale transformation: a comparison with the leading order quantum
correction to the gravitational potential and a comparison with the generalised
uncertainty principle. In both cases, we obtain the same functional form.
However, contrary to the widespread prediction of vacuum dispersion in quantum
gravity, numerous experiments have now definitively ruled out linear vacuum
dispersion beyond Planckian energy scales, and have now even constrained
quadratic dispersion. Motivated by these experimental constraints we seek to
reconcile quantum gravity with the absence of vacuum dispersion. We point out
that given a scale dependent geodesic distance, a scale dependent time interval
becomes essential to maintaining an invariant speed of light. We show how a
particular scale dependent time interval allows a photon's speed to remain
independent of its energy.
