  The requirement that particles propagate causally on non-trivial backgrounds
implies interesting constraints on higher-derivative operators. This work is
part of a systematic study of the positivity bounds derivable from time delays
on shockwave backgrounds. First, we discuss shockwaves in field theory, which
are infinitely boosted Coulomb-like field configurations. We show how a
positive time delay implies positivity of four-derivative operators in scalar
field theory and electromagnetism, consistent with the results derived using
dispersion relations, and we comment on how additional higher-derivative
operators could be included.
  We then turn to gravitational shockwave backgrounds. We compute the infinite
boost limit of Reissner-Nordstr\"om black holes to derive charged shockwave
backgrounds. We consider photons traveling on these backgrounds and interacting
through four-derivative corrections to Einstein-Maxwell theory. The inclusion
of gravity introduces a logarithmic term into the time delay that interferes
with the straightforward bounds derivable in pure field theory, a fact
consistent with CEMZ and with recent results from dispersion relations. We
discuss two ways to extract a physically meaningful quantity from the
logarithmic time delay -- by introducing an IR cutoff, or by considering the
derivative of the time delay -- and comment on the bounds implied in each case.
Finally, we review a number of additional shockwave backgrounds which might be
of use in future applications, including spinning shockwaves, those in higher
dimensions or with a cosmological constant, and shockwaves from boosted
extended objects.
