  Thermodynamic uncertainty relations (TURs) are recently established relations
between the relative uncertainty of time-integrated currents and entropy
production in nonequilibrium systems. For small perturbations away from
equilibrium, linear response (LR) theory provides the natural framework to
study generic nonequilibrium processes. Here we use LR to derive TURs in a
straightforward and unified way. Our approach allows us to generalize TURs to
systems without local time reversal symmetry, including, for example, ballistic
transport, and periodically driven classical and quantum systems. We find that
for broken time reversal, the bounds on the relative uncertainty are controlled
both by dissipation and by a parameter encoding the asymmetry of the Onsager
matrix. We illustrate our results with an example from mesoscopic physics. We
also extend our approach beyond linear response: for Markovian dynamics it
reveals a connection between the TUR and current fluctuation theorems.
