  Utilizing a coarse-graining method to convert hadronic transport simulations
of Au+Au collisions at SIS energies into local temperature, baryon and pion
densities, we compute the pertinent radiation of thermal dileptons based on an
in-medium $\rho$ spectral function that describes available spectra at
ultrarelativistic collision energies. In particular, we analyze how far the
resulting yields and slopes of the invariant-mass spectra can probe the
lifetime and temperatures of the fireball. We find that dilepton radiation sets
in after the initial overlap phase of the colliding nuclei of about 7 fm/c, and
lasts for about 13 fm/c. This duration closely coincides with the development
of the transverse collectivity of the baryons, thus establishing a direct
correlation between hadronic collective effects and thermal EM radiation, and
supporting a near local equilibration of the system. This fireball "lifetime"
is substantially smaller than the typical 20-30 fm/c that naive considerations
of the density evolution alone would suggest. We furthermore find that the
total dilepton yield radiated into the invariant-mass window of $M=0.3-0.7$
GeV/$c^{2}$, normalized to the number of charged pions, follows a relation to
the lifetime found earlier in the (ultra-) relativistic regime of heavy-ion
collisions, and thus corroborates the versatility of this tool. The spectral
slopes of the invariant-mass spectra above the $\phi$ mass provide a
thermometer of the hottest phases of the collision, and agree well with the
maximal temperatures extracted from the coarse-grained hadron spectra.
