  Recent X-ray observations of hot gas in the galaxy cluster MS 0735.6+7421
reveal huge radio-bright, quasi-bipolar X-ray cavities having a total energy
~10^{62} ergs, the most energetic AGN outburst currently known. We investigate
the evolution of this outburst with two-dimensional axisymmetric gasdynamical
calculations in which the cavities are inflated by relativistic cosmic rays.
Many key observational features of the cavities and associated shocks are
successfully reproduced. The radial elongation of the cavities indicates that
cosmic rays were injected into the cluster gas by a (jet) source moving out
from the central AGN. AGN jets of this magnitude must be almost perfectly
identically bipolar. The relativistic momentum of a single jet would cause a
central AGN black hole of mass 10^9 M_{sun} to recoil at ~6000 km s^{-1},
exceeding kick velocities during black hole mergers, and be ejected from the
cluster-center galaxy. When the cavity inflation is complete, 4PV
underestimates the total energy received by the cluster gas. Deviations of the
cluster gas from hydrostatic equilibrium are most pronounced during the early
cavity evolution when the integrated cluster mass found from the observed gas
pressure gradient can have systematic errors near the cavities of ~10-30%. The
creation of the cavity with cosmic rays generates a long-lasting global cluster
expansion that reduces the total gas thermal energy below that received from
the cavity shock. One Gyr after this single outburst, a gas mass of ~ 6 \times
10^{11} M_{sun} is transported out beyond a cluster radius of 500 kpc. Such
post-cavity outflows can naturally produce the discrepancy observed between the
cluster gas mass fraction and the universal baryon fraction inferred from WMAP
observations. (Abridged)
