  The Chandra X-ray Observatory is finding a large number of cavities in the
X-ray emitting intra-cluster medium which often coincide with the lobes of the
central radio galaxy. We propose high-resolution Sunyaev-Zel'dovich (SZ)
observations in order to infer the still unknown dynamically dominating
component of the radio plasma bubbles. This work calculates the thermal and
relativistic SZ emission of different compositions of these plasma bubbles
while simultaneously allowing for the cluster's kinetic SZ effect. As examples,
we present simulations of an Atacama Large Millimeter Array (ALMA) observation
and of a Green Bank Telescope (GBT) observation of the cores of the Perseus
cluster and Abell 2052. We predict a 5 sigma detection of the southern radio
bubble of Perseus in a few hours with the GBT and ALMA while assuming a
relativistic electron population within the bubble. In Abell 2052, a similar
detection would require a few ten hours with either telescope, the longer
exposures mainly being the result of the higher redshift and the lower central
temperature of this cluster. Future high-sensitivity multi-frequency SZ
observations will be able to infer the energy spectrum of the dynamically
dominating electron population in order to measure its temperature or spectral
characteristics. This knowledge can yield indirect indications for an
underlying radio jet model.
