  From the photoluminescence of SiC microcrystals uniformly covering a
rectangular mesa of the high transition temperature $T_c$ superconductor
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, the local surface temperature $T({\bm r})$
was directly measured during simultaneous sub-THz emission from the $N\sim10^3$
intrinsic Josephson junctions (IJJs) in the mesa. At high bias currents $I$ and
low bath temperatures $T_{\rm bath}\lesssim~35$ K, the center of a large
elliptical hot spot with $T({\bm r})> T_c$ jumps dramatically with little
current-voltage characteristic changes. The hot spot doesn't alter the
ubiquitous primary and secondary emission conditions: the ac Josephson relation
and the electromagnetic cavity resonance excitation, respectively. Since the
intense sub-THz emission was observed for high $T_{\rm bath}\gtrsim~50$ K in
the low $I$ bias regime where hot spots are absent, hot spots can not provide
the primary mechanisms for increasing the output power, the tunability, or for
promoting the synchronization of the $N$ IJJs for the sub-THz emission, but can
at best coexist non-mutualistically with the emission. No $T({\bm r})$ standing
waves were observed.
