  Advanced aneutronic fusion fuels such as proton-Boron$^{11}$ tend to require
much higher temperatures than conventional fuels like deuterium-tritium. For
electrons, the bulk plasma temperature can approach a substantial fraction of
the rest mass. In a mirror confinement system, where the electrons are confined
by an ambipolar potential of at least five electron temperatures, the tail
electrons which can escape the potential are fully relativistic, which must be
taken into account in calculating their confinement. In this paper, simple
estimates are employed to extend the scaling of the confinement time into the
relativistic regime. By asymptotically matching this scaling to known solutions
in the non-relativistic limit, accurate forms for the confinement time (and
thus the the ambipolar potential) are obtained. These forms are verified using
finite-element-based Fokker-Planck simulations over a wide range of parameters.
Comparing relativistic and nonrelativistic mirror-confined plasmas with the
same ratio of confining potential $|e\phi|$ to electron temperature $T_e$ and
the same mirror ratio $R$, the net result is a decrease in the confinement time
due to relativistic effects by a factor of $S \equiv
(1+15T_e/8m_ec^2)/(1+2|e\phi|/m_ec^2)$.
