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We consider a quantum field on a lukewarm black hole spacetime. We introducea new uniform approximation to the radial equation, constructed using anextension of Green-Liouville asymptotics. We then use this new approximation toconstruct the renormalized vacuum polarization in the Hartle-Hawking vacuum.Previous calculations of the vacuum polarization rely on the WKB approximationto the solutions of the radial equation, however the nonuniformity of the WKBapproximations obscures the results of these calculations near both horizons.The use of our new approximation eliminates these obscurities, enabling us toobtain explicitly finite and easily calculable values of the vacuumpolarization on the two horizons.

Extended Green-Liouville asymptotics and vacuum polarization for lukewarm black holes