*aus- (1)

"morning light, dawn," late 14c., from Latin Aurora, the Roman goddess of dawn, from PIE *ausus- "dawn," also the name of the Indo-European goddess of the dawn, from root *aus- (1) "to shine," especially of the dawn (source also of Greek ēōs "dawn").

"southern, of or pertaining to the south," 1540s, from Latin australis, from auster "south wind; south," from Proto-Italic *aus-tero- (adj.) "towards the dawn," from PIE *heus-tero- (source also of Sanskrit usra- "red; matutinal," usar-budh- "waking at dawn;" Greek aurion "tomorrow;" Lithuanian aušra "dawn;" Old Church Slavonic jutro "dawn, morning; tomorrow;" Old High German ostara "Easter"), from PIE root *aus- (1) "to shine," especially of the dawn.

The Latin sense shift in auster, if it is indeed the same word other Indo-European languages use for "east," for which Latin uses oriens (see Orient (n.)), perhaps is based on a false assumption about the orientation of the Italian peninsula, "with shift through 'southeast' explained by the diagonal position of the axis of Italy" [Buck]; see Walde, Alois, "Lateinisches etymologisches Wörterbuch," 3rd. ed., vol. I, p.87; Ernout, Alfred, and Meillet, Alfred, "Dictionnaire étymologique de la langue latine," 2nd. ed., p.94.

Or perhaps the connection is more ancient, and from PIE root *aus- "to shine," source of aurora, which also produces words for "burning," with reference to the "hot" south wind that blows into Italy. Thus auster "(hot) south wind," metaphorically extended to "south."