state capital vs metropolis

state capital and metropolis both are nouns.

Word	Noun	Adjective	Verb	Adverb
state capital	Yes	No	No	No
metropolis	Yes	No	No	No

As nouns, metropolis is a hypernym of state capital; that is, metropolis is a word with a broader meaning than state capital:

state capital: the capital city of a political subdivision of a country
metropolis: a large and densely populated urban area; may include several independent administrative districts

Other hypernyms of state capital include capital, city, urban center.

state capital (noun)	metropolis (noun)
the capital city of a political subdivision of a country	people living in a large densely populated municipality
	a large and densely populated urban area; may include several independent administrative districts