metropolis vs urban center

metropolis and urban center both are nouns.

Word	Noun	Adjective	Verb	Adverb
metropolis	Yes	No	No	No
urban center	Yes	No	No	No

As nouns, metropolis and urban center are synonyms defined as:

metropolis and urban center: a large and densely populated urban area; may include several independent administrative districts

Other synonyms of metropolis include city.

metropolis (noun)	urban center (noun)
people living in a large densely populated municipality	a large and densely populated urban area; may include several independent administrative districts
a large and densely populated urban area; may include several independent administrative districts