Using my sophisticated research method of searching for "Providence Baseball" occasionally on eBay, I found this 1857 newspaper, the May 9th copy of the New York based Porter's Spirit of the Times, which contains a short item on a new club:
BASE BALL AT PROVIDENCE -- We have received a notification of the formation of the Aurora Base Ball Club at this place, and in accordance with their name, the members meet from 5 to 7 o'clock in the morning. They have been out seven times since March, notwithstanding the pluvious state of the atmospheric phenomenon this season. Their President is Levi Starbuck, and Jas. V. Taylor Secretary and Treasurer. We hop they will have finer prospects to greet the blue-eyed goddess of the morn for the future.
First off, it it interesting that Timothy Hughes Rare and Early Newspapers chose to list this sixteen page newspaper based on a one-inch entry buried in the middle of it. Clearly, they know their market and a thing or two about baseball history.
This team is not generally known; my buddy Rick Stattler, who has extensively researched 19th century base ball in Providence for a decade, had never heard of the Aurora Club, until he independently noticed the listing on eBay. On the other hand, since this paper is a key primary source for early base ball, surely other historians had seen it.
What's really intriguing about this item is the possibility that the Aurora were playing the "New York Game," the direct precursor to modern ball, rather than the competing "Massachusetts Game" which was an evolutionary dead end. Peter Morris writes in A Game of Inches, on the first "Mass-circulated Rules" for baseball:
The Knickerbockers' rules appeared on December 6, 1856, in a publication called Porter's Spirit of the Times. This was an important step for a country that was still making the transition from oral communication to print. It enabled baseball to spread rapidly and also helped the game make a valuable ally in the burgeoning American newspaper industry. Tom Melville has concluded that "baseball was the first game Americans learned primarily from print," noting that town ball was "handed down from generation to generation orally" but that baseball was learned by reading "printed regulations."
So this suggests a storyline where the Aurora's founders read the New York Knickerbocker rules in Porter's, decide to take it up, and then send a letter to the paper saying, in effect, "we're trying out those rules you published." It is unlikely that there is any more evidence of the existence of the Auroras. I wouldn't be surprised if they never found another team to play against. Intra-club games were more common at that point; the Knickerbockers rarely played other clubs, for example.
So one way to build an indirect argument that the Auroras were playing the New York Game would be to show that Porter's does not write about Massachusetts Game teams, or specifies the difference when they do. Rick is an archivist at Harvard, so he has access to more copies of the paper and is going to look into it further. I did note that in a June 27, 1857 copy of Porter's they discuss the formation of the Tri-Mountain Club of Boston, which is generally regarded as the first New York Game team in New England. The paper does not, however, mention any of the Massachusetts Game teams in Boston, suggesting they're only interested in the New York Game. If the Auroras were playing the New York Game, they would predate the Tri-Mountain Club by two months, adding another little bragging point to Providence's baseball history.