One thing we do know for sure is that the Natatio outside swimming pool of the Baths of Caracalla was roofless and open to the sky. But did the Natatio really have these kinds of mirrors? Or is this just another much repeated but false internet rumour? Furthermore, if it is just a rumour or a myth, then how did this all start?
HOW THE MYTH STARTED
I realized that the subject of bronze mirrors existing in the Baths of Caracalla was a myth after contacting a few people who wrote articles discussing those mirrors. I discovered they all got their information from the same place - a Wikipedia article about the Baths of Caracalla. In this article, the bronze mirrors are mentioned in just one sentence in the whole article, shown below (as of Feb 2023):
After reading that statement from the Wikipedia article, I asked myself where did the person who wrote that statement get their information? And is what they wrote true and based on actual evidence of some kind?
See the  citation at the end of the Wikipedia sentence? That number represents where the person got their information from, which was a book. At the bottom of the Wikipedia article, that number is explained, as shown below in an actual image from the website:
So the citation above tells us that the person writing the Wikipedia statement got his information from a book called "Understanding Archtecture: Its Elements, History, and Meaning" written by Leland Roth in 2007. I examined the actual book (3rd edition) and found Mr. Roth's statement on page 270, where he wrote:
"To the north of the frigidarium was the swimming pool, the natatio, open to the sky but apparently additionally illuminated by bronze mirrors attached to metal fixtures overhead."
Notice how the word "apparently" is used, which means it is not 100% certain that this is accurate. Also, Mr. Roth does not provide any supporting references or citations for this statement that would tell us the source of his information. I thus contacted him and, based on his direction and my own research, I found a recent book that sheds much light on this subject, which I discuss below.
In 2021, a new book "Sandbows and Blacklight: Reflections on Optics" by Stephen R. Wilk, discusses the controversial subject of whether or not the Natatio of the Baths of Caracalla did have reflecting bronze mirrors.
Like myself, Mr. Wilk discovered the Wikipedia article statement about the mirrors. Again, just like me, he also contacted Mr. Roth, who then told him the source of his statement comes from a book titled "The Mute Stones Speak: The Story of Archaeology in Italy" by Paul MacKendrick.
From here on, it all gets very murky. In his book, Phil MacKendrick states that he got his information from a Mr. Plommer, who was very leery about the whole subject and gave his readers many warnings. Mr. Plommer claimed the dubious information was based on the writings of Erica Brodner, who apparently mistranslated an ancient statement regarding lattice work somewhere outside the Natatio area, which makes the reflecting mirror myth even more erroneous. Interestingly, Phil MacKendrick describes this whole bronze mirrors business as an "insoluble controversy", meaning it is impossible to solve.
Basically, it all boils down to the fact that there is no actual written evidence regarding the existence of reflecting bronze mirrors within the Baths of Caracalla. The quote below from Stephen R. Wilk's book (page 84) sums it all up perfectly, with many thanks to him for his excellent detective work in shedding light on this subject:
So we had a mistranslated passage that was cited, with misgivings, by Plommer, who was cited, again with warnings, by MacKendrick (who issued no warnings), who was cited by Roth, who was cited by Wikipedia (which gives the wrong room), which was cited by the internet.
So, after all is said and done, we have no actual reference to mirrors being used to direct sunlight into a Roman building.
Stephen R. Wilk, "Sandbows and Blacklight: Reflections on Optics"
In conclusion: absolutely, there were no reflecting bronze mirrors in the Baths of Caracalla; it is a myth started by a misunderstanding. In fact, it appears there were never any reflecting mirrors used in any Roman building. Hopefully, Mr. Wilk's book ends further debate and statements regarding whether or not the Natatio had reflecting bronze mirrors. But, frankly, I wish the baths did have those mirrors - it would have been wonderful for the bathers.