The thing is that whether or not a relic is 'authentic' is sometimes not the point, especially at the time when said relic came to its place of keeping. The linked answer does not go into modern considerations of authenticity, but it's directly relevant, and thus, OP, I commend to your attention u/WelfOnTheShelf's post on relics and their authenticity.

In order to answer this question, we have to establish what is meant by “authentic”. The gold standard would be multiple redundant narratives that place an object in its historical context, with a perfect example being a vintage Rolex watch accompanied by its purchase receipt, its certificate of authenticity, and the firsthand account of its purchase (I’m drawing this from a famous antiques roadshow clip). A profession evaluator would be able to confirm that the piece was made to Rolex standards by examining it directly, that the serial number on the watch matched the certificate, that the certificate was correctly formatted and printed, that the time and date of purchase matched the period that the watch was available for first sale, and that the reported location of purchase matched the distribution area of that serial number. The key here is that all of the individual elements can be independently evaluated, and if any single piece of information was misaligned, we would know immediately that the watch was inauthentic. Of course, we can never know for absolutely certain, because a sufficiently skilled forger could hypothetically create a watch to the same level of precision and fabricate all of the additional documentation, but it would be so unlikely that we can essentially rule that out.

Naturally, that level of scrutiny is impossible for events that happened so far in the past, so any authentication will have a significantly higher degree of doubt, but we can reach a level at which the academic consensus would probably agree. In this case it would mean artifacts whose age, chemical composition, craftsmanship, style, function, etc. are all appropriate to the era, which bear details that precisely match the details described in the Bible (most likely in the form of inscriptions of names, and dates), and whose location of discovery matches those details perfectly. Age, incidentally, is evaluated through a combination of radiocarbon dating, analysis of wear and degradation, stratigraphy (comparison with objects found at similar depth), chronological markers (date-stamped objects like coins that are found with the artifact), and seriation (statistical analysis of style, function, and craftsmanship).

All of this is to say that your question is unanswerable as phrased, because there are no Christian artifacts at all that we can say to be authentic with even a low degree of confidence. Every single example fails even a minimal level of scrutiny, or they have not been made available for authentication. To take a few examples: the shroud of Turin was shown by radiocarbon dating to have been manufactured hundreds of years after the life of Jesus, the scourging pillars of St. Peter and St. Paul have no evidence to support their use apart from legend, the chains of St Paul have no evidence to support their use as described, the head of John the Baptist has never been examined, and the crown of thorns has never been examined.

From the archaeological perspective, religious artifacts exist in the same space as myth and tradition. They are certainly capable of holding profound cultural and individual importance, but that importance is measured only in terms of their latter-day significance and not because they expand our understanding of the period from which they are claimed to originate.

EDIT: My initial answer was primarily directed at biblical relics, since anything much past that isn’t necessarily consistent across Christianity as a whole. However, since the question has been raised of more modern relics, it is worth addressing those.

We can, almost without question, know the origin of many, perhaps thousands, of relics from the last century or two, by which I mean we can say with a high degree of certainty that they are what they claim to be - clothing, hair, body parts, personal possessions etc. belonging to a person considered holy. However, this doesn’t really answer the question of authenticity.

It is intrinsic to the definition of a relic that the object itself be possessed of a degree of the holiness of its original possessor, which is usually measured by the object’s proximity to miraculous occurrences. Those events are typically considered to have been caused by the object, and their incidence is critical to recognition of the object as a relic in the first place.

If we are to include more modern relics in the question, it would seem that this latter aspect of authenticity is the truly interesting one. Unfortunately, this too fails even at a low degree of confidence. In the history of humanity, there has yet to be a single instance of an apparently miraculous event that has been shown to have been caused by divine intervention or some other supernatural force. We can say this with a high degree of certainty because any such event would be globally significant news. However, even despite this, modern relics retain an impressive capacity to generate awe in believers of their sect, and are absolutely worth recognizing in that regard.