What are the consequences?
Stronger Byzantine response against the Caliphate, I suspect. I think a POD of 602 is best: Maurice intended to make one of his sons Western Emperor I believe, but this is after Muhammad's birth, so butterflies will be more limited than otherwise. Firstly, Byzantium will still be exhausted by her wars against Persia, if these go ahead still for whatever reason, so the Arabs will likely conquer Syria and Egypt at some point, due to religious discontent etc. However, the lack of the enormous cash drain that was Italy means Constantinople can dedicate more resources into fighting back: Antioch and the Taurus may be held ITTL. North Africa beyond Cyrenaica is unlikely to be Arabised, since Carthage will be the heartland of a western state, even more so than Italy. Ultimately, we could end up with a multipolar Mediterranean: Arabic Egypt and Palestine, East Roman Anatolia, Syria and Greece, West Roman Italy and North Africa, and Frankish Iberia (getting creative with that one).
I don't think a surviving western roman empire can butterfly mohammed's founding of islam it will be less powerful.
I don't think so.What? A POD of a surviving WRE, which was dissolved almost a century before Mohammed's birth, would certainly butterfly him.
I don't think so.
Care to explain why not? I didn't understand the butterfly affect when I had recently joined, but it's a massive thing, and cannot be understated in alternative history.
Because it cannot or mostly not butterfly the events of Mohammed's birth and religion because the western roman empire might likely later merge with the frankish empire (if it is not butterfly) and it will be focused fighting the barbarians.
You've lost me here. The Franks only really invaded Gaul in the 470s, when central Roman authority was already largely legal fiction. So a surviving WRE is quite likely to butterfly away the Frankish Empire as we know it.
As for Mohammed. Let's say a Roman priest is born in Africa in 480, in a surviving WRE. He decides to go to Arabia to preach, where he meets Mohammed's grandfather. Mohammed's grandfather is converted to Christianity, and consequently marries someone else but Mohammed's grandmother. Now, multiply this scenario by a thousand times, a million times, and this is why Mohammed will be butterflied away by a surviving Western Roman state.
I agree with that but there is still a possibility with islam existing in that timeline i just wonder how would the caliphate fare in that timeline, the consequence of the Western Roman Empire and the caliphate coexisting is interesting to me.
Oh, Islam can certainly still exist: it may even be founded by a chap named Mohammed, but it won't be the same Mohammed as in OTL. Islam itself will likely be quite different in many ways.
However, your POD is still fairly inconcievable, I fear. With a surviving WRE, the Eastern Roman Empire won't have expended vast resources on reconquering Italy under Justinian, so even with the impact of the 542 plague (assuming that itself is not butterflied), the state will be in a substantially better structure than it was in OTL in seventh century. In addition to this, a surviving WRE means there'll be less desire for the Emperor in Constantinople to appease the Papacy, so a rather less alienated Monophysite East. This will help to fight off any Islamic attacks. The "Muslims" of TTL can still concievably do heavy damage, especially if the current Emperor is crap (quite plausible), but the trouble with crap Byzantine Emperors is that they tended to be overthrown relatively rapidly by a more competent one. Sorry about this, I'm really shitting on your question. My main point is though, that a surviving WRE makes any form of a Caliphate implausible, even allowing for butterflies.