WI Rome conquered Arabia during Trajan's reign?

What if Trajan chose to move on to Arabia after conquering Mesopotamia? Would this increase the viability of Mesopotamia as a roman province? And assuming that the invasion is a success and Arabia
is in roman arms until its fall, how would it affect Islam when - or if - it's created?
 
Trajan already had troubles holding Mesopotamia (and, while less so, Armenia), which was a really costly already: a deeper Roman presence in Arabia isn't impossible, but you'd rather need to butterfly away a lot of Trajan's wars against Parthians.
Long story short, any historical attempt by Romans to take Mesopotamia in its entirety ended to fail due to misunderstanding the military capacities of Parthians/Sassanians and being a bit too far removed from logistical basis, the province itself being quite difficult to defend along traditional borders and would require (critically in face of Parthians who had other resources at disposal) significant troops to hold. It's not impossible, but Trajan saw too big, too quickly and let details to his subordinates : it wasn't a net loss, but wasn't a stellar result either.

I think that a Roman Arabia would have its chances, and better ones, with someone else than Trajan, maybe in order to reinforce Roman strategic control of western Arabia. Note that it seems more and more likely that Rome was able to control or at least project its power in the southern coast and the hinterland further than it was considered some decades ago.
A WIP map before Trajan's death, to give an idea of regions involved.

A Roman Hedjaz seems quite possible giving right circumstances (a general Eastern African/Western Arabian strategy of establishing enforced alliances and relationships), and from there maybe some influence on Arabia Felix is possible, altough a bit too remote for really allowing a good provincial control. The key is to find a reason why Romans would prefer to provincialize the entiere western Arabia, rather than asserrt its dominance : you didn't have IOTL much political interest of Rome in these regions, and overall Roman presence in Red Sea was more of a Late Roman Empire/Eastern Roman Empire thing than late classical. But again, this could be changed relatively easily.
https://www.persee.fr/doc/topoi_1161-9473_1996_num_6_2_1695

As for Southern and Eastern Arabia...Not only it's definitely outside most Roman logistical and political reach, but it would ask for crippling Parthians well enough to make them accept the loss of what was essentially their traditional sphere of influence : at the very least, it would imply a war not only in Arabia proper, but almost certainly along Roman borders and Armenia (Persian empires generally didn't took very well Roman encroachment on petty-kingdoms and vassals) Even if Romans does manage to assert their dominance in Western Arabia, it likely means the continuation of petty-kingdoms supported/annexed by either empire in a conflict by proxy that would cyclically turn in full-blown wars. Except that whoever rules in Persia have a much easier time dealing in Arabia than Rome does have
 
The Romans ability to stretch further than the Hedjaz, you'd need two major things IMO.

1. To have the Arabian hinterland as your friend doing the actual fighting for you. I don't entirely know how to sell that alliance to be frank, rule over the Persian parts of Arabia? Not impossible, but still a hard fight, even with the Romans fighting in Mesopotamia.

2. Naval Projection - either allies in Somalia that have experience in sailing in this way, or basically a whole new endeavour as to my knowledge, their fleets never stretched far outside of the Red Sea at best. Now I think this is a great region for Roman Alternate History, with interesting sites for "Constantinoples of Africa", and the like, but fundamentally, it is an expensive endeavour that would primarily exist for attacking Persia in its flank. It is roughly as hard to control as Britain because of the distance, which complicates matters.

If we take that the idea that Romans bled gold to India as rebutted (as Pliny apparently said the Ganges was one of the best SOURCES of Roman Gold), then Trajan could as an aside, set up a fleet early in his reign to secure that, enabling the fleet to be used later on to push further around Arabia, allowing him to go further than clientelising Parthia, but also the Gulf.

Neither are easy, and you'd have to hope that the Romans can siphon enough taxes and tolls from trade in the Persian Gulf and Indian Ocean to pay for this fleet.

However, I reckon after Trajan, you'd basically lose the entire region. Either to rebels, or to the defacto independence of the Red Sea Fleet and Arabians. If you have the Empire divided, I can see Arabia being a seperate part, likely led from modern Yemen.

But overall, yeah - I could see it being a fast-track to Christianisation, butterflying away Islam entirely. Heck, it is early enough to butterfly away Christianity being embraced
 
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