Quote from Random.Capital:
The Empire didn't die - it moved its capital from Rome to Byzantium, where it continued its domination as the Roman Empire until the mid-11th century.
There is no "if" - the "eastern half" WAS the Roman Empire. Any distinction made is strictly a recent, modern one. Driven, primarily, by a Western bias.
To anybody living at the time, the Empire was the Empire.
I agree, but I will also take a greater more macro view:
The Romans considered themselves the inheritors of Alexander the Great's empire. This could be seen in many of the statues of the earlier emperors - the Julio-Claudians. Many imitated the busts of Alexander the Great, and in doing so, reached back to an older era to legitimize their rule and power. And in many ways, Rome and Greece were far more similar than the rest of the world.
When Constantine moved the capital to Byzantium, later Constantinople, there were two co-existing parts of the Roman Empire. Eventually Byzantium became the largest and wealthiest empire at the time as Rome fell. Even the Greeks/Byzantines during the Byzantium era referred to themselves as Romaioi(Romans).
After Byzantium fell, Eastern Europe went into a dark age, while Western Europe began it's Renaissance. The Italian city-states, Rome, international trade and banking by Florence, Milan, Venice... another century or so later the Spaniards ascended with their phenomenal gold finds in the Americas, to be replaced by Britain in power...
Then of course Anglo-Franco Napoleonic wars, replaced by Franco-Prussian-Wars, all the while Europeans continued colonizing Africa, the Middle East, South America, Australia, and Asia.
Then WWII knocked down the last Euro power - Great Britain to be replaced by the last Western Empire (IMHO) the USA.
Thus, since Alexander the Great to current times, about two and a half millenia, the West dominated the East more often than the other way around. That era is now over.
I hate to say this, but the US is the last Western Global power. What's next? Either cooperation, with the East having a greater say in international affairs and greater access to resources, or war.
We are now witnessing, real time, an accelerated and tremendous shift of wealth and power from East to West. This is historic and on a scale never before witnessed in human history.