In my world , instead of the world being discovered to be a finite planet during the age of discovery , humans discover that the world is infinite. Due to this , the age of discovery never ends , and with it colonization. The constant expansion of states goes on all the way up to the 21st century, with modern communication technology , transportation , etc.
My question is: how large could such a modern state get while still being controlled by a single central government?
Governments can extend their territory as far as they can act and react to things in a reasonable amount of time. A country like the US which has divisions of government would probably be what you'd expect most likely to arise due to the difference in communication and travel, so you'd have a local government and a more expansive federal government more than likely.
How big would these be? Basically Multiply the fastest attacker by 3 months...
7,200 km/h is the speed of the fastest attack. 3 months is 2,160 hours. That gives us a size of:
A country can be a diameter of 31,104,000 km. An area of 760,000,000,000,000 km^2 or 1,489,986 times the surface area of Earth.
the other idea is that countries can hold together so long as they have communication with it's furthest extent, without taking into account it's military assets, within a reasonable time frame. Let's say this is a month, though it might be as small as a week or even a day when pushed to extremes. We can communicate at the speed of light or just about so...
2,592,000 seconds in a month.
299,792,458 m/s is the speed of light.
1,554,124,102,272 km diameter is the max size of a country then. (It would take 25,000 years for our fastest jet to get there, so humans would clearly not be out this far.)
It would have an area of 1,900,000,000,000,000,000,000,000 km^2.
That's a surface area of 3,724,964,318,762,841 times that of Earth.
However, with that, I have to point out that Humans spread outward at about 1 to 4km per year, so going back to about 70,000 years ago when there was only about 10,000 humans on earth we only covered about an area with a diameter of 2,000 km. In the best conditions then humans would have only populated an area with a diameter of about 280,000 km.
So if your world is infinite, but follows the same or similar development as ours then the maximum size of a state is an area of 61,600,000,000km^2, roughly 121 times that of the surface area of Earth. And then they'd reach a place where no man has gone before and then you'd have to build up everything past that point. For that, look to the Colonial American era... but that's another question. At this point the question is less about could the "rule" the area to could they "control" the area. Humans would be sparse past that point that while you could patrol it and whatnot, you would miss a lot, especially with nothing built up. And policing really would be a nightmare because people would just keep moving out further and further. It might take years, but people could and would get beyond the borders of possible control, and even if they couldn't, far enough out and even a city would be missible. Patrols for many years would be along thing like the Oregon Trail where people are traveling to in great numbers, not over the whole of the area. For this scenario, I suggest a read of Terry Pratchett's Long Earth Series which discusses a similar idea that you might get something out of it.
The problem would really lie in defining what is meant by "state." I say this because it is that kind of structural problem which ultimately arises; when you get big enough, it becomes impossible for a central government to rule an empire.
Different empires through history have approached this problem in different ways, broadly-speaking. Let's look at a few:
The Romans enjoyed one of the oldest massive empires we know of; at its height, far before widespread literacy, mass media, or nationalist sentiment, the Roman Empire sprawled across three continents and commanded nearly half of the world's total human population. Worth mulling over a bit, I think. Anyway, this empire ruled from a central site (Rome), but to keep command of its far-flung peripheries, it delegated leadership to "client states" - that is, nominally in Egypt, for example, Egyptians might still rule at many levels of government, but there would be a Roman citizen near the top whose job it was to ensure that the wishes of the Empire were carried out by the local rulers. Later in its history, under immense pressure to deal with problems all along its very long borders, Rome split into two separate empires - the West and the East. This was, realistically, never going to be made whole again, because people are as people are, and no one would easily give up command of half the greatest empire the world had ever seen.
The Mongols came later and did things differently. They ended up controlling a much larger area of the earth, but not necessarily a larger part of the population; while a larger number of people possibly lived on the lands they claimed, the Mongols didn't really ask a whole lot of those populations once they owned them - they killed quite a few and did relatively little with them once they had them, save perhaps levy taxes, to be carried out by, again, local leadership, this time without even necessarily a Mongol citizen in place - instead the threat of the Mongol Horde was deemed sufficient to coerce behavior friendly to the empire. The Mongols ruled via these mobile armies first and foremost, and it makes clear that you can control vast territories and resources without "you" (that is, the citizens of your empire proper) being very large. The technology of the horse makes this possible for those willing to exploit it fully.
The British were exploiting another conveyance - the sailing ship - to rule their far-flung empire of colonial holdings - and again, the question is whether you want to consider all their lands and peoples properly part of the "British Empire" - because they might have "ruled" them, but under the auspices of things like the East India Company, a corporate affair that pretty much managed the Indian subcontinent for them for much of its history within the British Empire.
The British are fascinating because I think their empire in many ways models the problems a far-flung spacefaring empire would have to overcome - some of the greatest problems they experienced were the expense of projecting their power overseas, the loyalty of their colonies to their empire, and the communications troubles of such distant holdings. This is a polity that took an entire continent and couldn't do anything with it other than basically use it as a prison for a while. Again, whether we consider all Australia properly "owned and controlled" by the British at all points they claim to have owned it is debatable in my mind.
The United States of America, perhaps in response to the the problems of their British parents, had this idea that smaller confederated states should coexist under a federal, central government, which would effectively facilitate the cooperation of the state governments, which were themselves the thing seen more close to countries in their day. The "nation" of states which you sometimes hear the founding fathers of American government refer to encompasses this plurality, and indeed you can look at the diplomatic, military, and economic dealings of other countries and empires in the world with the early USA and see that there was considerable confusion about whether they were a group of independent states, or whether they were provinces of a larger federal government. That question eventually got resolved in a pretty nasty fight about a different but related problem, showing that even this method of controlling vast numbers of people and tracts of land has its pitfalls - namely, that if you enfranchise most of your highly heterogeneous population, there will be significant and empire-ending disagreements unless the confederation of states idea is abandoned in favor of a much more powerful central government. This then suffers the same problems as large, centralized empires of the past, but luckily America arises in a period of vastly improved conveyances (steam engines and telegraphs), and these technologies, along with high literacy and "universal" suffrage, allow it to mold a sense of national belonging throughout its population - enforcing its empire from the bottom up more than the top down. Indeed, this worked so well that it became the ultimate model of the modern nation-state; Europe especially quickly succumbed to this nationalism soon thereafter; nothing stands in the way of an empire whose entire citizenry is employed in the cause. This is an excellent reason to reconsider how big an empire truly is based on its method of control; though Mongolia might field a mighty army in terms of raw power and size compared to other feudal states, nothing compares to the kind of massive, million-strong citizen army France could field with its nationalist instincts in full swing, despite a much lower total population.
Ultimately, limits on coherent size will depend on the nature of the state in question. A "Mongolia" rule-through-fear and not-caring-ness might yield a very large empire in terms of size and resources, but Nationalist states with modern communications technologies would be the more powerful, more truly populous alternative, however much land they actually owned.
Finally, a note on technologies - at a certain point, you will run up against the problem of latency. Even with the internet, you are fighting, ultimately, the speed of light, and it becomes difficult to impossible to communicate with colonies separated from the federal control center without years-long lapses in judgment. The natural response to this will be to delegate, as the Romans did when they were dealing with the latencies inherent to foot and horse travel, and then from those small fractures will arise great schisms which will shake your nation-states apart one after the other, and the surface of your world will become an infinite Balkans, united only by the effective reach of light speed communications and the rate of change inherent to the forces of disunity they must struggle against to remain whole.
Empires might consist of many billions of people and many dozen Russias' worth of landmass, but it is difficult to imagine an order of magnitude larger than that working for very long.
Very large. World wide governments were impossible until modern transportation and communication. So if you want to find the limits of your empire find the limits of modern travel and communication.
The British were able to hold on to empire that was separated by months of sea travel. So take that as your limit but changes sea travel to air travel.
An airplane can cross the world in a few hours, obviously an empire that so big that it would take months to cross it by air would be several times larger than the Earth.
Of course governing an empire that large would be a super pain, but not impossible. Most likely the empire would be separated into large, mostly self-governing states, all loyal to a central government.
I think the problem lies on how often a powerful military can develop.
There are basically 3 parameters that should be taken into consideration:
The first two parameters are out of my knowledge to answer. I would summarize these 3 parameters into "time interval and duration that one state is externally militarily active in a given area". Referring to Wykes' answer and my own knowledge about Chinese history, in the Eurasian area, which is quite large, I only remember less than 10 times of a state extending its territory extensively. I do not have any actual values, but these parameters can help.
Let's not look at how far a state can invade first, but how far a state's technology can reach.
I think answering this question should be based on a theory: assuming all humans started evolving at the same time, due to different anthropological factors, some have developed modern technology first. For example, for a long time, most of the world outside Europe had the ability to handle electricity. This proves that this difference can be very great. However, in an infinitely large world, as long as we assume that a certain probability that a country has developed modern technology, there must be multiple countries that developed modern technology as well. Therefore, the only thing that matters is, how far are these countries apart? They must be far enough such that when one country has developed such technology, the other ones must not be close enough to know the technology.
In conclusion, from the technological perspective, I am against the answer that a country like this can be infinitely large. We only have one example on our Planet Earth, so there is nothing to fairly compare with. But assuming that Planet Earth is the median example of how human civilization should develop, if we have a planet with twice Planet Earth's size, technology would be developed at the same time at two halves of this planet. In a planet with x times Planet Earth's size, we can assume that x countries developed modern technology together.
From another thought, if our planet is infinitely large, exploration will never end, so explorers will never be satisfied and sit down and start exploiting what it has explored (just like Mongols). So maybe how globalization has developed in the past centuries (after Earth has been proved finite size) would not have happened in this way.
