Lots of wrong or incomplete answers here, understandably, because this is a complicated question. It’s easy to get caught up in the difficult physics but this question is reducible to a problem of geometry. When presented right, you can all but ignore the physics.
This is an open question with three possibilities and one of them must be true, including the possibility of a universe with a well defined center.
Scenario 1: The universe is infinite in size and has no boundary. In geometry you must have a boundary to have a center. Imagine that I ask you to find the center of a circle but tell you that the radius is infinite. Without a boundary no concept of a center can exist. There is currently no compelling evidence to say that the universe is *not* infinite in size. The limit of the observable universe may prevent ever falsifying scenario 1.
Scenario 2: The geometry of the universe is non-Euclidean. An analogy here would be if I asked you to find the center of the surface of the earth. While you can find the center of the earth, no concept of the center of the surface of the earth exists. People give hand-wavy explanations of this analogy with “higher dimensional” nonsense. If the universe is non-Euclidean then it follows the rules of non-Euclidean geometry. If the universe has a boundary but the geometry of the universe is not “flat,” then similar to the surface of the earth analogy it would make no sense to talk of a center of the universe. Again the limit of the observable universe rears it’s head. If the universe is finite and bounded but sufficiently large, even a closed or open universe would appear flat as far as the very limits of observation can tell. This scenario may also never be falsified.
Scenario 3: The geometry of universe is flat and the universe has a boundary. In this scenario there is a center. That’s an inevitable consequence of geometry. Yet again the size of the universe poses a problem. In this scenario, in order to know that there is a boundary and center, you need to be able to observe part of the boundary or at least effects of the boundary. Imagine being inside of a beach ball but being able to see only 1cm away. If you are near the edge of the beach ball you can deduce that you are in a boundary and determine exactly where the center is. If you are not near the edge you have no way of knowing if there is a boundary and thus no way of knowing that there must be a center. From all possible places you could be in that beach ball, most positions will not allow you to answer this question. If this scenario is true, it appears that we are not in a position to be able to observe part of the boundary and thus calculate where the center is.
The ultimate answer is unsatisfying. There may or may not be a center of the universe and even if there is (or isn’t!) we may not ever have any way to know.
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