Centre of the universe
Where is the centre of the Universe? Answered simply, this question is meaningless; depending on your interpretation there is no centre, or the centre is everywhere. For similar reasons, "where did the Universe begin" can also be answered with "everywhere". On the face of it, this sounds like a ridiculous answer to a simple and non-ridiculous question, but with some thought applied, it actually makes a lot more sense than an intuitive answer — as there are some ramifications of the Universe actually having a centre that few people notice.
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Historical views and "centres"
A "centre" is defined in many ways, some strict and some merely intuitive — but as we shall see, these ideas don't translate very well into the Universe. In order to have a centre, we need to have some sort of limit. The centre of a circle, for instance, can be located because it is the point equidistant from the boundary. The centre of a sphere can be located similarly, and for more complex shapes can be found by averaging out its shape or its mass, and locating the centre that way, as in with locating the centre of gravity of an object. So a centre has to be measured relative to something, otherwise it's merely an arbitrary co-ordinate.
Human views about the structure of the universe have evolved throughout time. This started with geocentrism, which states that the Earth is the centre of the universe, and this moved onto heliocentrism, which originally stated that it was the Sun at the centre — both these terms reference what orbits around what, so heliocentrism is still a valid viewpoint at the level of the solar system. All of these models had something at their centre, a fixed central point which everything else was referenced from and, more importantly, wasn't arbitrary. In these models, the Sun and the Earth were significantly important to act as centres because things revolved around them. The development of modern cosmology, however, has changed this view significantly.
Modern cosmology
Modern views of the universe posit a universe that develops as an expansion of spacetime (that is, space and time) itself. Because of this, the universe doesn't actually have a centre, because there is no point equidistant from the edges simply because there is no edge to speak of. It's not that the universe is a bubble of space inside something else — and expanding in the sense we might think of, like the volume of an inflating balloon — but space itself is expanding. All of this is happening in four dimensions, so it's a bit difficult for humans to wrap their three-dimensional heads around it, but this view is supported by observations of how radiation is distributed through the Universe and observations that have been used to confirm special and general relativity. This has led to the Standard Model of physics as we understand it today, and it all fits together quite nicely as a theory.
The idea that the Universe doesn't have a centre still baffles creationists and a few people such as flat Earth or fixed Earth advocates, who think that there must be some centre to the universe. After all, that the universe has a centre is an intuitive and common sense idea — it's just unfortunate that the Universe is bound by its own laws, and not the laws we merely think it should have. A misinterpretation of the inflationary models of the universe assumes that the Big Bang was a conventional explosion (even good pop science documentaries illustrate the Big Bang with visual explosions). The least accurate misunderstanding is that all space and time was here first and then matter and energy exploded into it, and so the source of this explosion would be the centre. This would make sense, but isn't an accurate picture of cosmology. It is space and time that are expanding in the Big Bang model; they didn't explode into anything at all.
The result of this is actually quite profound: everywhere is the centre of the Universe; space merely expanded to push the matter apart, and there is no boundary to it. The cosmic microwave background is often referred to as the remnants of the Big Bang, the initial blast of radiation from the event. This is often pictured as a map of the sky, as if it's a picture of the furthest reaches of the Universe, painted onto its edges — but again, this is a misinterpretation: the CMB is actually everywhere. The Big Bang happened at every point in the Universe.
By definition
You don't need to have qualifications in general relativity to conclude that the Universe has no centre. A centre is defined as a point equidistant from the ends of a line or the extremities of a figure. In order to have a centre, you need to have edges or boundaries to measure it from. The universe, however, has no edges or boundaries. This is because the universe is defined as everything that is and ever will be (not to confuse it with the observable Universe). If it had a boundary, this would necessitate something outside that boundary, and so it would cease to be the Universe by definition. This may sound like playing word games to get out of an awkward question, but is an important part of scientific understanding of the Universe. We need to be able to define the Universe and its boundaries (if any) before we can say anything useful about it. Such framing problems are an important distinction between string theory and loop quantum gravity, so have real applications in real science. So, even without all these observations to support the lack of a boundary (and therefore a lack of a centre) one can immediately conclude that the universe must not have a boundary (and therefore no centre).
One analogy to think of is to think about where the centre of the Earth's surface is. Not the centre of the Earth, of course, as that's a trivial concept; the Earth's surface forms a boundary from which we can derive a centre. The centre of the Earth's surface needs to be located on the surface. Neither is the centre of the Earth's surface the equator or the poles, as without the Earth's spinning, these locations are entirely arbitrary. The prime meridian is arbitrary and historical in nature, North and South poles can be interchanged, and elements of the axial tilt are measured externally to the Sun. There is nothing on the Earth's surface by which a non-arbitary point can be said to be the centre. There is only one conclusion; the entire concept of a "centre" is meaningless on an unbounded space.