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Does the universe have a fixed centre of mass?
If it does, doesn't it necessarily mean that every action of ours has to be balanced by a counteraction somewhere in the universe so as to neutralize the disbalance of mass?

As far as we know, the universe does not have a centre of mass because it does not have a centre. One of the basic assumptions we use when describing the universe is that, on average, it is the same everywhere. This is called the cosmological principle. While this is only an assumption, the evidence we have from observing the universe suggests that it is true.

This can seem a bit odd if you have the idea that the Big Bang happened at a point and the Big Bang blasted the universe outwards from that point. But the Big Bang did not happen at a point; it happened everywhere in the universe at the same time. For more on this, see Did the Big Bang happen at a point?

It is certainly true that every action of ours has to be balanced by a counteraction because this is just Newton's third law. If I apply a force on you then you apply an equal and opposite force on me, so if we were floating in space our combined centre of mass would not change. So while it does not make sense to ask about the centre of mass of the universe we can ask what happens on a smaller scale, and we find that unless some external force is being applied the centre of mass of a system cannot change.

The universe is not obeying a classical Newtonian physics, only locally Newton's laws hold. The universe as seen in the standard Big Bang model, obeys General Relativity.

This is a cut in the time dimension and one space dimension. At the line of the present universe, all points were at the beginning of the universe, and there can be no center of mass for the observable universe.

Visualize a balloon that starts inflating from a (0,0,0) point in space. At time t the surface is a sphere, and all points on the sphere were at the beginning (0,0,0). Is there a center of mass for the surface? All points are at the center of mass, because they are balanced by all other points.

The balloon is an analogue of the three dimensional space of the universe. In contrast to the balloon, the theory does not need to embed the universe in a higher dimension so as to start with a four dimensional space point. All points in our three dimensions were at the beginning of the Big Bang.

The universe is often modelled by the FLRW metric with the assumption of homogeneity and isotropy of space. If we for simplicity assume that there is no curvature $k=0$, and even if we pick a global coordinate system for space (which would artificially distinguish an origin, but remember: the universe has no center), then the center of mass is given as

$$bf R_rm COM ~=~ fracint_mathbbR^3 !d^3bf r~bf rint_mathbbR^3 !d^3bf r~1,$$

which is mathematically ill-defined. (A similar negative conclusion is reached in the case of curvature $k=pm 1$.)

Well, assuming classical mechanics(which is of course not applicable for our universe), of course our universe does have a fixed centre of mass.

I don't know which level you are in, but if you have the knowledge of general relativity(in which the idea of centre of mass seems pretty stupid) go through the other answers. But if you are somehow interested in the point of view of the classical physics too, you are welcome to read this answer.

See, centre of mass is primarily arises just from a statistical idea which later turns out to be of a greater significance, as the centre of mass of all bodies interacting with each other remain unaccelerated.

Wait, did i say ALL BODIES INTERACTING WITH EACH OTHER? Yes, indeed.

The idea is, any action we take involves more than one bodies. And of course you know that centre of mass remains unaccelerated.

Now, since all the possible centre of masses are unaccelerated, centre of mass of these masses will also remain fixed, giving the universe a fixed CM.

Now one may ask, SUPPOSE I'M PICKING UP A BOX FROM THE GROUND ONTO MY HEAD. WHO'S MOVING OTHER THAN THE BOX? HOW IS THE CENTRE OF MASS BALANCED HERE?

The answer is(sequentially), when you picked up that box you had to apply some amount of force on the box in order to provide an upward acceleration. The reaction from the box caused your weigh more. Earth felt force from your feet more than before. This caused the earth move just a bit away from a box. The Earth is really massive. So the shift on centre of mass is not so negligible. This balances the movement of he centre of mass of you, earth and the box.
So that C. M. remains fixed and your question is answered😊.

Thank you😊