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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?

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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.

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    $\begingroup$ @LenaDas you are correct that the universe is expanding and that means the distances between objects are increasing. However the universe is increasing only on average. Some bits of the universe are shrinking, for example the collapsing dust clouds that create stars, and some are neither shrinking nor expanding, like the Solar System. It is only when we average out all the expanding bits, all the shrinking bits and all the static bits that we end up with the overall average expansion. $\endgroup$ – John Rennie Apr 21 at 14:49
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    $\begingroup$ Anyhow, the Solar System is a static part of the universe so the expansion of the universe does not mean the Earth Sun distance is increasing. $\endgroup$ – John Rennie Apr 21 at 14:50
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    $\begingroup$ @LenaDas What John said; however, there will be observable effects - at some (very distant) point in the future, it will no longer be possible to observe the galaxies that aren't gravitationally bound. This will probably mean that it would not be possible to observe the expansion of the universe either. Also, we don't know if the universe is infinite or not, and we don't know if there's universe beyond what was "our" Big Bang - all we know is that some 15 billion years ago, everything we have evidence for was in a tiny volume. $\endgroup$ – Luaan Apr 21 at 15:41
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    $\begingroup$ "As far as we know the universe does not have a centre of mass because it does not have a centre." I don't quite understand how this is a reason. $\endgroup$ – JiK Apr 21 at 21:14
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    $\begingroup$ I'm inclined to agree with @JiK... What even is a "center" for arbitrary objects? The center of mass is well-defined, but even a volumetric center isn't really meaningful unless you just happen to have, e.g., a sphere... $\endgroup$ – Scott Apr 22 at 15:27
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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

$${\bf R}_{\rm COM} ~=~ \frac{\int_{\mathbb{R}^3} \!d^3{\bf r}~{\bf r}}{\int_{\mathbb{R}^3} \!d^3{\bf r}~1},$$

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

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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😊

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    $\begingroup$ The universe has no center and you can certainly not apply classical concepts (i.e. you need general relativity). $\endgroup$ – StephenG Apr 21 at 9:02
  • $\begingroup$ yeah that's true. i just thought the concept of GR would be still unknown to him/her(by seeing the question). Sorry for such stupid deed. $\endgroup$ – Ritam Dutta Apr 21 at 21:16
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    $\begingroup$ I don't think that adding "assuming classical mechanics" covers it. There's no point in talking about inapplicable physics under the pretext of targeting OP's background. $\endgroup$ – user191954 Apr 22 at 6:13
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    $\begingroup$ Do you think your classical physics answer would apply to an infinite universe? $\endgroup$ – PM 2Ring Apr 22 at 9:22
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    $\begingroup$ Some of this seems like it is addressing OP's point, but everything about the universe having a centre of mass requires some serious assumptions that seem to contradict our experience and observation. The problem is that you're trying to apply math that works on finite objects to something that doesn't really behave that way. $\endgroup$ – JMac Apr 26 at 13:26
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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.

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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.

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    $\begingroup$ The universe is 3 dimensional, and so is a balloon. We have no evidence whatsoever for any curvature in some higher dimension. But we do know that the balloon has a centre. $\endgroup$ – John Duffield Apr 21 at 8:50
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    $\begingroup$ @JohnDuffield The balloon is a two dimensional analogue embedded in our three dimensional space, to give an intuition of why there is no center to the surface. The general relativity obeying uiniverse is not embedded in a four dimensional space, as far as our modelin goes. The proof of the validity of the model is that it explains the observations up to now. $\endgroup$ – anna v Apr 21 at 13:59
  • $\begingroup$ @annav What is meant by "Earliest time visible with light" in the image attached? Does it mean that we've been able to see light coming from a point 380,000 years ago? And that it took light 380000 years to reach us and that any light beyond that has never reached us? $\endgroup$ – Tapi Apr 21 at 19:38
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    $\begingroup$ Lena, the times on the X axis, such as 380,000, are time after the origin, not time backward from the present. So that’s about 13 1/2 billion years ago. $\endgroup$ – prl Apr 21 at 21:57
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    $\begingroup$ Maybe we will be luckier with the gravitaitonal decoupling www-ucjf.troja.mff.cuni.cz/~iss2017/ISS2017_files/… and be able to probe the very beginning $\endgroup$ – anna v Apr 22 at 8:10

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