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## Main Question or Discussion Point

This is a rather long post, but I hope it will get interesting enough along the way to make you more and more interested to keep reading. Otherwise, never mind ;-)

I have a seemingly very simple question: why should an infinite universe tend to contract in the absence of a cosmological constant or dark energy? I've read some answers in various places, but they haven't really convinced me so far and I'd like to find out where my own reasoning is going off track (or, more unlikely, I might be right and hundreds of well known scientists might be wrong). I personally don't see any reason for a contraction at all.

In "A Brief History of Time", Stephen Hawking says that at one point astronomers thought an infinite, homogenously filled universe would not contract because matter has no central point to fall to. He then explains that this is "one of the pitfalls that you can encounter in talking about infinity", and that you should start out with a finite universe (which always contracts) and then add more mass uniformly around the initial part. Since a hollow sphere does not contribute to the field of gravity inside of it, the net effect of the added matter is zero. Since you can keep doing this up to infinity, the infinite universe should contract as well.

However (with all due respect, and once again, I may be completely wrong), I think he fell into a different pitfall of infinities: integrating a function that is not absolutely convergent over an infinite domain by choosing a convenient sequence of sets (concentric spheres) to approach infinity, while a different sequence may yield a different result.

If someone were to ask you what is the integral from -infinity to infinity of f(x) = x, the answer would be, at best, nuanced. If you consider this to be the limit of a sequence of integrals from -r ro r with r approaching infinity, the answer is clearly zero. However, if you consider it to be the integral from -r to r+1, the anser is suddenly infinity. You can even choose sequences of integration intervals, for example [-r, r+1/r], to get any value you like. This is an integral that is not absolutely convergent, and therefore has no definite value unless you agree on a specific method of integration for some particular purpose.

The same kind of flaw is present in Hawking's argument: he keeps adding mass uniformly in hollow spheres around the initial bunch of points, so that it all cancels itself out. But why should you? If you use non-spherical ovoids instead of spheres, you'll find different speeds of contraction in different directions, with less contraction in the direction of the longest axis of the ovoids. The universe might even expand in a particular direction if you use a seqence of coaxial cones with both the top and the base moving away as the cones grow bigger. Yet, these, too, will eventually include any part of the universe so they should be perfectly valid for integration to infinity if the result is to be well-defined.

I admit these approaches are all much less likely than Hawking's result, because of their asymmetry, but they do show that integration to infinity by using a specific sequence of sets is not guaranteed to give a correct result. Anyway, there's a more important flaw:

Why should you consider a sequence of sets of spheres around a single point to calculate the difference in gravitational acceleration between two different points? Since gravity travels at the speed of light, a more correct approach would be to compare the gravity acting on a point from a sequence of spheres around that point, with the gravity acting on a second point from a sequence of spheres around that other point. Clearly, then, the net effect should be zero. The last sphere to be considered is the one with a radius equal to the age of the universe times the speed of light. Even if you are wondering about the possibility of an infinitely old universe, I could argue that the result should remain valid since it is valid for any universe with a finite age, using the same kind of argument Hawking used for extrapolating from finite to infinite universes.

The only reason that finite universes contract is that, as my spheres of integration get bigger (centered around each point respectively), one of them will run into the edge of the universe on one side. The lack of gravity from that far side will completely explain the gravitational "attraction" between the two points. It has nothing to do with local attraction, but rather with the asymmetrical distribution of matter at a distance. In an infinite universe, no such asymmetry exists.

Since my argument might still not be totally convincing, I will now give a counterexample that (I think) disproves the soundness of extrapolation from finite to infinite universes. Consider this simple infinite "toy" universe, which does not resemble ours, but which I will only use to demonstrate the logical flaw in the argumentation.

Imagine a cartesian, Newtonian, non-relativistic universe called R^3. It goes on up to infinity in all directions, and you can consider any point to be its center. However, unlike our universe, it does have absolute speeds and accelerations. This pretty much corresponds to the Newtonian view of the universe. Now imagine that, for a convenient choice of coordinates, this toy universe has a stationary object of mass 1 on every integer lattice point.

Will this universe contract? In that case, you should be able to show at least one object that will start moving in some direction. But in what direction should it accelerate? There can be no preference for any particular direction since you can consider any point to be the center of the universe. Therefore, the only result can be that this toy universe remains static.

I can even do one better: remove any single object from the universe, and it will start expanding!

Of course I know that our universe does not resemble my toy universe (for one thing, the real universe does not have absolute speeds or even accelerations), but it only serves as a counterexample to make the point that "finite universes contract" does not necessarily imply "infinite universes contract". In the case of my toy universe, finite universes do contract (even with the same speed for any spherical universe) while the infinite version clearly does not. Yet, you might apply Hawking's argument to my toy universe and conclude that it must contract, which shows the logical unsoundness of the extrapolation from finite to infinite universes.

I know the real universe has a lot of properties that are different from my toy universe, and a few of them would invalidate the objections against contraction that I used in my example, but I can't see any reason why this should MAKE the real universe contract. Removing objections against contraction is not enough to show a cause for contraction.

Just to be clear, I'm not saying the universe without dark energy should definitely not contract, I'm just saying that extrapolation from finite universes to the infinite universe cannot, in itself, be used as an argument for contraction. There might be a different reason, maybe I just haven't heard of it yet.

Anyway, since our universe is relativistic and not Newtonian. let's look at this from a relativistic point of view. My knowledge on the subject is a bit limited (that's an understatement), but I'll give it a shot anyway:

If I understood correctly, gravity is really the curvature of space-time, which is caused by the presence of mass and can be calculated using differences in potential energy. If the gravitational potential decreases (curvature increases) from one point to another, you will be accelerated towards the latter (as measured by an observer at a distance in his frame of reference).

So let's look at the change in gravitational potential when we move from one place in the universe to another. As we start moving, we'll distance ourselves from certain objects while getting closer to others. On average, as long as we're not close to any particular object, the net change in gravitational potential seems to be zero since there's always roughly the same amount of matter in any direction from any vantage point. So why should there be any gravtiational pull between two points? I really don't see a cause for contraction, but I may be missing someting.

I could even go one step further (but now I'm really going out on a limb):

Distant objects are moving away from us at very high, "relativistic" speeds, and therefore have a higher mass. This could imply an increasing curvature of the universe at further distances, in our frame of reference. That in turn might explain the accelerating expansion of the universe! Of course this would be a subjective point of view, the aliens in that faraway galaxy would say that we are the ones being pulled away from them because of the higher curvature on our far side, but this would be one of those typical relativistic paradoxes in which two seemingly different perceptions of reality result in the non-contradictory result that we are simply being pulled apart. We'd disagree on a lot more things, like whose clock is faster, who's taller, etc... but it all magically works out just fine anyway.

So, did I just solve the mystery of the accelerating expansion of the universe? Or am I just a rambling, ignorant newbie? I assume the latter, but this is the best way to learn ;-)

Thanks,

Michel Colman

I have a seemingly very simple question: why should an infinite universe tend to contract in the absence of a cosmological constant or dark energy? I've read some answers in various places, but they haven't really convinced me so far and I'd like to find out where my own reasoning is going off track (or, more unlikely, I might be right and hundreds of well known scientists might be wrong). I personally don't see any reason for a contraction at all.

In "A Brief History of Time", Stephen Hawking says that at one point astronomers thought an infinite, homogenously filled universe would not contract because matter has no central point to fall to. He then explains that this is "one of the pitfalls that you can encounter in talking about infinity", and that you should start out with a finite universe (which always contracts) and then add more mass uniformly around the initial part. Since a hollow sphere does not contribute to the field of gravity inside of it, the net effect of the added matter is zero. Since you can keep doing this up to infinity, the infinite universe should contract as well.

However (with all due respect, and once again, I may be completely wrong), I think he fell into a different pitfall of infinities: integrating a function that is not absolutely convergent over an infinite domain by choosing a convenient sequence of sets (concentric spheres) to approach infinity, while a different sequence may yield a different result.

If someone were to ask you what is the integral from -infinity to infinity of f(x) = x, the answer would be, at best, nuanced. If you consider this to be the limit of a sequence of integrals from -r ro r with r approaching infinity, the answer is clearly zero. However, if you consider it to be the integral from -r to r+1, the anser is suddenly infinity. You can even choose sequences of integration intervals, for example [-r, r+1/r], to get any value you like. This is an integral that is not absolutely convergent, and therefore has no definite value unless you agree on a specific method of integration for some particular purpose.

The same kind of flaw is present in Hawking's argument: he keeps adding mass uniformly in hollow spheres around the initial bunch of points, so that it all cancels itself out. But why should you? If you use non-spherical ovoids instead of spheres, you'll find different speeds of contraction in different directions, with less contraction in the direction of the longest axis of the ovoids. The universe might even expand in a particular direction if you use a seqence of coaxial cones with both the top and the base moving away as the cones grow bigger. Yet, these, too, will eventually include any part of the universe so they should be perfectly valid for integration to infinity if the result is to be well-defined.

I admit these approaches are all much less likely than Hawking's result, because of their asymmetry, but they do show that integration to infinity by using a specific sequence of sets is not guaranteed to give a correct result. Anyway, there's a more important flaw:

Why should you consider a sequence of sets of spheres around a single point to calculate the difference in gravitational acceleration between two different points? Since gravity travels at the speed of light, a more correct approach would be to compare the gravity acting on a point from a sequence of spheres around that point, with the gravity acting on a second point from a sequence of spheres around that other point. Clearly, then, the net effect should be zero. The last sphere to be considered is the one with a radius equal to the age of the universe times the speed of light. Even if you are wondering about the possibility of an infinitely old universe, I could argue that the result should remain valid since it is valid for any universe with a finite age, using the same kind of argument Hawking used for extrapolating from finite to infinite universes.

The only reason that finite universes contract is that, as my spheres of integration get bigger (centered around each point respectively), one of them will run into the edge of the universe on one side. The lack of gravity from that far side will completely explain the gravitational "attraction" between the two points. It has nothing to do with local attraction, but rather with the asymmetrical distribution of matter at a distance. In an infinite universe, no such asymmetry exists.

Since my argument might still not be totally convincing, I will now give a counterexample that (I think) disproves the soundness of extrapolation from finite to infinite universes. Consider this simple infinite "toy" universe, which does not resemble ours, but which I will only use to demonstrate the logical flaw in the argumentation.

Imagine a cartesian, Newtonian, non-relativistic universe called R^3. It goes on up to infinity in all directions, and you can consider any point to be its center. However, unlike our universe, it does have absolute speeds and accelerations. This pretty much corresponds to the Newtonian view of the universe. Now imagine that, for a convenient choice of coordinates, this toy universe has a stationary object of mass 1 on every integer lattice point.

Will this universe contract? In that case, you should be able to show at least one object that will start moving in some direction. But in what direction should it accelerate? There can be no preference for any particular direction since you can consider any point to be the center of the universe. Therefore, the only result can be that this toy universe remains static.

I can even do one better: remove any single object from the universe, and it will start expanding!

Of course I know that our universe does not resemble my toy universe (for one thing, the real universe does not have absolute speeds or even accelerations), but it only serves as a counterexample to make the point that "finite universes contract" does not necessarily imply "infinite universes contract". In the case of my toy universe, finite universes do contract (even with the same speed for any spherical universe) while the infinite version clearly does not. Yet, you might apply Hawking's argument to my toy universe and conclude that it must contract, which shows the logical unsoundness of the extrapolation from finite to infinite universes.

I know the real universe has a lot of properties that are different from my toy universe, and a few of them would invalidate the objections against contraction that I used in my example, but I can't see any reason why this should MAKE the real universe contract. Removing objections against contraction is not enough to show a cause for contraction.

Just to be clear, I'm not saying the universe without dark energy should definitely not contract, I'm just saying that extrapolation from finite universes to the infinite universe cannot, in itself, be used as an argument for contraction. There might be a different reason, maybe I just haven't heard of it yet.

Anyway, since our universe is relativistic and not Newtonian. let's look at this from a relativistic point of view. My knowledge on the subject is a bit limited (that's an understatement), but I'll give it a shot anyway:

If I understood correctly, gravity is really the curvature of space-time, which is caused by the presence of mass and can be calculated using differences in potential energy. If the gravitational potential decreases (curvature increases) from one point to another, you will be accelerated towards the latter (as measured by an observer at a distance in his frame of reference).

So let's look at the change in gravitational potential when we move from one place in the universe to another. As we start moving, we'll distance ourselves from certain objects while getting closer to others. On average, as long as we're not close to any particular object, the net change in gravitational potential seems to be zero since there's always roughly the same amount of matter in any direction from any vantage point. So why should there be any gravtiational pull between two points? I really don't see a cause for contraction, but I may be missing someting.

I could even go one step further (but now I'm really going out on a limb):

Distant objects are moving away from us at very high, "relativistic" speeds, and therefore have a higher mass. This could imply an increasing curvature of the universe at further distances, in our frame of reference. That in turn might explain the accelerating expansion of the universe! Of course this would be a subjective point of view, the aliens in that faraway galaxy would say that we are the ones being pulled away from them because of the higher curvature on our far side, but this would be one of those typical relativistic paradoxes in which two seemingly different perceptions of reality result in the non-contradictory result that we are simply being pulled apart. We'd disagree on a lot more things, like whose clock is faster, who's taller, etc... but it all magically works out just fine anyway.

So, did I just solve the mystery of the accelerating expansion of the universe? Or am I just a rambling, ignorant newbie? I assume the latter, but this is the best way to learn ;-)

Thanks,

Michel Colman