}

1989/04/01 Arregi Bengoa, Jesus Iturria: Elhuyar aldizkaria

QUESTION. A finite or infinite universe?

QUESTION

A finite or infinite universe?

ELHUYAR. In issue 19 of the magazine ZIENTZIA ETA TEKNIKA, in the section "The Reader's Corner", by Jesús Arregi, I think an idea is wrong.

This article states that: If the gravity force of matter in the universe is sufficient to curb the speed of alienation and produce contraction, the universe is finite. But infinite.

I believe that if the gravity force of matter is sufficient to slow the speed of distance and produce contraction, the expansion will be slower and slower, becoming contraction, which makes the universe pulsable. If not, even if it has always been finite, its measures will increase.

Being the speed of light the largest in the universe and accepting the existence of the Hubble constant (expansion and basis of the “Big Bang” theory), the greatest distance between two points of the universe can reach 12.5 billion light years. See volume 11 Elhuyar, number 1. 1985 Past and future of the Universe. Section 3.

If the mass of the universe were not sufficient to produce contraction, the finitude of the kneading (its geometric measures) would be increasing, but finite still, even if that finitude had no limits.

We put a simple example: the set of natural numbers is infinite, but all natural numbers are finite.

Therefore, to conclude, we must not confuse the two concepts: the margins of the universe or the geometric limits (which do not exist), with the mathematical limits or limits of the measurements of the universe. The possible existence of the latter lies in the amount of invisible mass.

Antón del Campo

To understand the results obtained when applying the Theory of General Relativity to the whole universe, it is necessary to resort to situations of classical mechanics. Therefore, many times these analogies are extracted from their measures and errors are created because differences are not underlined. That is what often happens in the analogy between the evolution of universe expansion and the speed problem that one body needs to escape the gravity of another.

Let us think, to explain the second case, that we want to launch a pitcher to get out of Earth. When performing the test we are presented with three options depending on the speed we give the launcher. In the first, when the speed of the launcher is less than that necessary to overcome the attraction (which we call escape speed), the target is not achieved and the escape theme falls towards the attraction. In the other two, when the speed of the launcher is equal to the speed of escape or the first is greater, the objective is reached, having as only separator the difference of relative speeds between the bodies at the end of the process.

The parity between these conclusions and those derived from the cosmological analysis of the expansion of the universe is not total, and I believe that from that wrong equation you have deduced what you propose in your writing.

Within Classical Mechanics it is possible that what you propose will happen, that is, that the pitcher is a finite universe that expands forever with more energy than that corresponding to the escape speed. (In short, we must consider that the universe is formed only by the Earth and the launchers. The “Big Bang” would be the ignition of the last engines). This composition, however, is not fair, within relativistic cosmology. According to General Relativity, the effect of gravity is a consequence of the deformations or curvatures produced by matter in the tetradimensional space/time.

Therefore, the division of matter that plays against the expansion of the universe limits its form and dimensions. If the influence of gravity is in particular sufficient to curb expansion and produce contraction, the curvature is positive and the three spatial components of tetradimensional space/time will bend around themselves until it closes as the surface of a two-dimensional sphere. In this case the universe is finite, of course; otherwise, the closure of geodesics would be impossible. As for the fourth dimension, that is, time, and us, we can say that it has a beginning with the "Big Bang" and ends when the universe returns to singularity by contraction.

Therefore finitude is the peculiarity of space and time and it cannot happen that the time of a situation in finite space is infinite or vice versa. If the influence of gravity is not enough to produce contraction, the extension will always remain. The space/time curvature in this case is zero or negative and in any case the spatial dimensions are not closed. Therefore, they are infinite. Time is also like this, because although it starts with the "Big Bang" it has no end. We call these universes “open.”

As we have said, there are only two options:

• If the influence of gravity is greater than the limit of contraction, the universe is finite (also in time) and closed.
• If the influence of gravity is weaker than the border, the universe is infinite and open.

As we emphasized above, the particularities mentioned in each of the two options are intimately related and inseparable. On the contrary, the situation you propose requires that the spatial dimensions be infinite finite and temporal (a finite universe that constantly expands), and within General Relativity there is no place for this option: if space is finite also time, if time is infinite also space.

Therefore, I believe that the statement I quote is correct.

I agree with the calculation corresponding to the maximum distance between two points of the universe, but it must be borne in mind that this calculation corresponds to the observable universe and is useful both if it is finite and infinite.

You are right underlining the difference between geometric boundaries or margins and measurement limits. The difference is evident and therefore it is not necessary to mention the first to speak for the second time, as I have treated this time. In the previous answer I mentioned the two to underline the differences between the two and perhaps I did not achieve the goal.

Jesus Arrangi.

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