Is the mass of the universe finite (collection of objects)?

[William Henley]

The amount of matter gained as the observable universe increases slightly over cosmological time scales is something on the order of a rounding error in maybe the 4th decimal place, so while you are technically correct, it's a rather trivial nitpick and it's not clear to me that the reality of the situation is really what you had in mind. I got the impression from your statement that you believe there is some significant amount of matter being added.

I agree that it's not that significant but it is still relevant

 

[William Henley]

To be clear, the only way the observable universe is (sometimes) gaining mass is when we see farther and discover previously hidden regions. This "mass gain" if just the reflection of its "volume gain" which is strictly a result of perspective, not something "happening to the universe".

Yes I agree but the observable universe is still appearing to gain mass.

 

[phinds]

Perhaps an interesting would be: Suppose the mass of the entire universe was infinite, what implications would this at the moment of the Big Bang?

The implication would be that at the time of the singularity, things were already infinite and since the belief is getting stronger all the time that the universe IS spatially infinite, that IS the belief (but not confirmed)

 

[PeterDonis]

the observable universe is still appearing to gain mass.

If the expansion of the universe continues to accelerate (which, as far as we can tell, it will), this will not always be true; there will come a point when the acceleration moves objects out of our observable universe faster than the passage of time moves them in, so to speak. Again, if more discussion is desired on this, please start a new thread.

 

[PeterDonis]

Suppose the mass of the entire universe was infinite, what implications would this at the moment of the Big Bang?

None. The Big Bang is not a "moment"; correctly used, that term just refers to the hot, dense, rapidly expanding state of the early universe, not to an "initial singularity". The initial singularity is present in highly idealized models, but all that really tells us is that those models break down if we extrapolate them too far backward. The best answer to what came before the hot, dense, rapidly expanding state is that we don't know for sure; we have a number of hypotheses, but no conclusive tests to choose between them. An "initial singularity" is not one of them.

 

[Jon Richfield]

Perhaps an interesting would be: Suppose the mass of the entire universe was infinite, what implications would this at the moment of the Big Bang?

Seems to me that a more interesting question would be what an infinite universe would mean. Infinity isn't big, but beyond bog. In fact, it is so far beyond big that finity itself is more interesting IMO.
You doubt that?
Think of the biggest finite prime integer you can, expressed explicitly in some convenient positive positive integer base >3. Decimal or hex would be fine, but suit yourself. Get a few friends to help you think of bigger ones (none less than say 1000 significant digits need apply. Choose n numbers, n> 3, the more the merrier, but no prime that you choose is permitted to contain any digit smaller than three, or any string of more than thirty identical adjacent digits). Take them in some sequence and call them V1, V2...Vn and calculate P=V1^V2^V3^...Vn.
Long before this time you have an answer too large to fit into the volume of the observable universe even if the universe were filled solid with protons and each proton somehow held one digit. Now raise P to the P, and repeat that process P times. Next add the first P decimal digits of Pi, taken as expressing an integer. Multiply the result by the same number of decimal digits of the decimal expansion of the 23rd root of e, taken as expressing an integer. Call the result P'.
With negligible exceptions every FINITE number, in fact starting with P'+1, is larger than P', whose value you don't know because you ran out of observable universe to store your calculations in before you got past the first few steps. In fact, long before you got to the final P' you did not know a single digit of any of the numbers you were working with, let alone the final value of P'.
There is in fact very little you know about P' EXCEPT that P'<P'+1.
And that no one could tell you more because we have not world enough and time. We don't even know how long it would take light to reach our planet at the centre of our globularly packed number, starting from the outside.
Now, you will no doubt be asking what all this runaround is supposed to deliver, right?
Because asking about any REAL infinity is meaningless from a whole swadge of points of view. What attribute of an infinite (not just finitely large) universe could affect us where we are? Light from more than say 1e41 LY couldn't even reach us because of red shift, and even if it could, it would take too long to be of interest. Nor could any other physical signal. But a sphere of 1e41 LY filled with proton-sized numeric digits is vanishingly tiny compared with anything like P' or a brain large enough to read or calculate any number on such a scale.
And packing space so densely would be waaayyy beyond any kind of gravitational collapse, but my mind boggles at trying even to guestimate how far we'd have to expand our P' if we wished to prevent collapse.
And P' is negligibly small, remember...
Now, if you are not yet sick of that game, have fun playing it for as long as you like, but for my part I fall off the bus just trying to imagine what a modestly large finite universe would be like if P' is so tiny.
How could we even in principle tell the difference between our tiny P' notional universe and a modestly large one, let alone an infinite one, whatever that might mean?
In such terms just what do you see an infinite universe meaning, let alone existing?

 

[King Solomon]

I too have the same thought.

A universe comprising a finite set of objects would indeed be intriguing.

As mathematicians we know that there is no "Set of all Sets" so if our universe is finite, then there exists infinitely many finite proper supersets of our universe...!

 

[wabbit]

I too have the same thought.

A universe comprising a finite set of objects would indeed be intriguing.

As mathematicians we know that there is no "Set of all Sets" so if our universe is finite, then there exists infinitely many finite proper supersets of our universe...!

Surely you mean an infinite universe would be intriguing. Having infinitely many proper subsets is equivalent to being infinite

In any case an infinite universe can only be speculative, since we are finite beings and can only know a finite amount of information.

 

[phinds]

In any case an infinite universe can only be speculative, since we are finite beings and can only know a finite amount of information.

Hey, speak for yourself rabbit !

 

[wabbit]

Indeed :)

But against infinity even bigger creatures like dogs or humans are no match.
Even if we relax "knowing" to just "holding information somehow", using all the spin states of all the electrons in all the atoms inside us for starters, then the quantum states of the nuclei, and even the degrees of freedom of the gravitational field, my understanding is that quantum gravity puts a specific number on the maximum amount of information we can hold - more than what can fit in a hare's brain but less than an infinity

 

[phinds]

Well, shucks. That's disappointing. I was looking forward to maybe knowing as much as @Drakkith someday.

 

[King Solomon]

Surely you mean an infinite universe would be intriguing. Having infinitely many proper subsets is equivalent to being infinite

In any case an infinite universe can only be speculative, since we are finite beings and can only know a finite amount of information.

No, finite. I said "SUPERsets" not "subsets." There are a finite amount of subsets for any finite set (the power set).

And that itself is interesting, the Power Set of a Finite Universe would represent all possible collections of objects in this universe ( I assume that elementary particles are only things that qualify as an object), which implies that there are a finite amount of references for our universe (if our universe is finite).

Consider this (using the Power Set of a finite universe): There are (n² - n)/2 distinct relationships between individual pairs of objects (if universe consists of n objects), and there are 2ⁿ distinct shapes within the universe at all times (2ⁿ is the cardinality of the Power Set). Woah!

 

[Jon Richfield]

No, finite. I said "SUPERsets" not "subsets." There are a finite amount of subsets for any finite set (the power set).

Consider this (using the Power Set of a finite universe): There are (n² - n)/2 distinct relationships between individual pairs of objects (if universe consists of n objects), and there are 2ⁿ distinct shapes within the universe at all times (2ⁿ is the cardinality of the Power Set). Woah!

I don't buy that, KS.
It assumes that there is a single relationship between any two objects, whereas I see no simple limit to the number of possible relationships between two objects.
To me it seems perfectly conceivable that either the number of possible mutual (though not necessarily symmetrical) relationships between two arbitrary objects could be physically large, possibly even mathematically large, or even not uniquely (or precisely) definable. Furthermore, the power set members also are entities and therefore have mutual relationships that may not be limited to their component part sub-relationships.
It does not follow of course that the thus inflated power set would be infinite; In fact I believe that if none of those variables were infinite, the entire entity set very likely might be finite, even if inconveniently large to characterise or quantify.
However, if the volume of our space is finite, then I do not believe that the number of those possible relationships is infinite,because it seems to me that this would imply infinite information, and I do not believe that you could fit infinite information into a finite space. (Gravitational collapse and all that... )

 

[King Solomon]

I don't buy that, KS.
It assumes that there is a single relationship between any two objects, whereas I see no simple limit to the number of possible relationships between two objects.
To me it seems perfectly conceivable that either the number of possible mutual (though not necessarily symmetrical) relationships between two arbitrary objects could be physically large, possibly even mathematically large, or even not uniquely (or precisely) definable. Furthermore, the power set members also are entities and therefore have mutual relationships that may not be limited to their component part sub-relationships.
It does not follow of course that the thus inflated power set would be infinite; In fact I believe that if none of those variables were infinite, the entire entity set very likely might be finite, even if inconveniently large to characterise or quantify.
However, if the volume of our space is finite, then I do not believe that the number of those possible relationships is infinite,because it seems to me that this would imply infinite information, and I do not believe that you could fit infinite information into a finite space. (Gravitational collapse and all that... )

Sure, I don't buy it either given the way you interpreted it. Let me clarify what I had originally intended to say:

Given any particular type of relationship, such as velocity, there exists (n^2 - n)/2 distinct relationships of that type between all pairs of objects in a finite universe.

If we take any group (Group A) from a power set and find its average position and velocity relative to some other group (Group B) from a power set, then there exists 2^n-2 distinct relationships between the average positions and velocities of all other groupings of objects in the power set to the average position and velocity of the Union of Group A and B (the average position of Group A and B becomes the relative center of the universe, such that we can make Group A a single object and Group B an empty set, making a single object the relative center of the universe with 0 average velocity relative to itself).

Also, let's consider another implication of a universe of finite objects. If we let the set S represent the positions of all objects in the universe, then the universe's shape can always been reduced to a single polyhedron in accordance to Grunbaum's 1994 concerning the polyhedronization of S.

http://www.sciencedirect.com/science/article/pii/S0925772112000727

If the universe is a finite collection of objects, then question of "is the universe open, closed or flat" no longer applies. The answer would always be a polyhedron.

 

[Jon Richfield]

...

Also, let's consider another implication of a universe of finite objects. If we let the set S represent the positions of all objects in the universe, then the universe's shape can always been reduced to a single polyhedron in accordance to Grunbaum's 1994 concerning the polyhedronization of S.

http://www.sciencedirect.com/science/article/pii/S0925772112000727

If the universe is a finite collection of objects, then question of "is the universe open, closed or flat" no longer applies. The answer would always be a polyhedron.

Hmmm... I hadn't thought of that polyhedral thing, though it has a certain attraction. Mind you, It is not clear to me that universal shape means much in a closed universe, especially if we are speaking of more than three dimensions. It seems to me that in a closed and finite universe, most plausible views would suggest a universe full of dirty space (the dirt including things like hydrogen, stars, and planetary dust particles like Earth and their world lines, time being one of the dimensions, etc etc) much as a balloon would be full of gas, irrespective of its pressure.

Talking about universes is an activity fraught with flypapers for the unwary intellect, I am tempted to reflect...

 

[tadchem]

The observable universe has a finite volume.
Density is (locally) the ratio of mass to volume.
For the mass of the universe to be infinite, the local density of the universe must be (somewhere) infinite.
At the location where infinite density is to be found, the curvature of space-time (i.e. gravity) must also be infinite.
Since the space-time curvature follows an inverse-square law, if there is an infinite curvature *anywhere* there must be an infinite curvature *everywhere*.
[Infinity]/R^2 = [Infinity] for all real values of R.
There is a place where space-time curvature is finite.
Therefore space-time curvature is finite *everywhere*.

 

[PeterDonis]

For the mass of the universe to be infinite, the local density of the universe must be (somewhere) infinite.

Nobody is claiming that the mass of the observable universe is infinite.

space-time curvature follows an inverse-square law

No, it doesn't. Don't confuse Newtonian gravity with GR.

 

[Jon Richfield]

The observable universe has a finite volume.
Density is (locally) the ratio of mass to volume.
For the mass of the universe to be infinite, the local density of the universe must be (somewhere) infinite.
At the location where infinite density is to be found, the curvature of space-time (i.e. gravity) must also be infinite.
Since the space-time curvature follows an inverse-square law, if there is an infinite curvature *anywhere* there must be an infinite curvature *everywhere*.
[Infinity]/R^2 = [Infinity] for all real values of R.
There is a place where space-time curvature is finite.
Therefore space-time curvature is finite *everywhere*.

Do I detect a whiff of non sequitur?
I might assume that you have a cogent support for your first statement, though it seems to assume that the infinite mass of the universe is of the same or greater order than the volume of the universe. If the volume is of a greater order, you will have to explain why the infinity of mass must fill it. If the volume is in fact of the same order and its average density is everywhere the same on the same scale as we observe, then it would have infinite mass of the same order as the volume of space, without anywhere having infinite density (except possibly in some black holes somewhere FAIK).
Density is not mass, nor vice versa.

Please correct my misunderstandings.

 

[King Solomon]

Hmmm... I hadn't thought of that polyhedral thing, though it has a certain attraction. Mind you, It is not clear to me that universal shape means much in a closed universe, especially if we are speaking of more than three dimensions. It seems to me that in a closed and finite universe, most plausible views would suggest a universe full of dirty space (the dirt including things like hydrogen, stars, and planetary dust particles like Earth and their world lines, time being one of the dimensions, etc etc) much as a balloon would be full of gas, irrespective of its pressure.

Talking about universes is an activity fraught with flypapers for the unwary intellect, I am tempted to reflect...

Speaking of the shape of the universe, does the 1994 Polyhedronization proof apply to non-euclidean space?

Another interesting question: Assume the Big Rip is occurring, can set of points S , whose distances diverge to infinity in finite time, still be claimed to form a polyhedron?

I cannot tell you how often I lament that I was born too early. Then again, I'd also lament if I was born so late that every imaginable question had already been answered. Given a choice, I'd choose the former (our current situation). :)