Infinite Universe? Heaven?

[hecd2]

When someone says the universe is flat, does he mean that if we travel in a given direction we will get to the edge on the universe in a time “t”, while if we travel in a direction perpendicular to the first one we are going to get to the edge in a very small fraction of “t”?

And when someone says the universe is curved, does it mean that if we travel in a given direction, no matter which it is, we will get to the edge in a finite time (quite long but finite)?

As you can see from my post above, it is not thought that the Universe has an edge. The definition of a flat space, like a flat sheet, is one in which the angles of a triangle add up to 180 degrees and parallel lines never meet- this is also known as Euclidean geometry. On a positively curved space, like a sphere, the angles of a triangle add up to more than 180 degrees and parallel lines can meet (eg, meridians on a sphere are parallel at the equator and meet at the poles.) Similarly for negative curved spaces (except that angles of a triangle add up to less than 180 degrees). The examples of a flat sheet and a sphere are examples of 2 dimensional spaces, whereas space in the universe is 3-dimensional, but the principle is the same.

JuanFlorencio:

As for the expansion rate, I can imagine the sequence of increasing expansion rate, followed by a slowing down rate, then a stop, and then an accelerating contraction; but which physical principles would explain such behavior?

The slowing down and reversal of the Universe’s expansion would be caused by gravitational attraction of all the mass in the Universe to itself. For that to happen the mass-energy density in the Universe would have to be above a critical value.
JimG.:

Space is curved, as I understand it, by the presence of matter. So that a large star, for example curves the space around it. Gravity can perhaps be conceived as simply due to the curvature of space, but again I am no physicist or cosmologist, so I would recommend reading some better sources. In any case, there are no “edges” to the universe no matter its shape. The surface of the earth is a curved shape. You can travel in any direction and never come to an edge. But if it were an infinite flat surface, the same would be true.

This is broadly correct. The first thing to realise is that anything one says about General Relativity in words is only a rather faint echo of the actual theory which can only properly be expressed in mathematics - so bear that in mind when you read what I or anyone else says about it. Space is curved locally and globally by the presence of matter and energy (which are equivalent according to GR). The presence of mass-energy curves space-time - it is the curvature of the 4-dimensional manifold of space-time that gives rise to the effects of gravity. Objects in 4D space-time not under the influence of external forces follow geodesics (which are the curved space equivalents of straight lines in flat space) which appear to us as accelerations due to gravity.

 

[You]

can something infinite expand?

 

[hecd2]

We can observe that when a small body approaches a big one, it accelerates just as when it goes down along a slope (which is a curvature of a surface, or two-dimensional space). Then we can think (though we can’t imagine it) that also in the space of three dimensions there can be a kind of curvature (and we can fancy that it is produced by the presence of massive bodies): then, when bodies approaching a big one are accelerated, we could conceive that they are going down a three dimensional slope. Then, if that three dimensional curvature is produced by the presence of a massive body, such curvature at a given location does not imply that there are no more bodies beyond it. In other words, the space would be curved in many locations, depending on the presence of massive bodies here and there. As you suggest, the abundant curvatures of the space would not imply an edge of the universe.

There can be local curvature and global curvature. There is definitely local curvature due to the presence of large masses such as stars, galaxies and clusters. The cosmological question is whether the Universe is curved or flat overall. Measurements indicate that it is flat (within an error of about 0.4%). If the Universe is flat overall it could be infinite. However, there are some flat topologies which are finite and unbounded because they reconnect - for example a 3-torus. It comes a surprise to many people that the surface of a torus is flat but it is, because it satisfies the definition of a flat surface - angles of a triangle on a torus add up to 180 degrees and parallel lines never meet. A 3-torus is the 3 dimensional equivalent aof a familiar two dimensional torus (like a 3-sphere is the three dimensional equivalent of our familiar 2-sphere). So a 3-torus topolgy would be flat, finite and unbounded (no edge).

Also, if we fancy that the space has elastic properties it would be easy to understand that it is expanding now, and after a certain time it will stop and then contract. However, it would still be impossible to understand an accelerating expansion of it.

No-one know for sure what is causing the accelerating expansion, but it seems to behave as though there is a certain term in the equations of GR, which we call for now, dark energy. The dark energy acts like a pressure accelerating the expansion of space. The mass equivalence of the dark energy is also needed to get the critical mass-energy density for a flat universe.

Regarding the surface of the earth as an example, it is clear to me that we are on the edge already, and if we simply jump (that is to say, if we travel in a direction perpendicular to the surface), we go beyond the edge, don’t we? And if it were an infinite flat surface exactly the same would apply.

Not quite. The surface of the Earth is not an edge in the same sense because a 2-D surface like a sphere is undefined outside the surface. Some flat 2-D being living on a 2-D surface would not be aware that there is anything perpendicular to or beyond the surface. We can jump off the Earth because we are 3-D beings living in 3-D space. The edge of a 2-D surface is like the edge of a sheet of paper. A sphere has no edge.

Perhaps hecd2’s can tell us about those evidences of the overall flatness of the universe. Perhaps he can even explain in which sense the universe is flat. It would be interesting.

The evidence for the flatness comes mainly from the Cosmic Microwave Background. Two major experiments, WMAP and Planck, measured the angular size of certain features in the CMB. We know what angular size those features would be if the Universe is flat and, indeed, they turn out to be that size within an error of 0.4% (we are in effect checking what the angles in a cosmically large triangle add up to). Of course, the Universe could have a very slight negative or positive curvature smaller than our current ability to measure - the correct cautious statement is that the measurements are compatible with flatness. There is also the possibility that the OBSERVABLE universe is flat but is embedded in a larger space which is not flat - but we couldn’t ever test that so it is a pure speculation.

 

[yppop]

can something infinite expand?

No!!! Even though there are those that claim the universe is infinite even while still expanding.
yppop

 

[hecd2]

can something infinite expand?

I don’t see why not given that the cardinality of the real numbers on any finite number interval is the same (the cardinality of the continuum). Or to put it another way, n times infinity equals infinity where n is any finite number.

 

[Rolltide]

can something infinite expand?

Well, counterintuitively, there are some infinities that have been mathematically proven to be larger than other infinities. As an example of this, take the set of all whole numbers (…-2, -1, 0, 1, 2,…). This is an infinite set. However, if you take the set of all rational numbers (which includes every possible permutation of decimals), this has been proven to be a larger infinity than the first one.

It should also be noted that while we currently believe the universe to be flat, it is possible that the curvature of the universe is simply so slight that it is outside our range of measurement. This is similar to how the Earth looks flat until you get VERY far away, like being in orbit. If the universe does have a curvature, then it would be theoretically possible that if you traveled faster than light in a single direction, you would simply curve around (just like flying around the globe) and end up right back where you started. Keep in mind that if the universe IS curved that it is also still expanding, and getting ever larger and larger.

Another mind-bending fact to consider is that time is physically connected to space in a construct called “space-time”. The faster you travel, the slower time progresses in relation to the rest of the universe. The consequence of this is that for particles traveling at the speed of light (for example, photons), they lose all mass and time completely stops, making them for all intents and purposes functioning in eternity, experiencing all time simultaneously.

 

[You]

is infinity real. it sounds like a paradox. infinity stretches forever but at the same time it must be at the very end at the very same time but there is no end because it keeps stretching but it doesn’t stretch because its already there etc., etc.

 

[LogisticsBranch]

Hello
Interesting conversation. Here is an excerpt from a large document that I would like to add to the conversation:

U.S. Department of Energy
National Lab Scientists Win Nobel Recognition
October 6, 2011

*Science is all about opening eyes and expanding horizons. This week, Secretary Chu congratulated two scientists for their trailblazing work: Dr. Saul Perlmutter in the Energy Department’s Lawrence Berkeley National Laboratory, who was recently named the winner of the 2011 Nobel Prize in Physics, and Dr. Daniel Shechtman, who is currently an associate scientist at the Energy Department’s Ames Laboratory, for winning the 2011 Nobel Prize in Chemistry “for the discovery of quasicrystals.”

“[Dr. Perlmutter’s] groundbreaking work showed us that the expansion of the universe is actually speeding up, rather than slowing down," said Secretary Steven Chu, who was a 1997 Nobel Prize winner in Physics and former director at LBNL, in a statement congratulating the physicist. “Dr. Perlmutter’s award is another reminder of the incredible talent and world leading expertise America has at our National Laboratories. On a more personal note, I am delighted about this well-deserved recognition and to have worked with Saul during the time I spent at the Berkeley Lab.”

Berkeley Lab’s Dr. Perlmutter, who shared the Nobel Prize with Brian Schmidt and Adam Riess, specifically studied a special class of exploding stars known as Type Ia supernovae and discovered that the universe is expanding at an accelerating rate. These type of supernovae are all about the same brightness: They are thought to explode in about the same way, and so the dimmer they are, the more distant they are.

In studying distant supernovae, Dr. Perlmutter discovered that they weren’t slowing down as they moved away, but rather were zooming out at an accelerating rate. As Dr. Perlmutter found, that accelerating expansion seems to be happening on large scales, and it seems to be driven by a phenomena we now know as dark energy.

Scientists has measured that dark energy makes up some 70 percent of the mass-energy in the universe – a huge fraction of everything that’s out there. But they still don’t know what it actually is. As the Nobel Committee noted, “It is an enigma, perhaps the greatest in physics today.”

[snip]
*
energy.gov/articles/national-lab-scientists-win-nobel-recognition

https://www.energy.gov/articles/national-lab-scientists-win-nobel-recognition

Thank you.

 

[hecd2]

Well, counterintuitively, there are some infinities that have been mathematically proven to be larger than other infinities. As an example of this, take the set of all whole numbers (…-2, -1, 0, 1, 2,…). This is an infinite set. However, if you take the set of all rational numbers (which includes every possible permutation of decimals), this has been proven to be a larger infinity than the first one.

This isn’t quite right. The set of all rational numbers and the set of all integers is the same size. Perhaps you’re thinking of the set of all real numbers which is a larger infinity. The proof that the set of all integers and the set of all rational numbers are the same size is very clever and very elegant.

 

[yppop]

I don’t see why not given that the cardinality of the real numbers on any finite number interval is the same (the cardinality of the continuum). Or to put it another way, n times infinity equals infinity where n is any finite number.

hecd2
Yes, n x aleph(1) = aleph(1). which means that multiplying an infinite set of real numbers by any other number DOES NOT CHANGE the infinite set of real numbers which proves you can’t EXPAND an infinite set of real numbers that just happen to represent all the points of 3-dimensional continuous space. You then must conclude that if you can’t expand an infinite set of numbers, and since all the points in 3D space are in one to one correspondence with the numbers on the real number line then you can’t expand infinite space. And it is space that defines the dimensions of the universe, hence you can’t expand an infinite universe or the space of which it is composed…

The sequence of infinite sets, aleph(0), aleph(1), …aleph are advanced not by multiplication but by applying the concept of power sets.

The universe is finite and always will be no matter how big it becomes.

Yppop

 

[Bradski]

Aren’t you making the assumption that it was finite to start?

 

[hecd2]

hecd2
Yes, n x aleph(1) = aleph(1). which means that multiplying an infinite set of real numbers by any other number DOES NOT CHANGE the infinite set of real numbers which proves you can’t EXPAND an infinite set of real numbers that just happen to represent all the points of 3-dimensional continuous space. You then must conclude that if you can’t expand an infinite set of numbers, and since all the points in 3D space are in one to one correspondence with the numbers on the real number line then you can’t expand infinite space. And it is space that defines the dimensions of the universe, hence you can’t expand an infinite universe or the space of which it is composed…

The expansion of an infinite space does not entail changing the cardinality of the set of points in space.

Let’s assume space is continuous. Then at time t0 the set of points in an infinite space is bijective to the set of real numbers. If at t1 space has expanded by a factor of 2 or 10, or 10,000 or any arbitrarly large finite number then the set of points in space is still bijective to the set of real numbers.

If space is quantised then the same argument applies except that the set of points in space is countably infinite, the same as the set of natural numbers. This is a Hilbert Hotel.

Take the number line. There is an uncountably infinite set of real numbers on the number line between 0 and 1. If you expand the interval to be between 0 and 10,000 the real numbers on that interval are still uncountably infinite and are bijective to the real numbers on the 0 to 1 interval.

The sequence of infinite sets, aleph(0), aleph(1), …aleph are advanced not by multiplication but by applying the concept of power sets.

True, but irrelevant.

The universe is finite and always will be no matter how big it becomes.

Your conclusion does not follow from the maths.

 

[LogisticsBranch]

I hope our audience will review my message #27 on the previous page.

The universe is finite and always will be no matter how big it becomes.

Yppop

Hello dearie

I’ve been baking Christmas goodies for the whole neighborhood today! Trays upon trays of homemade cookies. Yummy! Now, I’m thinking about what you said. Hopefully, it will be helpful to everyone.

This is an excerpt from NASA
Webmaster: Britt Griswold
NASA Official: Dr. Edward J. Wollack
Page Updated: Friday, 01-24-2014

“We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe.”
map.gsfc.nasa.gov/universe/uni_shape.html
http://map.gsfc.nasa.gov/universe/uni_shape.html

NASA (National Aeronautics and Space Administration) is a highly reputable website!

 

[JuanFlorencio]

There can be local curvature and global curvature. There is definitely local curvature due to the presence of large masses such as stars, galaxies and clusters. The cosmological question is whether the Universe is curved or flat overall. Measurements indicate that it is flat (within an error of about 0.4%). If the Universe is flat overall it could be infinite. However, there are some flat topologies which are finite and unbounded because they reconnect - for example a 3-torus. It comes a surprise to many people that the surface of a torus is flat but it is, because it satisfies the definition of a flat surface - angles of a triangle on a torus add up to 180 degrees and parallel lines never meet. A 3-torus is the 3 dimensional equivalent of a familiar two dimensional torus (like a 3-sphere is the three dimensional equivalent of our familiar 2-sphere). So a 3-torus topology would be flat, finite and unbounded (no edge).

Fascinating! It seems to me then that those notions of curvature and flatness are generalized notions; and I wonder if the generalized notion of curvature is incompatible with the generalized notion of flatness. When you say that large masses curve the space(-time?) locally, does it mean that in the vicinity of these large masses the internal angles of a triangle do not sum up 180 degrees?

Also, what does it mean for the 3-torus to be unbounded? Does it mean that it cannot be “observed” from outside even though it is finite?

No-one know for sure what is causing the accelerating expansion, but it seems to behave as though there is a certain term in the equations of GR, which we call for now, dark energy. The dark energy acts like a pressure accelerating the expansion of space. The mass equivalence of the dark energy is also needed to get the critical mass-energy density for a flat universe.

I can see that I am missing a number of elements to be able to establish the correct relations… When you see that a body is accelerated as it approaches a large mass, you say (if I have “understood” correctly) that the space is curved there (referring to a generalized notion of curvature). You can look for the large mass which is causing the curvature and you will find it somewhere. On the other hand, if you notice that the universe is expanding more and more rapidly, instead of saying that the space becomes more and more curved there, you prefer to say that there is a dark energy acting like a pressure. I guess these would be generalized notions of energy and pressure, and not the notions that common university people might have. Is it?

Not quite. The surface of the Earth is not an edge in the same sense because a 2-D surface like a sphere is undefined outside the surface. Some flat 2-D being living on a 2-D surface would not be aware that there is anything perpendicular to or beyond the surface. We can jump off the Earth because we are 3-D beings living in 3-D space. The edge of a 2-D surface is like the edge of a sheet of paper. A sphere has no edge.

I realize that now! I am missing some level of imagination: How could I have imagined that I had to see myself as a flat being with a flat mind?

But then, do cosmologists conceive the universe as finite but unbounded because it reconnects? If so, it seems to me that the reconnection requires an extra dimension which implies a generalized exteriority in respect to the reconnected space. Also, I am very curious now to know which interactions led cosmologists to think that the universe is reconnected and unbounded?

The evidence for the flatness comes mainly from the Cosmic Microwave Background. Two major experiments, WMAP and Planck, measured the angular size of certain features in the CMB. We know what angular size those features would be if the Universe is flat and, indeed, they turn out to be that size within an error of 0.4% (we are in effect checking what the angles in a cosmically large triangle add up to). Of course, the Universe could have a very slight negative or positive curvature smaller than our current ability to measure - the correct cautious statement is that the measurements are compatible with flatness. There is also the possibility that the OBSERVABLE universe is flat but is embedded in a larger space which is not flat - but we couldn’t ever test that so it is a pure speculation.

There must be a lot of theory behind the interpretation of the CMB phenomena. When you say that you are checking the internal angles of a cosmically large triangle, I cannot figure out what that means. I can imagine that we can draw triangles on the surface of a torus or of a sphere, but you must be speaking of a physical triangle (one that you build somehow in the physical space) not of a mathematical triangle (one that you just conceive in your mind). How do you do that?

 

[JuanFlorencio]

I don’t see why not given that the cardinality of the real numbers on any finite number interval is the same (the cardinality of the continuum). Or to put it another way, n times infinity equals infinity where n is any finite number.

I don’t think the real numbers are a good example to realize if something infinite can expand or not, hecd2: you are not expanding the set of real numbers when you multiply its elements by any finite number, but it remains exactly the same. Besides, numbers are not “there”, but are just mental actions.

 

[JuanFlorencio]

The expansion of an infinite space does not entail changing the cardinality of the set of points in space.

Let’s assume space is continuous. Then at time t0 the set of points in an infinite space is bijective to the set of real numbers. If at t1 space has expanded by a factor of 2 or 10, or 10,000 or any arbitrarly large finite number then the set of points in space is still bijective to the set of real numbers.

If space is quantised then the same argument applies except that the set of points in space is countably infinite, the same as the set of natural numbers. This is a Hilbert Hotel.

Take the number line. There is an uncountably infinite set of real numbers on the number line between 0 and 1. If you expand the interval to be between 0 and 10,000 the real numbers on that interval are still uncountably infinite and are bijective to the real numbers on the 0 to 1 interval.
True, but irrelevant.

Your conclusion does not follow from the maths.

But precisely a definition of an infinite set is that it has a proper subset with which a bijective function or transformation can be established. Once you have considered the interval between 0 and 1, you can “expand” it to include the interval between 1 and 2, but by doing that you are not expanding the set of real numbers. Whatever interval you consider, it can be associated to an infinity of real numbers, but not all the real numbers will be associated to it: for example, the real numbers associated to the interval between 2 and 3 are not the same as those associated to the interval between 1 and 2. You cannot expand the set of real numbers, and you cannot expand the subset of real numbers associated to any interval.

 

[yppop]

I hope our audience will review my message #27 on the previous page.

Hello dearie

I’ve been baking Christmas goodies for the whole neighborhood today! Trays upon trays of homemade cookies. Yummy! Now, I’m thinking about what you said. Hopefully, it will be helpful to everyone.

This is an excerpt from NASA
Webmaster: Britt Griswold
NASA Official: Dr. Edward J. Wollack
Page Updated: Friday, 01-24-2014

“We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent; however, since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe.”
map.gsfc.nasa.gov/universe/uni_shape.html
http://map.gsfc.nasa.gov/universe/uni_shape.html

NASA (National Aeronautics and Space Administration) is a highly reputable website!

This is a typical pronouncement of the materialistic science community that can’t accept the possibility that God might exist as is suggested by that the fact that the universe had a beginning in a creation-like event called the big bang. Notice that the flatness of space only “suggests” that the Universe is infinite in extent; it doesn’t “mean” it is infinite. All they can “conclude” is that the universe is “larger” than what we observe.

If you believe the big bang, at one point in time the whole universe had expanded from a point-like object called the singularity to the size of a pumpkin. It was finite then and must still be finite and nothing can expand from the finite to the infinite.

The nature of the infinite is such that no matter how long and how far you traveled towards it you would not be any closer to it than when you first started. It is impossible to reach infinity. The only way the universe can be infinite is to have been infinite eternally and by that I mean there was no beginning, that the universe always existed, and that is contrary to the big bang theory and the defined dogma of the Catholic Church.

Yppop

 

[Rolltide]

This isn’t quite right. The set of all rational numbers and the set of all integers is the same size. Perhaps you’re thinking of the set of all real numbers which is a larger infinity. The proof that the set of all integers and the set of all rational numbers are the same size is very clever and very elegant.

Thank you for correcting me. The second I saw your post I realized I’d confused it.

 

[You]

but if the universe is expanding then you could draw lines backwards to a point which cannot logically become infinitely small. There is a limit on infinity in an expanding universe.

 

[JuanFlorencio]

but if the universe is expanding then you could draw lines backwards to a point which cannot logically become infinitely small. There is a limit on infinity in an expanding universe.

It seems proved that the universe is expanding. It’s infinity is a matter of speculation. It’s topology is a matter of speculation as well. That being said, if for the expanding universe you go mentally in reverse you could go for ever without ever stoping, asymptotically approaching a predetermined minimum size without ever reaching to it. Or you can always conceive a size which is smaller to any other given size, no matter how small it is. Then you would be asymptotically approaching zero without ever reaching to it. But it is because nothing prevents your mind from continuing a dividing operation for ever. However, physical division is a different thing: you cannot go for ever with it. Then it seems to me you are right.

Now, when I imagine the Big Bang, I cannot avoid imagining the initial particle in the middle of space. And if I focus my attention to that space, I can’t avoid thinking that it is infinite, because nothing prevents my mind from going always beyond a certain point (if I look at the particle there is always more space at my back). That happens to me even when I think about a finite universe: there is always room for a expanding universe in my mind. And all that is speculation too.