How can a big bang expand to an infinite size?

[Boltzmannbrain]

19 minutes ago, Airbrush said:

My next question is, can something finite in mass (or size) expand to an infinite size?  And please explain for me without advanced math.  Thank you.

Yes, if it expands fast enough.

[TheVat]

https://en.wikipedia.org/wiki/Kruskal's_tree_theorem

Really large number.

2 hours ago, Genady said:

 

 

What is "number TREE3"?

 

[Genady]

3 hours ago, Airbrush said:

My next question is, can something finite in mass (or size) expand to an infinite size?  And please explain for me without advanced math.  Thank you.

Can anyone explain how mathematicians know that the number TREE3 is finite?

 

1 hour ago, TheVat said:

https://en.wikipedia.org/wiki/Kruskal's_tree_theorem

Really large number.

 

A really large number is finite by definition.

[Genady]

On 12/18/2023 at 1:05 PM, Airbrush said:

can something finite in mass (or size) expand to an infinite size?  And please explain for me without advanced math.

Let me try to explain it with a bit of algebra.

In an expanding homogeneous isotropic universe, a distance between any two points - let's call them, galaxies - is proportional to a number, \(a(t)\), called scale factor, which increases with time, \(t\). So, for example, if a distance between some two galaxies at some moment is \(D\) then later, when \(a(t)\) is twice as large, the distance between these two galaxies is \(2D\). Thus, this distance increases with time as \(a(t)D\).

If the universe is finite, then there is a largest distance in it, which, just like any other distance, is proportional to \(a(t)\). Let's call it, \(a(t)L\). The only way for the \(a(t)L\) to become infinitely large is that \(a(t)\) becomes infinitely large. But, if \(a(t)\) becomes infinitely large, then distance between any two galaxies, \(a(t)D\), becomes infinitely large.

IOW, all galaxies become infinitely far from each other. We of course know that it isn't so. Thus, either the universe was finite and remains finite, or it was infinite to start with.

Edited December 20, 2023 by Genady

[Airbrush]

8 hours ago, Genady said:

Let me try to explain it with a bit of algebra.

In an expanding homogeneous isotropic universe, a distance between any two points - let's call them, galaxies - is proportional to a number, a(t) , called scale factor, which increases with time, t . So, for example, if a distance between some two galaxies at some moment is D then later, when a(t) is twice as large, the distance between these two galaxies is 2D . Thus, this distance increases with time as a(t)D .

If the universe is finite, then there is a largest distance in it, which, just like any other distance, is proportional to a(t) . Let's call it, a(t)L . The only way for the a(t)L to become infinitely large is that a(t) becomes infinitely large. But, if a(t) becomes infinitely large, then distance between any two galaxies, a(t)D , becomes infinitely large.

IOW, all galaxies become infinitely far from each other. We of course know that it isn't so. Thus, either the universe was finite and remains finite, or it was infinite to start with.

Good answer!  Thanks for the reply.

On 12/18/2023 at 9:10 AM, Genady said:

No, in a continuous process it cannot.

 

What is "number TREE3"?

Thank you

Whoops, I meant TREE(3).  They know it is finite, but I don't understand how they know that, since it blows past Graham's number like it's not even there.

Matter in the universe is probably finite, but maybe space by itself can stretch beyond that to infinity, assuming flat or hyperbolic universe?  A positively curved universe is certainly finite in mass and size.

Is "infinity," like a singularity, just an abstract idea?

Edited December 20, 2023 by Airbrush

[AIkonoklazt]

16 minutes ago, Airbrush said:

Infinity, like a singularity, is just an abstract idea?

At least mathematical infinity is abstract https://en.wikipedia.org/wiki/Actual_infinity

Okay, wait. It's abstract, period. https://bigthink.com/articles/infinity-is-not-real/

[Genady]

49 minutes ago, Airbrush said:

Is "infinity," like a singularity, just an abstract idea?

This question belongs to another forum 😉

[Genady]

On 12/20/2023 at 4:03 PM, Airbrush said:

A positively curved universe is certainly finite in mass and size.

Just to clarify, AFAIK, the concept of mass is not applicable to universe and thus it cannot be described as finite or infinite, regardless of the curvature.

[Airbrush]

15 hours ago, Genady said:

Just to clarify, AFAIK, the concept of mass is not applicable to universe and thus it cannot be described as finite or infinite, regardless of the curvature.

That is an amazing thing to consider, and a very new concept for me.  How do you know that?  When it curves around it ends up on a different dimension or groove in the record?

Edited December 23, 2023 by Airbrush

[Markus Hanke]

1 hour ago, Airbrush said:

That is an amazing thing to consider, and a very new concept for me.  How do you know that?

What Genady means is that the universe is a curved spacetime, as described by General Relativity. The thing now is that some concepts we are used to from old Newtonian physics do not straightforwardly translate to GR - and “mass” is one of them.  The question of “what is the mass associated with a given region of spacetime” has no simple answer; there are in fact several different notions of mass that apply to different sets of circumstances, so it really depends.

The underlying reasons for this is that the gravitational field in GR is self-coupling and thus itself a source of gravity (unlike in Newtonian gravity); but this type of energy cannot be localised, and is frame-dependent, so it is difficult to account for in an observer-independent way.

[Genady]

8 hours ago, Airbrush said:

That is an amazing thing to consider

As an illustration to the Markus' explanation above, consider this example from Gravitation by Misner, Thorn, and Wheeler:

[Airbrush]

10 hours ago, Markus Hanke said:

What Genady means is that the universe is a curved spacetime, as described by General Relativity. The thing now is that some concepts we are used to from old Newtonian physics do not straightforwardly translate to GR - and “mass” is one of them.  The question of “what is the mass associated with a given region of spacetime” has no simple answer; there are in fact several different notions of mass that apply to different sets of circumstances, so it really depends.

The underlying reasons for this is that the gravitational field in GR is self-coupling and thus itself a source of gravity (unlike in Newtonian gravity); but this type of energy cannot be localised, and is frame-dependent, so it is difficult to account for in an observer-independent way.

Thanks for the info.  That is so strange and surreal.  This is from your link:

"...general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined."

That seems obvious and makes perfect sense.  Hahaha.

[Boltzmannbrain]

4 hours ago, Genady said:

As an illustration to the Markus' explanation above, consider this example from Gravitation by Misner, Thorn, and Wheeler:

This is interesting, but I have no idea how this is an example of Markus' explanation.  Please explain.

[Genady]

2 hours ago, Boltzmannbrain said:

This is interesting, but I have no idea how this is an example of Markus' explanation.  Please explain.

This is an example of how

13 hours ago, Markus Hanke said:

some concepts we are used to from old Newtonian physics do not straightforwardly translate to GR

In this case, it is a concept of mass-energy contribution to a system from its parts.

[Markus Hanke]

12 hours ago, Airbrush said:

That is so strange and surreal.

Well, one must remember that Newtonian gravity is only a very simplified approximation that disregards all non-linearities, so it is perhaps not so surprising that some of its concepts turn out to be less general than we take them to be.

[Airbrush]

On 12/23/2023 at 9:20 PM, Markus Hanke said:

Well, one must remember that Newtonian gravity is only a very simplified approximation that disregards all non-linearities, so it is perhaps not so surprising that some of its concepts turn out to be less general than we take them to be.

When GR says "general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances,"

does than mean that mass is different depending on how close to the speed of light it is traveling thru space?  If so, I don't see how that would impact the concept of a finite amount of mass in the universe.  You mean an expansion of matter, contained within space, is moving at relativistic speeds, through space, so we don't know how massive it is because we don't know how fast it is moving?

[Genady]

54 minutes ago, Airbrush said:

When GR says "general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances,"

does than mean that mass is different depending on how close to the speed of light it is traveling thru space?  If so, I don't see how that would impact the concept of a finite amount of mass in the universe.  You mean an expansion of matter, contained within space, is moving at relativistic speeds, through space, so we don't know how massive it is because we don't know how fast it is moving?

No, it does not mean that. As Markus will clarify that, I think the following quote from Misner, Thorn, and Wheeler will help.

[KJW]

A fundamental problem with GR is that it is for the most part a local theory. That is, it deals with the various fields at individual points in spacetime. When one changes the coordinate system, the numerical values of the various components of the fields at the various points change according to how the coordinate system changed at those points. And the change in the components at one point is largely independent of the change in the components at another point due to the general nature of coordinate transformations. Thus, for many types of fields, it is impossible to create a total over a region of spacetime, quite simply because changing the coordinate system changes the total in a way that GR does not allow. And because there is no preferred coordinate system in GR, there is no way to say which value of the total over a region of spacetime is the correct value. So, it is impossible to have a total energy-momentum over the entire universe. Furthermore, regions in spacetime are four-dimensional, whereas we tend to think of three-dimensional regions and totals over three-dimensional regions, which also depend on the particular coordinate system.

 

[Markus Hanke]

10 hours ago, Airbrush said:

does than mean that mass is different depending on how close to the speed of light it is traveling thru space?

No. What it means is that the answer to the question of “how much mass/energy is in a region of spacetime” depends on what kind of geometry that region has. Depending on considerations such as symmetries (Killing fields), asymptotic flatness etc a certain definition may apply, while other definitions may not work. So one has to be very careful which one is to be used.

Note also that being in relative motion wrt to a gravitational source does not change the geometry of spacetime, it only changes the coordinate description of it.

[Airbrush]

10 hours ago, Markus Hanke said:

No. What it means is that the answer to the question of “how much mass/energy is in a region of spacetime” depends on what kind of geometry that region has. Depending on considerations such as symmetries (Killing fields), asymptotic flatness etc a certain definition may apply, while other definitions may not work. So one has to be very careful which one is to be used.

Note also that being in relative motion wrt to a gravitational source does not change the geometry of spacetime, it only changes the coordinate description of it.

This is an interesting explanation.  Thank you.  Can anyone state this in terms more understandable to non-experts?  What are "killing fields"?  How many geometries can a region of the universe have?

[Markus Hanke]

12 hours ago, Airbrush said:

This is an interesting explanation.  Thank you.  Can anyone state this in terms more understandable to non-experts?  What are "killing fields"?  How many geometries can a region of the universe have?

By “geometry” I mean how exactly the region of spacetime in question is curved. There’s mainly two considerations that are of relevance in this context here - does spacetime become approximately flat if we go far enough away (asymptotic flatness), or are there distant sources of gravity that need to be accounted for? And secondly - what symmetries does this spacetime have? Can we translate each point within this region a short distance along space and/or time, without affecting any of the physics? For example, each point in ordinary Schwarzschild spacetime can be shifted along a vector pointing - say - 1 second into the future, without changing anything about the physics of the system - it thus is said to admit a time-like Killing field, which is to say it is a stationary spacetime. Killing fields are one way to speak about symmetries in spacetime. And since continuous symmetries are by Noether’s theorem associated with conserved quantities, this will have an impact on how we define concepts such as energy-momentum across an extended region. Different definitions are available for different types of spacetime exhibiting different symmetries; and if you’re in a spacetime that’s complicated enough so that it has very few or no symmetries, then it may not even be possible to define its global mass in a meaningful way at all.

Does this make more sense?

[zapatos]

On 12/18/2023 at 11:25 AM, Boltzmannbrain said:

Yes, if it expands fast enough.

Precisely how fast is 'fast enough'?

[Boltzmannbrain]

1 hour ago, zapatos said:

Precisely how fast is 'fast enough'?

I already posted it.  It's a simple equation of many to choose from.  Something like 1/(1 - t) would work.  

[MigL]

On 12/20/2023 at 7:13 AM, Genady said:

In an expanding homogeneous isotropic universe, a distance between any two points - let's call them, galaxies - is proportional to a number, a(t) , called scale factor, which increases with time, t . So, for example, if a distance between some two galaxies at some moment is D then later, when a(t) is twice as large, the distance between these two galaxies is 2D . Thus, this distance increases with time as a(t)D .

Nice example, Genady.

And not to imply otherwise, but you only consider a linear scale factor as a function of time.
And I'm sure you know of many functions f(x), that tend to infinity as x approaches a specific finite value.

IOW, what if the scale factor is extremely non-linear ?with time 

[Genady]

1 hour ago, MigL said:

you only consider a linear scale factor as a function of time

I don't think so. I don't assume anything about how a depends on t . I only refer to how the distances depend on a .

Edited December 27, 2023 by Genady