The largest numbers

[studiot]

 

8 minutes ago, Markus Hanke said:

So how would you physically interpret the notion of “entropy” associated with a region of spacetime (ie a region on a semi-Riemannian manifold)? 

I wouldn't until I knew what sort of entropy you were talking about.

It is unfortunate that the same word (ie entropy) is used for both phenomena, unlike my example which clearly distinguishes.

 

Beltrami has a good discussion of different sorts of 'entropy'  in his book

What is Random ?

Chance and order in Mathematics and Life.

[Markus Hanke]

37 minutes ago, studiot said:

I wouldn't until I knew what sort of entropy you were talking about.

That’s a really good question, actually. In GR texts (which is what I am mostly familiar with) this is never really made any more explicit than connecting event horizon area to entropy - but the entire subject is generally treated under the heading “black hole thermodynamics”, and it is directly linked to the “temperature” of a black hole, so it stands to reason that it is the thermodynamic type of entropy that is in question here. Which again raises the issue just in what sense a region of completely empty spacetime should exhibit a property such as temperature…

[studiot]

Thermodynamic entropy is indeed a function of energy (and temperature)

It has units (dimensions) of energy per degree K.

Information entropy is dimensionless.

Probably the most interesting distinction is given by Caratheodory in his version of the 2nd Law.

 

Quote

Caratheodory Principle of Second Law of Thermodynamics states: “In every arbitrarily close neighborhood of a given initial state there exist states that cannot be approached arbitrarily closely by adiabatic processes”

As far as I know,

Information entropy admits no such restriction on the states and consequently does not require the second law.
Clearly since it it not directly concerned with energy it does not require the first law either.

 

Another commonly mixed up pair of terms is information and meaning.

Take the letters in alpha order A D E M N O R S

Form one of the 1024 (with replacement) possible 10 letter 'words'and you have a 1 in 1024 chance of getting RANDOMNESS.

How many more have meaning ?

 

This is rather like the difference in economics between price, cost and value or worth.

 

[sethoflagos]

44 minutes ago, studiot said:

Information entropy admits no such restriction on the states and consequently does not require the second law.
Clearly since it it not directly concerned with energy it does not require the first law either.

Are you sure about this? It sounds very Maxwell's Demonish.

In context any perfectly homogenous space carries no Shannon entropy because any measurement you make at any point always returns the same value. The information content is zero.

However the moment you discover a 'surprise' anomalous reading, something at that point is in a different state. Not only does that different state imply a different energy but you've acquired and stored the new information. Even if you've managed to acquire the information by reversible means, the stored data must at some point be deleted. 

As for Shannon entropy having no units, it's entirely reasonable to characterise thermodynamic entropy by quantities like S/R which are also dimensionless. 

 

[Genady]

5 minutes ago, sethoflagos said:

In context any perfectly homogenous space carries no Shannon entropy

Isn't Shannon entropy of one state 0 regardless of what kind of state it is, homogenous or not?

[sethoflagos]

I'd say not, but it does rather depend on what you mean by 'state'. A simple example maybe?

[studiot]

45 minutes ago, sethoflagos said:

Are you sure about this? It sounds very Maxwell's Demonish.

In context any perfectly homogenous space carries no Shannon entropy because any measurement you make at any point always returns the same value. The information content is zero.

However the moment you discover a 'surprise' anomalous reading, something at that point is in a different state. Not only does that different state imply a different energy but you've acquired and stored the new information. Even if you've managed to acquire the information by reversible means, the stored data must at some point be deleted. 

As for Shannon entropy having no units, it's entirely reasonable to characterise thermodynamic entropy by quantities like S/R which are also dimensionless. 

 

I don't follow your objection.

Please elaborate, perhaps with an example or two.

 

As regards units, since entroy difference refers to the difference between states in a space it follows that any system that possesses multiple states can have an entropy, defined by multiplying the statistical entropy of the states (which is a pure number) by the units of those states.

 

 

 

[Genady]

1 hour ago, sethoflagos said:

I'd say not, but it does rather depend on what you mean by 'state'. A simple example maybe?

No problem. Ignore my question. Thank you.

[sethoflagos]

1 hour ago, studiot said:

I don't follow your objection.

Please elaborate, perhaps with an example or two.

I thought that you might care to review your claim that information entropy was not subect to 2nd Law constraints after browsing through this paper:

https://www.physik.uni-kl.de/eggert/papers/raoul.pdf

I've attached a copy for your convenience.

Other relevant references are:

https://en.wikipedia.org/wiki/Maxwell's_demon

https://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory

I think some of the confusion lies in a tendency to think of information theory in purely abstract, mathematical terms when its application is very much a real world phenomenon. Shannon definitely framed it in terms of a physical link between sender and receiver.

In fact there seems to be a growing view that classical Clausius entropy and von Neumann entropy are simply special cases of the more general Shannon entropy. And the 2nd Law rules them all.

raoul.pdf

6 hours ago, Markus Hanke said:

... While it is interesting in itself that the total entropy of the system increases by orders of magnitude, what’s really surprising is that the resulting BH has finite, non-zero entropy at all - remember again that classical Schwarzschild spacetime is everywhere empty. 

Simply and somewhat sloppily put, entropy is a statistical property that reflects the number of ways the microstates of a system can be rearranged without affecting its overall macrostate. But a Schwarzschild BH is just empty spacetime - every point within that manifold is exactly the same as every other point, meaning no small local neighbourhood can be physically distinguished from any other small local neighbourhood. And because this is a classical model, the spacetime manifold is implicitly assumed to be smooth and differentiable everywhere (disregarding the singularity for now), or else the entire formalism of GR makes no sense.

.... It would seem to me that this is possible only if spacetime within such a volume is not in fact smooth and continuous everywhere - there needs to be some kind of non-trivial structure present at least in some subregion of the volume enclosed by the event horizon. The mere fact of geodesic incompleteness in and around r=0 doesn’t seem to account for this (a point singularity has no degrees of freedom, and isn’t part of the manifold in any case) - it would take a very non-trivial kind of micro-structure to yield an entropy of the magnitudes mentioned above.

So what is this micro-structure? I for one would dearly like to know…

Markus,

Can you briefly explain why we shouldn't expect to find a quark-gluon plasma at the heart of a black hole. There should be no problem in storing a huge amount of entropy in a small core of that if the uncertainty principle and extreme temperature is sufficient to resist collapse. 

[studiot]

2 hours ago, sethoflagos said:

I thought that you might care to review your claim that information entropy was not subect to 2nd Law constraints after browsing through this paper:

I am aware of the work of Prigogine and of Szilard.

But I am not aware that anyone has proven the need for either deletion or the actual physical embodyment of mathematics.
As far as I am concerned, the Mathematics existes and existed quite independently of any physical use or discovery.

I do not support the suggestion that symbols written say 1000 years ago should be erased or deleted, otherwise the equations of entropy that I write today, or even just conceive of but not write down, would somehow become unbalanced. In my opinion they are totally unconnected.

 

But we are getting further and further from the OP question which was about large numbers and my observation that since there are no limits to the number of point in a region of space we can assign a different number to each one without limit. We simply need to follow a Peano curve 1,2,3,4,5... to accomplish this.

Since one one the basic tenets of number theory is that we can take any number proposed as the largest and increase it by various means. We can repeat this indefinitely.

Each repetition we will  uniquely identify a new largest 'number'.

 

[geordief]

45 minutes ago, studiot said:

 my observation that since there are no limits to the number of point in a region of space.....

That is not self evident ,is it?

I am predisposed to believe  that a point in space only exists when something happens there.

Is that wrong?

[Genady]

21 minutes ago, geordief said:

a point in space only exists when something happens there

So, if I point to a point and observe that nothing happens there, that point does not exist?

[geordief]

25 minutes ago, Genady said:

So, if I point to a point and observe that nothing happens there, that point does not exist?

How can you point to a point  without something happening there?

You can only see it ,for example if a photon has reflected off it.

 

If I just follow the direction of your finger the point could be anywhere between the end of your finger and  ,say the moon.

[Genady]

4 minutes ago, geordief said:

If I just follow the direction of your finger the point could be anywhere between the end of your finger and  ,say the moon.

I could point with two fingers to a point on the intersection of the two directions.

[geordief]

15 minutes ago, Genady said:

I could point with two fingers to a point on the intersection of the two directions.

I am getting boxed in but I could respond by saying that your two fingers are pointing at something that (if it did exist) no longer does.

Plus on grounds of accuracy we would need to be talking about two lasers and they would have to be produced until they met -where if we saw that intersection (which we wouldn't) that point would again be in the past and probably inaccurate anyway.

Do you not like (or just accept as a valid pov) my assumption (maybe an act  of faith) that points need to be a site of interaction to be called points?(otherwise they could be called something like "meta points")

Edited January 24, 2023 by geordief

[Genady]

1 minute ago, geordief said:

Do you not like (or just accept as a valid pov) my assumption (maybe an act  of faith) that points need to be site of interaction to be called points?(otherwise they could be called something like "meta points")

Yes, I do not like or accept as a valid pov this assumption. I don't see any justification for such restriction.

[geordief]

1 minute ago, Genady said:

Yes, I do not like or accept as a valid pov this assumption. I don't see any justification for such restriction.

OK.That is just the way I see things.

[MigL]

Just to clarify my position, Studiot ...

Mathematics treats lines, surfaces, volumes and higher dimensional manifolds as infinitely sub-divisible.
Physics, on the other hand, may describe a space-time which is quantized and has a smallest possible value.

Now Mathematics will allow you to treat andrepresent those many sub-divisions, no matter how large their number, but if you want to 'count' possible events in a section of sace-time, Physics may have some constraints as to the maximum number of those events..

Perhaps Geordief should indicate which viewpoint he wishes to consider.
( I will excuse your thinking like a Mathematician, if you excuse my thinking like a Physicist 🙂 )

[geordief]

On 1/23/2023 at 6:22 PM, geordief said:

 

"

By the way ,(and back to the OP) if events and particles are very closely related(particles  only manifesting when interacting) and particles are not local but spread out like waves,then maybe events likewise are spread out and so are not finite even in a finite system?"

@MigL I said the above earlier in the thread.

 

If that was the case would that mean particles could be continuous throughout  a finite system?

  Does that clarify  what you are asking from me or do I have a foot in both camps?

 

Edited January 25, 2023 by geordief

[Markus Hanke]

12 hours ago, sethoflagos said:

Can you briefly explain why we shouldn't expect to find a quark-gluon plasma at the heart of a black hole.

Because that violates the singularity theorems. Once you have an event horizon, the formation of a gravitational singularity within the region enclosed by it is always inevitable, at least in classical GR. 

But that notwithstanding, I was specifically referring to a classical Schwarzschild BH, where we have \(T_{\mu \nu}=0\) everywhere by definition - there are no distributions of energy-momentum of any kind, anywhere in this scenario. The event horizon in this type of spacetime encloses a region that is completely empty (at least classically), and yet you can still associate thermodynamic entropy with this black hole. So the question arises: thermodynamics of what, exactly? There are no constituents or “states” to this system at all, in the classical picture.

[studiot]

15 hours ago, geordief said:

That is not self evident ,is it?

I am predisposed to believe  that a point in space only exists when something happens there.

Is that wrong?

If it doesn't exist, how can you identify it to demonstrate that it doesn't exist ?

Further how can you tell that nothing happens there ?

 

These are really similar questions to those Genady has already put.

 

12 hours ago, MigL said:

Just to clarify my position, Studiot ...

Mathematics treats lines, surfaces, volumes and higher dimensional manifolds as infinitely sub-divisible.
Physics, on the other hand, may describe a space-time which is quantized and has a smallest possible value.

Now Mathematics will allow you to treat andrepresent those many sub-divisions, no matter how large their number, but if you want to 'count' possible events in a section of sace-time, Physics may have some constraints as to the maximum number of those events..

Perhaps Geordief should indicate which viewpoint he wishes to consider.
( I will excuse your thinking like a Mathematician, if you excuse my thinking like a Physicist 🙂 )

Indeed so, but Riemann has already taken that into account. Translation from German by Spivak, adapted by McCleary.

Quote

Notions of of quantity are possible only when there exists a general concept that allows different realizations. Depending upon whether or not a continuous transition of instance can be found between any two of them, the realizations form a continuous or a discrete manifold; individual instances in the first case are called points and in the latter case elements of the manifold.

So was the birth of Topology and topological manifolds.

 

[geordief]

7 minutes ago, studiot said:

If it doesn't exist, how can you identify it to demonstrate that it doesn't exist ?

Further how can you tell that nothing happens there ?

1  you don't(need to)

2 ditto

But I do accept  that  unless there are consequences to my "claim" then it is just an "article of faith"  and part of my hard drive.

 

It is easy I think to show that your argument has all the advantages of consequential weight.

[studiot]

2 minutes ago, geordief said:

1  you don't(need to)

2 ditto

But I do accept  that  unless there are consequences to my "claim" then it is just an "article of faith"  and part of my hard drive.

 

It is easy I think to show that your argument has all the advantages of consequential weight.

So you haven't really shown that they 'don't exist' -  whatever that means.

I don't know of any human who has tried and failed to move from one 'real ' point to another through the empty space between them, where you seem to claim there are no points.

 

As a matter of interest two different scientific definitions of the word 'event' have been introduced in this thread.

One is the Physics (relativistic) definition which means point in space, occupied or not and regardless of 'whether anything happens there '

The other is the statistics definition which seems to fit how you intend to use the word.

This does indeed define points in 'event space' as an individual instance of something happening.

Event space is by nature discrete, though we postulate underlying abstract continuous spaces in some cases.

 

[geordief]

22 minutes ago, studiot said:

I don't know of any human who has tried and failed to move from one 'real ' point to another through the empty space between them, where you seem to claim there are no points.

To be fair,I was claiming that something has to happen at those "points"

If someone moves across the room there is no shortage of points where things are happening

A practically infinite number.But the starting and ending point need ,in my mind to be the site of a real event and not part of a mathematical model

 

Those mathematical points do exist ,but are of a different nature (I am not claiming proof of any kind ,just that it is how I like to see those things)

 

[studiot]

5 minutes ago, geordief said:

To be fair,I was claiming that something has to happen at those "points"

If someone moves across the room there is no shortage of points where things are happening

A practically infinite number.But the starting and ending point need ,in my mind to be the site of a real event and not part of a mathematical model

 

Those mathematical points do exist ,but are of a different nature (I am not claiming proof of any kind ,just that it is how I like to see those things)

 

So what about my shadow ?

That occupies points where, by definition, nothing happens.

And the shadow is made of nothing, again by definition.

 

Also if you are going to claim that the points are but abstract and only come into existence when something happens then you also need to establish whether or not the points and their nature depend in any way upon the something that happens.

A real can of worms.