I have a real issue with infinities. The problem i have with this, is how can an infinite number of shapes fit in a finite space. The other problem i have with this is how can you find the area of an infinite number of shapes. Area should equal infinity.

Are you familiar with Zeno's Paradox? His one-dimensional version is probably the clearest & simplest take on the problem, & the first. Around 450 BC.

If you are not familiar with it, Zeno said something like this.

You are standing in your room, & you draw a line one meter in front of you.

Then you take a first step of 1/2 the distance toward the line (this first step is 1/2 meter long).

Then you take a second step of 1/2 of the REMAINING distance to the line (this step is then 1/4 meter long).

Then you take a third step of 1/2 of the then-remaining distance (now we're down to 1/8 meter).

And you just keep on stepping like this.

Since, with each step, you always leave some space between you & the line, you can always take another step.

So you can fit an infinite number of steps across a distance that is just one meter long in total. [You can do it because the steps just get smaller & smaller.]

And if you happen to know how to do the sum of a geometric series, you can easily calculate that the sum of the infinite number of steps does add up to just one meter.

It gets messier with more dimensions, but the principle is no different.

I read somewhere that, after propounding this theory for some time, he dislocated his shoulder in an accident. Someone who helped him said that this was impossible, because the arm part of his joint would have had to move half the dislocation distance, followed by half the remainder, followed by....
You get the rest.

I read somewhere that, after propounding this theory for some time, he dislocated his shoulder in an accident. Someone who helped him said that this was impossible, because the arm part of his joint would have had to move half the dislocation distance, followed by half the remainder, followed by....
You get the rest.

:-)
If you want a real good laugh, read Terry Pratchett's book 'Pyramids'. The character Xeno in the kingdom of Djelibeybi throws tortoises while trying to shoot them with arrows, then argues with his mates Copolymerand Pthagonal. There Nature not only abhors a vacuum, but also the chuck keys of electric drills. ;-)

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

you can easily calculate that the sum of the infinite number of steps does add up to just one meter.

The following is a terrible cheat/kludge which would horrify pure mathematicians because ( amongst other things ) it's assumes the existence of the limit you want to calculate, but try this :
S = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .....
S/2 = 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ..... ( divide both sides by 2 )
S/2 + 1/2 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..... ( add 1/2 to both sides )
S/2 + 1/2 = S ( substitute what you started with )
1/2 = S - S/2 ( subtract S/2 from both sides )
1/2 = S/2 ( simplify )
1 = S ( bingo )
This 'works' for this example, but easily leads to absurdities in others. :-(

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

I'd like to address one of the many points in your original question that hasn't been discussed yet.

In general, when you see infinite numbers coming out of a physical theory, it means something has gone wrong with the theory. (Here I'm talking about observable numbers, not something like imagining the surface area of the Earth as an infinite number of infinitely small squares because the observable total area is finite.)

What went wrong, then? Physical theories that have been around a while (like general relativity) are still around because they do a good job of predicting the observed (finite) numbers under some conditions. So when they predict an infinite number, like GR predicts a point of infinite "density" at the beginning of the universe or the center of a black hole, it's not that the theory is just garbage.

All theories have simplifying assumptions. The whole point of a physical theory is to predict an outcome without having to include everything in the universe. So you make some approximation that is valid under some conditions and move on.

In the case of the infinity you get at the Big Bang or inside a black hole in GR, the culprit is probably the assumption that spacetime is continuous. (That is, you can chop an interval into smaller and smaller distances infinitely.) This assumption is pretty good even if there is a minimum distance scale, as long as you are looking at things happening on much larger scales. But from quantum mechanics it seems likely that there is a minimum scale. The universe today is much, much larger than that scale, so GR is fine. But when you go back into the past far enough that the universe was that small, neglecting that minimum scale as GR does gets you into trouble.

People have been trying for a long time to combine quantum mechanics and GR to produce something that looks like GR at large distances but respects that minimum distance scale, but it's hard. Loop quantum gravity, an approach which has been making great strides here at Penn State, is starting to produce models of the big bang and black holes which avoid the infinity you were talking about. String theorists have come up with several ways of doing that, though I get different answers from different string theorists.

I'd like to address one of the many points in your original question that hasn't been discussed yet.

In general, when you see infinite numbers coming out of a physical theory, it means something has gone wrong with the theory. (Here I'm talking about observable numbers, not something like imagining the surface area of the Earth as an infinite number of infinitely small squares because the observable total area is finite.)

What went wrong, then? Physical theories that have been around a while (like general relativity) are still around because they do a good job of predicting the observed (finite) numbers under some conditions. So when they predict an infinite number, like GR predicts a point of infinite "density" at the beginning of the universe or the center of a black hole, it's not that the theory is just garbage.

All theories have simplifying assumptions. The whole point of a physical theory is to predict an outcome without having to include everything in the universe. So you make some approximation that is valid under some conditions and move on.

In the case of the infinity you get at the Big Bang or inside a black hole in GR, the culprit is probably the assumption that spacetime is continuous. (That is, you can chop an interval into smaller and smaller distances infinitely.) This assumption is pretty good even if there is a minimum distance scale, as long as you are looking at things happening on much larger scales. But from quantum mechanics it seems likely that there is a minimum scale. The universe today is much, much larger than that scale, so GR is fine. But when you go back into the past far enough that the universe was that small, neglecting that minimum scale as GR does gets you into trouble.

People have been trying for a long time to combine quantum mechanics and GR to produce something that looks like GR at large distances but respects that minimum distance scale, but it's hard. Loop quantum gravity, an approach which has been making great strides here at Penn State, is starting to produce models of the big bang and black holes which avoid the infinity you were talking about. String theorists have come up with several ways of doing that, though I get different answers from different string theorists.

Hope this helps,
Ben

As I understand it, there are really two ways of avoiding the kinds of infinities that show up both in extreme cases of GR, and in attemps to create a quantum field theory of gravity. The first is, as you describe, discretizing spacetime - which is done in Loop Quantum Gravity. The other is to move away from the idea of point particles, so spacetimes is still continuous, but at the same time no interactions of any type can take place at a single point. This is what is done by String Theory. I don't know that there's any reason to choose one of these methods a priori; although, to me the string theory idea seems a little more natural.

It is, however, my understanding that LQG has a great deal of trouble getting its predictions in line with already accepted phenomenology such as Hawking radiation.

It is, however, my understanding that LQG has a great deal of trouble getting its predictions in line with already accepted phenomenology such as Hawking radiation.

This site, Loop Quantum Gravity, says â€œ...Among the most significant results obtained are: (i) The computation of the physical spectra of geometrical quantities such as area and volume, which yields quantitative predictions on Planck-scale physics. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler's intuition of spacetime 'foam'...â€?

Haven't had time to look at the whole site yet, but it sure sounds like the proper approach...

... interesting that LQG doesn't address where the mass of the fundamental particles comes from... Does string theory address that issue directly? Are there any theories that deal with this issue?

Also, one point not addressed in the original post, regarding how it could be possible to fit an infinite amount of mass into a single point in space; happy to hazard a guess: the mass would have to be in the form of energy, in the form of light (not visible light, but infinitely high frequency, infinitely small wavelength)...

... one point not addressed in the original post, regarding how it could be possible to fit an infinite amount of mass into a single point in space; happy to hazard a guess: the mass would have to be in the form of energy, in the form of light (not visible light, but infinitely high frequency, infinitely small wavelength)...

Some sort of multi-mega-stack of bosons?

I have made this letter longer than usual because I lack the time to make it shorter.Blaise Pascal

Hmm... would there necessarily have to be more than one? Not that there couldn't be many... I'm wondering how much (energy) would need to be in a single point before a stability as a fundamental particle is reached, and what's the nature of the stability that a proton should have more mass (energy) than an electron... or doesn't that arise from the nature of integer (or half integer) spin? I recall reading that a particle composed of 5 quarks was theorized, and has subsequently been observed... So can enough energy be in one place at the same time to form an event horizon? It can knot... :) Or is it not proper to think of a 'loop' as some kind of event horizon?

## RE: I have a real issue

)

Are you familiar with Zeno's Paradox? His one-dimensional version is probably the clearest & simplest take on the problem, & the first. Around 450 BC.

If you are not familiar with it, Zeno said something like this.

You are standing in your room, & you draw a line one meter in front of you.

Then you take a first step of 1/2 the distance toward the line (this first step is 1/2 meter long).

Then you take a second step of 1/2 of the REMAINING distance to the line (this step is then 1/4 meter long).

Then you take a third step of 1/2 of the then-remaining distance (now we're down to 1/8 meter).

And you just keep on stepping like this.

Since, with each step, you always leave some space between you & the line, you can always take another step.

So you can fit an infinite number of steps across a distance that is just one meter long in total. [You can do it because the steps just get smaller & smaller.]

And if you happen to know how to do the sum of a geometric series, you can easily calculate that the sum of the infinite number of steps does add up to just one meter.

It gets messier with more dimensions, but the principle is no different.

ADDMP

## I read somewhere that, after

)

I read somewhere that, after propounding this theory for some time, he dislocated his shoulder in an accident. Someone who helped him said that this was impossible, because the arm part of his joint would have had to move half the dislocation distance, followed by half the remainder, followed by....

You get the rest.

## RE: I read somewhere that,

)

:-)

If you want a real good laugh, read Terry Pratchett's book 'Pyramids'. The character Xeno in the kingdom of Djelibeybi throws tortoises while trying to shoot them with arrows, then argues with his mates Copolymerand Pthagonal. There Nature not only abhors a vacuum, but also the chuck keys of electric drills. ;-)

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## RE: you can easily

)

The following is a terrible cheat/kludge which would horrify pure mathematicians because ( amongst other things ) it's assumes the existence of the limit you want to calculate, but try this :

S = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .....

S/2 = 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ..... ( divide both sides by 2 )

S/2 + 1/2 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ..... ( add 1/2 to both sides )

S/2 + 1/2 = S ( substitute what you started with )

1/2 = S - S/2 ( subtract S/2 from both sides )

1/2 = S/2 ( simplify )

1 = S ( bingo )

This 'works' for this example, but easily leads to absurdities in others. :-(

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## Hockeyguy, I'd like to

)

Hockeyguy,

I'd like to address one of the many points in your original question that hasn't been discussed yet.

In general, when you see infinite numbers coming out of a physical theory, it means something has gone wrong with the theory. (Here I'm talking about observable numbers, not something like imagining the surface area of the Earth as an infinite number of infinitely small squares because the observable total area is finite.)

What went wrong, then? Physical theories that have been around a while (like general relativity) are still around because they do a good job of predicting the observed (finite) numbers under some conditions. So when they predict an infinite number, like GR predicts a point of infinite "density" at the beginning of the universe or the center of a black hole, it's not that the theory is just garbage.

All theories have simplifying assumptions. The whole point of a physical theory is to predict an outcome without having to include everything in the universe. So you make some approximation that is valid under some conditions and move on.

In the case of the infinity you get at the Big Bang or inside a black hole in GR, the culprit is probably the assumption that spacetime is continuous. (That is, you can chop an interval into smaller and smaller distances infinitely.) This assumption is pretty good even if there is a minimum distance scale, as long as you are looking at things happening on much larger scales. But from quantum mechanics it seems likely that there is a minimum scale. The universe today is much, much larger than that scale, so GR is fine. But when you go back into the past far enough that the universe was that small, neglecting that minimum scale as GR does gets you into trouble.

People have been trying for a long time to combine quantum mechanics and GR to produce something that looks like GR at large distances but respects that minimum distance scale, but it's hard. Loop quantum gravity, an approach which has been making great strides here at Penn State, is starting to produce models of the big bang and black holes which avoid the infinity you were talking about. String theorists have come up with several ways of doing that, though I get different answers from different string theorists.

Hope this helps,

Ben

## RE: Hockeyguy, I'd like to

)

As I understand it, there are really two ways of avoiding the kinds of infinities that show up both in extreme cases of GR, and in attemps to create a quantum field theory of gravity. The first is, as you describe, discretizing spacetime - which is done in Loop Quantum Gravity. The other is to move away from the idea of point particles, so spacetimes is still continuous, but at the same time no interactions of any type can take place at a single point. This is what is done by String Theory. I don't know that there's any reason to choose one of these methods a priori; although, to me the string theory idea seems a little more natural.

It is, however, my understanding that LQG has a great deal of trouble getting its predictions in line with already accepted phenomenology such as Hawking radiation.

## RE: It is, however, my

)

This site, Loop Quantum Gravity, says â€œ...Among the most significant results obtained are: (i) The computation of the physical spectra of geometrical quantities such as area and volume, which yields quantitative predictions on Planck-scale physics. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler's intuition of spacetime 'foam'...â€?

Haven't had time to look at the whole site yet, but it sure sounds like the proper approach...

## ... interesting that LQG

)

... interesting that LQG doesn't address where the mass of the fundamental particles comes from... Does string theory address that issue directly? Are there any theories that deal with this issue?

Also, one point not addressed in the original post, regarding how it could be possible to fit an infinite amount of mass into a single point in space; happy to hazard a guess: the mass would have to be in the form of energy, in the form of light (not visible light, but infinitely high frequency, infinitely small wavelength)...

## RE: ... one point not

)

Some sort of multi-mega-stack of bosons?

I have made this letter longer than usual because I lack the time to make it shorter. Blaise Pascal

## RE: Some sort of

)

Hmm... would there necessarily have to be more than one? Not that there couldn't be many... I'm wondering how much (energy) would need to be in a single point before a stability as a fundamental particle is reached, and what's the nature of the stability that a proton should have more mass (energy) than an electron... or doesn't that arise from the nature of integer (or half integer) spin? I recall reading that a particle composed of 5 quarks was theorized, and has subsequently been observed... So can enough energy be in one place at the same time to form an event horizon? It can knot... :) Or is it not proper to think of a 'loop' as some kind of event horizon?