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A failure of euclidean geometry, and some relativity
By Brad Rodgers (P1930) on Wednesday, January 26, 2000 - 09:56 pm:

I have recently noticed a series of problems that seem to occur within euclidean geometry. And, I can find no seemingly logical way to avoid a failure of the system of euclidean geometry. So, I must ask for help and ask that you find out what mistakes I have made in my study.

I first wish to outline euclid's errors at an infinite level. Generally, when one thinks of parallel lines, one thinks that they would never intersect. But, would not these lines intersect at infinty. The short answer is YES. If one takes such a result and "plugs it in" the law of sines, one gets a theorem that looks like this.

Where: a=infinite side of a triangle
b=other infinite side of a triangle
c=a finite side of a triangle

a/ sinA = b/ sinB = c/ sinC

¥/1 = ¥/1 = 1/ sinC

and if A=90, B=90, C=0

Therefore ¥= 1/0

There is much more to this paper, but I would much rather write it once I have heard your thoughts on this section of the paper.Thanks.

By Dave Sheridan (Dms22) on Wednesday, January 26, 2000 - 10:40 pm:

There is no problem here. As far as I am aware, Euclidean geometry has no known inconsistencies. It is an unfortunate consequence of logic that we can not use the axioms to tell whether the theory is consistent, but we have no evidence to suggest that it is not.

Infinity is a concept. It is not a number. Let me repeat that. Infinity does not behave like normal numbers. If it did, you would find that infinity+1 existed, and more interestingly, so would infinity-1. The former of these can be made sense of with the use of ordinal arithmetic, but the latter is meaningless.

When we speak of infinity, it is just a convenient shorthand for what we call limiting behaviour. Any expression you have seen which involves the infinity symbol is actually claiming something different. Let me demonstrate.
A cat sits at the end of a table. Every second it jumps half the distance to the end of the table. The cat only reaches the other end after an infinite number of seconds.
This statement is not well formed, because it implies that infinity is a number. What we actually mean is that the cat can be arbitrarily close to the end of the table, but it takes longer and longer to be within smaller distances of the end. If you simply "plug infinity" into the time of this problem, you do find that the cat has reached the end of the table. That does not justify doing so though. Often, just replacing an x with an infinity symbol will give you lots of problems.

However, one of the earliest ways of thinking about what infinity "is" was to define infinity as 1/0. The problem is that you expect this to obey the laws of addition and multiplication. It does not, because they only apply to finite numbers.

I defy you to draw a triangle with infinite sides. There is no "point at infinity" in Euclidean geometry, explicit or implicit, just as there is no number infinity on the real line. So you can't make parallel lines meet.

In summary, you can't use infinity like other numbers. It represents a limit, no more and no less.

Let me give one final example, which demonstrates that we can't just "plug infinity in" when we feel like it.

Here's a sequence of numbers.
X1=1
X2=1-1
X3=1-1+1
X4=1-1+1-1
and so on.
What is the value of Xinfinity?

Well, since we can pair off every +1 with a -1, we conclude that Xinfinity=0. That is, we write
Xinfinity=(1-1)+(1-1)+(1-1)+...=0.
However, we could also write
Xinfinity=1+(-1+1)+(-1+1)+(-1+1)+...=1.
Therefore algebra itself doesn't work!

The actual answer is that Xinfinity does not exist. We can't just plug infinity into the subscript, because that assumes that there is a unique value which could be taken. So it was not the laws of arithmetic which are to blame, it is our assumption that infinity can be used like any other number.

I hope this has been helpful. If you don't understand any of this, or would like to know more, please write back. If you have any problems with Euclidean geometry which do not involve using infinity like a real number, I'd be interested in hearing about them.

-Dave

By Brad Rodgers (P1930) on Thursday, January 27, 2000 - 01:13 am:

Thank you, this has been of great help. As I am only 14, I am denied access to many resources to check some of my work. But, I do have one other part which I must add to my work. If one considers the reciprocal of infinity, 1/ ¥, You will notice that this number is equal to 0. We can show this by observing the fact that a decimal place really means divide the number by ten to the power of the amount of places the number goes out past the decimal place and you have the denominator of a fraction whose numerator is 1( this is extremely wordy and if you do not grasp what I am saying, which I almost don"', go on- you probably know what a decimal place means-This is phrased very poorly.) Therefore 1/¥=1-.9(repeating). Thus 1/¥=1/1-(1/9+8/9)=0. Couldn't this 1/¥ ,which equals 0, number multiply so that an area becomes larger than its supposed value, 1 dimension becomes 2, and a line becomes a curve. And, if not, how would anything form from a system starting with points as its base.

By Dave Sheridan (Dms22) on Thursday, January 27, 2000 - 11:54 am:

I'm glad you appreciate this discussion! Right, onto your next point. You can think of 1/infinity as equal to 0, since this is well defined. Let me explain.
Infinity represents the limit as numbers get as large as possible. So look at natural numbers, and see how 1/n behaves for larger and larger numbers n. It's consistent - 1/n becomes smaller and smaller, so that if you want it to be smaller than 0.000000000001 (say), you can find a large number m so that 1/n is smaller than this number for every n>m. Can you tell in this case what m should be? It's this consistency which allows us to say that the limit of 1/n as n becomes larger and larger ("n tends to infinity") is equal to 0. We may write this as 1/infinity=0, but that's just shorthand for the above statement.

In this sense, you might say that infinity*0=1 (since infinity "is" 1/0 or from above, 0 "is" 1/infinity). You'd be wrong...depending on what you mean by it. Unfortunately, even infinity isn't safe when mathematicians are around. There are various infinities (infinitely many of them in fact!) all of which are "infinite". I won't go into that though. Unless you want me to.

So, how do we get anything from a system which starts with points? At this point, I'm not sure that you know what the axioms of Euclidean geometry are. Please tell me what you understand about this system, and I'll explain if you've been told a simplified version, or whatever. In "real" Euclidean geometry, there are both points and lines, which obey certain laws. For example, there is a unique line through any two points. So we already have 1 dimensional objects. You can make a curve from many lines, and I'll go into the topology of this if you like.

I don't understand how you can multiply an area into a bigger one though. Remember, multiplying anything by 0 reduces it to zero. You can't then divide by zero to get the number you first thought of - that would be "plugging infinity in" to an equation without checking that it first worked. Remember, 1/0 can be thought of as infinity, so dividing by 0 is similar to multiplying by infinity. This would of course cause problems - multiply anything by infinity and you'll get infinity (except perhaps zero). I hope that makes sense.

Give me a concrete demonstration of something you don't understand, and I'll try to explain how it actually works.

Hope to continue this soon,

-Dave

By Brad Rodgers (P1930) on Thursday, January 27, 2000 - 11:27 pm:

As I have been told, a line forms from an infinite series of points; however it is becoming increasingly apparent to me that this is not the case. Please let me know if the above statement is correct. I think that it must be.

However, here is a completely different thought. If we were to take this argument into the realm of physical science, it would be impossible for a universe to arise from nothing, and there would have to be a limit on the minimum area that can be achieved in our universe. Hence quantum mechanics is not the only thing that "ruins" spacetime at very small level, and our new theory of quantum gravity must involve a minimum area of this sort.

And, I think that even multiply 0 by infinity can equal infinity:

¥/¥=¥x¥/¥=¥

By Dave Sheridan (Dms22) on Friday, January 28, 2000 - 12:12 am:

A line can be thought of as infinitely many points, but there are uncountably many (you may think of a line as the real numbers, each point being a number). Thus it isn't a sequence of points; it's far more than that. This is an easy way to think of it. But Euclidean geometry does not state that this is the case; it is merely something you can make sense of using the theory. Lines are objects, points are objects, and there are laws about how objects behave. That's it. Any more than that is an inference.

Brad, I don't believe you can cancel in that way. Here, let me show you that 2=1:
2*0=1*0.
Therefore, dividing by zero, 2=1. That is, essentially, your argument. Spot the flaw.

I'll write more tomorrow.

-Dave

By Brad Rodgers (P1930) on Friday, January 28, 2000 - 01:11 am:

Yea, that's because I used infinity as an integer, which is incorrect. The point is that when infinity is used, like 0, to multiply, all defineability is lost . But, if a line, thus line segment, has infinite points, then a line can just naturally-because infinity has different values which( I think) are equivalent- add more points to it and become another dimension. Perhaps I am missing something though.

Also on another subject matter, can you reccomend a good book to learn calculus from- so far I have not been able to find one and I cannot take that course for four more years.

Thanks for your patience,

Brad.

By Dave Sheridan (Dms22) on Friday, January 28, 2000 - 11:26 am:

In that case, I agree. You can't put infinity into any equation and just expect it to work.

Ok, infinity doesn't have different values. The problem is that your current definition of "value" is unsuitable for dealing with infinity. It works fine for finite numbers, but breaks down if you plug infinity in. So the only way to deal with infinity is to look at cardinality (or ordinality, which is even more complicated). How many integers are there? How many even integers are there? The answer to both questions is "countably infinitely many", ie the same cardinality. That is to say, our new definition of value says that despite the feeling that there should be "half" as many even integers as integers themselves, half of infinity is still infinity so there are the same amount. How do we show this? We simply show that we can pair off every integer with an even integer (by multiplying by two). Since we can pair off every one of them simultaneously, they have the same cardinality.

Right, now you need to know that the cardinality of a line is the same as the cardinality of the plane. That is, if you're one dimensional or two dimensional, you have the same cardinality. So you don't need to "add" more points, just rearrange the ones you have. Clearly this is an odd concept, and to be able to do this requires some very strange operations which are not physically possible, but mathematically they're feasible (and indeed there's a result which says that you can take a sphere, break it up into finitely many pieces, reassemble these pieces after rotating and translating them, and you'll have two spheres with identical volume to the original one... Something out of nothing!)

How do you show that an extra dimension doesn't increase things? I'll give you the general idea but it's a little complicated. First of all, when dealing with cardinals, we find that multiplication works again! There are a couple of extra rules though. When you have an infinite cardinal c and an integer n, nc=c. Similarly, if you have two infinite cardinals, c and d, where d>c (one will always be larger than the other unless they're the same cardinal), then cd=d. Similarly, dd=d (infinity×infinity=infinity, but this preserves the cardinal we're dealing with). The only thing which we can't do is 2d. This will always produce a larger cardinal than d (and you shouldn't be surprised to learn that 2d is the same as 3d or even dd). Proving this is not too difficult when you have the correct framework, but I won't go into that. If you believe me on this, the result is easy. The cardinality of the real line is c (say), so what's the cardinality of the plane? Well, every point on the line is associated with a whole line - that's what we mean by (x,y) co-ordinates. So there are c×c points in the plane. But c×c=c, so there are the same "number" of points in the plane as there are in the line. Similarly, any number of dimensions up to c of them (try to understand what c dimensions even means!) will have the same number of points.

Ok, I'd better stop there in case I'm really confusing you! I'll explain in more detail if you like - let me know what you don't understand.

Hope this remains interesting,

-Dave

By Brad Rodgers (P1930) on Friday, January 28, 2000 - 06:37 pm:

I think actually understand this. But the question I don't understand is why are the equations for gravity and other forces different for different dimensions if mathematically they can be considered to be the same. I believe that to avoid this , especially in the physical universe, a system of a minimum area, instead of points, is required. But, in a pure mathematical sence, such a thing is not needed. But, maybe I am missing something again - but if not I would appreciate it if you would explain this concept to a physics prof (if you know one) to further the notion of a new theory needing a minimum area.

Thanks,

Brad

By Brad Rodgers (P1930) on Friday, January 28, 2000 - 06:41 pm:

By the way, I would like if you would go into more detail- but I think I understand this because I have read Hilbert's (I think that's his name) work.

By Dave Sheridan (Dms22) on Saturday, January 29, 2000 - 10:51 am:

This is where you learn that cardinality is not the whole story. Would you say that the positive real numbers are identical to the real line? Well, the positive numbers have an endpoint (0) but the whole real line has no such point. This is some extra "structure" which the positive numbers have. Likewise, the rationals have the same cardinality as the integers, yet we can list the integers in a list of increasing size. We can't do that for the rationals, although we can list them (at least theoretically).

So, what are we missing? Lots acutally. Size is just the start. Things like lines have an inherant ordering - choose a direction on a line segment, and say that one point is "bigger" than another if it's further along in that direction. This ordering is the one you're used to for the real numbers. However, the same ordering can't be applied to the plane; this has extra structure. Ordinal numbers represent this, and they act just like finite numbers.

For example, suppose that w is the ordinal corresponding to the set of natural numbers. Then the ordinal w+1 corresponds to a set consisting of the natural numbers, and an extra object which is considered to be larger than any number (call it infinity if you like). This set does not obey the normal laws of arithmetic of course; we can't use infinity as if it were another number. But we can consider the set as different to the set of natural numbers which shares the cardinality.

So in a sense, infinity+1 does exist, and is different from infinity. It all depends how much structure you allow. Can you guess what 1+infinity corresponds to, and which ordinal that is? (Don't worry if this is all way above your head though)

So Physics does need to determine dimensions, since they contain information which we'd lose if we merely looked at size. Also, we can't manipulate physical objects in the way we'd need to (mathematically) to be able to change their dimension - it would require separating points and the current belief is that there aren't even points in nature at all (they're just a convenient way of regarding things) so we couldn't even develop the technology to do that.

Also, dimensions are preserved by physical laws, which often helps you to find which laws are feasible and which can't ever be the case. Have you met dimensional analysis yet? Basically, you look at the units on both sides of an equation and they have to match for that equation to make any sense at all. This gives you a hint as to when to use a square law, or a cubic one for example.

-Dave

By Brad Rodgers (P1930) on Monday, January 31, 2000 - 01:42 am:

OK-I have a few questions. First, what is an ordinal?- I think this might help in the understanding of your above paper. Also, if points could "rearrange" (mathematically) we would need a mechanism to prevent this- a minimum area. So even though our new idea of spacetime requires these because of the the incoherence of relativity and quantum mechanics, we would need this minimum area anyway. Basically I am just trying to show that physicists are on the right track with a minimum area theory.

Also though when I was working thinking of relativity, wouldn't the mechanics of it force the "infinite area" around a black hole to act as an integer thus destroying any framework of laws inside the black holes horizon off?

By Dave Sheridan (Dms22) on Monday, January 31, 2000 - 11:17 am:

The ordinals are a different way of looking at numbers. We've discussed cardinals, which measure the "size" of a set in some way. Now, ordinals represent the "order type" of a set. That is, if you can compare elements of your set, you get an ordering; two sets have the same ordinal if their ordering is the same. Some examples are necessary.
The ordinal corresponding to a set with 3 elements is the number 3. Anything with 3 elements which you put into an ordering will have the same ordering as (0,1,2).
The ordinal representing the natural numbers is w. If you add the number -1 to this set, this doesn't affect the ordering (there's still a smallest element, and you can map the set into the natural numbers by adding one to each element; this preserves the orer). However, if you add "infinity" to the set, this gives a new ordering (everything is less than infinity; previously there was no "top" element) so the ordinal changes to w+1.
The integers have neither a top nor a bottom element. You can view this set as two copies of the natural numbers (positive and negative) which do not overlap; this leads you to conclude that the order type is w for each, or in total w+w=w2. Note that I don't write 2w, which is the order type of the even natural numbers (which turns out to be equal to w).
[Note that there is an error here, which Dan corrects later. - The Editor}
Ordinals behave differently from cardinals. You may add them and multiply, but neither of these operations commutes; that is n+m is not necessarily equal to m+n and similarly for nm and mn. However, it is of course true for finite ordinals.

How do you generate ordinals? This is the wonderful thing which is preserved from finite numbers. You can add one to any ordinal to produce a new one. That is w+1, (w+1)+1=w+2, etc are all separate ordinals. You can also take limits, in the sense that wherever you see a 1, 2, 3, ... you can replace it with an w to get a new ordinal. So this tells us that w+1, w+2, ... , w+w all exist. I've already said that we call w+w the ordinal w2. SImilarly, you get w3, w4, ... ww (which is w2) and so on. You can generate every ordinal with these operations. Interestingly, we may extend the ideas of induction to work for ordinals as well (called transfinite induction) which is a powerful technique for number theory and logic.

Let me know if you still have queries about ordinals. Now I still don't understand the minimum area criterium. Please explain that to me. However, it does not solve the problem you envisage. How much difference does it make to remove one point from the plane? Well, the area is unchanged. In fact, you can remove any countable set of points without changing the area; some uncountable sets as well, but not all of them. This is part of measure theory which is a whole other discussion (see the "Sequence of consecutive integers" Open Discussion on nrich where I've tried to explain lots about measure theory and its relation to probability).

As for your final question, perhaps that is the case; I'm not sure I understand your terminology. Certainly, physical laws break down at the event horizon and it's impossible to predict what happens in there. This is true already, before any modifications to the theories are required.

-Dave

By Sean Hartnoll (Sah40) on Monday, January 31, 2000 - 02:14 pm:

Hi Brad. A book I found useful to learn calculus from was "Teach Yourself Calculus", written by P. Abbot and M.E. Wardle, and published by Hodder and Stoughton. It is part of a series of Teach Yourself Books.

- Sean

By Dave Sheridan (Dms22) on Monday, January 31, 2000 - 11:51 pm:

Oops. Apologies for a mistake in the last posting of mine. I stupidly wrote that the integers have as an ordinal w2. This is clearly not true, since they are not well-ordered. One of the main points about ordinals is that they are comparable, so a set of order w2 should have a set of order w followed by a set of order w. The integers do not fulfill this criterion; the negative numbers with ordering "<" do not have a least element so do not have order type w. I was slighty careless with that example.

The integers do not have an ordinal, since they do not have a least element; however, every set has a cardinal so at least that was right. For a set of order type w2, think instead of a set of red natural numbers and blue ones. Every red number is automatically thought of as "less than" any blue number. Otherwise, we use the normal comparison. This set has order type w2, since the least element is red zero, and there are clearly two distinct orderings of type w within the set.

Apologies for the error... I'll try to be more careful in future (and hope I didn't make a mistake just now!)

-Dave

By Brad Rodgers (P1930) on Wednesday, February 2, 2000 - 02:08 am:

My belief that a minimum area is required rests not in the subtraction of points, but in the addition of points. If points A and B are on opposite ends of a line segment, and this line segment became a 2 dimensional entity through pure mathematical "force" the distance between A and B would change; instead of simply being seperated by the vertice of a square, they would be seperated by the diagonal of a square. And, thus all laws of gravity using distance as a determining factor(ei: all of them) would be "thrown off".

I think I am getting ordinals though; so thanks. But, I am still not sure I understand them completely- Let me review: w represents the natural numbers. It can only be changed by adding infinity to it. But, I am still not understanding the multiplication of ordinals and some of the addition of ordinals. But I think that this may be because I am still thinking in the frame of mind that I use in algebra.

I am not sure that time has existed forever. There would be a point in which time completely stopped and that would be the beginning of time(poor phrasing again, sorry). I don't think that all the values before that would add up to be infinity - just approaching it. But I am not well aquainted with calculus- so I may be wrong. But, anyway our measure of time, as we are inside the universe, is that of the universe; so if the universe uniformly"changed its measure of time,we would too. So unless we are observing through hyperspace , our universe is about 12 billion years old.

And because photons are massless, relativistic mass increase( the gamma factor, I think) does not really cause any problems with them. It is purely a mathematical tool- not a real life entity( I may be wrong here). But if they had mass it would. It is just that I think that the infinite predictions for the distance from a horizon to a singularity should be embraced as meaning that chaos becomes infinite inside of a black hole- not as an error in general relativity, as many think.

By Dave Sheridan (Dms22) on Thursday, February 3, 2000 - 10:30 am:

I still don't think the "minimum area" concept is necessary. Once you've imposed laws like gravity, you assume that the system is unaffected by anything else, and see how it evolves. Points can't spontaneously rearrange themselves and none of the physical laws you've introduced would cause them to be so weird that the whole changes its dimension (that's one of the requirements of a valid physical law really - believe me, the rearrangement would have to be really weird).

Here's a way of thinking about addition of ordinals. If you have a set of ordinality a and a set of ordinality b, you can construct a set of ordinality a+b simply by colouring the second one blue and the first one red, and using the order "all red things are smaller than blue things; otherwise we can already compare them". So for example, 1+w consists of red 0 followed by blue natural numbers. What's the order type of this? Well, if we called the red 0 a blue -1 instead, we'd have the same ordering as the natural numbers (just add 1 to everything, and this preserves order) so the set has ordinality w. So 1+w=w. However, if we have a red set of natural numbers and blue zero, we can't map this back into the natural numbers whilst preserving the ordering, since we have a largest element. So w+1 is not w. Try to work out the ordinality of some sets you know - like the integer lattice (ie both x and y axes are natural numbers, giving you a grid - but only the positive quadrant).

There's a similar way to think of multiplying ordinals, but I won't go into that now.

-Dave

By Sean Hartnoll (Sah40) on Thursday, February 3, 2000 - 12:19 pm:


Hi Brad.

When you are talking about relativity, you have to be very careful to take into account which reference frame ("point of view") you are in. So for example, although from my frame on Earth it looks like time comes to an end at the event horizon of a black hole, this is not true from the frame of someone actually falling into the black hole. Someone falling in could pass the event horizon without noticing (well, apart from getting turned into spaghetti), although she would find it impossible to go back. She would also fall right the way to the singularity in a finite time.

Another point is that in general relativity the concept of "force" isn't really used. All you have is matter and spacetime. The way it works is that matter tells spacetime how to curve (from ten equations called Einstein's field equations) and then the curvature of spacetime tells matter how to move (particles move along the shortest possible paths, called geodesics). There isn't a force law like there is in Newtonian gravity, all there is is geometry. Although it is certainly true that spacetime gets curved it will never happen that a point becomes a line or a line an area, these are completely different types of changes.

Finally, people have tried to come with theories involving the idea of a minimum area. However, they have not been particularly successful. Some of the immediate difficulties are obvious: because space and time are all mixed up, a minimum area requires a minimum time also. Because different observers see different things, then the mimimum area cannot be absolute. However, there is a small length scale, the planck length, at which small "quantum fluctuations" make it hard talk about the very concept of space. It is in this domain that a new quantum theory of gravity is needed.

A very good book about general relativity is Kip Thorne's "Black Holes and Time Warps: Einstein's Outrageous Legacy". The book I mentioned earlier on Calculus is also important: calculus is the language used in all of theoretical physics.

Hope this helps your ideas!

Sean

By Brad Rodgers (P1930) on Thursday, February 3, 2000 - 10:08 pm:

Yes, I have the book "Black Holes and Time Warps". It is indeed a good book. But it shows very well, according to general relativity, that the area surrounding a pre-black hole becomes infinite as the singularity gains enough mass to become a black hole- at least according to our reference point. This would cause a small area to become infinite; in fact even a line surrounding the black hole would become infinite, as there are infinite infinities in infinity( another one of my poorly phrased sentences- I really can't do better). Perhaps this is why nature has enshrouded this area with a cover.

As far as the idea of a minimum area criterion, perhaps I should leave the idea of changing dimensions to demonstrate my point. As you said, a sphere could rearrange itself into two spheres. Provided the same logic holds for a simple line, a line of distance y could rearrange itself into a line of distance 2y. And as a point would be a minimum area, it would be the measure of all things. So, the distance light travels in a second could be anything, a serious contradiction of modern physics. I think that if mathematics expresses the physical world then a mathematically possible way of "manuevering" these points in such a manner would translate into a physical way of doing so.

By Brad Rodgers (P1930) on Wednesday, February 9, 2000 - 10:25 pm:

Is the above correct?

Thanks,

Brad.

By Sean Hartnoll (Sah40) on Wednesday, February 9, 2000 - 10:54 pm:

I'll get back later about the collapsing star, as I have to read a bit first to give you a good answer.

As for the minimum area, two points. Firstly, I don't know what you mean by "rearrange". I certainly don't think a sphere could "rearrange" itself into two spheres. Secondly, when you say that the existence of a minimum area would mean that the speed of light could be anything you are forgetting that the mimimum area would also imply a minimum time. These would be related in such a way that the speed of light would always be the same (i.e. the minimum time would be the time it would take light to cross the minimum area).

It is hard to speak accurately about general relativity without using the mathematics it is framed in (tensor analysis), this is the only way to state exactly what curvature of spacetime really is. To this end I would repeat that the best thing you could do with a view to understanding general relativity better would be to learn as much calculus as possible.

More later,

Sean

By Sean Hartnoll (Sah40) on Wednesday, February 9, 2000 - 10:55 pm:

I'll get back later about the collapsing star, as I have to read a bit first to give you a good answer.

As for the minimum area, two points. Firstly, I don't know what you mean by "rearrange". I certainly don't think a sphere could "rearrange" itself into two spheres. Secondly, when you say that the existence of a minimum area would mean that the speed of light could be anything you are forgetting that the mimimum area would also imply a minimum time. These would be related in such a way that the speed of light would always be the same (i.e. the minimum time would be the time it would take light to cross the minimum area).

It is hard to speak accurately about general relativity without using the mathematics it is framed in (tensor analysis), this is the only way to state exactly what curvature of spacetime really is. To this end I would repeat that the best thing you could do with a view to understanding general relativity better would be to learn as much calculus as possible.

More later,

Sean

By Anton Ilderton (Abi20) on Thursday, February 10, 2000 - 01:52 pm:

Hello. About the spheres; if you assume the Axiom Of Choice (which is basically given an infinite set of sets, you can pick one item from each set), then the Banach-Tarski `paradox' states that you can cut a sphere into finitely many parts (which aren't the everyday shapes you might be expecting) and then you can rearrange them(whereby we mean translate and rotate the shapes, standard euclidean stuff) to form two spheres each the same size of the orginial.

The proof involves some heavy Group Theory, but this kind of thing has little to do with General Relativity.

Anton.

By Dan Goodman (Dfmg2) on Thursday, February 10, 2000 - 01:57 pm:

One other note about the spheres, you can find the proof here and it isn't as difficult as you might think, although a course or two in group theory is essential.

By Anton Ilderton (Abi20) on Thursday, February 10, 2000 - 02:10 pm:

Referring to one of the above messages (infinite distance from an event horizon to a singularity); an object which has passed beyond the event horizon of a black hole, irrespective of whatever it then does, however it then tries to move, in a finite time as measured by itself it reaches the singularity.

To put this in perspective, suppose you go and stick your finger into a black hole. Your finger is crushed by the singularity in finite proper time, but from your point of view (outside the event horizon) it takes infinitely long to put your finger beyond the event horizon, so you never experience it. Which is nice.

Anton.

By Sean Hartnoll (Sah40) on Thursday, February 10, 2000 - 02:52 pm:

Okay, granted that a rearrangement of a sphere into two spheres is possible. But the gravitational field is not going to produce a Banach-Tarski rearrangement.

Sean

By Brad Rodgers (P1930) on Sunday, February 13, 2000 - 09:57 pm:

Please excuse the latency of my reply; for some reason I couldn't access the boards portion of your site for a few days.
When I said that a line segment of points could rearrange into a line segment of twice the length of the original, I meant that since infinity can double itself without a mathematical change, a line segment (measurement= infinite points) would be equivalent to a line segment of twice its length. And because a point would be the measure of all things, it would cause a line segment of 2y to be equivalent to y. Thus light could travel 2y in 1 second, or it could travel 1y in 1 second. Perhaps this just shows that we can't measure the universe in points; but wouldn't this implie that points aren't the minimum area. Thus something without an actual area would be required for a minimum area.

It also seems to me that the reason that the individual in the horizon would reach the singularity in a finite period of time is because of his measuring of time being infinite. This would allow for him to go an infinite distance in a normal amount of time. I'm not sure if this is right, but I think it is.

Also, I have had trouble finding that book; do you know where a good place to find it is.

Thanks,

Brad.

By Sean Hartnoll (Sah40) on Monday, February 14, 2000 - 12:35 am:

Hi,

About the book. Full details are:

Teach Yourself Calculus, 4th Ed
Abbot, P., Neill, Hugh
£8.99

Series Title: Teach Yourself
Publisher: Hodder & Stoughton Publishers
Binding: Paperback
Publication Date: 06-AUG-1997
ISBN: 0340701609

You should be able to find it at most (large) bookshops in the maths section. Otherwise you
could call Blackwells bookshop at 01865 261381 and they send it within a day.

Looking at a line as an infinte number number of inifinitely small points isn't usually very helpful, because you end up with things like Zeno's paradoxes, one of which says an arrow cannot move because to get from one point to another it has to go through infinitely many points which would take an inifinite time (can you see how to solve this?). You should see it as a continuum which you can parametrise: i.e. you can label each point on a line segment, but that point is just something that divides the line into what is before it and what is after, it doesn't have any extension in itself. If you look at it like this, I think the problems you are suggesing don't arise.

About the black hole. You are looking at the problem as though our external perspective is the "right" one whilst the person falling somehow travels an infinite distance in a finite time. The thing is, all viewpoints are equally valid (this is the principle of relativity) and for the person falling in there really is only a finite distance, and this is completely real fact, not an illusion.

Sean

By Brad Rodgers (P1930) on Monday, February 14, 2000 - 11:12 pm:

I think I understand the point concept; It is starting to make sense to me. Also on the black holes- I'm seeing that because the time allows for light to travel a distance in finite time, there must not be an infinite distance.

But, I am still convinced that general relativity predicts and firmly shows that chaos becomes infinite inside of a black hole from our perspective.But, once one has entered the black it runs by perfectly normal mechanics. What I believe this means is that in ordrer for general relativity to hold up, there can be no naked singularities. This also may mean that as an object enters the horizon( and if points do exist) he may be able to see this chaos. He would be able to do so because several rays of light would be orbiting the singularity at a point differenciating between chaos and mechanics. Wouldn't this mean that if points are the building block of the universe, then at the point of the entire universe contracting to a singularity, the known laws of physics would change to any form( at least concerning the distance used to determine the amount of natural force exerted on an object at a certain distance). I'm sure I am missing something here, but I am not sure what it is.

Along with this the idea of quantum foam as being probabalistic would have to be dismissed. instead everything would have a probability of 1infinite(or 0)%. Thus everything would cease to have form from our viewpoint. But once the singularity is entered everything is as if it is normal. Again this makes no sense; but I think that it's true- so long as the singularity causes our perception of the space around it to be infinite.

Also, about the book, can I find it in America. And is that number toll free from America.

By Sean Hartnoll (Sah40) on Tuesday, February 15, 2000 - 12:10 am:


I don't about the book in the States. The number is not toll free from America, and it'll take ages to get to you, so another approach is required. You can get it from Amazon over the internet, go to

click here for Calculus at Amazon

I don't know what you mean by "chaos".

Sean

By Brad Rodgers (P1930) on Wednesday, February 16, 2000 - 06:37 pm:

By chaos, I mean that a distance cannot be predicted. Thus the "quantum foam"( In this case it is moreso a way for space-time to manipulate itself without contradicicting laws) that resides within a black hole would be unlimited in its form. Therefore, inside of the horizon, from our point of view, their are infinite possibilities of the topology. This, in my mind, is infinite chaos. But, seemingly, this quantum foam disappears when one enters the horizon. I believe, that if the above idea is right, this shows that naked singularities are not allowed. If they are, there would be several different sets of probabilities of what is happining- a contradiction of even quantum mechanics.

But also, if the above is correct, there would be infinite possibilities of what is happening. I believe that this would mean that each possibility would have a 0% chance of occuring( because this is a pure manipulation- not one caused by the placing of particles). And of course, this would mean that from our viewpoint the area inside of a black hole has no form. I am very confused about this result, but can find no flaw in my reasoning. The only way I could think of avoiding this is if the probabilities of a form occuring take the form of {1/2,1/4,1/8,....}or something of the like. Either that or there is a flaw in my initial logic, or einstien's theory- I'd place my bet on the first.

Please discount the second paragraph in my board before this- I realized that if a photon has volume, there is no problem.

Brad

By Sean Hartnoll (Sah40) on Wednesday, February 16, 2000 - 07:46 pm:


Firstly, you have to be more careful in talking about probabilities. It is wrong to say that because there are infinite possibilities then each possibility has a 0% chance of happening. This is like saying there are infinitely many places I could be at this moment, therefore I am nowhere.

To resolve this properly you need calculus, but the basic idea is that if the variable can take values x, where x is a real number (like my position), then what you have to consider is the probability of the variable being between x and x+dx, where dx is a small increment. This probability will not be zero. This may seem like we haven't really found what we wanted, but remember that dx can be as small as we like.

I don't think that you can call the quantum foam "more" or "less" chaotic. It just is what it is, random fluctuations in spacetime because when we get small enough then small quantum fluctuations in mass produce fluctuations in the gravitational field, i.e. spacetime.

Sean

By Brad Rodgers (P1930) on Friday, February 18, 2000 - 02:41 am:

OK- I see what you are saying; but, it still doesn't resolve all of my problem. By showing that this type of spacetime can exist inside of a black holes horizon, it leads one to believe that this region of spacetime could entirely isolate itself through this type of curvature. This leads to all sorts of problems. If you don't see what type of topology I am trying to show, I can try to scan a picture in- it is somewhat hard to describe. This form, however, leads one to believe that naked singularities are possible, as a sigularity could isolate itself, thus gravitational field, while still leaving a small amount of light to escape revealing( for a short time) the singularity as it just began.

Thanks for all your patience

Brad

By Brad Rodgers (P1930) on Sunday, February 20, 2000 - 03:44 am:

Is the above right?

By Sean Hartnoll (Sah40) on Sunday, February 20, 2000 - 03:39 pm:

Send in a picture.

Sean

[Unfortunately numerous attempts to do this failed. - The Editor]

By Brad Rodgers (P1930) on Monday, February 21, 2000 - 10:38 pm:

I have to apologize; it appears my scanner isn't working. Tomorrow I will see if I can use my school's scanners to scan an image in. But, in case my consequences disprove this theory, I'll go ahead and state them

I have found that it leads one to believe that naked singularities are possible if the evaporation of Black Holes is possible. But it does arrive at the interesting conclusion that a singularity can break off from the rest of the universe, forming a new baby universe of it's own. This is posible because there would be gravity still pulling on the object from the newly formed universe. Also note that an infalling observer would arrive at this newly formed universe.

By Sean Hartnoll (Sah40) on Tuesday, February 22, 2000 - 12:35 am:

I look forward to seing your picture. Just a few words on what a theory is though. A physical theory involves mathematics in order to be precise. It is not enough to argue qualitatively, or if you do it should be backed up by a solid mathematical understanding and intuition. What you are talking about is indeed the subject of some speculation (formation of so called "baby universes", perhaps you have read the essay by Hawking with this title?) and the naked singularities you describe a similar to the singularity of the big bang (which would have been "naked"). However, to be able to produce a sceintific argument about this you need to show how it happens within the context of the mathematics of the theory. This is why maths is so important, it is a way of making precise the things you want to say, and often it has surprises in store.

By Brad Rodgers (P1930) on Tuesday, February 22, 2000 - 10:07 pm:

[Failed attempt to scan picture.]

In this, the standard black hole's topology is our current understanding. New Black Hole Topology is my theory. This theory is backed by mathematics in that relativity specifies that the radius around black holes is infinite in comparison to the circumference. By understanding that this is not changed by an area "bulging in", it is very feasible to see that a portion of a black hole can isolate itself. And, as from our perspective there is an infinite distance to this cutoff, the force of gravity will continue for an amount of time until vaccum flucuations cause the hole to evaporate. As far as the cutoff of our universe to the singularity, this region would be finite and thus shrink and form a new universe. if you have any questions about the logic used to deduce this picture, or of the writings on the image, please write back. If you understand my logic and see a flaw in, write back as well. However, if you see no flaw in my reasoning, please tell of this to a physics proffesor at your university

By Sean Hartnoll (Sah40) on Tuesday, February 22, 2000 - 10:47 pm:

I couldn't get to see your document. However, what you are describing is something that people have studied, somewhat speculatively. The point I must insist on is that this does not constitute a theory. If you want to say something about what's happening with black holes in a scientific way you need to express mathematically what is happening to the spacetime, and here mathematically means tensor analysis, Einstein's field equations and topology. It is not enough to draw some pictures that look plausible, unless you have an underlying understanding of the maths I mentioned in the previous sentence. It is really good that you're thinking about these things and I'll be happy to answer your questions but the best thing you could do is learn maths. Calculus is a good place to start.

Sean

By Brad Rodgers (P1930) on Tuesday, February 22, 2000 - 11:14 pm:

So this is really just speculation moreso than a theory. I think I can see why.But does that neccessarily prove my idea wrong. Or can it still exist without an equation to show things about it.
Just to let you know, I have just purchased that book and have started to learn Calculus.



Thanks,

Brad

By Sean Hartnoll (Sah40) on Tuesday, February 22, 2000 - 11:36 pm:

I would like to see your picture, as I said, I am happy to clarify conceptual difficulties you may have. The more you have thought beforehand the easier you will find it to learn general relativity and quantum mechnics when it comes to it. You will find in the maths a whole new and powerful way of expressing the things you are already trying to think. The reason equation are important, aside form being very natural, is that they give numerical results and predictions that can be tested experimentally. You can also say whether a page of maths is right or wrong, which you can't really do with anything else. The maths itself, however, is useless without ideas.

Sean

By Brad Rodgers (P1930) on Sunday, February 27, 2000 - 01:20 am:

Ok, I've tried uploading the picture several times and it doesn't appear to be working. So here's a different aproach: albeit rather crude. In the following lines ' stands for for nothing. It is a space filler. A . stands for a point in a curve. Unless otherwise noted all the curves extend to infinity

CURRENTLY THOUGHT BLACK HOLE TOPOLOGY
.....''''''.....''''''''''''''''''''''''''''''''''''''''.''.''''''''''''''''''''''''''''''''''''''''''''''.''.''''''''''''''''''''''''''''''''''''''''''''''.''.''''''''''''''''''''''''''''''''''''''''''''''.''.''''''''''''''''''''''''''''''''''''''''


MY THINKING

.......''''''''''.......''''''''''''''''''''''''''''''''''.''''''.''''''''''''''''''''''''''''''''''''''''''.''''''.''''''''''''''''''''''''''''''''''''''''''.''''''.-infinite'''''''''''''''''''''''''
''''''''.''''''.''''''''''''''''''''''''''''''''''''''''''''.''.'''''''''''''''''''''''''''''''''''''''''''''''..'''''''''''''''''''''''''''''''''''''''''''''''.''.'''''''''''''''''''''''''''''''''''''''''''''.''''.'''''''''''''''''''''''''''''''''''''''''''''.''.'''''''''''''''''''''''''''''''''''''''''''''''..-black hole ends'''''''''''''''''''''
'''''''''''''''(not infinite)

This is really rather speculative, but there is some good logical backing to it. If I haven't already said it, I'm sorry for this approach, very very sorry. When reading this keep in mind that the lines for the type-up are only 51 spaces long. So rearrange these lines if you want; or I could try to mail a picture to you if you could give me an address to a department at your university. It would only take a week or so.

By Brad Rodgers (P1930) on Friday, April 7, 2000 - 06:20 pm:

Sorry about not getting my scanner to work( or my school's for that matter, but I think I may have another solution. if you have access to the newsmagazine Time, you can find a picture to my idea in the "Will we discover another Universe?" segment of the Visions section. It is illustration 2 under reality check. this leads me to believe that this has been discovered before; however, in the magazine, it says that this theory is still speculative. But in the past few weeks, I have read a good deal of that calculus book and a number of set theory books and believe that this theory is very related to Cantor's Continuum Hypothesis and may be able to show this very easily.

By Sean Hartnoll (Sah40) on Saturday, April 8, 2000 - 04:26 pm:

I would be interested to hear what you have to say about Cantor's Continuum Hypothesis in relation to physics. Which edition of Time was it?
However, remember that the whoever wrote the Time article would have had a lot of maths to back up his pictures (there is a good reason why general relativity is not taught until the third or fourth year of university). Maths is the fundamental difference between popular science and real science.

Sean

By Brad Rodgers (P1930) on Saturday, April 15, 2000 - 02:36 am:

It is the April 10, 2000 edition. It is part of the visions series and entitled "In the Future we will...". I do not see any page numbers though. I believe that this is related to the CH in that if their are an infinite number of points in a body of length, and their are an infinite number of these bodies of length in another body of length, are their uncountably infinite points in the larger body of lengths? I am not sure that this is the CH, but to the best of my ability, this what I made the word contortions in my book out to be. Anyway, if there are uncountably infinite points, parametrisis becomes impossible. Therefore, I believe (but am not sure of) it would be possible for the space-time leading to the singularity to add length to itself and then "pinch off" and form an isolated universe. This should be what the picture shows. I am sorry that I am unable to provide the tensor analysis to show that this would be uncountably infinite points as I am just getting through learning normal Calculus now.

Brad

By Brad Rodgers (P1930) on Friday, June 16, 2000 - 07:23 pm:

I think I may have finally developed the mathematics to back my no-point conjecture up. First assume that the universe is flat(therefore doesn't curl) and is made of points. If these are true, then the sset of points in the Universe, U, is U=c , where c is the continuum. But, Godel has shown that c=alephx where x is undefined. This (I think) means that parametrising is impossible to do. How then would anyone measure the number of points in the universe. If they measured it once, then its area would expand or decrease to another aleph above 0. This would mean that the universe could mathematically increase its area. This leads to all sorts of odd problems. As these are not typically seen, I believe that points do not exist.Perhaps though, I am still not fully understanding.

Brad

By Michael Doré (P904) on Saturday, June 17, 2000 - 07:40 am:

Brad - a few points here:

If the assumptions you made at the start are true, the cardinality of the set of points in the universe is the cardinality of the irrational set of numbers. This is fixed. So although you don't know x where the set is alephx, this doesn't mean the points can suddenly increase in cardinality. It just means it is unknown how many cardinalities are below it, but this doesn't say anything about its actual absolute size.

In fact all lengths/areas/volumes/n-D spaces have the same cardinality of points. Independent of the size of that space. It is the irrational cardinality. (There's a nice easy proof of this.) But this means it's possible to take (for instance) a sphere and re-arrange the points to form two spheres provided you accept the axiom of choice. (This is very hard to prove though.) But the kind of transformations in operation here cannot be done physically. The sort of transformations I'm talking about (perhaps not the specific ones in operation here) are for instance:

Take all points which are an irrational length from the centre of the sphere and remove them only.

I hope you can see why this is a problem physically. Between any two numbers there are infinitely many irrational and rational numbers.

So anyway, changing in area has nothing to do with changing in cardinality, or changing the aleph if you prefer. If you did increase the cardinality it would be impossible to fit the points in ANY sort of space, no matter how big, or how many dimensions.

I'm not sure what you mean by parametrising is impossible.

Anyway, as it happens I think the latest QM theories don't have points in the universe. Length is quantised by the Planck length (if I remember correctly, this length compared to an atomic radius is like comparing an atomic radius to the radius of the Earth, approximately). Planck's time quantises time - and Planck's length / Planck's time = speed of light. However I'm not sure exactly what this means - maybe points do appear still in the wave calculations.

Yours,

Michael

By Brad Rodgers (P1930) on Saturday, June 17, 2000 - 05:18 pm:

Yes, I have read quite a bit on the new quantum field theories. I do not think that points appear at all in at least one of these theories(string theory). I am just thinking that perhaps it is not only an option in our theory to have points, it is an impossibilty.

I am still not sure I entirely understand you here though. Suppose, for example, that we chose one line in the universe to measure. To be entirely accurate, we chose to count the points found in it. We realized before doing so that it would be infinity, but there are still different infinities which it could be. Suppose that we classified it as aleph(1). However, when our friends measure, they find that they count aleph(2). Surely something is going wrong. We don't notice these sort of jumps when we are simply measuring distance without points(even infinite distance). It therefore seems that for a system to be consistent within our world, it must not include points.

I think I may understand what is wrong with this, but am still not sure. If c=aleph(x) then does this mean that aleph(1)=aleph(2). But, it still seems to me that in geometry, where everything implies measurement, we can't define anything by points. We cannot measure anything by points. If we cannot do either of these, then points should simply be not allowed to enter our system.

Brad

By Michael Doré (P904) on Sunday, June 18, 2000 - 03:34 pm:

I think there are two important points here.

1) It is true that if we let c = set of points in a Euclidean space, we cannot determine x where c = alephx. But this doesn't mean the size of c can change, only it's not possible to know how many other infinite sets there are below c in size. So we don't see the amount of points on a line changing all the time or anything like that.

2) The amount of points an object has is not directly related to its length/area/volume etc. A 1cm ruler has the same amount of points as a 4-D space, stretching out to infinity in all directions. This is because the set theory definition of amount of points is to do with whether a one-one correspondence can be established. Two sets of points are said to have the same amount of points (or same cardinality) if you can establish a one-one mapping between the two sets. You can always do this for any set of points in Euclidean geometry, so we say that all lines/areas/volumes etc have the same amount of points.

If you want a proof of that last statement, then let me know.

Yours,

Michael

By Brad Rodgers (P1930) on Sunday, June 18, 2000 - 08:53 pm:

I think I am understanding, however, if the amount of points in all structures is the same, then how can we consider points to be a valid part of geometry.

Also, isn't the ordinality of points in two different structures different. However, when one switches to an infinite distance made of points, we cannot rely on ordinality as the order is uncountable. The cardinality can also not be told apart. Therefore, in infinite distance points are an invalid system.

Maybe the proof you speak of would help me understand better, but I think I am understanding why cardinality is the same for all structures.

Thanks,

Brad