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grav
2006-Jul-07, 01:30 AM
I have read that a non-Euclidean geometry can form triangles that are less or greater than 180 degrees and that the value for pi (more specifically, the ratio of the circumference of a circle to its diameter) can also change. This much is true. But I have also read that this may mean that our universe may be non-Euclidean and that the value for pi may be a result of this. In other words, if it were flat in this case, it might change to something else, perhaps exactly three. So the only reason it is not exactly three, then, would be because space is curved.

I have also read implications, including in this forum, that the value of pi might also change from one region of the universe to another. That somehow the space-time continuum curves differently for different regions, probably having something to do with the apparent curving of light through denser regions or something of this nature. I can demonstrate that pi is a universal constant and in a round-about way show implications that the universe is also flat.

If the value of pi changed even slightly from region to region in the universe, then the value that is obtained in any particular region becomes arbitrary. That is to say, it can have any value at all (but would depend, of course, upon the characteristics for that particular region). It might be three, or slightly greater, or much smaller. And it wouldn't just jump from value to value for different regions, but would change ever so gradually with the topography, so all possible values in between are accounted for, and pi can be represented by any number whatsoever depending on what region it is measured in, which makes it purely arbitrary.

Okay. So here's my argument. If the value of pi were a purely arbitrary number which depends on the topography of a region, then no true formula can be made to obtain its value. In other words, no regular pattern of calculations could be made that would exactly equal pi, so that its value can be found to billions of digits, which it is. The odds of the value of pi varying against this, then, is at least ten to the order of billions, which is tremendous. Many such formulas exist to find pi to any digit. This means that its value is not any arbitrary one, but very much a universal constant. This should also be the case for the universe in general, although it is much more difficult to explain, much less prove. But if my thinking is right, though, and pi is the ratio for a flat space or topography only, and it is a constant for the universe as a whole, then the universe should also be flat, otherwise pi would have some other arbitrary value that depends on its topography.

As far as I can see, the only argument that can be presented against this is the possibility that the value of pi as we know it is wrong. That is to say, the only true way to find the ratio of the circumference to the diameter is to actually measure them firsthand. To do this, we must first create a perfect circle somehow, and then a perfect measuring device. We assume with the formulas that we derive for pi that it will follow these formulas to the finest degree. So the billions of digits we produce by using these formulas is with the assumption that the value of pi follows the specific pattern by which the formulas are found, and that it does not vary in the slightest. I personally, however, believe this to be the case.

WaxRubiks
2006-Jul-07, 02:08 AM
Pi is an exact number in mathematics and can be found from a formula like this

http://img62.imageshack.us/img62/3656/piformula9pu.png (http://imageshack.us)

which is the Leibnez formula (http://en.wikipedia.org/wiki/Leibniz_formula_for_pi)

it is not subject to change and is not derived from experiments.

The ratio of the circumference to the diameter of a circle drawn on a bit of paper might vary(I suppose) but that is not how pi is calculated..
I think space is considered near flat in this part of the Universe due to the Universe being so big.

Tensor
2006-Jul-07, 03:18 AM
Pi is defined as the ratio of the diamter to the circumference in Euclidean Geometry. Other Geometries can and will have a different value for that ratio (in which case, it is not called Pi). Here (http://www.daviddarling.info/encyclopedia/N/non-Euclidean_geometry.html) is a good summary of the differences between the different geometries. I'm not sure why you think it's value is arbitrary though. The value of the ratio is dependent on the shape and amount of curvature. We could find different formulas for each of the amount and shape of the curvatures.

Ronald Brak
2006-Jul-07, 03:27 AM
My friend is a wizard and changed the value of pi to zero. Unfortunately he made a circle with his thumb and finger and his hand disappeared.

grav
2006-Jul-07, 04:04 AM
I'm not sure why you think it's value is arbitrary though.
Maybe I didn't phrase my post correctly (looking back, I'm sure of it). I'm saying that the value of pi is not arbitrary. If it were, we could not find exact formulas that produce its value. What I am also saying is that if the value of pi changed from region to region of space depending on the topography of space-time, then this would not be true and pi could have any value whatsoever depending on the characteristics of that particular topography. Furthermore, I'm saying that this might also be true of the universe as a whole, whereas the value of pi only works with the exact formulas that describe it in a flat space (Euclidean geometry) so that a non-Euclidean geometry of the universe should provide a different value which would probably be something arbitrary (depending on the topography, which could take on about any non-Euclidean form imaginable ) and therefore most likely could not be matched by any specific formula by which to calculate it.

Tensor
2006-Jul-07, 04:13 AM
What I am also saying is that if the value of pi changed from region to region of space depending on the topography of space-time,

I'll take this as you mean Pi is the ratio of the diameter to the circumference. Although, to be techical, Pi is only defined in Euclidean space.

then this would not be true and pi could have any value whatsoever depending on the characteristics of that particular topography.

True

Furthermore, I'm saying that this might also be true of the universe as a whole, whereas the value of pi only works with the exact formulas that describe it in a flat space (Euclidean geometry) so that a non-Euclidean geometry of the universe should provide a different value which would probably be something arbitrary (depending on the topography, which could take on about any non-Euclidean form imaginable )

True, here too.

and therefore most likely could not be matched by any specific formula by which to calculate it.

Why exactly do you say this. For any particular combination of curvature and shape, I'm sure there could be a formula. Why not? That we don't have one for any other geometry, dependent on the shape and amount of curvature, doesn't mean we couldn't find one. As of right now, there really isn't a reason to try and find one of those formulas.

grav
2006-Jul-07, 05:19 AM
Quote:
Originally Posted by grav
and therefore most likely could not be matched by any specific formula by which to calculate it.

Why exactly do you say this. For any particular combination of curvature and shape, I'm sure there could be a formula. Why not? That we don't have one for any other geometry, dependent on the shape and amount of curvature, doesn't mean we couldn't find one. As of right now, there really isn't a reason to try and find one of those formulas.
It could be possible, but the odds would be tremendous, beyond tremendous even. The ratio of integers to fractions is infinite. And the ratio of rational numbers to irrational numbers is also infinite. Pi is an irrational number. The odds of matching a formula to describe any arbitrary number, with any number of possible digits in any order whatsoever is beyond imaginable. Even if we came up with a formula for such a number which correctly predicted the first billion digits, it would surely be purely by chance and the billion and first number might differ. Try this. Take the number three. Now add an infinite number of random digits as the fraction and see if you can come up with a precise formula for the resulting number. No need to try it, you can't. Pi, on the other hand, is apparently very precise. Even though it requires infinite equations to produce it, it can be found accurately to any number of digits. This can only be possible for a completely unvarying topography (exactly zero curve).

WaxRubiks
2006-Jul-07, 05:27 AM
Now add an infinite number of random digits as the fraction and see if you can come up with a precise formula for the resulting number. No need to try it, you can't. Pi, on the other hand, is apparently very precise. Even though it requires infinite equations to produce it, it can be found accurately to any number of digits. This can only be possible for a completely unvarying topography (exactly zero curve).

you could come up with an equally accurate formula for the ratio of the diameter to circumference on a sphere.

grav
2006-Jul-07, 05:54 AM
you could come up with an equally accurate formula for the ratio of the diameter to circumference on a sphere.
But in order to do this, you would need to figure in the angle at which it is measured from a plane through the center. The ratio for any particular circumference on a sphere to its diameter from some angle through its center (which is what I'm assuming you mean) will give any arbitrary value from zero to pi depending on exactly where it is measured. If we don't know where it is measured, we have a purely arbitrary number and we can't find the formula, yet pi is known exactly regardless. This can only be the case for a flat topography, where the formula remains the same no matter what because the angle is always zero.

WaxRubiks
2006-Jul-07, 06:08 AM
If you draw a circle on a sphere with a compass, the diameter is the measurement from the hole where the compass spike went and the circle ,taking the path of the curve of the sphere to get to the circle. All you need is a formula for the real radius of the circle(in 3D) as a product of the diameter and then you can plug that into the existing formulas for pi and voil&#224; you have a formula for the ratio on a non-flat plane.

Tensor
2006-Jul-07, 06:39 AM
But in order to do this, you would need to figure in the angle at which it is measured from a plane through the center. The ratio for any particular circumference on a sphere to its diameter from some angle through its center (which is what I'm assuming you mean) will give any arbitrary value from zero to pi depending on exactly where it is measured.

Grav, it's called Differential Geometry (http://en.wikipedia.org/wiki/Differential_geometry). It can tell you how much, and where the manifold (GR models spacetime as a manifold) is curved.

If we don't know where it is measured, we have a purely arbitrary number and we can't find the formula,

We could find a formula for any combination of curvature and shape. If the curvature and shape in two different locations are the same, the ratio would be the same. BTW, in different geometries, the ration can be either greater or less than Pi, depending the on curvature and shape.

yet pi is known exactly regardless.

Because Pi is defined for Euclidean space. Well it's not known exactly, after all, it's an irrational number.

This can only be the case for a flat topography, where the formula remains the same no matter what because the angle is always zero.

For the ratio in Euclidean space, yes.

Stealth Poster
2006-Jul-07, 10:02 AM
Pi is an exact number in mathematics and can be found from a formula like this

http://img62.imageshack.us/img62/3656/piformula9pu.png (http://imageshack.us)

which is the Leibnez formula (http://en.wikipedia.org/wiki/Leibniz_formula_for_pi)

it is not subject to change and is not derived from experiments.

The ratio of the circumference to the diameter of a circle drawn on a bit of paper might vary(I suppose) but that is not how pi is calculated..
I think space is considered near flat in this part of the Universe due to the Universe being so big.

This series converges very slowly only. It also changes value if you add the same terms in another order :D

grav
2006-Jul-07, 10:45 PM
If you draw a circle on a sphere with a compass, the diameter is the measurement from the hole where the compass spike went and the circle ,taking the path of the curve of the sphere to get to the circle. All you need is a formula for the real radius of the circle(in 3D) as a product of the diameter and then you can plug that into the existing formulas for pi and voil&#224; you have a formula for the ratio on a non-flat plane.
But in order to know the formula for the value of pi in a curved universe, you must already know the curvature to begin with in order to "plug it in". There might be a way to find this curvature, however, if it really exists.

In my last post I thought of the ratio for pi as found through the sphere. This is wrong. The idea is that if space has curvature, all two-dimensional measurements must be taken across the surface of the sphere and its interior would then be an extra dimension which creates the curve and is untraversable. This would be similar to the "ant on a tube" analogy for superstring theory. So if we were to draw a circle on the surface of a spherically curved space, the diameter would actually be the distance across the circle which is curved with the sphere.

If the circle which is drawn on the sphere is very small in comparison to the size of the sphere, the ratio will be very nearly pi, so that space would appear nearly flat because the universe is so big, just as you stated in your first post. If the drawn circle is bigger, the value for pi decreases. The largest circle that can be drawn on the sphere would be around its "equator", in which case the diameter would become half of the circumference of the sphere, and its ratio would be two.

Now here's how we can tell if the universe has curvature. The value for pi as we know it would be for very small circles compared to the size of the universe. In this case, space would appear to be almost flat. As circles get larger, the value for pi decreases. All possible numbers between pi and two must be accounted for with increased size, which would make pi an arbitrary number for an arbitrary size (since we don't know the curvature of the universe). However, pi can be obtained using very specific equations where its digits can be found to any degree desired. So it is not arbitrary. And it has only one specific value, so this must be the value for the smallest circles (in what appears to be flat space). In other words, the value for pi is the absolute limit for small circles, where two would be the limit for larger ones.

Now let's take a circle which is drawn at an arbitrary size. Let's say it is about 1/10^26 the size of the universe. This means that it will contain about one part in 10^26 parts of its curvature. This number is very large and it would seem that space must definitely appear very flat under such circumstances. But let's see how this would effect the actual value for pi as it is measured by this circle we have drawn. If its curvature contains only about one part in 10^26 of that of the entire universe, then the value of pi as it is actually measured should be off by this amount as compared to the value given by the formulas for that of perfectly flat space. But for the number for pi itself, this would really only mean that the value of pi is found correctly to about 26 digits. After that, it would be much different. A much larger circle with a curvature of about 1/10^10 parts would be off after about the tenth digit. An extremely small circle with a size of 1/10^100 of that of the universe, however, would be found correct to 100 digits.

The problem with such measurements, however, is how do we draw a perfect circle and then measure it so precisely? This would have to be done physically because the formulas only derive the value under ideal circumstances (for when space is flat). If such a measurement could be made to a degree to where we could tell to what digit a signifant difference lies, we could possibly then tell not only the curvature for our region but the size of the universe as well. If the universe is not spherical, but some other odd curvature, we could tell this by measuring the ratio for pi in many regions of the universe (but of course this won't happen anytime soon). But by comparing different sizes of circles, we might also obtain an idea of its general shape.

However, all in all, if the value of pi is to remain unchanged and its formulas are to describe it for any region of the universe at any time for all of its digits regardless of the size of the circle or any other circumstances, then the universe is infinite in size and space is necessarily flat. But even if we were capable of physically measuring the ratio of pi for a circle to 1000 digits and find that it is identical with the formulated value for pi to this degree, we could still always say that the curvature of the universe, then, must actually be much smaller than this but still not quite zero as for flat space. So we would never really know for sure.

Tensor, I guess this would be the response to your last post as well. I hope this post is a little more satisfactory than the last one.

snarkophilus
2006-Jul-07, 11:36 PM
The ratio of integers to fractions is infinite.

Nope. The integers and rationals map one to one.

And the ratio of rational numbers to irrational numbers is also infinite. Pi is an irrational number. The odds of matching a formula to describe any arbitrary number, with any number of possible digits in any order whatsoever is beyond imaginable.

Even worse than simply being irrational, it is transcendental. However, we still have a formula for it. The nice thing about calculating numbers is that there is no chance involved. Sure, if we got pi simply by guessing the first number, then checking, then guessing the second, and so on, we'd never ben able to determine it. But as it stands, we have algorithms to calculate it, including a nice digit extraction algorithm.

By your probability argument, I can state that it is impossible to add 2 and 3. Why is that? There are an infinite number of integers, so by guessing them at random and comparing to the sum of 2 and 3, I'll probably never find 5 in there. Yet, 5 exists.

This can only be possible for a completely unvarying topography (exactly zero curve).

This is simply not true. Try finding the value of the pi equivalent on the surface of a sphere. It's not that hard. (You may wish to fix the size of the sphere and the size of the triangle/circle/whatever you are measuring.) I think it will be quite illuminating for you.

Fortis
2006-Jul-07, 11:43 PM
But in order to know the formula for the value of pi in a curved universe, you must already know the curvature to begin with in order to "plug it in".
And this is what the field equations of GR gives you. :)

(More strictly you obtain the Einstein tensor, and from this you can get to the curvature if you wanted to...)

Caveat: This obviously assumes that GR is correct ;)

grav
2006-Jul-08, 12:38 AM
Quote:
Originally Posted by grav
The ratio of integers to fractions is infinite.

Nope. The integers and rationals map one to one.

By this I meant that after the number two, for example, there are an infinite number of fractions that lie between two and three. After three, the same thing applies for between three and four, and so on. One could also say that for just the rational fractions, we could use the integer two as the numerator and infinite possibilities for integers can be used for the denominator. The same is true for three, and so on.

By your probability argument, I can state that it is impossible to add 2 and 3. Why is that? There are an infinite number of integers, so by guessing them at random and comparing to the sum of 2 and 3, I'll probably never find 5 in there. Yet, 5 exists.

The argument only applies to arbitrary numbers. I don't consider two, three, and five as arbitrary. They are just as precise as the calculated value for pi. That is the point. If we were to find that some geometrical ratio with an exact "flat space" value of five should change in a curved universe, then the argument would be the same. If we were to find the value to be actually physically measured at 5.000000000123221... instead, then we would know the universe is curved and we could probably determine its characteristics by examining this latter result. We would do this by finding the curvature that would allow for the measured difference.

Quote:
This can only be possible for a completely unvarying topography (exactly zero curve).

This is simply not true. Try finding the value of the pi equivalent on the surface of a sphere. It's not that hard. (You may wish to fix the size of the sphere and the size of the triangle/circle/whatever you are measuring.) I think it will be quite illuminating for you.
This is another point. We would already have to know the size of the sphere (or the size of the universe) in order to do this. And the result would not be equal to pi unless space is flat. If we were to actually physically measure the value of pi to be different than the calculated value, however, we could do the reverse of what you say to find the curvature of the universe and its size.

grav
2006-Jul-08, 12:49 AM
Quote:
Originally Posted by grav
But in order to know the formula for the value of pi in a curved universe, you must already know the curvature to begin with in order to "plug it in".

And this is what the field equations of GR gives you.

(More strictly you obtain the Einstein tensor, and from this you can get to the curvature if you wanted to...)

Caveat: This obviously assumes that GR is correct

The field equations of GR may (or may not) be based on non-Euclidean geometry, but not the other way around. But if this were the case, what value for pi would the formulas of GR give us?

grav
2006-Jul-08, 01:21 AM
Let's look at this from a different perspective. If three-dimensional space is curved, then the universe must necessarily be contained within a larger space of four or more dimensions in order to give at least one extra dimension for this curvature to take place. The dimension vectors for distance, velocity, and acceleration are the same for each additional dimension. In one dimension, the distance, for example, is d=(x^2)^1/2. For two dimensions, it is (x^2+y^2)^1/2. For three, it's (x^2+y^2+z^2)^1/2. We can see a pattern here. For four, it would be (x^2+y^2+z^2+j^2)^1/2, where j is the value of the vector as it would be in the fourth spatial dimension. For more dimensions, we just continue this process since each would be directed perpendicularly to the others. The problem, here, however, is that the first three vectors contain the full amount of these quantities. Gravity, for instance, is felt in all three of these vectors, but with nothing left over, so that it is completely accounted for within the first three. This means that the value for the vectors of any other dimensions we might add to the list would be zero. It would be the same as if they didn't exist at all, and probably don't.

publius
2006-Jul-08, 02:50 AM
For four, it would be (x^2+y^2+z^2+j^2)^1/2, where j is the value of the vector as it would be in the fourth spatial dimension. For more dimensions, we just continue this process since each would be directed perpendicularly to the others.

Grav,

That is only for a Euclidean 4D geometry. You've got to start thinking more abstractly. True, to "visualize" 3D space curving, we compare it to a 2D surface in 3D and imagine the 3D space curving in 4 Euclidean dimensions.

However, why? You can construct any non-Euclidean space you want with a different norm ("length") than Euclidean space. Think more abstractly -- you can define a "metric", which basically tells you how to calculate norms, and you don't even worry about it having to exist in a higher Euclidean space.

Space-time is non-Euclidean. The norm (flat norm, that is) is r^2 - (ct)^2, not the sum, and it can be negative, positive, or zero. That's not anything like Euclidean.

In GR, the "metric" is something that modifies that Minkowski norm, and makes the expression something else that depends on the curvature. You don't even worry, nor need to, about some higher dimensional space in which that curves. In fact, because of the negative sign on the time coordinate, flat space-time is pretty strange anyway. You can't imagine a real space with a length element that behaves that way, really. And then, with GR, it curves on top of that.

-Richard

Ufonaut99
2006-Jul-08, 02:53 AM
I'm sure I'm missing something here. The Liebnitz formula gives a precise value for pi, which is the ratio of the circumference of a circle to its diameter in a flat plane. This calculation is exact and precise, and will always give the same result regardless of the topology of any universe any being happens to live in.

Now, as soon as you talk about actually performing a measurement, then yes, you will get a value that may differ from the calculated value, and that difference will show how your space is curved. But that won't change the result of the formula.

To take a concrete example, suppose you had a heavy lead pellet 1cm across, and a rubber sheet. You put the pellet in the sheet, and find the sheet drops 5cm. You now get a flat plane of glass and lay it across the sheet, and draw a circle where the glass meets the sheet (ie. where the sheet rises to meet the glass).

Now, the circumference of the circle on both the glass and the sheet is the same. However, measuring the radius of the circle reveals pi on the pane of glass, but a "flat-man" on the sheet would measure greater than pi on the sheet (eg. it could be 4). From this, our flatman can deduce how much his "space" is curved. However, the flat-man could still work out the formula above and still get pi.

Same thing in the universe. If Earth was an ideal sphere, and we drilled right through it, we'd find the radius diameter is longer than we'd expect from measuring the circumference, and so deduce how much space is curved. Doesn't change the value of pi, though.

grav
2006-Jul-08, 03:12 AM
RobA,

You are thinking about it correctly. The value of pi as found by the formula will not change. That is the flat space derivation. If we were to actually physically measure the value of pi (that is, the ratio of the circumference of a circle to its diameter, not the result of the formula) differently than the result of the formula, then we would know such a curvature exists.

grav
2006-Jul-08, 03:33 AM
Publius,

Your formula for r^2-(ct)^2 looks similar to something I am now working on where (ct)^2-r^2=(vt)^2. It is a combination of vectors for gravity aberration (time delay for gravity) and the relativity for centrifugal force (from the thread, Is Rotation Absolute? (http://www.bautforum.com/showthread.php?t=43314)). For one vector I get a formula similar to that for the Doppler shift and another is similar to that for relativity. I am still going through it to get a better understanding of what is happening and how it might also apply to electric and magnetic forces. I will post it soon. I'd be interested in your opinion of it.

<<<<<>>>>>

This thread is beginning to sound too much like common sense. I started by stating that if the formula for pi is true for our part of the universe, then it is true for all parts since such a formula cannot be found to any number of digits for any arbitrary number. But that does not mean that our actual measurement of the ratio does not differ to some degree depending on the curvature of space. I personally do not believe that such a curvature exists and that the formulas for relativity should also work for a flat geometry. I guess the only way to know for sure (about the curvature) is to find a way to precisely measure the physical ratio for pi, but pi has infinite digits to account for, and a measuring device will always have some error.

snarkophilus
2006-Jul-08, 11:01 AM
The argument only applies to arbitrary numbers. I don't consider two, three, and five as arbitrary. They are just as precise as the calculated value for pi.

I'm not certain what you mean. If pi is "precise," then absolutely any number is precise. I'm just saying that you can't say things about values that can be calculated algorithmically, using only vague statistical arguments.

Besides that, I don't think 2, 3, or 5 are special in any way. They are as much numbers as 6/19 or pi or erf(16).

For reference, you said:

The odds of matching a formula to describe any arbitrary number, with any number of possible digits in any order whatsoever is beyond imaginable.

But suppose you wanted to try using your argument using integers. It still falls apart, because there are an infinite number of integers that could possibly be a solution. It's not just that the value of pi is unlikely: if things are random (required for a statistical argument like this), then every possible number is equally unlikely. If it's not random (and it's not), then only one value is likely at all, and that value can be calculated. Although it's impossible to say if it can be calculated to arbitrary precision, differential geometry is your friend, and in most metrics I imagine that it can.

The term "beyond imaginable" says it all, I think. But just because you can't imagine it doesn't mean that no one else can. :)

This is another point. We would already have to know the size of the sphere (or the size of the universe) in order to do this. And the result would not be equal to pi unless space is flat. If we were to actually physically measure the value of pi to be different than the calculated value, however, we could do the reverse of what you say to find the curvature of the universe and its size.

No, you don't need to know the size. You can parametrize it if you like, using the ratio of the size of your object to the size of the sphere. Two measurements with different objects on the sphere will give you enough info to figure out the size of the sphere relative to the ratio of those two objects (actually, this is true for any uniformly curved surface). I think you've suggested something like this. However, you are thinking about it in terms of changing the value of pi, rather than determining the size of your object with respect to the curvature of space (or vice versa): they are very different things.

Anyway, do the calculation, and leave pi at its conventional value, just for comparison's sake. You'll understand then. Remember: pi is a mathematical construct only. It doesn't exist in the real world, any more than any other number exists. There's only an imperfect association between those numbers and reality.

Fortis
2006-Jul-08, 11:58 AM
Besides that, I don't think 2, 3, or 5 are special in any way. They are as much numbers as 6/19 or pi or erf(16).
And there is that nice reductio ad absurdum proof that there are an infinite number of 'special' numbers. :)

montebianco
2006-Jul-08, 02:15 PM
Nope. The integers and rationals map one to one.

Well, yes, there exists such a map. But there also exists maps in which each integer maps to many rationals. Of course, there are also maps in which each rational maps to many integers :)

grav
2006-Jul-08, 03:53 PM
snarkophilus,

By arbitrary I mean a string of random digits after the decimal point. For the value of pi on a sphere, one could select any arbitrary size for the diameter of the circle as compared to the diameter of the sphere. The ratio would come out somewhere between two and pi for this. So instead of taking some abitrary diameter for the circle, let's set our initial parameters for an arbitrary value for pi between 2 and 3.14159... It is the same difference. In this case, it would be like taking the number two and adding any random digits at all that we wish beyond the decimal point. This will give us some arbitrary number between two and three that will match the value of pi for some diameter on the surface of the sphere.

The problem becomes that for any such arbitrary number, we cannot always find a formula for just any random digits, although this may truly be a purely philosophical point. Sure, if we originally knew the curvature of the universe or its diameter, we could then plug either value in to get the value of pi as measured by the Liebnitz formula or any other related one, but until we know one or the other, we cannot do this. For example, let's say that we draw a circle and the curvature of the universe is such that we find the ratio for pi to be 3.107384659... We would then be forced to find a formula for that number instead. There is no way we would know what the actual value for pi should be and we don't originally know the curvature or the size of the universe, so we would be stuck with trying to determine a formula for an obscure number and we wouldn't know the difference. Lucky for us the universe is at least flat enough to initially find the value for pi to a large degree (at least six digits) whereas we can then find the measured value for pi to a large degree in order to compare the formulas to. The formulas, then, are based on a flat geometry only, but might vary after the tenth or twentienth or one hundreth digit, depending on the curvature of space.

As far as being capable of finding a formula for just any arbitrary number, this would depend on whether or not an identifiable pattern exists in the number to begin with. The numbers two, three, and five are special because they all contain an infinite number of zeros after the decimal point. This identifies a pattern. I don't believe a formula can be found for rambling off random digits after the decimal place (unless the "random" digits themselves were derived from a formula). I have, however, developed a method for finding the formula for any set of numbers or digits that follow such a pattern that does not include irrationals or geometric progressions. Using this, I can find a formula for pi for any number of digits desired when the digits are already known. This is because pi is an irrational number but its digits are not. But the number of terms in this case would be equal to the number of digits themselves so that an infinite equation would still result for the full extent of pi. This is because all irrational numbers require infinite equations. But that doesn't mean that a pattern does not exist, otherwise the derivation of even such infinite equations would be impossible.

But let's make things easier on ourselves. Let's try drawing a different shape using only straight lines. A square or triangle, for example. When drawn very small upon the surface of our sphere (which is only one such possibility for the curve but is the simplest to imagine), the shapes are close to that for flat space, and are the way we formulate them in Euclidean geometry. But now let's make them bigger. As they come close to the actual size of the sphere, they all become circular. By the time we fit them over the "equator", they are perfectly round, and the perimeter of our shape now matches that of the circumference of the sphere. This would be, then, because all "straight" lines are really curved somewhat in this non-Euclidean geometry. This is where GR fits in. The equations of GR would basically translate, in my opinion, to mean that we could not "tell" that the lines are curved, so all appears the same as it would for perfectly flat space. That is to say, whatever way we were to measure the sides of our drawn figure, we would not know that an actual curve existed. If we measure it, with a ruler, the ruler would also curve in the same proportion, and no difference can be seen. If we measure it with light, the path of light would curve and we would have the same result. In this case, the value for pi will always be measured the same under any circumstances and we could never know otherwise. Of course, we could then say the same thing that caused Einstein to write his special relativity equations to begin with. That is, if we cannot measure a difference (as it was with the results of the ether), then why believe that it exists at all? In other words, if we cannot "feel" the effects of such a curvature, whereas it doesn't play a role in the actual physical universe as far as anyone can tell, then it is the same as if it does not exist for us, so why presume it to exist at all?

TravisM
2006-Jul-08, 04:23 PM
With a quick skim through the begining of this thread, I didn't see this bit rasied:
The value of things like h-bar would change in any area of the universe where the topology varied. i.e.: physics would change. Not likely.

So, with the idea that phyiscs are the same from one region to another, the entire universe would probably take on only one value of "pi"... It might be that the exact value for pi is influenced slightly by the local geometry of space, but the mathematical definition for pi is pretty much concrete.

Trav

Aristocrates
2006-Jul-08, 08:19 PM
grav, geometry is a branch of mathematics, and is thus not dependant on the local or overall shape of the universe in which is it practiced. Thus, pi, which is defined only for Euclidean geometry, would have the same value whether the universe is flat, or shaped like three donuts and a squirrel. Since it is an ideal value in mathematics, and not a measured value from physics, it really isn't any surprise that it can be approximated through such simple formulae as shown above.

Practical measurements do show that the circumference to diameter ratio in this part of the universe is as close to pi as we can tell, but that only shows that the universe is fairly flat locally. If this were not the case, it would still say nothing about the nature of pi, but only the local shape of space.

Fortis
2006-Jul-08, 09:58 PM
With a quick skim through the begining of this thread, I didn't see this bit rasied:
The value of things like h-bar would change in any area of the universe where the topology varied. i.e.: physics would change. Not likely.
The curvature dependence of physical laws is dealt with, in GR, through the strong principle of equivalence, which roughly says that if you can express a physical law in tensor notation in SR, then it looks exactly the same in a local inertial frame of a curved spacetime. This is how we can do quantum field theories in classical curved spacetimes.

I'm still struggling to understand what grav is trying to get at here. He appears to be saying that we can't say anything about physics (or at least geometry) in a curved spacetime, and if so, then he is not correct (within the current standard model.)

snarkophilus
2006-Jul-09, 08:35 AM
The problem becomes that for any such arbitrary number, we cannot always find a formula for just any random digits, although this may truly be a purely philosophical point.

Here is part of the problem. You assume that the digits are random, but they are not. You can calculate the number directly, to any precision you like, no matter the curvature of space and the size of object you are measuring. This is, of course, with the caveats that space is continuous and otherwise mathematically well-behaved.

I really don't understand what you mean when you talk of infinite equations, but I assume you are talking about infinite summations, and treating pi as as infinite series of coefficients (the digits) multiplied by powers of 10. Furthermore, you claim that for every number, it is not possible to find a general equation to predict these coefficients. There are two problems with this. 1) You have no proof that this is the case. 2) The lack of a simple digit extraction algorithm (what you are suggesting) does not imply that a number can't be computed to arbitrary precision, and the digit extracted that way.

For instance, I don't know of a simple way to find the 1000th digit of 2.6pilog32, but I can still find the value of that digit with enough time. Furthermore, since in every simple geometry (that is, spaces with constant curvature), a formula exists to find the angles within a triangle (for instance), it is possible to find any digit of the value you seek for any object within that space. I wanted you to actually do that math on the sphere so you could discover that.

Actually, there's something neat in this. In spaces with constant curvature, the limit of the interior angles of a triangle as the size of that triangle tends to zero is pi (as it is normally defined). *This is true for all such spaces.* However, I think there is a class of spaces with non-constant curvature for which this is not true. I have to think about those spaces some more, but at first glance they appear to be very interesting objects. (Edit: upon further investigation, it appears that a non-pi limit can only occur around neighbourhoods in the metric space that are non-differentiable. I doubt there's any real world analogue.)

This would be, then, because all "straight" lines are really curved somewhat in this non-Euclidean geometry.

Not really. In terms of that geometry, they are straight. In terms of an Euclidean space into which that object is embedded, they are curved.

In other words, if we cannot "feel" the effects of such a curvature, whereas it doesn't play a role in the actual physical universe as far as anyone can tell, then it is the same as if it does not exist for us, so why presume it to exist at all?

Well, you can feel the effects of spatial curvature. You're stuck to the Earth now because of it. Gravitational lensing is a more extreme example; it wouldn't really work without curved space (or a change in spatial "density," if such a thing makes sense). The problem with your analysis in this section is that you are assuming measurement of an object (a triangle, say) which is the size of the whole universe (the sphere). Furthermore, you assume that all objects within the universe are that size, which is clearly not the case.

grav
2006-Jul-09, 08:18 PM
I still don't seem to be coming across correctly. :wall:

By the measurement of pi, I mean something like actually taking a piece of string and running it around a cylindrical object and again across its diameter and comparing lengths. By the formulas for pi, I mean the mathematics that are required to give us the number for it.

When referring to an arbitray number, I am not saying that the calculated value for pi is arbitrary. It is defined quite precisely. That is kind of the point. If the measured value for pi is found to be precisely that of the calculated value, then the universe is perfectly flat everywhere since the odds of it being just a local phenomenom without a single digit of difference between the two compared values is unthinkable.

By infinite equations, I do in fact mean infinite sums. I should have stated that better. :o

As for finding the 1000th digit of 2.6pilog32, we have already defined the pattern in this case with the given. So all we have to do is perform the calculations that will provide us with the 1000th digit. This is also not the same as spouting off random digits.

I know that it is possible to find a formulation for the digits of pi within a curved space that is not arbitrary. But the reason it would not be arbitrary in this case is because we already know the definition of the curve and/or the size of the sphere (or other). But if we were to begin with the measured value for pi and did not originally know the curve, the number would appear arbitrary because we would not know what the actual value should be. In other words, if we were to actually measure pi at 3.071..., we would not have found the formula for pi because we would not know what we should really be looking for. If someone did manage to come up with a formula for it (3.14159...), they would not know to relate it to the actual measurement because it is so far off, and no relation to a curvature could otherwise be made. The only reason we know better than that is because the measured and formulated values are so close, which means either space is flat, or the local curvature is very small. Since we now have the actual formula for what pi should be for a flat topography, we can compare it to the measured value to check for curvature. The only problem with this would be that the very theory that says that such a curvature should exist (GR) also says (as far as I can tell) that relativity works in such a way that such a curvature can never be seen or measured. I'm not sure about this, however, so it may or may not be true. :neutral:

The purpose of SR and GR is that motion in any frame of reference is equivalent. This would probably mean that all bodies should appear to be travelling in a straight line within their own frame of reference. If this is the case, then the equations of GR should provide for this. That would mean that the curvature of the universe would actually be undetectable because of the effects of relativity (at least within our own frame of reference). But if this were the case, then there is no proof of the curvature of space, except that the equations for GR require it, which is kind of a catch-22.

In case anybody decides that gravity itself is a proof of curved space, this is not the case (in my opinion). Push gravity provides a much better mechanism. This does not mean that the equations for GR are invalid, however. They do a very good job with the perihelion of Mercury. I intend to eventually show that the same equations can be used within a flat geometry. It's all just a matter of how we perceive the concepts of relativity. As it stands, it contains too many confusing factors and paradoxes. I believe a new theory can be made which utilizes most of the same equations but within a flat space and in a much clearer way. :lol:

Fortis
2006-Jul-09, 09:42 PM
I know that it is possible to find a formulation for the digits of pi within a curved space that is not arbitrary. But the reason it would not be arbitrary in this case is because we already know the definition of the curve and/or the size of the sphere (or other). But if we were to begin with the measured value for pi and did not originally know the curve, the number would appear arbitrary because we would not know what the actual value should be. In other words, if we were to actually measure pi at 3.071..., we would not have found the formula for pi because we would not know what we should really be looking for.
Just thought it worth mentioning that for a general curved space, it is unlikely that you would even obtain the relationship

circumference = k*radius, where k is a constant.

It is more likely that k would be a function of radius, rather than being a constant. (For example, consider what that function looks like on the 2-d space defined by the surface of a sphere.)

grav
2006-Jul-10, 12:05 AM
Just thought it worth mentioning that for a general curved space, it is unlikely that you would even obtain the relationship

circumference = k*radius, where k is a constant.

It is more likely that k would be a function of radius, rather than being a constant. (For example, consider what that function looks like on the 2-d space defined by the surface of a sphere.)
That's true. That's how we would identify the curvature, by comparing the discrepency between the measured and formulated values of pi at different radii. After doing so, we could then find the curve for a particular radius, and we would know how it must be placed on the sphere (or other) to produce this curve, whereby we would find the size of the universe. I just realized, however, that if the universe contains no measurable boundaries, then determining its size should not be possible. Yet it would be if the universe had a measurable curvature. Therefore, I suppose this would be an additional argument that the universe has no measurable curvature. In other words, it is flat. (Or else, GR must provide that we cannot measure this curvature, as said before, in which case it is still as if it does not exist, and might as well not, since it would then have no influence on reality as we know it since the universe would still appear flat regardless.)

Nereid
2006-Jul-12, 08:19 AM
[snip]

In case anybody decides that gravity itself is a proof of curved space, this is not the case (in my opinion). Push gravity provides a much better mechanism. This does not mean that the equations for GR are invalid, however. They do a very good job with the perihelion of Mercury. I intend to eventually show that the same equations can be used within a flat geometry. It's all just a matter of how we perceive the concepts of relativity. As it stands, it contains too many confusing factors and paradoxes. I believe a new theory can be made which utilizes most of the same equations but within a flat space and in a much clearer way. :lol:And may I wish you all the very best in your endeavour to work out this "push gravity" idea.

If, when you're done, you find that there is no observable difference (with GR), even in principle, then we will be able to add another chapter to the meta-holographic principle book.

OTOH, if you do find something that's testable, in principle, in the way of a difference between "push gravity" and GR, then we can move on to the next step.

In the meantime, is there anything to discuss, in the narrow sense of the stated purpose of this ATM section of BAUT (http://www.bautforum.com/showthread.php?t=32864)? I mean, for example, I could say something like "As it stands, GR contains essentially no confusing factors or paradoxes. I believe a successful, new "push gravity" theory canNOT be made", but then all we'd be left with is two "believes" and a difference of opinion wrt "confusing factors and paradoxes", right? :razz:

antoniseb
2006-Jul-12, 09:44 AM
Hi grav, I don't know if this was stated in the middle of any of the longer posts above but...

If the universe is curved, then the ratio of the diameter and circumference of a circle will depend on the diameter of the circle, and potentially on where that circle is drawn if space is not uniformly curved. As such small circles will all have that ratio vanishingly close to pi, and cosmically large circles, where measuring this ratio would be darn near impossible would have different values.

Grey
2006-Jul-12, 06:32 PM
The problem with such measurements, however, is how do we draw a perfect circle and then measure it so precisely? This would have to be done physically because the formulas only derive the value under ideal circumstances (for when space is flat). If such a measurement could be made to a degree to where we could tell to what digit a signifant difference lies, we could possibly then tell not only the curvature for our region but the size of the universe as well. If the universe is not spherical, but some other odd curvature, we could tell this by measuring the ratio for pi in many regions of the universe (but of course this won't happen anytime soon). But by comparing different sizes of circles, we might also obtain an idea of its general shape.This is true, and so far our best measurements have indeed shown the universe to be flat. Actually, there are some more clever techniques that astronomers use to determine the curvature of space by observing distant objects, since we have pretty serious limits on how large we can make circles, how precisely we can measure them, and how easy it is to make certain that they are in fact perfect circles. Those techniques suggest that the universe is darn close to flat as well, at least globally. Actually, though, the data leans toward a very slight positive curvature. Just to assume that it must be flat would be rather premature at this point, though. Moreover, the fact that the overall curvature is flat or very close to it does not necessarily mean that there are not regions of space with local curvature.

I intend to eventually show that the same equations can be used within a flat geometry. It's all just a matter of how we perceive the concepts of relativity. As it stands, it contains too many confusing factors and paradoxes. I believe a new theory can be made which utilizes most of the same equations but within a flat space and in a much clearer way.It is always possible to change a description using curved space with a description the uses flat space, provided that you also add in "stretching and shrinking fields" that would change the lengths of all measuring devices and modify the rates of clocks. You could do that, but it would probably not be simpler and clearer than general relativity. On the other hand, you cannot duplicate the results of general relativity in flat space without allowing objects to change size in response to what would be curved spacetime in general relativity.

grav
2006-Jul-13, 12:56 AM
If, when you're done, you find that there is no observable difference (with GR), even in principle, then we will be able to add another chapter to the meta-holographic principle book.

OTOH, if you do find something that's testable, in principle, in the way of a difference between "push gravity" and GR, then we can move on to the next step.

I'm not sure if by meta-holographic principle you mean that any concepts which result in identical formulas as relativity may then be indistiquishable from relativity itself, or if you mean that any formulas for flat space might just be those for three dimensions only but others might exist for within extra dimensions. In either case, however, you would be right. The trick would be, as you have stated, to find a way to distinguish between any alternative concepts and relativity to begin with. This could prove rather difficult since the concepts behind relativity are rather vague. What is space-time made of that allows it to warp? How is mass specifically effected by this? What is it about time that allows it to dilate, not just in an illusionary sense? How exactly would the curvature of the universe (once it is found) specifically figure into the theory of relativity? These are questions which need to be answered. If Einstein had presented his theories on this forum, I'm sure all you would have at least demanded this much from him before fully accepting the theory.

So even if I were to come up with a theory that explains many things in detail according to a combination of an ether, push gravity, and tired light theory, all of which are considered against the mainstream and intangible, with the particles which tie them all together, the neutrino, it would be considered proof of these things to the people that support them, but further proof of relativistic formulas for the ones that support relativity, especially since relativity contains many of the same formulas as that of an ether, but with a completely different way of thinking about it. It is almost like a "negative" ether.

What is really the difference between calling it an ether and calling it space-time, anyway? The results are virtually the same. So the difference lies in one respect to the application of a mechanism. The "substance" which causes these effects are very similar to that of a neutrino. Gravity requires that it is capable of penetrating large quantities of matter, so it be very small, neutral in charge, and very abundant in space. The tired light theory requires that it be very small and penetrating as well. And the same with the ether. The only particle I know of with these qualities is the neutrino. I have even found its size to be on the order of the Plank length, which ties it in with both relativity and quantum theory. That doesn't mean that another similar particle doesn't exist, but it should demonstrate many of the properties normally associated with neutrinos.

As far as I know, the only theories that require the curvature of space are relativity and the Big Bang. The ether can take the place of SR (but with slightly different equations which produce a null result, whereby it is simply in equilibrium), push gravity that of GR, and tired light that of the redshift as proof of the Big Bang. Even after all this is done, however, I will still unfortunately have to provide proof that this can only occur within flat space, in order to satisfy the multi-dimensional enthusiasts. This may prove exceedingly difficult.

grav
2006-Jul-13, 01:31 AM
antoniseb,

Your statement has been covered and you appear to have a clear comprehension of it. How do you think this relates to relativity and what are your thoughts on the applicability of it as relativity (and the Big Bang) would require?

Grey,

What data are you referring to that implies a slightly positive curve? Do you think it's accurate?

<<<<<>>>>>

I've been thinking about it. Doesn't both the theories of relativity and the Big Bang theory require that the curvature of space should be the same in every direction and from every frame of reference? If this is the case, then the only possible shape for the curvature that meets these requirements is that of a sphere. So that much would be definite, right? Only a sphere is perfectly symmetrical from every direction. But then some points within the sphere would be different than others depending on how they are related to its center and edge. So the sphere can have no center or edge. But this is only for three-dimensional spheres. So now we require another dimension. Einstein said that if we were to travel "around" the universe in a straight line, we would end up where we started. But this would mean the universe has a definite size, since we could multiply our velocity in relation to a starting point by the time it took to travel this distance. So this goal must be unattainable, so that the universe is expanding at the speed of light, which we cannot attain in order to surpass the "boundary" to begin with. But now it is found to be accelerating. What, faster than the speed of light? Some very far away galaxies already show a >z redshift, which should not be possible. What does this mean for relativity?

It could be explained easily with tired light, however. The energy decreases over a distance, but it does this in proportion to its original energy. So after some distance is travelled, some of its energy is depleted, and it loses less energy in proportion. Therefore, the relation of the loss of energy to distance would be smaller from very far away, which is the same as the evidence of the acceleration of expansion. Also, light from distant galaxies could easily show a >z redshift this way. Which makes more sense?

antoniseb
2006-Jul-13, 02:33 AM
How do you think this relates to relativity and what are your thoughts on the applicability of it as relativity (and the Big Bang) would require?

I don't especially see it relating to relativity or the big bang in ways that I can think through. Any change that you might claim happens to formulas because some circles might have radiuses the wrong size for their circumferences during the inflation period, or close to the event horizons of small primordial black holes (which I don't have any reason to think ever existed), can well be non-existant for all I know. It will be hard to tell, since the largest meaningful circle for determinging something like this is probably the that of an atom. The last time the universe was curved enough for that size circle to produce a detectably altered pi was perhaps microseconds after the beginning of the universe (a time about which I claim no knowledge).

grav
2006-Jul-13, 02:41 AM
Well, if relativity requires a curvature of space, then how is that incorporated into the formulas for it? The most common formula for most relativistic effects is based on the ratio (1-v2/c2)1/2. I suppose that the curvature could be incorporated by something like (1-k*v2/c2)1/2, where k is the curvature in some region of the universe (k=1 for flat space). But then the value of k would become a form of relativity for relativity, and doesn't explain the initial requirement for curvature to begin with, within its initial context. So we must ask ourselves, what part of this formula should change depending on the curvature of space? The value of v cannot, for this is the relative velocity for a source and observer as compared in flat space. It is therefore the given, a variable in itself. So what is left? The speed of light.

So the measure of the speed of light may depend on the curvature of space. But this is also seen for an ether, where its pressure and density may change from region to region. Since the "sound" speed depends on these conditions, the speed of light is also variable, where P=Dc2/3. However, regardless of its ability to change in value, the factor of three demonstrates that it exists in three-dimensional, flat space. I originally came across this formula in relation to others in The Grand Puzzle. I didn't recognize its significance right away, but eventually read something to a similar effect in a book called Modern Cosmologies (I think). Until then, I didn't have too much problem considering that the universe (space) might be curved.

grav
2006-Jul-13, 11:45 PM
Okay. In my last post, I used the formula, (1-k*v2/c2)1/2 as an example, just an arbitrary possible for demonstration, where k would equal 1 for flat space. But I have realized that since the speed of light is the only possible variable in Einstein's equations, then this arbitrary formula should not be too far off. In keeping with k=0 for flat space, the formula should probably read [1-v2/(j/k)2]1/2, where the greatest attainable speed of an object really depends on the value of k and j and j/k=c for our universe. For a flat universe (k=0), this would be j/k=j/0=infinity, so there is no greatest speed. Any object may have any speed imaginable. The effects of relativity in flat space would then become (1-0)1/2=1, so relativity wouldn't exist in flat space. So in order for the theories of relativity to incorporate the curvature of space, the speed of light would probably depend on some original speed associated with the expansion of the universe divided by its curvature. If we call this original speed j, then the speed of light would be j/k. The formula I just proposed, however, would provide the same formulas for relativity regardless of whether the curvature is negative or positive. The speed limit becomes negative for a negative curve, though. Any other suggestions for the formula, or should we conclude that the curvature for the universe should always be positive?

It may seem strange that I am attempting to identify a curvature for the universe and associated properties when I don't even believe in such a thing. But I have learned that it is just as valuable to learn what can't be the case in order to trim down the possibilities. Furthermore, it will tell me what I should or shouldn't be looking for. Sometimes formulas can still be applied with just another way of thinking about them. Also, it is a good exercise that helps to understand how a theory stands (or doesn't). And finally, if I run into an impossible situation (paradox), then there is no further need to consider that theory unless it can somehow be overcome to satisfaction.

Grey
2006-Jul-14, 11:50 AM
Okay. In my last post, I used the formula, (1-k*v2/c2)1/2 as an example, just an arbitrary possible for demonstration, where k would equal 1 for flat space. But I have realized that since the speed of light is the only possible variable in Einstein's equations, then this arbitrary formula should not be too far off. In keeping with k=0 for flat space, the formula should probably read [1-v2/(j/k)2]1/2, where the greatest attainable speed of an object really depends on the value of k and j and j/k=c for our universe.Actually, the math describing the curvature is rather more complex than that. Here (http://www.math.ucr.edu/home/baez/gr/) is a good brief introduction, and it also has links to a number of texts. If you want to understand how the math of general relativity works, there's really no alternative to hitting the books.

mugaliens
2006-Jul-14, 09:55 PM
Of course it is - it's dimentionless and entirely dependant upon local conditions, whereas most other constants require locally observed constants to remain constant over vast distances, something I don't think is viable.

grav
2006-Jul-16, 02:02 AM
Actually, the math describing the curvature is rather more complex than that.
Oh, I'm sure it is. But what I'm really getting at here is that the only variable that can change due to curvature in relavistic equations would be the speed of light. v is already a variable and depends upon how we define our measurements within a particular frame of reference. In other words, v is the observed flat space measurement which relates the formula [1-(v/c)2)1/2 for relativistic effects. Once we have gone through all of the necessary steps to aquire a measure of a curvature for the universe, we can then relate it to flat space by some ratio, either constant or depending on distance and/or velocity. This should then relate to the speed of light with some constant or formula for the ratio k, which would then be part of the equation for relativity.

grav
2006-Jul-16, 02:27 AM
Quote:

Originally Posted by grav
For four, it would be (x^2+y^2+z^2+j^2)^1/2, where j is the value of the vector as it would be in the fourth spatial dimension. For more dimensions, we just continue this process since each would be directed perpendicularly to the others.

Grav,

That is only for a Euclidean 4D geometry. You've got to start thinking more abstractly. True, to "visualize" 3D space curving, we compare it to a 2D surface in 3D and imagine the 3D space curving in 4 Euclidean dimensions.

Space-time is non-Euclidean. The norm (flat norm, that is) is r^2 - (ct)^2, not the sum, and it can be negative, positive, or zero. That's not anything like Euclidean.

I think I see what you mean. I have noticed that even in this forum the value for the fourth term (time) is usually believed to be simply ct. But this cannot be the case. If it were, the total velocity (this distance divided by the time t) , when all terms are simply added together, would give us a value greater than c. But c should always be our hypotenuse for all vectors put together, right? In other words, it is always the resultant vector. This would mean the equation should read (x^2+y^2+z^2+j^2)^1/2=ct. j, for the fourth term, would then be j=((ct^2-x^2-y^2-z^2)^1/2. Correct? Furthermore, if x, y, and z represent some distance travelled over time t as we observe it in three dimensional space, where vt=(x^2+y^2+z^2)^1/2, then the formula will read j=((ct)^2-(vt)^2)^1/2. To then find a ratio for the value of j to the total sum of vectors, ct, we simply divide both sides by ct and get a ratio of j/ct=(1-(v/c)2)1/2, the formula for relativity. This, then, would be the ratio of the value of the fourth vector in curved space to the total value of all of the vectors combined.

grav
2006-Jul-16, 03:50 AM
Okay. So in the last post, we found that the ratio of the fourth term (j) to the total resultant vector (ct) is j/ct=(1-(v/c)^2)^1/2 (if this is correct). This is for relativity. But the Big Bang theory also predicts such a curve. But in this case, it would be due to the expansion of the universe. If the universe is presently expanding at the rate of H, the total expansion for a distance in some time t would be 1+Ht. This would give us a total expansion of (1+Ht)(x^2+y^2+z^2) in time t, as compared to the relativistic value of ct=(x^2+y^2+z^2+j^2)^1/2. If these two effects are one and the same (caused by the same curvature), then they should be equivalent. In this case, (1+Ht)^2(x^2+y^2+z^2)=(x^2+y^2+z^2+j^2). Solving for j, we have j=[((1+Ht)^2-1)(x^2+y^2+z^2)]^1/2. Substituting vt=(x^2+y^2+z^2)^1/2, we get j=(vt)[(1+Ht)^2-1]^1/2. Finally, dividing by the resultant vector, we get j/ct=(v/c)[(1+Ht)^2-1]^1/2.

Now let's compare formulas. We had j/ct=(1-(v/c)^2)^1/2 for relativity and we now have j/ct=(v/c)[(1+Ht)^2-1]^1/2 for the Big Bang. Setting these equal, we find the relationship (1-(v/c)^2)^1/2=(v/c)[(1+Ht)^2-1]^1/2. Squaring both sides, we get 1-(v/c)^2=(v/c)^2[(1+Ht)^2-1], which becomes 1-(v/c)^2=[(v/c)(1+Ht)]^2-(v/c)^2, so 1=[(v/c)(1+Ht)]^2. Reducing further, we find (v/c)(1+Ht)=1.

This, then, is definition of the curvature of spacetime as it must be for both relativity and the Big Bang to operate from the same curve. It is simply (v/c)(1+Ht)=1. As we can see, if the velocity is small, the curvature over some distance travelled is large. If v approaches c, the curvature becomes small. At v=c, (1+Ht)=1, so the curvature is zero. This is because the distance we have defined is vt, so the faster we travel this distance, the less time the universe has had to expand. If we were to travel at the speed of light (v=c), it would appear to us that this distance has been traversed instantaneously within that frame of reference, and the curvature is zero. I guess you could say that we were travelling with the curve (or with the expansion).

Once again, I am not saying that I believe in the curvature of spacetime. I believe spacetime to simply be the neutrino medium and gravity and the redshift of light to be produced by this same medium. The formulas provided by the effects, however, might be similar. So I am attempting to follow this path to its natural conclusion. It's kind of like starting at the end and working my way backwards. But thinking backwards is what gave me the neutrino gravity theory to begin with, when no other form of attraction seemed to work. Maybe I can connect the dots somewhere in the middle.

north
2006-Jul-16, 05:46 AM
Okay. So in the last post, we found that the ratio of the fourth term (j) to the total resultant vector (ct) is j/ct=(1-(v/c)^2)^1/2 (if this is correct). This is for relativity. But the Big Bang theory also predicts such a curve. But in this case, it would be due to the expansion of the universe. If the universe is presently expanding at the rate of H, the total expansion for a distance in some time t would be 1+Ht. This would give us a total expansion of (1+Ht)(x^2+y^2+z^2) in time t, as compared to the relativistic value of ct=(x^2+y^2+z^2+j^2)^1/2. If these two effects are one and the same (caused by the same curvature), then they should be equivalent. In this case, (1+Ht)^2(x^2+y^2+z^2)=(x^2+y^2+z^2+j^2). Solving for j, we have j=[((1+Ht)^2-1)(x^2+y^2+z^2)]^1/2. Substituting vt=(x^2+y^2+z^2)^1/2, we get j=(vt)[(1+Ht)^2-1]^1/2. Finally, dividing by the resultant vector, we get j/ct=(v/c)[(1+Ht)^2-1]^1/2.

Now let's compare formulas. We had j/ct=(1-(v/c)^2)^1/2 for relativity and we now have j/ct=(v/c)[(1+Ht)^2-1]^1/2 for the Big Bang. Setting these equal, we find the relationship (1-(v/c)^2)^1/2=(v/c)[(1+Ht)^2-1]^1/2. Squaring both sides, we get 1-(v/c)^2=(v/c)^2[(1+Ht)^2-1], which becomes 1-(v/c)^2=[(v/c)(1+Ht)]^2-(v/c)^2, so 1=[(v/c)(1+Ht)]^2. Reducing further, we find (v/c)(1+Ht)=1.

This, then, is definition of the curvature of spacetime as it must be for both relativity and the Big Bang to operate from the same curve. It is simply (v/c)(1+Ht)=1. As we can see, if the velocity is small, the curvature over some distance travelled is large. If v approaches c, the curvature becomes small. At v=c, (1+Ht)=1, so the curvature is zero. This is because the distance we have defined is vt, so the faster we travel this distance, the less time the universe has had to expand. If we were to travel at the speed of light (v=c), it would appear to us that this distance has been traversed instantaneously within that frame of reference, and the curvature is zero. I guess you could say that we were travelling with the curve (or with the expansion).

Once again, I am not saying that I believe in the curvature of spacetime. I believe spacetime to simply be the neutrino medium and gravity and the redshift of light to be produced by this same medium. The formulas provided by the effects, however, might be similar. So I am attempting to follow this path to its natural conclusion. It's kind of like starting at the end and working my way backwards. But thinking backwards is what gave me the neutrino gravity theory to begin with, when no other form of attraction seemed to work. Maybe I can connect the dots somewhere in the middle.

grav

it is known that the further apart that quarks get, the stronger gluon attraction becomes between quarks. interesting, is not? , I thought so any way.( it took time for me to find the response) they disagreed with me for this reason below.

No, it's a very interesting subject, but has nothing to do with gravity. Quark and gluon interactions are described by something called gauge theory, specifically the SU(3) gauge theory "quantum chromodynamics". Gravity is described by a diffeomorphism-invariant theory called "general relativity". They have certain similarities, like how both are fundamentally based on symmetries and invariances, but they are also very different.

I'm not sure that I agree with this argument yet but there it is.

particle physics.

Grey
2006-Jul-16, 07:37 AM
Oh, I'm sure it is. But what I'm really getting at here is that the only variable that can change due to curvature in relavistic equations would be the speed of light. v is already a variable and depends upon how we define our measurements within a particular frame of reference. In other words, v is the observed flat space measurement which relates the formula [1-(v/c)2)1/2 for relativistic effects. Once we have gone through all of the necessary steps to aquire a measure of a curvature for the universe, we can then relate it to flat space by some ratio, either constant or depending on distance and/or velocity. This should then relate to the speed of light with some constant or formula for the ratio k, which would then be part of the equation for relativity.Again, not that simple. If spacetime has a non-Euclidean metric, distances and times will both be measured differently. So both v and c will be measured differently by a local observer than they would be measured by a distant observer, though they will both vary in such a way that any local measurement of the speed of light will always arrive at the same value of c.

Fortis
2006-Jul-16, 01:26 PM
I think I see what you mean. I have noticed that even in this forum the value for the fourth term (time) is usually believed to be simply ct. But this cannot be the case. If it were, the total velocity (this distance divided by the time t) , when all terms are simply added together, would give us a value greater than c. But c should always be our hypotenuse for all vectors put together, right? In other words, it is always the resultant vector. This would mean the equation should read (x^2+y^2+z^2+j^2)^1/2=ct. j, for the fourth term, would then be j=((ct^2-x^2-y^2-z^2)^1/2. Correct? Furthermore, if x, y, and z represent some distance travelled over time t as we observe it in three dimensional space, where vt=(x^2+y^2+z^2)^1/2, then the formula will read j=((ct)^2-(vt)^2)^1/2. To then find a ratio for the value of j to the total sum of vectors, ct, we simply divide both sides by ct and get a ratio of j/ct=(1-(v/c)2)1/2, the formula for relativity. This, then, would be the ratio of the value of the fourth vector in curved space to the total value of all of the vectors combined.
Grav, the magnitude of the 4-vector velocity is always equal to -1 (in units where c=1).

grav
2006-Jul-16, 02:07 PM
Again, not that simple. If spacetime has a non-Euclidean metric, distances and times will both be measured differently. So both v and c will be measured differently by a local observer than they would be measured by a distant observer, though they will both vary in such a way that any local measurement of the speed of light will always arrive at the same value of c.
Yes. I agree. That is what I mean. The formula for relativity relates in such a way that the speed of light remains constant in any frame of reference. I myself believe that the Doppler shift and the pressure and density of the neutrino medium provide for this, although it does not account for real time dilation. But in order for this to work in the context of the curvature of space, the value for the speed of light should also depend originally on the amount of the curvature and the distance travelled (or observed from). This is because the curvature is different for different distances, as the value of pi would be different for different size circles. It can also be related by a velocity as travelled over this distance, as I have done. While travelling a distance, however, the universe is also expanding, so this is incorporated as well. We come up with a result that also depends on the time of travel, so that travelling a distance at a velocity of c would appear instantaneous, with zero time and zero curvature, where one would then be travelling with the curve and the expansion of the universe. This, then, must be what light does according to these theories. The distance, time, and velocity that are used in the formula are only how an observer would view them from flat space, but the perception of them change within curved spacetime. The formula then incorporates them into a curved spacetime provided by relativity and the expansion of the universe.

grav
2006-Jul-16, 02:20 PM
Grav, the magnitude of the 4-vector velocity is always equal to -1 (in units where c=1).
It is true that physics nowadays usually determines the value of c as 1, which may shorten formulas and keep them less cluttered, but I prefer to spell mine out completely and even write them over and over again to make sure I have a full comprehension. Eliminating c in equations only makes them more confusing to me since I also like to observe the units involved. But this does not mean that its value cancels out completely or that it no longer contributes. It almost appears that this is what you are saying. In order to drop the value of c, we must still include it with our value for time. So instead of using the speed of light and units of seconds for the time, for example, we would now express the distance in terms of light-seconds.

grav
2006-Jul-16, 02:25 PM
grav

it is known that the further apart that quarks get, the stronger gluon attraction becomes between quarks. interesting, is not? , I thought so any way.( it took time for me to find the response) they disagreed with me for this reason below.

I'm not sure that I agree with this argument yet but there it is.

particle physics.
I also have a simple theory about nuclear force that is incorporated into the theory of gravity by the neutrino medium. I do not have time to post it now, however, but will post it here as soon as I can (later today). Maybe you can help me work out the details. I must warn you, though. I am particle physics illiterate. I have read books on it from cover to cover and examined every formula. And after doing so, I can say for certain now, that I didn't understand a single word of it. The formulas read like Greek to me, and most of them are.

[EDIT-Maybe my simple ideas about nuclear force are worth discussion after all. I will start a thread in general science called Theory on nuclear force.]

Fortis
2006-Jul-16, 11:38 PM
It is true that physics nowadays usually determines the value of c as 1, which may shorten formulas and keep them less cluttered, but I prefer to spell mine out completely and even write them over and over again to make sure I have a full comprehension. Eliminating c in equations only makes them more confusing to me since I also like to observe the units involved. But this does not mean that its value cancels out completely or that it no longer contributes. It almost appears that this is what you are saying. In order to drop the value of c, we must still include it with our value for time. So instead of using the speed of light and units of seconds for the time, for example, we would now express the distance in terms of light-seconds.
The use of c=1 was incidental to what I was trying to get across.

If you want to calculate the magnitude of the 4-velocity, you just need to calculate du/dtau, where u is the position 4-vector, and tau is the proper time of the particle that you are considering. I leave it as an exercise for the student to show that the magnitude of the 4-velocity (leaving the c in) is simply -c. ;) :)

grav
2006-Jul-17, 12:46 AM
The use of c=1 was incidental to what I was trying to get across.

If you want to calculate the magnitude of the 4-velocity, you just need to calculate du/dtau, where u is the position 4-vector, and tau is the proper time of the particle that you are considering. I leave it as an exercise for the student to show that the magnitude of the 4-velocity (leaving the c in) is simply -c. ;) :)
I'm quite not sure about everything you just said, but you seem to be implying that the fourth term is a negative quantity. So instead of considering ct to be the resultant velocity for all four vectors, I guess I should look at how relativity effects normal (flat) geometry. We can start by writing the formula for the distance travelled at a velocity v in time t as vt=d=(x^2+y^2+z^2)^1/2. Next we can determine how relativity effects these quantities. Apparently both distance and time will be effected by the velocity between a source and observer (an observer travelling at this velocity from a starting point or an observer at the starting point), so we get v[t*(1-(v/c)^2)^1/2]=d*(1-(v/c)^2)^1/2=(1-(v/c)^2)^1/2(x^2+y^2+z^2)^1/2. Setting this relativistic distance equal to that for the four vectors of spacetime, we get (x^2+y^2+z^2+j^2)^1/2=(1-(v/c)^2)^1/2(x^2+y^2+z^2)^1/2. Squaring both sides and reducing, we find j^2=-(v/c)^2(x^2+y^2+z^2). Substituting d=vt=(x^2+y^2+z^2), we end up with j^2=-d^4/(ct)^2, so that j=id^2/ct, where i=(-1)^1/2. So the fourth term in this case isn't positive or negative, but complex. Normally this would tell me that the formula is incorrect, since an imaginary number as the result tells us that this coordinate would lie outside of normal geometry, but in the case of relativity, I guess this is kind of the point.