# timespace, a quantum analog to spacetime

#### steveupson

My understanding of science is that it is a process. Skepticism plays an important role, but the doubt should be reserved for the theories, without prejudice. What we are presenting is a block diagram of a theory of everything, and it is based on some previously unknown mathematics that we have stumbled across.

The problem is completely unlike anything else that exists in mathematics. Its solution is an identity that defines a family of functions. Because this single identity defines the behavior of each member of the family of functions, together with the behavior of the entire family, it is unique in mathematics (as far as we have been able to determine.) The identity only exists in three dimensions and it falls apart when all of the angles exist in a single plane. This is not unique, as there are other identities that are associated with spherical trigonometry that also require three dimensions.

Each member of the family of functions presents a representation of direction as a two-dimensional curve. Since area is a scalar value, the area under the curve expresses direction as a quantity or a number, not as a ratio or vector as is the usual method.

3D identity (tridentity) in Euclidean 3-space

Weâ€™ve prepared a proof of the equation, but it has not yet been peer reviewed. If there are any issues with the math, then that would definitely have an impact on our views about the block diagram. But until such time as the math has been proven to be correct or incorrect, we would like an opportunity to argue in support of some new interpretations of the significance of the underlying principles on which Euclidean geometry is based. Until now, direction has never been represented as a mathematical quantity. The ability to do this has some pretty significant implications. We refer to direction as a base quantity, or in other words a quantity from which other derived quantities are based. For example, time and direction can be combined to express frequency.

Due to relativity and the equivalence principle, there has to be some way to reconcile the universe in which direction is a quantity. One of the first things to acknowledge is the concept of scale. At the human scale, direction and distance have a relationship with one another that allows us to calculate the ballistic paths of objects and the orbits of planets. On the astronomical scale, this does not seem to be the case. Distances and directions no longer add up, as is indicated by the wagon wheel rotation of galaxies.

At the quantum scale, direction completely dominates distance. The radii become so tiny that they are almost insignificant, while at the same time, a turn is still a turn. There is no commensurate change of scale for direction. This fact only becomes significant because direction is also a quantity, in addition to being a ratio in orthogonal systems. When we look at direction as a ratio, it naturally changes scale when distance changes scale. This isnâ€™t the case with our observations. Our observations tell us that direction does not stay scaled with distance, or in other words, it is also a quantity in its own right. The math that we are presenting shows how direction can be represented as a quantity. It is only a quantity in three dimensions so there is no equivalent expression that can be made using plane geometry.

When we try to understand a photon under these circumstances, there is a package of directions or orientations that accompany that particle. It has its own system of direction, as does every other particle. These directions that accompany the photon theoretically extend indefinitely in all directions without regard to distance or Lorentz or c. When the photon passes through the double-slit, some of its self-contained direction passes through the alternate opening and reacts with the photon on the other side of the slit in order to create a wave pattern.

As for entanglement and superposition, these can also be explained in a similar manner. Since direction commutes (according to the mathematics), the entangled pair end up traveling in the same direction (according to their own system of directions) and they are also traveling at the same speed. In any other reference frame (any frame that does not include their own self-contained system of direction), the two entangled particles appear to be in separate places in spacetime, but to the entangled pair, they are at one and the same place in timespace. Timespace, if it maintains its symmetry with spacetime, will have its own geometry and its own equations that will be based on direction instead of distance. In this alternative geometry, c will have no impact and time will only relate to periodic events.

Atomic wave function can be explained as several of these local self-contained systems of direction in superposition with one another. There's plenty of observational data available to show whether or not this is the case. The only thing missing has been the hunk of math that will allow for the quantification of direction. This is what we have stumbled across.

One of the basic distinctions that should be made while considering this theory is between the mathematics that exist and the speculation for which there is no existing support. The way that this particular theory lays out, the mathematics are proven (as far as we can tell) and there should no longer be any doubt that direction is also a quantity (in addition to being the vector that we all know and love.) The speculation is all about how this fact would impact the math that underlies scientific observations.

We need help with proving/disproving this theory. Weâ€™re not quite sure how to go about it. The underlying math seems very solid. Any help or criticism will be greatly appreciated.

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#### Maschke

My understanding of science is that it is a process.
True. So is cooking, skydiving, and walking the dog.

... it is based on some previously unknown mathematics that we have stumbled across.
"Crank shields up full strength."

"Aye Cap'n I'm givin' 'er all she's got!"

I don't mean to be inhospitable to your thread. And it is possible that you have indeed stumbled upon some "previously unknown mathematics." Yitang Zhang was working in a Subway sandwich shop when he made an astonishing breakthrough in number theory. I'm just saying the odds are against it.

I might suggest that if you want grizzled veterans of the crank wars to take your post seriously, that you dial back the grandiose claims. Not that they are necessarily false; but rather that they are almost surely indicative of crankery to follow. And I mean almost surely in the technical sense. The odds are zero that you found something new. But it's possible. Like reaching your hand into a bowl filled with the real numbers and pulling out a rational.

The problem is completely unlike anything else that exists in mathematics.
Please stop. Can you at least see, objectively, that even if you are telling the truth -- that you've really found something new -- can you at least see that many readers will stop taking you seriously?

Because this single identity defines the behavior of each member of the family of functions, together with the behavior of the entire family, it is unique in mathematics (as far as we have been able to determine.)
What do you mean by unique in mathematics? Do you mean unique, as in, "There is a unique Abelian group of order 6," or, "There is a unique integer strictly between 4 and 6." Or do you mean something else? Like the idea is somehow unique or ... something.

The identity only exists in three dimensions and it falls apart when all of the angles exist in a single plane. This is not unique, as there are other identities that are associated with spherical trigonometry that also require three dimensions.
Ok. I don't know much spherical trig. Can you just tell us the equation?

Each member of the family of functions presents a representation of direction as a two-dimensional curve. Since area is a scalar value, the area under the curve expresses direction as a quantity or a number, not as a ratio or vector as is the usual method.
I don't know what that means. A direction in n-space is given by a family of colinear vectors regardless of length. Is that what you're doing?

I skipped this part, I hope you'll forgive me.

Weâ€™ve prepared a proof of the equation, but it has not yet been peer reviewed.
That's ok. The peer review system is broken anyway.

If there are any issues with the math, then that would definitely have an impact on our views about the block diagram. But until such time as the math has been proven to be correct or incorrect, we would like an opportunity to argue in support of some new interpretations of the significance of the underlying principles that Euclidean geometry is based on. Until now, direction has never been represented as a mathematical quantity.
That isn't true. Direction is represented as a quantity in math all the time.

The ability to do this has some pretty significant implications. We refer to direction as a base quantity, or in other words a quantity that other derived quantities are based on. For example, time and direction can be combined to express frequency.
This doesn't sound out of the ordinary to me.

Due to relativity and the equivalence principle
Aha. Failure to distinguish math from physics. I see where this is going. Done reading.

At the quantum scale ...
There is no quantum scale in math. It's real numbers all the way down.

ps -- Skimmed your paper. I don't doubt that you have gained some insight into spherical trigonometry. I think if you try to describe that rather than claim this is something brand new in math, you probably have something interesting.

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• 3 people

#### steveupson

That isn't true. Direction is represented as a quantity in math all the time.
Really? And you accuse me of making outrageous claims?

on edit >>> can we please try to avoid the recriminations, the personal attacks, the ad hominem arguments and have a serious discussion of THIS theory and THESE claims? You sound like a hack.

second edit >>> there seems to be some confusion. I mean to say the scalar quantity or magnitude of direction. that should be obvious from the context

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#### steveupson

From the proof:

I. Introduction

There are two general types of trigonometric identities; those that can be expressed in two dimensions and those which need three dimensions. Identities which require three dimensions are limited to spherical trigonometry. The identity discussed here expresses the tangent to a small circle on a sphere, so it doesnâ€™t really fit the definition of spherical trigonometry. Traditionally, spherical trigonometry relies exclusively on great circles and arc segments of great circles. There are no other identities that describe the geometry of small circles on a sphere.

â€œThe origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook Spherical trigonometry for the use of colleges and Schools. This book is now readily available on the web. The only significant developments since then have been the application of vector methods for the derivation of the theorems and the use of computers to carry through lengthy calculations.â€ 

The identities which require three dimensions will be called tridentities in order to distinguish them from all of the other mathematical identities which can be expressed in other forms than exclusively in Euclidean 3-space. An example of a known tridentity is (Chapter XVII (Chapter XIV) â€“ Arcs Drawn to Fixed Points on the Surface of a Sphere (Todhunter)):
$$\displaystyle cosTA^2+cosTB^2+cosTC^2=1$$
Note that the tridentity describes a characteristic of great circle arcs and that great circle arcs donâ€™t occur in two dimensions. This satisfies the requirements of tridentity; an identity that requires three dimensions in order to be expressed. Perhaps another way to view this distinction is that this specific tridentity requires a spherical triangle with three right angles. This can only exist in three dimensions.

The tridentity which is the topic of this paper is different in that it describes the slope of the tangent to a small circle on a sphere. The function must have dihedral angles in 3D in order to be an identity. Therefore, this satisfies the definition in that the tridentity is only valid when a dihedral angle exists.

.......

The new small circle tridentity:

$$\displaystyle (\cos\frac{\phi}{2}\,\sin\frac{\lambda}{2})^2 +(\cot\frac{\phi}{2}\,\cos\alpha)^2=1$$

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#### SDK

This looks super cranky. But at least its not another disproof of Cantor, or a proof of Collatz or FLT. Carry on.

• 3 people

#### topsquark

Math Team
Really? And you accuse me of making outrageous claims?

on edit >>> can we please try to avoid the recriminations, the personal attacks, the ad hominem arguments and have a serious discussion of THIS theory and THESE claims? You sound like a hack.

second edit >>> there seems to be some confusion. I mean to say the scalar quantity or magnitude of direction. that should be obvious from the context
You refer to scalars so I would have assumed you ran into the concept of a vector at some point. Not an outrageous claim at all.

Not to get pedantic but what is the "magnitude" of a "direction?" Do you mean magnitude in a given direction?

I have to admit that I, too, left the conversation when it got around to QM. For example: At the quantum scale, direction completely dominates distance. What the heck does that even mean? I suppose if you wanted to you could define a set of directions for basis vectors in the Hilbert space spanned by the system but what does that have to do with distance? Distance is not typically a quantity that QM works with... QM and distance don't play together well due to the Heisenberg principle.

-Dan

• 1 person

#### steveupson

I don't know what that means. A direction in n-space is given by a family of colinear vectors regardless of length. Is that what you're doing?
There is a method for graphing $$\displaystyle \alpha={\cot}^{-1 }(\cos\upsilon\tan{\sin}^{-1}(\frac{\sin\frac{\lambda}{ 2}}{ \sin\upsilon}))$$ while holding $$\displaystyle \upsilon$$ constant. This method produces a 2D curve. If the curve is normalized (compensated for the difference in size between the radius of the small circle and the radius of the sphere) for all values of $$\displaystyle \upsilon$$, in a manner (conceptually) similar to the method used in order to normalize the unit vector, then you will have a smooth function that produces a curve that varies in contour from the sine curve to a square curve.

Understand that we are not considering direction to be a vector quantity in this new geometry. That requires distance (a separate base quantity) and we don't need distance in order to use this scheme.

#### steveupson

You refer to scalars so I would have assumed you ran into the concept of a vector at some point. Not an outrageous claim at all.

Not to get pedantic but what is the "magnitude" of a "direction?" Do you mean magnitude in a given direction?

I have to admit that I, too, left the conversation when it got around to QM. For example: At the quantum scale, direction completely dominates distance. What the heck does that even mean? I suppose if you wanted to you could define a set of directions for basis vectors in the Hilbert space spanned by the system but what does that have to do with distance? Distance is not typically a quantity that QM works with... QM and distance don't play together well due to the Heisenberg principle.

-Dan
That's the question that should be asked and answered. That's the point of this thread. We claim to have stumbled across a formula for the magnitude of direction in space. If we're wrong, someone should try and show us where the error is. Isn't that the way science works?

If you think about it, you'll admit that something has always been a little hinky when it comes to a turn in physics. They are used from time to time but they are never defined. And we know a half turn is not the same thing as whole turn divided by two. Why is that? We believe that we have cracked that particular code. In three dimensional space there is a magnitude that can be defined that is associated with the relative difference between orientations of two directions.

#### Maschke

on edit >>> can we please try to avoid the recriminations, the personal attacks, the ad hominem arguments and have a serious discussion of THIS theory and THESE claims?
I made no personal attacks and no ad hominem arguments. An ad hominem is an argument "directed against a person rather than the position they are maintaining." I have not done anything of the sort.

I pointed out (correctly) that your post included strong crank indicators. These included claiming you have made a brand new discovery in math; and confusing math with physics. Those are behaviors. Those are things you wrote. I never commented on any aspect of your personhood; only on what you wrote.

#### steveupson

...And it is possible that you have indeed stumbled upon some "previously unknown mathematics."
Exactly, this why we are here.