novaBBS - sci.physics.relativity - Re: Euclidean Relativity

It was nearly 20 years ago that I started posting my critique of Special Relativity. At the time, I introduced the loxodrome spiral as a model for units. It has the all-important property that the effective radius of a loxodrome spiral embedded in the surface of the unit sphere is a function of relative velocity, through a tilt angle. By rotating the tilt angle, the arclength of the spiral is scaled by the secant of the tilt angle, or more familiarly, by γ, the Lorentz factor. For γ = sec(tilt), relative velocity, v = c sin(tilt) = β. At the time, I was trying to assemble a model of relativity that was more intuitive than the mainstream version. I had no grievance with it, other than its inscrutability.

Over time, I have acquired information that changed all that. No big secret.. The Lorentz Transform is a symmetric matrix with real elements. Someone asked me what were the eigenvalues. I had to look it up. But when I got the eigenvalues, of course, the next step is eigenvectors. And when I got them, I realized how wrong special relativity really is. I want to share that information. For years physics has stood in the way of progress that they didn't sanction. There is a whole list of things they have said we can't do. It's possible that they accidentally got some right, but most of them are rubbish, based on lies.

This conclusion is based on eigenvector decomposition. I will go into inordinate detail later, but for now, what matters is that the eigenvectors are the bisectors of the angles between the time and space axes. While the time and space axes are variable in direction, their bisectors are fixed, thanks to the symmetry of the Lorentz matrix. It's a simple argument based on Euclidean geometry. Two lines which are not parallel intersect at a single point. At this point, the two lines form 4 quadrants. The sum of the angles of any two adjacent quadrants is Pi, because they are supplementary angles. If we bisect the quadrants, the sum of the same adjacent pairs is cut in half as well. The angle between the bisectors is therefore Pi/2, independent of the angle between the original axes. But before we even get into any details, we start with two axes that are invariant with respect to relative velocity. Technically, there is only enough information to fix the direction of the eigenvector. Euclidean geometry has more to say about the magnitude.

Without going through the derivation, it is a fact that the eigenvalues of the general Lorentz matrix are a pair of reciprocal exponentials, 1/e^w amd e^w. They are both real. The corresponding lines are the equations of the eigenvectors. The first is ct-r=0, or ct=r. The second is ct+r=0 or ct=-r. For future reference, we note that the two eigenvectors are linear combinations of the time and space vectors, and are embedded in the same plane. The product of the slopes of the two eigenvectors is -1, confirming that the eigenvectors, which are perpendicular where they cross, and indefinitely far away are a rigid grid ala gedanken experiment. More recently, Sir Hermann Bondi introduced k-calculus, a more intuitive approach to relativity. The k in his k-calculus was a "fundamental ratio" according to him. According to its behavior, k is an eigenvalue. So Bondi's book derives special relativity from an abstract ratio. Bondi uses radar measurements to provide unambiguous values. This mirrors eigenvector analysis, because the two eigenvectors have the worldlines of photons. Maybe lidar instead of radar. Point is, distance is the time it takes light to get there. In any case, Bondi did not pursue the eigenvector analysis.

What he missed is that coordinates on an eigenvector axis are actually his radar measurements. We now have two edge vectors that are perpendicular, so we can define the magnitude of the cross-product as the product of the magnitudes of the two edges. Physically, the cross-product represents the area of a rectangle defined by a vertex. In eigenspace, the point projects light rays to each of the axes, which are also light rays. The area defined by the light-rectangle is logically related to the Euclidean coordinates. The coordinates in eigenspace are (Σ‚Δ). These are equivalent to (ct+r,ct-r). Then the cross-product is (ct+r)(ct-r) = c²t²-r² = s², the Einstein Interval. Both Euclidean and eigenspace unit dimension lines are hyperbolas, but not merely rotated. The transform maps the magnitude² of the 1D unit vector to the 2D area vector. The vertex moves from (1,0) to (1,1). The semi-major axis of the Euclidean hyperbola has the same magnitude² as the area of the light-square formed by the vertex of the eigenspace hyperbola. The semi-major axis is not the same length, but the invariant area does not care about the length of the diagonal, or even the shape of the rectangle.

Here is how it affects special relativity. Einstein postulated the invariance of the speed of light with respect to the relative velocity of either its source or its observer. The use of radar techniques like Bondi's gives every point on the light ray grid invariant coordinates. But Einstein also pronounced that time dilates and space contracts physically. In eigenspace, all the information about relative velocity is collected into an eigenvalue, and the eigenvector itself is invariant. So, it isn't physical, it's an illusion. As an illusion, it readily explains how multiple different observers get different measurements of the same events. But this option was suggested to Einstein back in the day, and he rejected it. Instead, we had to give up common sense and rigorous logic, because "That's relativity for you."

It has been pointed out that the sum of two eigenvectors is not generally also an eigenvector. So, they argue that there is no reason to expect invariance from a combination, which both time and space vectors are, and neither one is invariant. But the issue is that in eigenspace, we can factor out 100% of the effects of relative velocity, leaving both eigenvectors intact. The effect that Einstein called physical is an illusion. Whatever his rationale, he apparently didn't test it in eigenspace. Since Minkowski did not invent his own spacetime until several years after Einstein first published relativity, Einstein had to operate under the paradigm of Euclidean geometry. In Euclidean geometry, he contradicted himself. So, Minkowski custom engineered a geometry in which the contradiction vanished. Over the course of subsequent posts, Euclidean geometry will be used to explain special relativity. In Euclidean geometry, the Pythagorean identity holds, along with other properties.

We have long distinguished time and space, both informally and formally. So there was significant resistance to the idea that the Lorentz Transform blended time and space into each other. So, there is some logical reason to treat time and space the same way (but still different). In the Lorentz boost, the equation that defines a unit of time or space in a relatively moving frame is the same for both the unit of time and the unit of distance. And then we learn that gravitational time dilation is the same as relativistic time dilation. Except that gravitational time dilation only affects time, not space. So, the two units that are treated identically in special relativity are treated quite differently in general relativity. I propose that the difference is a result of the symmetry of the geometry that is associated with the effect. The eigenspace geometry uses symmetry axes, in which each one has the same proportions of time and space. A perfect reason for the two units to be handled identically. They go through the transform together. The Einstein conditions of simultaneity and coincidence selectively zero out one of the components of the symmetry combination. For an effect to be selectively applied to time, the implication is that the unit axes are time and space, rather than symmetry axes. The point is, if Minkowski geometry is best for general relativity, it would appear to not be best for special relativity.

The fact that Minkowski geometry is embedded in the minds of mainstream physics as being necessary for special relativity does not apply here. Euclidean and Minkowski are equivalent. The fact that you can represent special relativity by either one is unremarkable. For example, it is possible to model a Cartesian grid in spherical coordinates, or a spherical grid with rectangular coordinates. Just because Minkowski geometry is the mainstream model does not make it the best choice. Physics does not apply special relativity to low velocity situations, because the effect of the relativistic terms is unmeasurable. If Minkowski geometry is necessary for general relativity, why should we use it for special relativity if Euclidean geometry is sufficient? Another paper I ran across extended Euclidean geometry to general relativity. And, even if that's doable, it isn't necessarily the best fit. Stay tuned for more examples.

On Friday, October 15, 2021 at 8:27:24 PM UTC-7, tgca...@gmail.com wrote:
> It was nearly 20 years ago that I started posting my critique of Special Relativity. At the time, I introduced the loxodrome spiral as a model for units. It has the all-important property that the effective radius of a loxodrome spiral embedded in the surface of the unit sphere is a function of relative velocity, through a tilt angle. By rotating the tilt angle, the arclength of the spiral is scaled by the secant of the tilt angle, or more familiarly, by γ, the Lorentz factor. For γ = sec(tilt), relative velocity, v = c sin(tilt) = β. At the time, I was trying to assemble a model of relativity that was more intuitive than the mainstream version.. I had no grievance with it, other than its inscrutability.
>
> Over time, I have acquired information that changed all that. No big secret. The Lorentz Transform is a symmetric matrix with real elements. Someone asked me what were the eigenvalues. I had to look it up. But when I got the eigenvalues, of course, the next step is eigenvectors. And when I got them, I realized how wrong special relativity really is. I want to share that information. For years physics has stood in the way of progress that they didn't sanction. There is a whole list of things they have said we can't do. It's possible that they accidentally got some right, but most of them are rubbish, based on lies.
>
> This conclusion is based on eigenvector decomposition. I will go into inordinate detail later, but for now, what matters is that the eigenvectors are the bisectors of the angles between the time and space axes. While the time and space axes are variable in direction, their bisectors are fixed, thanks to the symmetry of the Lorentz matrix. It's a simple argument based on Euclidean geometry. Two lines which are not parallel intersect at a single point. At this point, the two lines form 4 quadrants. The sum of the angles of any two adjacent quadrants is Pi, because they are supplementary angles.. If we bisect the quadrants, the sum of the same adjacent pairs is cut in half as well. The angle between the bisectors is therefore Pi/2, independent of the angle between the original axes. But before we even get into any details, we start with two axes that are invariant with respect to relative velocity. Technically, there is only enough information to fix the direction of the eigenvector. Euclidean geometry has more to say about the magnitude.
>
> Without going through the derivation, it is a fact that the eigenvalues of the general Lorentz matrix are a pair of reciprocal exponentials, 1/e^w amd e^w. They are both real. The corresponding lines are the equations of the eigenvectors. The first is ct-r=0, or ct=r. The second is ct+r=0 or ct=-r. For future reference, we note that the two eigenvectors are linear combinations of the time and space vectors, and are embedded in the same plane. The product of the slopes of the two eigenvectors is -1, confirming that the eigenvectors, which are perpendicular where they cross, and indefinitely far away are a rigid grid ala gedanken experiment. More recently, Sir Hermann Bondi introduced k-calculus, a more intuitive approach to relativity. The k in his k-calculus was a "fundamental ratio" according to him. According to its behavior, k is an eigenvalue. So Bondi's book derives special relativity from an abstract ratio. Bondi uses radar measurements to provide unambiguous values. This mirrors eigenvector analysis, because the two eigenvectors have the worldlines of photons. Maybe lidar instead of radar. Point is, distance is the time it takes light to get there. In any case, Bondi did not pursue the eigenvector analysis.
>
> What he missed is that coordinates on an eigenvector axis are actually his radar measurements. We now have two edge vectors that are perpendicular, so we can define the magnitude of the cross-product as the product of the magnitudes of the two edges. Physically, the cross-product represents the area of a rectangle defined by a vertex. In eigenspace, the point projects light rays to each of the axes, which are also light rays. The area defined by the light-rectangle is logically related to the Euclidean coordinates. The coordinates in eigenspace are (Σ‚Δ). These are equivalent to (ct+r,ct-r). Then the cross-product is (ct+r)(ct-r) = c²t²-r² = s², the Einstein Interval. Both Euclidean and eigenspace unit dimension lines are hyperbolas, but not merely rotated. The transform maps the magnitude² of the 1D unit vector to the 2D area vector. The vertex moves from (1,0) to (1,1). The semi-major axis of the Euclidean hyperbola has the same magnitude² as the area of the light-square formed by the vertex of the eigenspace hyperbola. The semi-major axis is not the same length, but the invariant area does not care about the length of the diagonal, or even the shape of the rectangle.
>
> Here is how it affects special relativity. Einstein postulated the invariance of the speed of light with respect to the relative velocity of either its source or its observer. The use of radar techniques like Bondi's gives every point on the light ray grid invariant coordinates. But Einstein also pronounced that time dilates and space contracts physically. In eigenspace, all the information about relative velocity is collected into an eigenvalue, and the eigenvector itself is invariant. So, it isn't physical, it's an illusion. As an illusion, it readily explains how multiple different observers get different measurements of the same events. But this option was suggested to Einstein back in the day, and he rejected it. Instead, we had to give up common sense and rigorous logic, because "That's relativity for you."
>
> It has been pointed out that the sum of two eigenvectors is not generally also an eigenvector. So, they argue that there is no reason to expect invariance from a combination, which both time and space vectors are, and neither one is invariant. But the issue is that in eigenspace, we can factor out 100% of the effects of relative velocity, leaving both eigenvectors intact.. The effect that Einstein called physical is an illusion. Whatever his rationale, he apparently didn't test it in eigenspace. Since Minkowski did not invent his own spacetime until several years after Einstein first published relativity, Einstein had to operate under the paradigm of Euclidean geometry. In Euclidean geometry, he contradicted himself. So, Minkowski custom engineered a geometry in which the contradiction vanished. Over the course of subsequent posts, Euclidean geometry will be used to explain special relativity. In Euclidean geometry, the Pythagorean identity holds, along with other properties.
>
> We have long distinguished time and space, both informally and formally. So there was significant resistance to the idea that the Lorentz Transform blended time and space into each other. So, there is some logical reason to treat time and space the same way (but still different). In the Lorentz boost, the equation that defines a unit of time or space in a relatively moving frame is the same for both the unit of time and the unit of distance. And then we learn that gravitational time dilation is the same as relativistic time dilation. Except that gravitational time dilation only affects time, not space. So, the two units that are treated identically in special relativity are treated quite differently in general relativity. I propose that the difference is a result of the symmetry of the geometry that is associated with the effect. The eigenspace geometry uses symmetry axes, in which each one has the same proportions of time and space. A perfect reason for the two units to be handled identically. They go through the transform together.. The Einstein conditions of simultaneity and coincidence selectively zero out one of the components of the symmetry combination. For an effect to be selectively applied to time, the implication is that the unit axes are time and space, rather than symmetry axes. The point is, if Minkowski geometry is best for general relativity, it would appear to not be best for special relativity.
>
> The fact that Minkowski geometry is embedded in the minds of mainstream physics as being necessary for special relativity does not apply here. Euclidean and Minkowski are equivalent. The fact that you can represent special relativity by either one is unremarkable. For example, it is possible to model a Cartesian grid in spherical coordinates, or a spherical grid with rectangular coordinates. Just because Minkowski geometry is the mainstream model does not make it the best choice. Physics does not apply special relativity to low velocity situations, because the effect of the relativistic terms is unmeasurable. If Minkowski geometry is necessary for general relativity, why should we use it for special relativity if Euclidean geometry is sufficient? Another paper I ran across extended Euclidean geometry to general relativity. And, even if that's doable, it isn't necessarily the best fit. Stay tuned for more examples.

There is the derivation then for what is contrived.

What this looks like to me is "light falls away, these light
rays are what fall out in whatever changes occur, what
that it's not kinetic energy's bound to them, but only the
influence on the geodesy, which works out the image, and
also, any what would change the image, as what in real time,
the image as "lensing over the focus", as a convention like
"right hand convention is what we say and let's not talk about
phase, in this convention, as what besides the 'linear' part of
this image, changes in any way with respect to the travel time
of light, and changes in imagery, how both mass-energy equivalence
or Einstein and GR and if so GTR, and, SR and here not so STR as where
this is "General Relativity" and "Special Relativity", as defined under
mass-energy equivalence in the local _and_ global_ frame, and the
light imagery that is local everywhere or for where STR is instead a
special case for electromagnetism [globally] that is a neat simpler theory,
with a GR for the usually kinetic, GTR where it is so is as well either
usually zero or usually what results GR, and, SR locally".

Click here to read the complete article

Subject	Author
Euclidean Relativity	Tom Capizzi
Old cretin? New cretin	Dono.
Re: Old cretin? New cretin	Tom Capizzi
Re: Old cretin? New cretin	carl eto
Re: Old cretin? New cretin	Dono.
Re: Old cretin? New cretin	carl eto
Re: Old cretin? New cretin	Tom Capizzi
Re: Old cretin? New cretin	Dono.
Re: Old cretin? New cretin	Tom Capizzi
Re: Euclidean Relativity	Ross A. Finlayson

Sex is like air. It's only a big deal if you can't get any.

tech / sci.physics.relativity / Re: Euclidean Relativity