This may seem a bit of a stretch, but the focus of this series is the eigenvector decomposition in Euclidean geometry. The symmetry transform which generates the eigenvector coordinates from spacetime coordinates is behind each example.

In the previous post, we discovered that the 45 degree spiral through the origin on the equator has points all of which relate their latitude and longitude increments according to the gudermannian function of longitude arc. Unlike spherical triangles using great circle arcs, loxodromes have cosine and sine projections which correspond to latitude and longitude arcs. The spiral has an arc length proportional to πR sec(θ). In the case of the 45 degree spiral, sec(θ) = √2. Equal projections of sin(45) = cos(45) = √2/2, times √2, results in units of πR, the arclength of a meridian. This means that a 45 degree spiral has equal arclengths in latitude and longitude. And in superposition with various mirror images, the sum of two diagonal spirals is the meridian arc, and the difference between the same spirals is a latitude arc of the same length at the equator. The diagonal spiral is the inverse Mercator projection of an eigenvector of the Lorentz matrix.
Here's the interesting thing. The meridians map to vertical lines on the map, and the parallels are horizontal lines. Their spiral sum and spiral difference are the two 45 degree diagonal spirals. On the map, they are straight lines that bisect the parallel and meridian axes. If we assume that the scale of the map is adjusted to be 1:1 on the equator, we can ignore the drawing of the map and just focus on the linear measurement. If the scaling of the spiral arcs is 1:1 at the equator, then the actual distance (on the map) to the origin is the coordinate. And the axes on the map follow the same transform rule as the spirals on the globe. But in the transform from spiral to map axes, there is no change of velocity, so it is not a Lorentz Transform. Nor is there any change of velocity for the transform from rectangular to diagonal coordinate systems.

It is the same transform I use to switch from (ct,r) coordinates to (Σ,Δ) coordinates. A critic recognized the transform and told me it was well-known as the Minkowski light-cone transform. Then he detailed how the relationships I cited were incorrect, because the axes of the light-cone were not Minkowski-perpendicular. All well and good, but totally irrelevant. Because I was following the trail of real eigenvectors, not light-cone vectors that can't be real eigenvectors. For one thing, the addition or subtraction of scaled copies of any two vectors results in a composite which is in the plane of the initial vectors, and is just another vector in the plane. Real eigenvectors do not form 3 dimensional structures like light-cones. I have also learned that this sum and difference algorithm is at the heart of a 6-cyclic set of functions, possibly related to the Standard Model. In any case, he made several mistaken assumptions. First, because I was using a symmetry transform to produce real eigenvectors, he assumed I was using Minkowski geometry, and he assumed I was talking about light-cones. This transform is the smallest order set of a recursively defined spectral analysis by square wave. It does not belong to Minkowski, it belongs to Walsh, as a neutral observer, whose spectral analysis sorts information by symmetries. It relies heavily on the concept of holographic superposition.

According to Wikipedia:
https://proofwiki.org/wiki/Groups_of_Order_6
I don't mean to comment on the proof itself, but the article refers to the symmetry group of the equilateral triangle. I reproduce the Cayley table here, because I am going to reveal a hidden symmetry. Since the order of the rows and columns has no effect on the validity of the table, I will now perform some elementary row and column swaps:

e p q r s t
q e p t r s
p q e s t r
r t s e q p
s r t p e q
t s r q p e

e p q t s r
q e p s r t
p q e r t s
r t s p q e
s r t q e p
t s r e p q

e p q t s r
q e p s r t
p q e r t s
t s r e p q
s r t q e p
r t s p q e

e t p s q r
t e s p r q
q s e r p t
s q r e t p
p r q t e s
r p t q s e

0 1 2 3 4 5
1 0 3 2 5 4
4 3 0 5 2 1
3 4 5 0 1 2
2 5 4 1 0 3
5 2 1 4 3 0

These are the 6-cyclic maps, already in modulo arithmetic. This is a Cayley table for column index – row index, pre modulo arithmetic:
0 1 2 3 4 5
-1 0 1 2 3 4
-2-1 0 1 2 3
-3-2-1 0 1 2
-4-3-2-1 0 1
-5-4-3-2-1 0

Caley table for column index + row index, pre modulo:
0 1 2 3 4 5
1 2 3 4 5 6
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 A

If we use the parity table as a mask:
E O E O E O
O E O E O E
E O E O E O
O E O E O E
E O E O E O
O E O E O E

The index of the product of a row and column pair uses the first table if it is an even parity cell, and the second table if it's odd:
0 O 2 O 4 O
O 0 O 2 O 4
-2 O 0 O 2 O
O-2 O 0 O 2
-4 O-2 O 0 O
O-4 O-2 O 0

0 1 2 3 4 5
1 0 3 2 5 4
-2 3 0 5 2 1
3-2 5 0 1 2
-4 5-2 1 0 3
5-4 1-2 3 0

Modulo 6:
0 1 2 3 4 5
1 0 3 2 5 4
4 3 0 5 2 1
3 4 5 0 1 2
2 5 4 1 0 3
5 2 1 4 3 0

And this is the Cayley table of the symmetry group of the equilateral triangle. Modulo arithmetic is cyclic, so the index can be viewed as the number of increments in a circle. Then the entire table can be condensed into a simple equation. Let the column index be i, the row index be j, and the product have an index of k. Then, k(π/3) =
((i-(-1)^(i+j)*j)mod6)(π/3)
While the Cayley table of the symmetry group of the equilateral triangle represents the product of 6 discrete states, the index can be taken as a rotation angle that is actually a smooth linear function. While the states of the symmetry group are distinct, they can be viewed as uniformly spaced samples of a continuous angular argument. The point is, many varied groups obey this same table, from the group of 6 trig functions and the cross-ratio to the group of 3x3 permutation matrices and the Standard Model. This is of special interest, because the 6 discrete states can be incrementally transformed into each other. They are like whole numbers on the real number line. The Cayley table describes the composition of the whole numbers while the analog equation does the same thing, but defines every fractional point between the whole numbers. And it reduces the 6*6 pairs of discrete products to a single continuous analog formula, that is a function of two perpendicular, cyclic indices.
The trig function group is quite interesting. The generic members of the group are λ,λ/(λ-1),(λ-1)/λ,1/λ,1/(1-λ),1-λ. These map to the trig functions:
λ = cos²(θ) = sech²(w)
λ/(λ-1) =-cot²(θ)=-csch²(w)
1/(1-λ)=csc²(θ)=coth²(w)
1/λ=sec²(θ)=cosh²(w)
(λ-1)/λ=-tan²(θ)=-sinh²(w)
(1-λ)=sin²(θ)=tanh²(w)

The expressions in λ can be written as a set of bilinear transforms:
(1*λ+0)/(0*λ+1), (1*λ+0)/(1*λ-1), (0*λ+1)/(-1*λ+1),
(0*λ+1)/(1*λ+0), (1*λ-1)/(1*λ+0), (-1*λ+1)/(0*λ+1)

Scaling numerator and denominator does not change the transform:
(1*λ+0)/(0*λ+1), (-1*λ+0)/(-1*λ+1), (0*λ-1)/(1*λ-1),
(0*λ+1)/(1*λ+0), (-1*λ+1)/(-1*λ+0), (1*λ-1)/(0*λ-1)

Now, if we list the coefficients:
1 0 0 1
-1 0-1 1
0-1 1-1
0 1 1 0
-1 1-1 0
1-1 0-1

If we list the 6 equally spaced samples of the analog function, 2/√3 sin(k(π/3)), and shift each row by +π/3:
0 1 1 0-1-1
1 1 0-1-1 0
1 0-1-1 0 1
0-1-1 0 1 1
-1-1 0 1 1 0
-1 0 1 1 0-1

Finally, if we leave the 1s as is and swap the entries for 0 and -1, we get:
-1 1 1-1 0 0
1 1-1 0 0-1
1-1 0 0-1 1
-1 0 0-1 1 1
0 0-1 1 1-1
0-1 1 1-1 0

Comparing these columns to the coefficients above, we see that the 1st coefficient is the 3rd column, 2nd coefficient is the 5th column, 3rd coefficient is the 6th column and the last coefficient is the 2nd column. These are phase offsets of +2π/3, -2π/3, -π/3 and π/3. All 4 coefficients can be described by a single sequence with these phase shifts.
The general bilinear transform has 4 coefficients, 3 of which can be chosen independently. If we set 3 initial conditions, we should be able to find a unique bilinear transform to generate the three specific cases. But the bilinear transform is valid everywhere between as well.
So, if the transform is (L*x+I)/(J*x+K), and the three conditions are (L*1+I)/(J*1+K) = 1, (L*0+I)/(J*0+K) = -1 and (L*-1+I)/(J*-1+K) = 0, we find L+I=J+K, I=-K and I-L=0. Then I=L, K=-L and 2L=J-L or J=3L.. So, in terms of the common factor, L, the coefficients of a bilinear transform that maps the sine function to the coefficients, I=L, J=3L and K=-L. Since L is arbitrary, let it be 1, then I=1, J=3 and K=-1:
(1*x+1)/(3*x-1)
as a check, when x=1, the transform is 2/2=1. When x=0,
1/-1=-1 and when x=-1, 0/-4=0. The bilinear transform does map the coefficients properly, so we can now express the entire 6x6 table of bilinear transforms operating on each other into a single analog equation. The equation takes any input and projects 1 of 6 trig functions of it, depending on a master rotation angle. The dummy variable, x, is the analog function 2/√3 sin(φ+k(π/3)). The complete bilinear transform of the coefficients is:
(2/√3 sin(φ+k(π/3))+1)/(2√3 sin(φ+k(π/3))-1)

We can trim the leading coefficient as follows:
(sin(φ+k(π/3))+√3/2)/(3*sin(φ+k(π/3))-√3/2)

Then the 4 coefficients of the 6-cyclic bilinear transform are:
(sin(φ+2π/3)+√3/2)/(3*sin(φ+2π/3)-√3/2)
(sin(φ-2π/3)+√3/2)/(3*sin(φ-2π/3)-√3/2)
(sin(φ-π/3)+√3/2)/(3*sin(φ-π/3)-√3/2)
(sin(φ+π/3)+√3/2)/(3*sin(φ+π/3)-√3/2)

φ is a linear, continuous central angle. When it takes on the value of 1 of the 6 indices, kπ/3, the resulting set of coefficients is the bilinear transform for that trig projection. Since k is defined by another analog equation, this one transform embodies the entire 6x6 Cayley table, as an analog function of a real k. Admittedly, this is a somewhat convoluted derivation. I'm not sure that the bilinear function has a physical meaning, but the point is that the discrete states of the 6 group members are continuously deformable into each other. This has implications for the Standard Model. Not only does each state operate on every other state to produce another member of the group, but they all continuously deform into each other. The precise nature of the transform is not as significant as the fact that these discrete states are uniformly spaced samples of a 6-cyclic analog function. To recreate the 6 discrete states, we merely replace φ by kπ/3, with k = {0,1,2,3,4,5}. The bilinear transform coefficients become:
(sin((k+2)π/3)+√3/2)/(3*sin((k+2)π/3)-√3/2)
(sin((k-2)π/3)+√3/2)/(3*sin((k-2)π/3)-√3/2)
(sin((k-1)π/3)+√3/2)/(3*sin((k-1)π/3)-√3/2)
(sin((k+1)π/3)+√3/2)/(3*sin((k+1)π/3)-√3/2)

The reader is invited to confirm that each value of k corresponds to the same set of coefficients as listed above mapping a bilinear transform to a trig projection. All 6 of both sets of trig functions are driven by the same argument and that argument is either the rapidity or its gudermannian, depending on whether we want the set of hyperbolic functions or the set of circular functions. This table applies to a wide variety of 6-cyclic entities. For example, it is the same table that defines the matrix product of the 6 3x3 permutation matrices as well:

│1 0 0│ │1 0 0│ │0 1 0│ │0 1 0│ │0 0 1│ │0 0 1│
│0 1 0│ │0 0 1│ │0 0 1│ │1 0 0│ │1 0 0│ │0 1 0│
│0 0 1│ │0 1 0│ │1 0 0│ │0 0 1│ │0 1 0│ │1 0 0│

│1 0 0│ │1 0 0│ │0 1 0│ │0 1 0│ │0 0 1│ │0 0 1│
│0 0 1│ │0 1 0│ │1 0 0│ │0 0 1│ │0 1 0│ │1 0 0│
│0 1 0│ │0 0 1│ │0 0 1│ │1 0 0│ │1 0 0│ │0 1 0│

│0 0 1│ │0 1 0│ │1 0 0│ │0 0 1│ │0 1 0│ │1 0 0│
│1 0 0│ │1 0 0│ │0 1 0│ │0 1 0│ │0 0 1│ │0 0 1│
│0 1 0│ │0 0 1│ │0 0 1│ │1 0 0│ │1 0 0│ │0 1 0│

│0 1 0│ │0 0 1│ │0 0 1│ │1 0 0│ │1 0 0│ │0 1 0│
│1 0 0│ │1 0 0│ │0 1 0│ │0 1 0│ │0 0 1│ │0 0 1│
│0 0 1│ │0 1 0│ │1 0 0│ │0 0 1│ │0 1 0│ │1 0 0│

│0 1 0│ │0 0 1│ │0 0 1│ │1 0 0│ │1 0 0│ │0 1 0│
│0 0 1│ │0 1 0│ │1 0 0│ │0 0 1│ │0 1 0│ │1 0 0│
│1 0 0│ │1 0 0│ │0 1 0│ │0 1 0│ │0 0 1│ │0 0 1│

│0 0 1│ │0 1 0│ │1 0 0│ │0 0 1│ │0 1 0│ │1 0 0│
│0 1 0│ │0 0 1│ │0 0 1│ │1 0 0│ │1 0 0│ │0 1 0│
│1 0 0│ │1 0 0│ │0 1 0│ │0 1 0│ │0 0 1│ │0 0 1│

Again, the reader is invited to verify that the matrix multiplication follows the same Cayley table. Note that the determinant matches the parity of the grid position.
The 4 coefficients of the bilinear transform have an isomorphism with 2x2 matrices:
│ L I │ │ │ 1 0│ │-1 0│ │0 -1│ │ 0 1│ │-1 1│ │1 -1│
│ J K│ │ │ 0 1│ │-1 1│ │1 -1│ │ 1 0│ │-1 0│ │0 -1│
────┼──────────────────────────────
│ 1 0│ │ │ 1 0│ │-1 0│ │0 -1│ │ 0 1│ │-1 1│ │1 -1│
│ 0 1│ │ │ 0 1│ │-1 1│ │1 -1│ │ 1 0│ │-1 0│ │0 -1│
────┼──────────────────────────────
│-1 0│ │ │-1 0│ │ 1 0│ │ 0 1│ │0 -1│ │1 -1│ │-1 1│
│-1 1│ │ │-1 1│ │ 0 1│ │ 1 0│ │1 -1│ │0 -1│ │-1 0│
────┼──────────────────────────────
│-1 1│ │ │-1 1│ │ 0 1│ │ 1 0│ │1 -1│ │0 -1│ │-1 0│
│-1 0│ │ │-1 0│ │ 1 0│ │ 0 1│ │0 -1│ │1 -1│ │-1 1│
────┼──────────────────────────────
│ 0 1│ │ │ 0 1│ │-1 1│ │1 -1│ │ 1 0│ │-1 0│ │0 -1│
│ 1 0│ │ │ 1 0│ │-1 0│ │0 -1│ │ 0 1│ │-1 1│ │1 -1│
────┼──────────────────────────────
│0 -1│ │ │0 -1│ │1 -1│ │-1 1│ │-1 0│ │ 1 0│ │ 0 1│
│1 -1│ │ │1 -1│ │0 -1│ │-1 0│ │-1 1│ │ 0 1│ │ 1 0│
────┼──────────────────────────────
│1 -1│ │ │1 -1│ │0 -1│ │-1 0│ │-1 1│ │ 0 1│ │ 1 0│
│0 -1│ │ │0 -1│ │1 -1│ │-1 1│ │-1 0│ │ 1 0│ │ 0 1│
LK-IJ = ±1

So far, we have explored 1, 2, 3 and 4 dimensional structures that all obey the same 6-cyclic Cayley table. The next one is a combination. The cross-ratio applies to the distances between 4 collinear points. In a vector space, these lengths are the vector difference between the coordinates of the points in n-dimensions. This example uses scalars. After all, scalars are just points on the real number line, and as such, they are collinear. But the length between them is just the difference of the two scalar endpoints. Given 4 scalars, LIJK, the cross-ratio is (L-I)(J-K)/(L-J)(I-K) = p. With 4 scalars, there are 4! = 24 permutations. The effect this has on the cross-ratio depends on the permutation. Out of 24 combinations, there are 6 different functions of p, and each one has 4 copies that are invariant to the permutation. The 6 functions are the same 6 that began this section, and the 4 copies are the symmetrical 4-phase permutations:
ILKJ: (I-L)(K-J)/(I-K)(L-J) = (L-I)(J-K)/(L-J)(I-K) = p
JKLI: (J-K)(L-I)/(J-L)(K-I) = (L-I)(J-K)/(L-J)(I-K) = p
KJIL: (K-J)(I-L)/(K-I)(J-L) = (L-I)(J-K)/(L-J)(I-K) = p

The set of 3! permutations are:
LIJK: (L-I)(J-K)/(L-J)(I-K) = P
LIKJ: (L-I)(K-J)/(L-K)(I-J) = P/(P-1)
LKIJ: (L-K)(I-J)/(L-I)(K-J) = (P-1)/P
LJIK: (L-J)(I-K)/(L-I)(J-K) = 1/P
LJKI: (L-J)(K-I)/(L-K)(J-I) = 1/(1-P)
LKJI: (L-K)(J-I)/(L-J)(K-I) = 1-p

And each of these is invariant with respect to the 4 phase permutations. The point is, all of these different structures obey the same 6-cyclic Cayley table, which is derived from the symmetry transform that also generates the eigenvectors of Euclidean geometry, and the light-cone axes of Minkowski geometry. It is incorrect to identify this transform AS the light-cone transform, since that is only one of many applications. And further, since the Cayley table can be condensed into a single equation in two independent variables, all of the different structures do so as well.
I included this section of math to show how powerful the 2 dimensional symmetry transform is. Later, this symmetry algorithm will be expanded by powers of 2. To map the entire set of isotopes in units of quarks requires 18 powers of 2. I can only imagine what it might reveal about hidden symmetries. In any case, the universe in which all physics takes place is Euclidean. To paraphrase Noether, behind every conservation law is a symmetry. If we examine symmetry in a Euclidean universe, we could conceivably discover new conservation laws. When I brought up Euclidean geometry in another forum I was told it was off-topic in a thread about relativity. By showing that anything remotely related to relativity fits in Euclidean geometry, I hope to disprove their hypothesis.

On Saturday, October 23, 2021 at 7:21:23 PM UTC-7, tgca...@gmail.com wrote:
> A critic recognized the transform and told me it was well-known as the Minkowski
> light-cone transform. Then he detailed how the relationships I cited were incorrect...

Right.

> but I was following the trail of real eigenvectors, not light-cone vectors that can't be
> real eigenvectors.

Are you claiming that the lightlike eigenvectors of the 1+1 dimensional Lorentz transformation discussed in the other thread -- that you ran away from -- are not "real eigenvectors"?

Your messages are getting progressively spammier and spammier, as the ratio of total number of characters divided by the number having anything to do with relativity approaches infinity.

> When I brought up [lunatic ramblings referring to] Euclidean geometry in another
> forum I was told it was off-topic in a thread about relativity.

Imagine that.

> By showing that anything remotely related to relativity fits in Euclidean geometry,
> I hope to disprove their hypothesis.

Look, the space of special relativity is trivially Euclidean (happy?), whereas the relations of 3+1 dimensional space-time are not. And none of this comes close to hinting at any explanation of your claim that special relativity (and/or its premesis) is wrong.

Is there any possibility that you will ever get to the point? By all appearances, you don't seem to have a point at all.

On Saturday, October 23, 2021 at 7:21:23 PM UTC-7, tgca...@gmail.com wrote:
> snip utter insanity<

OK,

So, aside from being totally insane about "proving" relativity wrong you also suffer from a very severe case of graphomania. You need to seek medical help, pronto. On the other hand, can you also prove general relativity wrong? You are pretty much done with special relativity.

On Sunday, October 24, 2021 at 12:34:41 AM UTC-4, Dono. wrote:
> On Saturday, October 23, 2021 at 7:21:23 PM UTC-7, tgca...@gmail.com wrote:
> > snip utter insanity<
>
> OK,
>
> So, aside from being totally insane about "proving" relativity wrong you also suffer from a very severe case of graphomania. You need to seek medical help, pronto. On the other hand, can you also prove general relativity wrong? You are pretty much done with special relativity.

Sorry, I gave up feeding trolls.

On Sunday, October 24, 2021 at 7:27:40 AM UTC-7, tgca...@gmail.com wrote:
> The eigenvalues have not been shown in any of the other replies.

It is self-evident and trivial (in units with c=1) that the Lorentz transformation maps lightlines to lightlines, meaning the eigenvectors of the Lorentz transformation (in one space dimension, standard form) are (1,+-1), corresponding to the lightlines in the two directions (i.e., x=t and x=-t), and of course the corresponding eigenvalues are the Doppler factors sqrt[(1+-v)/(1-+v)]. This is high school stuff.

> and its determinant is (γ-λ)²-(βγ)² = 0. This expands to...

Well, there's no reason to expand it and solve the quadratic, because this equation immediately gives (γ-λ)²=(βγ)² and therefore we have γ-λ = +-βγ and hence λ=γ(1-+β), which of course is the approaching and receding Doppler factors sqrt[(1+-β)/(1-+β)].

> There you have it. Eigenvalue, 1/e^w, eigenvector, ct=r. Or, eigenvalue, e^w, eigenvector, ct=-r.
> These are the diagonals of the spacetime plane.

Well, you're using "r" instead of "x", which is slightly weird, but yes, the intervals with invariant "direction" in the x,t plane (i.e., the eigenvectors) are the lightlines, t=x and t=-x. This is self-evident and trivial.

So, I ask again, what's your point? Remember, in your very first post you said "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." Okay, so like every high school student you now know the eigenvalues and eigenvectors of the Lorentz transformation, but how does this support your claim that special relativity is "wrong"? And how do these elementary trivialities of special relativity constitute a "Euclidean relativity" "alternative" to special relativity?

On Thursday, October 21, 2021 at 8:21:49 PM UTC-7, tgca...@gmail.com wrote:
> The object has a single length at rest. Everything else is an illusion.

No, the object does indeed have a specific spatial length L in terms of the inertia-based coordinates in which it is at rest, but it also has spatial length L*sqrt(1-v^2/c^2) in terms of the inertia-based coordinate system in which it is moving (along its length) at the speed v. Neither of these is an illusion. They are both objectively verifiable facts.

> You simply cannot measure distance unless your relative velocity is 0.

That's not true at all. Conceptually we can construct a rectangular grid of standard rulers at rfest in any given frame with standard clocks located at each node, and we can synchronize the clocks inertially, meaning we synchronize them in such a way that inertia is isotropic in terms of these measures of position and time. This is an inertia-based coordinate system. An object moving through this grid has the speed dx/dt in terms of the x,t coordinates of the grid and clocks, and its spatial length is -- by definition -- the difference between the x coordinates of the ends of the object at equal values of the t coordinate. The object need not have 0 velocity relative to the grid. Space and time coordinate systems would not be very useful if, as you imagine, they applied only to stationary objects!

> And then you attribute the contracted length to the motion of the object's
> coordinate system.

That has no relation to anything I've said. First, I explained that, already in the 1880's it was known that the equi-potential surfaces around a moving charge contract by the factor sqrt(1-v^2/c^2) in the direction of motion.. Second, I clarified for you the distinction between passive and active transformations. Yes, if an object in a single state of motion is described in terms of two different systems of coordinates, the object is (obviously) not undergoing any physical change, and the difference in descriptions is due to the difference in coordinate systems... which by itself is not physically significant. The reason we attribute physical significance to Lorentz transformations is because they match the effects of active transformations, i.e., the descriptions of a solid object (at equilibrium, after modding out the time translation phase shifts, etc.) in two different states of motion in terms of a single coordinate system. This is obviously undergoing a physical change (e.g., the phase relations between the ends of the object change, even in Born rigid motion), but the configurations are congruent because the laws of physics are locally Lorentz invariant.

> Relativity says it has exactly the same contracted length when it is NOT moving, and
> it is the observer who is.

There is no absolute distinction between being "moving" or "stationary", but yes, if you start with two solid rods, A and B, each of spatial length L and initially at rest in S, each of these rods has spatial length L*sqrt(1-v^2/c^2) in terms of an inertial coordinate system S' moving at speed v in terms of S. Now, if you gently accelerate rod B until it is at rest in S', the spatial length of A in S is still L, and the spatial length of B in S' is now L, and the spatial lengths of A in terms of S' and of B in terms of S are both L*sqrt(1-v^2/c^2). Do you disagree with any of this?

> You are so convinced that only the approved version of relativity can
> possibly be correct, you simply reject any alternative out of hand.

What "alternative" are you referring to? You said special relativity is wrong and entails contradictions, but when I press you to identify any contradiction or what is wrong, you just run away. Phrases like "approved version" are just silly. Special relativity consists of local Lorentz invariance, which is the proposition that the equations of physics take the same homogeneous and isotropic form in terms of systems of coordinates related by Lorentz transformations. You've agreed to this, so I don't know what "alternative" you are talking about, nor why you continue to insist that special relativity is wrong and/or entails contradictions. Can you clarify?

> If my proposition is false, it will inevitably lead to a contradiction.
> If you're correct, show me that contradiction.

What is your proposition? As far as I can see, your only relevant proposition is "special relativity is wrong", but then you contradict this by agreeing that inertia-based coordinate systems are indeed related by Lorentz transformations. So what "proposition" are you referring to? I'll be happy to critique your "proposition" if you can tell me clearly what it is. Are you just talking about speaking in terms of rapidity rather than velocity?

On Sunday, October 24, 2021 at 10:48:26 AM UTC-4, Townes Olson wrote:
> On Thursday, October 21, 2021 at 8:21:49 PM UTC-7, tgca...@gmail.com wrote:
> > The object has a single length at rest. Everything else is an illusion.
>
> No, the object does indeed have a specific spatial length L in terms of the inertia-based coordinates in which it is at rest, but it also has spatial length L*sqrt(1-v^2/c^2) in terms of the inertia-based coordinate system in which it is moving (along its length) at the speed v. Neither of these is an illusion. They are both objectively verifiable facts.
>
> > You simply cannot measure distance unless your relative velocity is 0.
>
> That's not true at all. Conceptually we can construct a rectangular grid of standard rulers at rfest in any given frame with standard clocks located at each node, and we can synchronize the clocks inertially, meaning we synchronize them in such a way that inertia is isotropic in terms of these measures of position and time. This is an inertia-based coordinate system. An object moving through this grid has the speed dx/dt in terms of the x,t coordinates of the grid and clocks, and its spatial length is -- by definition -- the difference between the x coordinates of the ends of the object at equal values of the t coordinate. The object need not have 0 velocity relative to the grid. Space and time coordinate systems would not be very useful if, as you imagine, they applied only to stationary objects!
>
> > And then you attribute the contracted length to the motion of the object's
> > coordinate system.
>
> That has no relation to anything I've said. First, I explained that, already in the 1880's it was known that the equi-potential surfaces around a moving charge contract by the factor sqrt(1-v^2/c^2) in the direction of motion. Second, I clarified for you the distinction between passive and active transformations. Yes, if an object in a single state of motion is described in terms of two different systems of coordinates, the object is (obviously) not undergoing any physical change, and the difference in descriptions is due to the difference in coordinate systems... which by itself is not physically significant. The reason we attribute physical significance to Lorentz transformations is because they match the effects of active transformations, i.e., the descriptions of a solid object (at equilibrium, after modding out the time translation phase shifts, etc.) in two different states of motion in terms of a single coordinate system. This is obviously undergoing a physical change (e.g., the phase relations between the ends of the object change, even in Born rigid motion), but the configurations are congruent because the laws of physics are locally Lorentz invariant.
>
> > Relativity says it has exactly the same contracted length when it is NOT moving, and
> > it is the observer who is.
>
> There is no absolute distinction between being "moving" or "stationary", but yes, if you start with two solid rods, A and B, each of spatial length L and initially at rest in S, each of these rods has spatial length L*sqrt(1-v^2/c^2) in terms of an inertial coordinate system S' moving at speed v in terms of S. Now, if you gently accelerate rod B until it is at rest in S', the spatial length of A in S is still L, and the spatial length of B in S' is now L, and the spatial lengths of A in terms of S' and of B in terms of S are both L*sqrt(1-v^2/c^2). Do you disagree with any of this?
>
> > You are so convinced that only the approved version of relativity can
> > possibly be correct, you simply reject any alternative out of hand.
>
> What "alternative" are you referring to? You said special relativity is wrong and entails contradictions, but when I press you to identify any contradiction or what is wrong, you just run away. Phrases like "approved version" are just silly. Special relativity consists of local Lorentz invariance, which is the proposition that the equations of physics take the same homogeneous and isotropic form in terms of systems of coordinates related by Lorentz transformations. You've agreed to this, so I don't know what "alternative" you are talking about, nor why you continue to insist that special relativity is wrong and/or entails contradictions. Can you clarify?
>
> > If my proposition is false, it will inevitably lead to a contradiction.
> > If you're correct, show me that contradiction.
>
> What is your proposition? As far as I can see, your only relevant proposition is "special relativity is wrong", but then you contradict this by agreeing that inertia-based coordinate systems are indeed related by Lorentz transformations. So what "proposition" are you referring to? I'll be happy to critique your "proposition" if you can tell me clearly what it is. Are you just talking about speaking in terms of rapidity rather than velocity?

Keep reposting your same nonsense and you will be a spammer in addition to a troll.

On Sunday, October 24, 2021 at 7:27:40 AM UTC-7, tgca...@gmail.com wrote:
> The eigenvalues have not been shown in any of the other replies.

It's self-evident and trivial (in units with c=1) that the Lorentz transformation maps lightlines to lightlines, meaning the eigenvectors of the Lorentz transformation (in one space dimension, standard form) are (1,+-1), corresponding to the lightlines in the two directions (i.e., x=t and x=-t), and of course the corresponding eigenvalues are the Doppler factors sqrt[(1+-v)/(1-+v)].

> and its determinant is (γ-λ)²-(βγ)² = 0. This expands to...

There is no reason to expand it and solve the quadratic, because this equation immediately gives (γ-λ)²=(βγ)² and therefore we have γ-λ = +-βγ and hence λ=γ(1-+β), which of course is the approaching and receding Doppler factors sqrt[(1+-β)/(1-+β)].

> There you have it. Eigenvalue, 1/e^w, eigenvector, ct=r. Or, eigenvalue, e^w, eigenvector, ct=-r.
> These are the diagonals of the spacetime plane.

Well, you are using "r" instead of "x", which is slightly weird, but yes, the intervals with invariant "direction" in the x,t plane (i.e., the eigenvectors) are the lightlines, t=x and t=-x. This is self-evident and trivial.

So, I ask again, what is your point? Remember, in your very first post you said "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.." Okay, so like every high school student you now know the eigenvalues and eigenvectors of the Lorentz transformation, but how does this support your claim that special relativity is "wrong"? And how do these elementary trivialities of special relativity constitute a "Euclidean relativity" "alternative" to special relativity?

On Sunday, October 24, 2021 at 4:35:10 PM UTC-4, Townes Olson wrote:
> On Sunday, October 24, 2021 at 7:27:40 AM UTC-7, tgca...@gmail.com wrote:
> > The eigenvalues have not been shown in any of the other replies.
> It's self-evident and trivial (in units with c=1) that the Lorentz transformation maps lightlines to lightlines, meaning the eigenvectors of the Lorentz transformation (in one space dimension, standard form) are (1,+-1), corresponding to the lightlines in the two directions (i.e., x=t and x=-t), and of course the corresponding eigenvalues are the Doppler factors sqrt[(1+-v)/(1-+v)].
> > and its determinant is (γ-λ)²-(βγ)² = 0.. This expands to...
> There is no reason to expand it and solve the quadratic, because this equation immediately gives (γ-λ)²=(βγ)² and therefore we have γ-λ = +-βγ and hence λ=γ(1-+β), which of course is the approaching and receding Doppler factors sqrt[(1+-β)/(1-+β)].
> > There you have it. Eigenvalue, 1/e^w, eigenvector, ct=r. Or, eigenvalue, e^w, eigenvector, ct=-r.
> > These are the diagonals of the spacetime plane.
> Well, you are using "r" instead of "x", which is slightly weird, but yes, the intervals with invariant "direction" in the x,t plane (i.e., the eigenvectors) are the lightlines, t=x and t=-x. This is self-evident and trivial.
>
> So, I ask again, what is your point? Remember, in your very first post you said "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." Okay, so like every high school student you now know the eigenvalues and eigenvectors of the Lorentz transformation, but how does this support your claim that special relativity is "wrong"? And how do these elementary trivialities of special relativity constitute a "Euclidean relativity" "alternative" to special relativity?

To Townes:
Your exact words, again, "Okay, so like every high school student you now know the eigenvalues and eigenvectors of the Lorentz transformation,". I call bullshit. I took AP courses in high school and I didn't encounter eigenvalues and eigenvectors until 2nd year college. And even then, it was not connected to the Lorentz matrix. And the vast majority of students never take any course in linear algebra. You want to fool me into taking you seriously and you stoop to fabrications to discredit me. Fool me once, shame on you. Fool me twice, shame on me.

On Sunday, October 24, 2021 at 5:16:32 PM UTC-7, tgca...@gmail.com wrote:
> > So, I ask again, what is your point? Remember, in your very first post you said "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." Okay, so like every high school
> > student you now know the eigenvalues and eigenvectors of the Lorentz transformation,
> > but how does this support your claim that special relativity is "wrong"? And how do
> > these elementary trivialities of special relativity constitute a "Euclidean relativity"
> "alternative" to special relativity?
>
> I call bullshit. I took AP courses in high school and I didn't encounter eigenvalues and
> eigenvectors until 2nd year college. And even then, it was not connected to the Lorentz
> matrix.

Huh? The subject here is not the age at which various people learn about eigenvalues, the subject is your startling claim that when you learned about eigenvalues it made you "realize how wrong special relativity really is". That's how you began your very first message, and that has been the theme of all your messages, and I'm trying to elicit from you some clarification as to what you think is wrong with special relativity, and how your awareness of eigenvalues led you to the conclusion that special relativity is wrong. I can't see any connection, so I'm asking you for clarification.

> ...you stoop to fabrications to discredit me.

Wait... discredit you? By pointing out that eigenvalues and eigenvectors of linear operators, including Lorentz transformations, are very well known? How does this discredit you? Were you claiming to have discovered them?

> The eigenvalues have not been shown in any of the other replies.

Well, they weren't mentioned because they weren't particularly relevant, but since you are interested, it's well-known that the Lorentz transformation maps lightlines to lightlines (in a sense, that is its defining property!), meaning the eigenvectors of the Lorentz transformation (in one space dimension, standard form) are (1,+-1), corresponding to the lightlines in the two directions (i.e., x=t and x=-t), and of course the corresponding eigenvalues are the Doppler factors sqrt[(1+-v)/(1-+v)]. Do you disagree with this? Or do you claim to have discovered this?

> and its determinant is (γ-λ)²-(βγ)² = 0. This expands to...

There's no reason to expand it and solve the quadratic, because this equation immediately gives (γ-λ)²=(βγ)² and therefore we have γ-λ = +-βγ and hence λ=γ(1-+β), which of course is the approaching and receding Doppler factors sqrt[(1+-β)/(1-+β)]. I ask again: Do you disagree with this?

> There you have it. Eigenvalue, 1/e^w, eigenvector, ct=r. Or, eigenvalue, e^w, eigenvector, ct=-r.
> These are the diagonals of the spacetime plane.

Well, you're using "r" instead of "x", which is slightly weird, but yes, the intervals with invariant "direction" in the x,t plane (i.e., the eigenvectors) are the lightlines, t=x and t=-x. Again, this is all obvious and well-known.

None of this explained why you think special relativity is wrong. You made some comments that I addressed in another post (about reciprocal length contraction, etc.), but your misunderstanding about that were cleared up, and you disengaged from that discussion. So this leaves us still not knowing why you think special relativity is wrong. Can't you just tell me?

Tom Capizzi <tgcapizzi@gmail.com> wrote:

> To Townes:
> Your exact words, again, "Okay, so like every high school student you now
> know the eigenvalues and eigenvectors of the Lorentz transformation,". I
> call bullshit. I took AP courses in high school and I didn't encounter
> eigenvalues and eigenvectors until 2nd year college. And even then, it
> was not connected to the Lorentz matrix. And the vast majority of
> students never take any course in linear algebra. You want to fool me
> into taking you seriously and you stoop to fabrications to discredit me.
> Fool me once, shame on you. Fool me twice, shame on me.
>

I agree that this is not something a high school student typically learns.
It is however something that a math or physics student learns in first or
second year of college. Diagonalization as a skill in linear algebra is a
basic and oft-repeated exercise.

So I’m trying to understand whether you think your dabbling with
eigenstates and eigenvalues is novel and a revelatory exercise in special
relativity.

--
Odd Bodkin -- maker of fine toys, tools, tables

On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:
> Tom Capizzi <tgca...@gmail.com> wrote:
>
> > To Townes:
> > Your exact words, again, "Okay, so like every high school student you now
> > know the eigenvalues and eigenvectors of the Lorentz transformation,". I
> > call bullshit. I took AP courses in high school and I didn't encounter
> > eigenvalues and eigenvectors until 2nd year college. And even then, it
> > was not connected to the Lorentz matrix. And the vast majority of
> > students never take any course in linear algebra. You want to fool me
> > into taking you seriously and you stoop to fabrications to discredit me..
> > Fool me once, shame on you. Fool me twice, shame on me.
> >
> I agree that this is not something a high school student typically learns..
> It is however something that a math or physics student learns in first or
> second year of college. Diagonalization as a skill in linear algebra is a
> basic and oft-repeated exercise.
>
> So I’m trying to understand whether you think your dabbling with
> eigenstates and eigenvalues is novel and a revelatory exercise in special
> relativity.
>
> --
> Odd Bodkin -- maker of fine toys, tools, tables

To Townes:
<... crickets chirping ...>

To bodkin:
The response was prompted by ,
"Let's see what you are capable of doing: calculate the eigenvalues of the Lorentz transform.
Once you did that, find the eigenvectors. Put up or shut up."
I should ignore this troll, too, but eigenvalues are integral to the discussion of Bondi k-calculus and eigenvector decomposition, and none of the other responses listed them correctly.

I clearly did not invent them, but I've never seen them applied to Lorentz before. Some have posted drivel about light-cone axes. The eigenvalues of a standard Lorentz matrix, like the eigenvectors themselves, are real. The elements of the transform matrix that map rectangular Euclidean axes to diamond eigenvector axes are all real, as well. The linear combination of two vectors in a flat plane is another vector embedded in the same plane. This statement does not apply to light-cone axes, which require 3 dimensions, and go off the plane. You can call them real eigenvectors, but they are NOT. They may be eigenvectors of some matrix, but not a standard Lorentz matrix. And it clearly matters, because Euclidean eigenvectors are perpendicular while light-cone axes are not. In addition, the dot and cross products are both clearly defined in Euclidean geometry. In that alternative Minkowski geometry, dot products of units with themselves are 0, and the dot product of the light-cone axes is NOT zero, because they are partially parallel. And there is no meaning to the cross-product because the axes are not perpendicular. So to the thick-headed cranks who keep asking what is different, the very fact that it is Euclidean geometry is what makes it different. Since the dot and cross product are valid in Euclidean geometry, the area defined by any point on the Euclidean eigenvector grid is Σ*Δ = s², the Einstein interval. In Minkowski geometry, this is not true, because they angle between the vectors depends on relative velocity. But it is true in Euclidean geometry, and in Euclidean geometry, which is what Einstein had to use in his first paper, he contradicted himself. Not only is s² invariant, but the coordinates, Σ and Δ, are each individually invariant with respect to relative velocity, being measured by light rays. And in Euclidean geometry, these eigenvectors are truly perpendicular, and stay that way, regardless of relative velocity. Such things are not possible in Minkowski geometry. Einstein asserted the invariance of the speed of light and he asserted the dilation of time and the contraction of length, things which are contradicted by a grid of light lines, which do not change for relative velocity. In Minkowski geometry, these contradictions vanish, at the price of rubber units that flutter around in the wind. In Euclidean geometry, the paradox is resolved by the fact that both time dilation and length contraction are pure illusions, illusions we can measure, but illusions nonetheless. Nothing actually shrinks, but the effect of relativistic spacetime causes the illusion all by itself.

Tom Capizzi <tgcapizzi@gmail.com> wrote:
> On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:
>> Tom Capizzi <tgca...@gmail.com> wrote:
>>
>>> To Townes:
>>> Your exact words, again, "Okay, so like every high school student you now
>>> know the eigenvalues and eigenvectors of the Lorentz transformation,". I
>>> call bullshit. I took AP courses in high school and I didn't encounter
>>> eigenvalues and eigenvectors until 2nd year college. And even then, it
>>> was not connected to the Lorentz matrix. And the vast majority of
>>> students never take any course in linear algebra. You want to fool me
>>> into taking you seriously and you stoop to fabrications to discredit me.
>>> Fool me once, shame on you. Fool me twice, shame on me.
>>>
>> I agree that this is not something a high school student typically learns.
>> It is however something that a math or physics student learns in first or
>> second year of college. Diagonalization as a skill in linear algebra is a
>> basic and oft-repeated exercise.
>>
>> So I’m trying to understand whether you think your dabbling with
>> eigenstates and eigenvalues is novel and a revelatory exercise in special
>> relativity.
>>
>> --
>> Odd Bodkin -- maker of fine toys, tools, tables
>
> To Townes:
> <... crickets chirping ...>
>
> To bodkin:
> The response was prompted by ,
> "Let's see what you are capable of doing: calculate the eigenvalues of
> the Lorentz transform.
> Once you did that, find the eigenvectors. Put up or shut up."
> I should ignore this troll, too, but eigenvalues are integral to the
> discussion of Bondi k-calculus and eigenvector decomposition, and none of
> the other responses listed them correctly.
>
> I clearly did not invent them, but I've never seen them applied to
> Lorentz before. Some have posted drivel about light-cone axes. The
> eigenvalues of a standard Lorentz matrix, like the eigenvectors
> themselves, are real. The elements of the transform matrix that map
> rectangular Euclidean axes to diamond eigenvector axes are all real, as
> well. The linear combination of two vectors in a flat plane is another
> vector embedded in the same plane. This statement does not apply to
> light-cone axes, which require 3 dimensions, and go off the plane. You
> can call them real eigenvectors, but they are NOT. They may be
> eigenvectors of some matrix, but not a standard Lorentz matrix. And it
> clearly matters, because Euclidean eigenvectors are perpendicular while
> light-cone axes are not. In addition, the dot and cross products are both
> clearly defined in Euclidean geometry. In that alternative Minkowski
> geometry, dot products of units with themselves are 0, and the dot
> product of the light-cone axes is NOT zero, because they are partially
> parallel. And there is no meaning to the cross-product because the axes
> are not perpendicular. So to the thick-headed cranks who keep asking what
> is different, the very fact that it is Euclidean geometry is what makes
> it different. Since the dot and cross product are valid in Euclidean
> geometry, the area defined by any point on the Euclidean eigenvector grid
> is Σ*Δ = s², the Einstein interval. In Minkowski geometry, this is not
> true, because they angle between the vectors depends on relative
> velocity. But it is true in Euclidean geometry, and in Euclidean
> geometry, which is what Einstein had to use in his first paper, he
> contradicted himself. Not only is s² invariant, but the coordinates, Σ
> and Δ, are each individually invariant with respect to relative velocity,
> being measured by light rays. And in Euclidean geometry, these
> eigenvectors are truly perpendicular, and stay that way, regardless of
> relative velocity. Such things are not possible in Minkowski geometry.
> Einstein asserted the invariance of the speed of light and he asserted
> the dilation of time and the contraction of length, things which are
> contradicted by a grid of light lines, which do not change for relative
> velocity. In Minkowski geometry, these contradictions vanish, at the
> price of rubber units that flutter around in the wind. In Euclidean
> geometry, the paradox is resolved by the fact that both time dilation and
> length contraction are pure illusions, illusions we can measure, but
> illusions nonetheless. Nothing actually shrinks, but the effect of
> relativistic spacetime causes the illusion all by itself.
>

Inner and outer products do exist in hyperbolic 4-spaces, though the
geometric interpretations differ from those in Euclidean 3-spaces. In
general, an outer product expressed in matrices and vectors is
X x Y = YX^T - XY^T, and in 4-dimensions, this will resolve to a 6-vector
equivalent to the 6 independent components of the antisymmetric matrix
you’ll get explicitly. The outer product doesn’t correspond to any useful
physical properties that I’m aware of, but this should not be a surprise.
The inner products of 4-vectors correspond to Lorentz invariants that are
different than what one sees in 3-dimensional Euclidean physics also. For
example. Invariant mass is the self-inner-product of the momenergy
4-vector.

If you haven’t seen this kind of treatment before, you could try the first
couple of chapters of Sean Carroll’s book on GR.

--
Odd Bodkin -- maker of fine toys, tools, tables

Op 26-okt.-2021 om 15:22 schreef Tom Capizzi:
> On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:
>> Tom Capizzi <tgca...@gmail.com> wrote:
>>
>>> To Townes:
>>> Your exact words, again, "Okay, so like every high school student you now
>>> know the eigenvalues and eigenvectors of the Lorentz transformation,". I
>>> call bullshit. I took AP courses in high school and I didn't encounter
>>> eigenvalues and eigenvectors until 2nd year college. And even then, it
>>> was not connected to the Lorentz matrix. And the vast majority of
>>> students never take any course in linear algebra. You want to fool me
>>> into taking you seriously and you stoop to fabrications to discredit me.
>>> Fool me once, shame on you. Fool me twice, shame on me.
>>>
>> I agree that this is not something a high school student typically learns.
>> It is however something that a math or physics student learns in first or
>> second year of college. Diagonalization as a skill in linear algebra is a
>> basic and oft-repeated exercise.
>>
>> So I’m trying to understand whether you think your dabbling with
>> eigenstates and eigenvalues is novel and a revelatory exercise in special
>> relativity.
>>
>> --
>> Odd Bodkin -- maker of fine toys, tools, tables
>
> To Townes:
> <... crickets chirping ...>
>
> To bodkin:
> The response was prompted by ,
> "Let's see what you are capable of doing: calculate the eigenvalues of the Lorentz transform.
> Once you did that, find the eigenvectors. Put up or shut up."
> I should ignore this troll, too, but eigenvalues are integral to the discussion of Bondi k-calculus and eigenvector decomposition, and none of the other responses listed them correctly.
>
> I clearly did not invent them, but I've never seen them applied to Lorentz before.

Exercise 2 of chapter 1, section 1, in
https://books.google.com/books?id=rxVoCgAAQBAJ&pg=PA9

Exercise 16 in
https://books.google.com/books?id=kzeiyDgckvIC&pg=PA207

More examples galore with google books...
How many years did you say were studying special relativity?

Dirk Vdm

El martes, 26 de octubre de 2021 a las 10:22:28 UTC-3, tgca...@gmail.com escribió:
> On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:

Einstein asserted the invariance of the speed of light and he asserted the dilation of time and the contraction of length, things which are contradicted by a grid of light lines, which do not change for relative velocity. In Minkowski geometry, these contradictions vanish, at the price of rubber units that flutter around in the wind. In Euclidean geometry, the paradox is resolved by the fact that both time dilation and length contraction are pure illusions, illusions we can measure, but illusions nonetheless. Nothing actually shrinks, but the effect of relativistic spacetime causes the illusion all by itself.

What you call "illusions" have very real measurable and physical effects (see gravitational redshift and blueshift, gravitational time dilation, twin paradox, etc. etc.).

On Tuesday, 26 October 2021 at 16:24:02 UTC+2, Paparios wrote:
> El martes, 26 de octubre de 2021 a las 10:22:28 UTC-3, tgca...@gmail.com escribió:
> > On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:
>
> Einstein asserted the invariance of the speed of light and he asserted the dilation of time and the contraction of length, things which are contradicted by a grid of light lines, which do not change for relative velocity. In Minkowski geometry, these contradictions vanish, at the price of rubber units that flutter around in the wind. In Euclidean geometry, the paradox is resolved by the fact that both time dilation and length contraction are pure illusions, illusions we can measure, but illusions nonetheless. Nothing actually shrinks, but the effect of relativistic spacetime causes the illusion all by itself.
> What you call "illusions" have very real measurable and physical effects (see gravitational redshift and blueshift, gravitational time dilation, twin paradox, etc. etc.).

So do angels pushing planets along these crystal rings.
In the meantime in the real world, however, GPS clocks
keep measuring t'=t, just like all serious clocks always did.

On Tuesday, October 26, 2021 at 6:22:28 AM UTC-7, crank tgca...@gmail.com wrote:

> To bodkin:
> The response was prompted by ,
> "Let's see what you are capable of doing: calculate the eigenvalues of the Lorentz transform.
> Once you did that, find the eigenvectors. Put up or shut up."
> I should ignore this troll, too, but eigenvalues are integral to the discussion of Bondi k-calculus and eigenvector decomposition, and none of the other responses listed them correctly.
>

I posted this question to you, demented old fart. Your calculations contained a laughable error. Which I pointed out. Find a different hobby.

> They may be eigenvectors of some matrix, but not a standard Lorentz matrix.

You are digging yourself deeper, do you want a shovel?

On Tuesday, October 26, 2021 at 11:31:39 AM UTC-4, Dono. wrote:
> ...nothing of relevance...

To Dirk:
Thanks for the cites, but I'm too busy to study the whole book right now. Too bad neither book was published before I started college courses. But with your familiarity, you should have no trouble identifying where something I wrote runs afoul of the text. Page number, please?

On Tuesday, October 26, 2021 at 10:24:02 AM UTC-4, Paparios wrote:
> El martes, 26 de octubre de 2021 a las 10:22:28 UTC-3, tgca...@gmail.com escribió:
> > On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:
>
> Einstein asserted the invariance of the speed of light and he asserted the dilation of time and the contraction of length, things which are contradicted by a grid of light lines, which do not change for relative velocity. In Minkowski geometry, these contradictions vanish, at the price of rubber units that flutter around in the wind. In Euclidean geometry, the paradox is resolved by the fact that both time dilation and length contraction are pure illusions, illusions we can measure, but illusions nonetheless. Nothing actually shrinks, but the effect of relativistic spacetime causes the illusion all by itself.
> What you call "illusions" have very real measurable and physical effects (see gravitational redshift and blueshift, gravitational time dilation, twin paradox, etc. etc.).

That's kind of my point. Just because something is measurable does not mean it is not also an illusion. In fact, reality itself is a measurable illusion. The only difference between stationary observations of the illusion of reality and relatively moving observations of the illusions of relativity is that the complex phase angle for the stationary observer is 0, and the projection cosine is 1. When there is relativistic velocity involved, the phase angle is not 0, and the cosine projection is less than 1. The sine projection defines this phase angle as v/c = sin(angle). With this definition, the Lorentz factor, γ, equals the secant of the phase angle, which is 1/cosine, or cosine = 1/γ. And that's all there is to relativity.

El martes, 26 de octubre de 2021 a las 15:12:53 UTC-3, tgca...@gmail.com escribió:
> On Tuesday, October 26, 2021 at 10:24:02 AM UTC-4, Paparios wrote:

> > What you call "illusions" have very real measurable and physical effects (see gravitational redshift and blueshift, gravitational time dilation, twin paradox, etc. etc.).

> That's kind of my point. Just because something is measurable does not mean it is not also an illusion. In fact, reality itself is a measurable illusion. The only difference between stationary observations of the illusion of reality and relatively moving observations of the illusions of relativity is that the complex phase angle for the stationary observer is 0, and the projection cosine is 1. When there is relativistic velocity involved, the phase angle is not 0, and the cosine projection is less than 1. The sine projection defines this phase angle as v/c = sin(angle). With this definition, the Lorentz factor, γ, equals the secant of the phase angle, which is 1/cosine, or cosine = 1/γ. And that's all there is to relativity.

Both SR and GR are geometrical models. Hafele-Keating experiments, showed that flown atomic clocks elapsed times (whose ticking rates do not change while moving relative to the control clock) were different to the elapsed time of the ground atomic clock in total agreement with GR.
The geometry of spacetime clearly affects the elapsed time of clocks.

Op 26-okt.-2021 om 19:11 schreef Tom Capizzi:
> On Tuesday, October 26, 2021 at 11:31:39 AM UTC-4, Dono. wrote:
>> ...nothing of relevance...
>
> To Dirk:

I did not write the above.

> Thanks for the cites, but I'm too busy to study the whole book right now. Too bad neither book was published before I started college courses. But with your familiarity, you should have no trouble identifying where something I wrote runs afoul of the text. Page number, please?
>
>
>

There is no need to study whole books.
I did not ask you to study whole books.
You should find a way to properly reply to *anyone* under *their*
messages, and thus to *me* under *my* messages.

Dirk Vdm

Tom Capizzi <tgcapizzi@gmail.com> wrote:
> On Tuesday, October 26, 2021 at 10:24:02 AM UTC-4, Paparios wrote:
>> El martes, 26 de octubre de 2021 a las 10:22:28 UTC-3, tgca...@gmail.com escribió:
>>> On Monday, October 25, 2021 at 9:06:00 AM UTC-4, bodk...@gmail.com wrote:
>>
>> Einstein asserted the invariance of the speed of light and he asserted
>> the dilation of time and the contraction of length, things which are
>> contradicted by a grid of light lines, which do not change for relative
>> velocity. In Minkowski geometry, these contradictions vanish, at the
>> price of rubber units that flutter around in the wind. In Euclidean
>> geometry, the paradox is resolved by the fact that both time dilation
>> and length contraction are pure illusions, illusions we can measure, but
>> illusions nonetheless. Nothing actually shrinks, but the effect of
>> relativistic spacetime causes the illusion all by itself.
>> What you call "illusions" have very real measurable and physical effects
>> (see gravitational redshift and blueshift, gravitational time dilation,
>> twin paradox, etc. etc.).
>
> That's kind of my point. Just because something is measurable does not
> mean it is not also an illusion. In fact, reality itself is a measurable
> illusion. The only difference between stationary observations of the
> illusion of reality and relatively moving observations of the illusions
> of relativity is that the complex phase angle for the stationary observer
> is 0, and the projection cosine is 1. When there is relativistic velocity
> involved, the phase angle is not 0, and the cosine projection is less
> than 1. The sine projection defines this phase angle as v/c = sin(angle).
> With this definition, the Lorentz factor, γ, equals the secant of the
> phase angle, which is 1/cosine, or cosine = 1/γ. And that's all there is to relativity.
>

To this, there is a very important physical principle. If a hypothesis
involves no distinct *measurable* consequences, then it is for all intents
and purposes useless in physics, though it may be entertaining
metaphysically. As an example of this, back when people thought neutrinos
were massless, the only interaction that neutrinos would ever engage in was
the weak interaction. But the weak interaction is inherently left-handed.
So only left-handed neutrinos participate. The hypothesis then arise about
whether there could be right-handed neutrinos as well, if only for the sake
of symmetry. But right handed neutrinos have NO interaction they can
participate in. The have no mass, so gravity is out. No charge, so
electromagnetism is out. No color, so the nuclear strong force is out. And
the are right-handed so the weak interaction is out. They are called
“sterile neutrinos” because though they might (invisibly) exist, they
participate in no interactions at all and so cannot have no bearing in any
theory which purports to explain nature. They therefore have no rightful
place in any physical theory, because their existence is not testable in
any way. It is literally a useless idea.

In much the same way, you are proposing a length of a moving object that
CANNOT be measured and makes no DISTINCTIVE predictions compared to
relativity. It is a useless idea in physics, even if you find it somehow
appealing on some other non physical grounds.

--
Odd Bodkin -- maker of fine toys, tools, tables

On Tuesday, 26 October 2021 at 20:44:22 UTC+2, bodk...@gmail.com wrote:

> To this, there is a very important physical principle. If a hypothesis
> involves no distinct *measurable* consequences, then it is for all intents
> and purposes useless in physics

like 2+2=4, for instance.

On Tuesday, 26 October 2021 at 20:32:55 UTC+2, Paparios wrote:
> El martes, 26 de octubre de 2021 a las 15:12:53 UTC-3, tgca...@gmail.com escribió:
> > On Tuesday, October 26, 2021 at 10:24:02 AM UTC-4, Paparios wrote:
>
> > > What you call "illusions" have very real measurable and physical effects (see gravitational redshift and blueshift, gravitational time dilation, twin paradox, etc. etc.).
>
> > That's kind of my point. Just because something is measurable does not mean it is not also an illusion. In fact, reality itself is a measurable illusion. The only difference between stationary observations of the illusion of reality and relatively moving observations of the illusions of relativity is that the complex phase angle for the stationary observer is 0, and the projection cosine is 1. When there is relativistic velocity involved, the phase angle is not 0, and the cosine projection is less than 1. The sine projection defines this phase angle as v/c = sin(angle). With this definition, the Lorentz factor, γ, equals the secant of the phase angle, which is 1/cosine, or cosine = 1/γ. And that's all there is to relativity.
> Both SR and GR are geometrical models. Hafele-Keating experiments, showed that flown atomic clocks elapsed times (whose ticking rates do not change while moving relative to the control clock) were different to the elapsed time of the ground atomic clock in total agreement with GR.
> The geometry of spacetime clearly affects the elapsed time of clocks.

In the meantime in the real world, however, GPS clocks keep
measuring t'=t, just like all serious clocks always did.

On Tuesday, October 26, 2021 at 2:32:55 PM UTC-4, Paparios wrote:
> El martes, 26 de octubre de 2021 a las 15:12:53 UTC-3, tgca...@gmail.com escribió:
> > On Tuesday, October 26, 2021 at 10:24:02 AM UTC-4, Paparios wrote:
>
> > > What you call "illusions" have very real measurable and physical effects (see gravitational redshift and blueshift, gravitational time dilation, twin paradox, etc. etc.).
>
> > That's kind of my point. Just because something is measurable does not mean it is not also an illusion. In fact, reality itself is a measurable illusion. The only difference between stationary observations of the illusion of reality and relatively moving observations of the illusions of relativity is that the complex phase angle for the stationary observer is 0, and the projection cosine is 1. When there is relativistic velocity involved, the phase angle is not 0, and the cosine projection is less than 1. The sine projection defines this phase angle as v/c = sin(angle). With this definition, the Lorentz factor, γ, equals the secant of the phase angle, which is 1/cosine, or cosine = 1/γ. And that's all there is to relativity.
> Both SR and GR are geometrical models. Hafele-Keating experiments, showed that flown atomic clocks elapsed times (whose ticking rates do not change while moving relative to the control clock) were different to the elapsed time of the ground atomic clock in total agreement with GR.
> The geometry of spacetime clearly affects the elapsed time of clocks.

What's your point? I agree that the elapsed time of clocks is measurable, and that it is the real projection of invariant complex time. As such, it varies with relative velocity. I make no claim about GR.