This is the latest installment of a critical analysis of Einstein's signature work, special relativity. As a work in progress, new developments continue to arise. Over the past few months, I have been posting on other media, where I find the comments to be more intelligent, with none of the ad hominem attacks that are so common here. One nitwit here claimed he couldn't understand how the dot product could be a measurement protocol. How informative. In the other forum, a PhD commented that the dot product could not be used in the way that I suggested, but that he could show me a way to extend it so that it applied to an arbitrary number of dimensions. Unfortunately, he never went any further than that, so I can't comment on the accuracy of his remarks. But, it is true that the dot product behaves differently in Euclidean and Minkowski geometries, so I researched further. The first thing that became clear is the dot product is a special case of the more general inner product. And the inner product itself comes in real and complex varieties. Looking into this distinction more closely, I found that the complex inner product based on the square root of -1 had useful properties. Unfortunately, they were not suitable for relativity.

The definition of the complex inner product of two arbitrary vectors, x = a+bi and y = c+di, is <x,y> = xy* = (a+bi)(c-di) = (ac+bd) + (bc-ad)i. Normally, the dot product is a real scalar, but the complex inner product is not. It is also not commutative, as <y,x> = yx* = (c+di)(a-bi) = (ca+db) + (da-cb)i. Comparison of the results shows that the transpose of the factors is the conjugate of the product. While the first term, which is the dot product, is the same, the imaginary term is negated. So, there is a clear distinction between the real and the complex inner product. This in itself is not a problem, but one becomes clear after substituting polar coordinates for the Cartesian ones.

In polar coordinates, the scalars become a = |x| cos(θ), b = |x| sin(θ), c = |y| cos(θ') and d = |y| sin(θ'). We still have two arbitrary vectors, and if we plug their components into the inner product, <x,y> = (|x| cos(θ) * |y| cos(θ') +
|x| sin(θ) * |y| sin(θ')) + (|x| sin(θ) * |y| cos(θ') - |x| cos(θ) * |y| sin(θ'))i |x||y|((cos(θ)*cos(θ')+ sin(θ)*sin(θ')) + (sin(θ)*cos(θ')-cos(θ)*sin(θ'))i) |x||y|(cos(θ-θ')+sin(θ-θ')i) = |x||y|e^(θ-θ')i.
This is perfectly normal polar notation, with the complex exponential phase the difference between the angles of the factors. The two components also match the geometrical definitions of the dot and cross product, with θ-θ' the included angle. This is all well and good if you are interested in rotation around an axis, but it is not suitable for relativity. Since velocity can be specified in terms of a parametric angle, superficially this seems ok. However, in relativity, velocity addition is non-linear, and so is angle addition. It is easy to contrive a situation where the two velocity vectors are large and in opposite directions, such that the combination of their angles amounts to more than Pi/2. This is prohibited, because at Pi/2, v = c.

If we change from circular polar to hyperbolic polar, a = |x| cosh(w), b = |x| sinh(w), c = |y| cosh(w) and d = |y| sinh(w). Then the inner product becomes
<x,y> = (|x| cosh(w) * |y| cosh(w') +|x| sinh(w) * |y| sinh(w')) +
(|x| sinh(w) * |y| cosh(w') - |x| cosh(w) * |y| sinh(w'))i = |x||y|cosh(w)*cosh(w')+sinh(w)*sinh(w')) +
(sinh(w)*cosh(w')-cosh(w)*sinh(w'))i) |x||y|(cosh(w+w')+sinh(w-w')i).
There is no way to put the product into standard exponential form, because the sum and difference arguments are actually orthogonal.

However, as we are working outside the box, there is another option, the hypercomplex inner product. If you google this, you get lots of hits, but in google's inept way, it treats all search key words as or'd, so each hit has at least one key word. But it won't narrow the search by anding the key words. I could not find a reference for hypercomplex inner product, so I had to invent one. Using the complex inner product as a template, all I had to do was change the imaginary unit. You could say that hypercomplex means more than 1 imaginary unit. The quaternions have 3 imaginary units, but their only difference is spatial orientation. In any case, quaternions have only 4 components, which doesn't take us out of the box, since each of the 3 imaginary units behaves the same as the complex inner product above. To get out of the box, we need to replace the real scalars of the quaternion with complex scalars. Then, since linear combinations of the components produce the same 4 components, a new imaginary unit, h, had to be introduced. It is much like the i, j, and k units of the quaternion, but in combination with them, first, it is commutative, and second, their products do not repeat the i, j and k directions. This is a description of the biquaternions, as opposed to the octonions, both 8 dimensional hypercomplex numbers. So the biquaternion has 8 coordinates for its 8 units, {1, i, j, k, h, hi, hj, hk}. The reason I chose biquaternions instead of octonions is that all 7 imaginary units of the octonion have squares that equal -1, while (hi)² = (hj)² = (hk)² = +1. This is consistent with (hi)² = h²i² = (-1)(-1) = +1. The hypercomplex inner product then becomes:
<x,y> = xy* = (a+bhi)(c-dhi) = (ac-bd(hi)²)+(bc-ad)hi = (ac-bd)+(bc-ad)hi. At this point, if we substitute circular polar coordinates, we get the same kind of mismatch as above. But if we use hyperbolic polar coordinates, the game changes. Let a = |x| cosh(w), b = |x| sinh(w), c = |y| cosh(w') and d = |y| sinh(w'). Then, <x,y> = |x||y|((cosh(w-w')+sinh(w-w'))hi) = |x||y| e^(w-w')hi.

This is correct form for hyperbolic polar coordinates. It follows from the following identities:
e^w = cosh(w) + sinh(w)
e^wh = cosh(wh) + sinh(wh) = cos(w) + sin(w)h
e^wi = cosh(wi) + sinh(wi) = cos(w) + sin(w)i
e^wj = cosh(wj) + sinh(wj) = cos(w) + sin(w)j
e^wk = cosh(wk) + sinh(wk) = cos(w) + sin(w)k
e^whi = cosh(whi) + sinh(whi) = cosh(w) + sinh(w)hi
Then, e^whi*e^(-whi) = (cosh(w) + sinh(w)hi)(cosh(w) - sinh(w)hi) cosh²(w) - sinh²(w)(hi)² = cosh²(w) - sinh²(w) = 1.

Since the product is the same form as the factors, we can ask what happens when we take the hypercomplex inner product of the product with itself. For a simple hypercomplex vector, a and b are the same as above, but |y| = |x|, c = a and d = b. The inner product is (a*a-b*b) + (b*a-a*b)hi = a²-b² |x|²cosh²(w) - |x|²sinh²(w) = |x|². So, <<x,y>,<x,y>> |x||y|((cosh(w-w')+sinh(w-w'))hi) * |x||y|((cosh(w-w')-sinh(w-w'))hi) |x|²|y|²(cosh²(w-w') - sinh²(w-w')(hi)²) = |x|²|y|²(cosh²(w-w') - sinh²(w-w')) |x|²|y|² = (ac-bd)² - (bc-ad)²
If we require that neither x nor y be the zero vector, we can divide both sides of this equation by |x|²|y|², yielding 1 = (ac-bd)²/(|x|²|y|²) - (bc-ad)²/(|x|²|y|²)
Term by term comparison tells us that cosh(Δw) = (ac-bd)/(|x||y|) and sinh(Δw) = (bc-ad)/(|x||y|), for some arbitrary Δw. If we start with the same equation and move the negative term to the other side of the equal sign,
|x|²|y|² + (bc-ad)² = (ac-bd)². In hyperbolic coordinates, a and c are always positive and always greater than b and d, so this term can never be zero. Divide through by the RHS, and the result is:
|x|²|y|²/(ac-bd)² + (bc-ad)²/(ac-bd)² = 1, the equation of the unit circle. By symmetry, cos(Δθ) = |x||y|/(ac-bd) and sin(Δθ) = (bc-ad)/(ac-bd). Comparison with the previous case indicates that
cosh(Δw) = (ac-bd)/(|x||y|) = 1/cos(Δθ) = sec(Δθ) = γ.
This is one of the 6 identities of the gudermannian of Δw. The rest are:
sinh(Δw) = (bc-ad)/(|x||y|) = tan(Δθ) = βγ
tanh(Δw) = (bc-ad)/(ac-bd) = sin(Δθ) = β
sech(Δw) = (|x||y|)/(ac-bd) = cos(Δθ) = 1/γ
csch(Δw) = (|x||y|)/(bc-ad) = cot(Δθ) = 1/βγ
coth(Δw) = (ac-bd)/(bc-ad) = csc(Δθ) = 1/β

This gives us the relationships between each hyperbolic trig function and its corresponding circular function. Since the squares of these 12 functions are members of a 6-group, {λ, λ/(λ-1), 1/(1-λ), 1/λ, (λ-1)/λ, (1-λ)} all 12 can be determined from a single function, without even knowing the relationship between the angles. But we are interested in the relationship between the angles. These are 6 different ways to express it. But if we differentiate each with respect to either angle, the result in all cases is the same differential equation, dw/dθ = sec(θ) or dθ/dw = cos(θ). (Since Δw and Δθ are both arbitrary labels, we can drop the Δ.) The first form can be integrated directly by moving dθ from the LHS to the RHS, dw = sec(θ) dθ. Then, w ln(sec(θ)+tan(θ)), or e^w = sec(θ)+tan(θ). And, e^-w = sec(θ)-tan(θ). Further,
½(e^w+e^-w) = cosh(w) = sec(θ) and ½(e^w-e^-w) = sinh(w) = tan(θ), confirming the identities cited above.

The second form is even more interesting. It tells us that for equal small increments of w, the increment of θ is its cosine projection, based on the current value of θ. As w ranges from 0 to a final value, θ is the integral of all these cosine projections. Using the same diffeq, we can make it integrable by substituting sech(w) for cos(θ). Then the integral of sech(w) dw is θ = arctan(sinh(w)), or tan(θ) = sinh(w), one of the above identities. If we carry the integration out to w = infinity, we find that the value of θ at that limit is Pi/2, and sinh(∞) = tan(Pi/2) or tanh(∞) = sin(Pi/2), implying that c sinh(∞) = c tan(Pi/2) and c tanh(∞) = c sin(Pi/2). Infinite Proper velocity maps to measurable lightspeed. The speed of light is the limit of the cosine projections of Proper velocity as rapidity approaches infinity. This is why the speed of light is invariant and the same for all observers regardless of relative velocity. Infinity is the same everywhere. And compared to infinity, any finite Proper velocity is essentially zero.. It is an intrinsic, mathematical property of infinity and has nothing to do with physics, nor does it require time dilation and length contraction to make it true. Einstein's postulate is irrelevant. It is just a restatement of the math. The properties of apparent time dilation and length contraction are artifacts of the math, too.

On Sunday, October 23, 2022 at 8:37:26 PM UTC-4, Tom Capizzi wrote:
> This is the latest installment of a critical analysis of Einstein's signature work, special relativity. As a work in progress, new developments continue to arise. Over the past few months, I have been posting on other media, where I find the comments to be more intelligent, with none of the ad hominem attacks that are so common here. One nitwit here claimed he couldn't understand how the dot product could be a measurement protocol. How informative. In the other forum, a PhD commented that the dot product could not be used in the way that I suggested, but that he could show me a way to extend it so that it applied to an arbitrary number of dimensions. Unfortunately, he never went any further than that, so I can't comment on the accuracy of his remarks. But, it is true that the dot product behaves differently in Euclidean and Minkowski geometries, so I researched further. The first thing that became clear is the dot product is a special case of the more general inner product. And the inner product itself comes in real and complex varieties. Looking into this distinction more closely, I found that the complex inner product based on the square root of -1 had useful properties. Unfortunately, they were not suitable for relativity.
>
> The definition of the complex inner product of two arbitrary vectors, x = a+bi and y = c+di, is <x,y> = xy* = (a+bi)(c-di) = (ac+bd) + (bc-ad)i. Normally, the dot product is a real scalar, but the complex inner product is not. It is also not commutative, as <y,x> = yx* = (c+di)(a-bi) = (ca+db) + (da-cb)i. Comparison of the results shows that the transpose of the factors is the conjugate of the product. While the first term, which is the dot product, is the same, the imaginary term is negated. So, there is a clear distinction between the real and the complex inner product. This in itself is not a problem, but one becomes clear after substituting polar coordinates for the Cartesian ones.
>
> In polar coordinates, the scalars become a = |x| cos(θ), b = |x| sin(θ), c = |y| cos(θ') and d = |y| sin(θ'). We still have two arbitrary vectors, and if we plug their components into the inner product, <x,y> = (|x| cos(θ) * |y| cos(θ') +
> |x| sin(θ) * |y| sin(θ')) + (|x| sin(θ) * |y| cos(θ') - |x| cos(θ) * |y| sin(θ'))i =
> |x||y|((cos(θ)*cos(θ')+ sin(θ)*sin(θ')) + (sin(θ)*cos(θ')-cos(θ)*sin(θ'))i) =
> |x||y|(cos(θ-θ')+sin(θ-θ')i) = |x||y|e^(θ-θ')i.
> This is perfectly normal polar notation, with the complex exponential phase the difference between the angles of the factors. The two components also match the geometrical definitions of the dot and cross product, with θ-θ' the included angle. This is all well and good if you are interested in rotation around an axis, but it is not suitable for relativity. Since velocity can be specified in terms of a parametric angle, superficially this seems ok. However, in relativity, velocity addition is non-linear, and so is angle addition. It is easy to contrive a situation where the two velocity vectors are large and in opposite directions, such that the combination of their angles amounts to more than Pi/2. This is prohibited, because at Pi/2, v = c.
>
> If we change from circular polar to hyperbolic polar, a = |x| cosh(w), b = |x| sinh(w), c = |y| cosh(w) and d = |y| sinh(w). Then the inner product becomes
> <x,y> = (|x| cosh(w) * |y| cosh(w') +|x| sinh(w) * |y| sinh(w')) +
> (|x| sinh(w) * |y| cosh(w') - |x| cosh(w) * |y| sinh(w'))i = |x||y|cosh(w)*cosh(w')+sinh(w)*sinh(w')) +
> (sinh(w)*cosh(w')-cosh(w)*sinh(w'))i) =
> |x||y|(cosh(w+w')+sinh(w-w')i).
> There is no way to put the product into standard exponential form, because the sum and difference arguments are actually orthogonal.
>
> However, as we are working outside the box, there is another option, the hypercomplex inner product. If you google this, you get lots of hits, but in google's inept way, it treats all search key words as or'd, so each hit has at least one key word. But it won't narrow the search by anding the key words. I could not find a reference for hypercomplex inner product, so I had to invent one. Using the complex inner product as a template, all I had to do was change the imaginary unit. You could say that hypercomplex means more than 1 imaginary unit. The quaternions have 3 imaginary units, but their only difference is spatial orientation. In any case, quaternions have only 4 components, which doesn't take us out of the box, since each of the 3 imaginary units behaves the same as the complex inner product above. To get out of the box, we need to replace the real scalars of the quaternion with complex scalars. Then, since linear combinations of the components produce the same 4 components, a new imaginary unit, h, had to be introduced. It is much like the i, j, and k units of the quaternion, but in combination with them, first, it is commutative, and second, their products do not repeat the i, j and k directions. This is a description of the biquaternions, as opposed to the octonions, both 8 dimensional hypercomplex numbers. So the biquaternion has 8 coordinates for its 8 units, {1, i, j, k, h, hi, hj, hk}. The reason I chose biquaternions instead of octonions is that all 7 imaginary units of the octonion have squares that equal -1, while (hi)² = (hj)² = (hk)² = +1. This is consistent with (hi)² = h²i² = (-1)(-1) = +1. The hypercomplex inner product then becomes:
> <x,y> = xy* = (a+bhi)(c-dhi) = (ac-bd(hi)²)+(bc-ad)hi = (ac-bd)+(bc-ad)hi. At this point, if we substitute circular polar coordinates, we get the same kind of mismatch as above. But if we use hyperbolic polar coordinates, the game changes. Let a = |x| cosh(w), b = |x| sinh(w), c = |y| cosh(w') and d = |y| sinh(w'). Then, <x,y> = |x||y|((cosh(w-w')+sinh(w-w'))hi) = |x||y| e^(w-w')hi.
>
> This is correct form for hyperbolic polar coordinates. It follows from the following identities:
> e^w = cosh(w) + sinh(w)
> e^wh = cosh(wh) + sinh(wh) = cos(w) + sin(w)h
> e^wi = cosh(wi) + sinh(wi) = cos(w) + sin(w)i
> e^wj = cosh(wj) + sinh(wj) = cos(w) + sin(w)j
> e^wk = cosh(wk) + sinh(wk) = cos(w) + sin(w)k
> e^whi = cosh(whi) + sinh(whi) = cosh(w) + sinh(w)hi
> Then, e^whi*e^(-whi) = (cosh(w) + sinh(w)hi)(cosh(w) - sinh(w)hi) =
> cosh²(w) - sinh²(w)(hi)² = cosh²(w) - sinh²(w) = 1.
>
> Since the product is the same form as the factors, we can ask what happens when we take the hypercomplex inner product of the product with itself. For a simple hypercomplex vector, a and b are the same as above, but |y| = |x|, c = a and d = b. The inner product is (a*a-b*b) + (b*a-a*b)hi = a²-b² =
> |x|²cosh²(w) - |x|²sinh²(w) = |x|². So, <<x,y>,<x,y>> =
> |x||y|((cosh(w-w')+sinh(w-w'))hi) * |x||y|((cosh(w-w')-sinh(w-w'))hi) =
> |x|²|y|²(cosh²(w-w') - sinh²(w-w')(hi)²) = |x|²|y|²(cosh²(w-w') - sinh²(w-w')) =
> |x|²|y|² = (ac-bd)² - (bc-ad)²
> If we require that neither x nor y be the zero vector, we can divide both sides of this equation by |x|²|y|², yielding 1 = (ac-bd)²/(|x|²|y|²) - (bc-ad)²/(|x|²|y|²)
> Term by term comparison tells us that cosh(Δw) = (ac-bd)/(|x||y|) and sinh(Δw) = (bc-ad)/(|x||y|), for some arbitrary Δw. If we start with the same equation and move the negative term to the other side of the equal sign,
> |x|²|y|² + (bc-ad)² = (ac-bd)². In hyperbolic coordinates, a and c are always positive and always greater than b and d, so this term can never be zero. Divide through by the RHS, and the result is:
> |x|²|y|²/(ac-bd)² + (bc-ad)²/(ac-bd)² = 1, the equation of the unit circle. By symmetry, cos(Δθ) = |x||y|/(ac-bd) and sin(Δθ) = (bc-ad)/(ac-bd). Comparison with the previous case indicates that
> cosh(Δw) = (ac-bd)/(|x||y|) = 1/cos(Δθ) = sec(Δθ) = γ.
> This is one of the 6 identities of the gudermannian of Δw. The rest are:
> sinh(Δw) = (bc-ad)/(|x||y|) = tan(Δθ) = βγ
> tanh(Δw) = (bc-ad)/(ac-bd) = sin(Δθ) = β
> sech(Δw) = (|x||y|)/(ac-bd) = cos(Δθ) = 1/γ
> csch(Δw) = (|x||y|)/(bc-ad) = cot(Δθ) = 1/βγ
> coth(Δw) = (ac-bd)/(bc-ad) = csc(Δθ) = 1/β
>
> This gives us the relationships between each hyperbolic trig function and its corresponding circular function. Since the squares of these 12 functions are members of a 6-group, {λ, λ/(λ-1), 1/(1-λ), 1/λ, (λ-1)/λ, (1-λ)} all 12 can be determined from a single function, without even knowing the relationship between the angles. But we are interested in the relationship between the angles. These are 6 different ways to express it. But if we differentiate each with respect to either angle, the result in all cases is the same differential equation, dw/dθ = sec(θ) or dθ/dw = cos(θ). (Since Δw and Δθ are both arbitrary labels, we can drop the Δ.) The first form can be integrated directly by moving dθ from the LHS to the RHS, dw = sec(θ) dθ. Then, w =
> ln(sec(θ)+tan(θ)), or e^w = sec(θ)+tan(θ). And, e^-w = sec(θ)-tan(θ). Further,
> ½(e^w+e^-w) = cosh(w) = sec(θ) and ½(e^w-e^-w) = sinh(w) = tan(θ), confirming the identities cited above.
>
> The second form is even more interesting. It tells us that for equal small increments of w, the increment of θ is its cosine projection, based on the current value of θ. As w ranges from 0 to a final value, θ is the integral of all these cosine projections. Using the same diffeq, we can make it integrable by substituting sech(w) for cos(θ). Then the integral of sech(w) dw is θ = arctan(sinh(w)), or tan(θ) = sinh(w), one of the above identities. If we carry the integration out to w = infinity, we find that the value of θ at that limit is Pi/2, and sinh(∞) = tan(Pi/2) or tanh(∞) = sin(Pi/2), implying that c sinh(∞) = c tan(Pi/2) and c tanh(∞) = c sin(Pi/2). Infinite Proper velocity maps to measurable lightspeed. The speed of light is the limit of the cosine projections of Proper velocity as rapidity approaches infinity. This is why the speed of light is invariant and the same for all observers regardless of relative velocity. Infinity is the same everywhere. And compared to infinity, any finite Proper velocity is essentially zero. It is an intrinsic, mathematical property of infinity and has nothing to do with physics, nor does it require time dilation and length contraction to make it true. Einstein's postulate is irrelevant. It is just a restatement of the math. The properties of apparent time dilation and length contraction are artifacts of the math, too.
>
> Now we are in a position to solve the riddle of relativistic mass. It was long known that relativistic momentum is γmv, and it was incorrectly assumed that the Lorentz factor applied to the mass. But mass is a relativistic invariant of the Lorentz transform of 4-momentum. The factor belongs to the velocity, not the mass. From the above analysis of the gudermannian, we can see that it is not the mass which is getting heavier. It is the velocity which is becoming more and more hypercomplex as it gets faster, and because of the phase angle introduced by the gudermannian of the rapidity, less and less of the increment of rapidity is projected parallel to the path, and lore and lore of it is projected perpendicular, until, at light speed, all of it is projected perpendicular to the path and no further increase in velocity is possible. The hypercomplex inner product tells us exactly what percentage is projected at any specific velocity.
>
> Conservation of momentum also plays a part. We know that Newtonian momentum is mv, and we know that relativistic momentum is γmv. Now we know that the Newtonian projection is the literal cosine projection of the hypercomplex total. Where there is a cosine projection, there must be a sine projection. It must be perpendicular to the cosine projection. And the square of its magnitude is the difference of the squares of the relativistic and the Newtonian momenta. Conservation requires that it go somewhere, even if we can't measure it. I propose that it is toroidal rotation around the smaller circular cross-section. Rotation around the major circle of the toroid is angular momentum. Rotation around the smaller cross-section is perpendicular to both the major circle and the axis of rotation through the donut hole.. It is easy to visualize with a toy slinky. This toroidal rotation is also angular momentum, but it is perpendicular to the usual angular momentum, and has no net linear component. But when a particle is smashed into a target, all this internally stored momentum is released, and returned to the surroundings. Relativistic mass never was, and in no way represents a legitimate alternative point of view. No legitimate course should be teaching this nonsense.
>
> I can hear some of you saying that's nice, but it's all abstract math. What does it have to do with physics? See if this answers your question. The hypercomplex vector represents the relative velocity of the origins of two arbitrary frames, travelling along the same path. The hypercomplex inner product essentially computes the relative rapidity between them, hence the relative velocity. As we have seen, the inner product is also a hypercomplex vector, so we can define such a vector between any two origins. Now suppose we are interested in the apparent coordinates of an arbitrary point in one frame, which is not the origin. To be considered in the same frame of reference, it must have the same hypercomplex vector as the origin. Otherwise it is moving relative to the origin. To find its coordinates in the other frame, we simply take the hypercomplex inner product of the arbitrary point and the phase vector resulting from the comparison of the origins. So, if we evaluate <ct+rhi,γ+βγhi>, where γ = cosh(w-w') and βγ = sinh(w-w'). Then, <ct+rhi,γ+βγhi> = (ct+rhi)(γ-βγhi) = (γct-βγr(hi)²)+(γr-βγct)hi =
> (γct-βγr)+(γr-βγct)hi .= ct'+r'hi. Comparing terms, ct' = γct-βγr, while
> r'hi = (γr-βγct)hi. In matrix format:
> │ ct'│ │ γ -βγhi││ct │
> │r'hi│ ̿ │-βγhi γ││rhi│
> If we conjugate the phase factor, and swap the primes, as we do when switching frames of reference, <ct'+r'hi,γ-βγhi,> = (ct'+r'hi)(γ+βγhi) =
> (γct'+βγr'(hi)²)+(γr'+βγct')hi = (γct'+βγr')+(γr'+βγct')hi = ct+rhi. Comparing terms, ct = γct'+βγr', while rhi = (γr'+βγct')hi. In matrix format:
> │ ct│ │γ βγhi││ct' │
> │rhi│ ̿ │βγhi γ││r'hi│
> Buried in the heart of the hypercomplex inner product is the Lorentz transform and its inverse. Is that physics enough for you? And if we take the inner product of the coordinates of an arbitrary event with itself, <ct+rhi,ct+rhi> = (ct+rhi)(ct-rhi) = c²t²-r²(hi)² = c²t²-r²
> <ct'+r'hi,ct'+r'hi> = (ct'+r'hi)(ct'-r'hi) = c²t'²-r'²(hi)² = c²t'²-r'² =
> (γct-βγr)²-(γr-βγct)² = (γ²c²t²-2γctβγr+β²γ²r²)-(γ²r²-2γrβγct+β²γ²c²t²) =
> (γ²c²t²-β²γ²c²t²)-(γ²r²-β²γ²r²) = (c²t²-r²)(γ²-β²γ²) = c²t²-r² = s²
>
> That's all for now.

Tom Capizzi <tgcapizzi@gmail.com> wrote:

> Same post as before, but a few minor typos corrected. Is there a way to
> edit a post other than replacing it?

Yes, there is 'Supersedes' and 'Replaces' for that.
Many servers don't honour it,

Jan
(but no one will read your ramblings anyway, try to edit down)

On Monday, October 24, 2022 at 5:31:43 AM UTC-4, J. J. Lodder wrote:
> Tom Capizzi <tgca...@gmail.com> wrote:
>
> > Same post as before, but a few minor typos corrected. Is there a way to
> > edit a post other than replacing it?
> Yes, there is 'Supersedes' and 'Replaces' for that.
> Many servers don't honour it,
>
> Jan
> (but no one will read your ramblings anyway, try to edit down)

You can call it rambling. I call it thoroughness. Don't care if you are only interested in superficial generalizations. It takes you a lot less time to read than it takes me to write. On my website, I convert a lot of the details to pop-ups so the details are there but not on top. Usenet is not that sophisticated.