I see your point, but I wonder, isn't that the whole point of Minkowski spacetime, that time is different from space? In any case, the discussion is focused on those light-like vectors. In the context of this being an isomorphism, the dot product is not the only way to determine causal...
But my point was that the light-cone axes are not the same as the eigenvector axes. For one thing, the surface of the light-cone encloses a volume, so that any arbitrary point not on the surface requires 3 coordinates. The eigenvector plane has no depth. I understand your argument that the...
I've been thinking about this information all week, and I believe there is a misunderstanding. When I used the transform pair, it was in the context of invariance of the magnitude in transforming from a vector to a bivector hyperbolic coordinate system. In point of fact, the standard Lorentz...
"A(OQ→)=(U→×V→)=−2(UtVx−UxVt)=−2((Qt+Qx2)(−Qt−Qx2)−(Qt+Qx2)(Qt−Qx2))=Qt2−Qx2=ΔsQ→2."
The second line was clipped in the post. I see it now after pasting it. There is still a problem, however. Trying to refer to the paper in the link, and it says I don't have access unless I buy it.
I was with you up to " (although of course \tilde u \cdot \tilde u = 0 and \tilde v \cdot \tilde v = 0 )." In eigenspace, the eigenvectors remain perpendicular to infinity. u dot v = 0, everywhere. Usually, it is u x u = 0 and v x v = 0. Is this a typo, or is it because of the light-cone...
As I may have mentioned, there is no boost that can transform to eigenvector axes. They are the asymptotes of the hyperbolic grid. I was looking at an earlier post, and there was a graphic of several rectangles in your section on clock diamonds. Each of the rectangles have 3 corners on the axes...
It would be appropriate to note that light-cone coordinates are the eigenvector coordinates. While the symmetry transform specifies the size of the coordinates, it is silent on the physical interpretation. The transform maps the symmetry coordinates back to the same plane. But the representation...
I understand that there is no absolute rest frame. It is just awkward to talk about an eigenvector space which is at rest relative to whatever inertial frame was already chosen to be the reference frame. You know that you can't even talk about comparative measurements unless two inertial frames...
I agree with your sentiments about hyperbolic functions. It appears to me that the universe prefers hyperbolic coordinates. The gudermannian function can be implemented with a classic Greek geometry ruler-and-compass construction. Through simple geometric operations, it is easy to demonstrate...
I agree that the author was probably unaware of the fact. In fact, I'd wager that most physics "experts" are unaware of that fact. I would go into more detail, but PF moderators seem to think that mathematical analysis is somehow "personal speculation", and gave me non-expiring demerits for...
The k-factor in Bondi's k-calculus is an eigenvalue of the Lorentz matrix which boosts relative velocity from rest to βc. Since k is never 0, we can divide both numerator and denominator by k, resulting in β = (k-1/k)/(k+1/k) = 2 sinh(w)/2 cosh(w) = tanh(w), because k = e^w. From common...
Everything that I've read about k-calculus, including Bondi's book, overlooks the fact that k-calculus is an incomplete eigenvector decomposition. The k-factor itself is just an eigenvalue of the Lorentz matrix, and the radar measurements are an integral part of the decomposition, because the...
Are you aware that the Bondi approach is a half-hearted attempt at eigenvector decomposition? The k-factor in Bondi k-calculus is, in fact, an eigenvalue of the Lorentz matrix. The radar measurements are another feature of eigenvector analysis, because the eigenvectors of the 2x2 Lorentz matrix...
To address some of the objections that have been raised in this discussion, I have gone back to the source and re-read Einstein's derivation of Special Relativity. I have not yet located a copy of the revision which included the train experiment with multiple stationary observers. Nor have I...