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Trysse
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In the meantime: What is your answer to the question:
Why Is Minkowski Spacetime Non-Euclidean?
Why Is Minkowski Spacetime Non-Euclidean?
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Are you asking why we believe it to be non-Euclidean, or why it is?Trysse said:In the meantime: What is your answer to the question:
Why Is Minkowski Spacetime Non-Euclidean?
Ah - that’s a reasonable understanding as well. The metric of Minkowski spacetime is by definition ##ds^2=-dt^2+dx^2+dy^2+dz^2## (to within a sign convention) and that’s not the metric of a Euclidean space.Dragon27 said:I would have personally understood the question as "Why a certain mathematical space (a Minkowski spacetime) can't be considered Euclidean" (essentially, by definition) rather than "Why is the spacetime of our world, whose structure we've established empirically, isn't Euclidean".
More the first. However, I am not interested in generalised answers "why we believe" or "why physicists believe". I am interested to see, what individual people take as a good explanation respectively as a good answer to the question. I am interested to see what people take as a meaningful explanation. Or what they think is necessary for a meaningful explanation.Nugatory said:Are you asking why we believe it to be non-Euclidean, or why it is
Do you know what Euclidean and non-Euclidean mean?Trysse said:I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?
For that you will want Taylor and Wheeler’s “Spacetime Physics” - the first edition is altogether satisfactory and available free online.Trysse said:I am interested to see, what individual people take as a good explanation respectively as a good answer to the question. I am interested to see what people take as a meaningful explanation. Or what they think is necessary for a meaningful explanation.
The most common mental image is probably the ubiquitous Minkowski space-time diagram. Getting comfortable with these is an essential first step in understanding relativity.I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?
Why, may I ask, do you prefer this sign convention over the other one?Nugatory said:Ah - that’s a reasonable understanding as well. The metric of Minkowski spacetime is by definition ##ds^2=-dt^2+dx^2+dy^2+dz^2## (to within a sign convention) and that’s not the metric of a Euclidean space.
I have no preference, just didn’t see the need to type out both forms.Thadriel said:Why, may I ask, do you prefer this sign convention over the other one?
Convenience is probably the best answer I could have imagined. Like choosing your axes so that they line up with the acceleration in an inclined plane problem.Ibix said:If, today, you are primarily interested in calculating elapsed times, picking +--- will cut down the number of modulus signs under your square roots. If, tomorrow, you are interested in distancess, -+++ is better for the same reason.
This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.Trysse said:I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?
This simply isn't true. What you always demand we do is try to find a way to draw an undistorted representation of Minkowski space on a Euclidean space. That is impossible, yes, but that does not make it unimaginable any more than the non-Euclidean nature of the surface of the Earth makes it unimaginable. You just can't imagine it the way you want to.student34 said:This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.
Well, it was imaginable to Hermann Minkowski!student34 said:This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.
student34 said:Trysse said:I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?
This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.
May be he was not a human.PeroK said:Well, it was imaginable to Hermann Minkowski!
The autonomous AI must think very lowly of us when choosing such an obvious first name as Herr Mann thinking we will not see through it …martinbn said:May be he was not a human.
student34 said:Trysse said:I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?
This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.
At that time, Klein had a glimpse of the mathematics of spacetime geometry,robphy said:I believe this is referencing the fact that the Galilean and Lorentz Transformations are not periodic.from Torretti's Philosophy of Geometry from Riemann to Poincare, p 129 [via Google books]
In his posthumous Lectures on Non-Euclidean Geometry (1926) Klein briefly examines the other four degenerate cases. He does not pay much attention to the resulting geometries because angle-measure in them is not periodic - a fact that, in Klein's opinion, makes them inapplicable fo the real world, since "experience shows us that a finite sequence of rotation [about an axis of a bundle of planes] finally takes us back to our starting point". [Torretti references Klein's lectures [in German], p. 189]
It's possible that Klein (in the 1890s) in his study of hyperbolic and elliptical geometry could have uncovered, by analogy, the mathematics of special relativity before Einstein (1905) and Minkowski (1907).
My personal opinion is that this "Why?" question is too fundamental to be answered. Fundamental "Why?" questions cannot be answered, because there is nothing more fundamental out of which to construct an answer. Maybe someday (local) Minkowski spacetime will be emergent from a more fundamental theory, but not today.Trysse said:I am not interested in generalised answers "why we believe" or "why physicists believe". I am interested to see, what individual people take as a good explanation respectively as a good answer to the question. I am interested to see what people take as a meaningful explanation. Or what they think is necessary for a meaningful explanation.
I want to know, what people think when they say "space-time is non-euclidean". Do they have a mental image in which space-time is represented?
Is there a groan emoji?PeroK said:old Klein in a new bottle
Vanadium 50 said:Is there a groan emoji?
At least Klein's tombstone doesn't mention being famous for a "bottle".PeroK said:So, you could say that Minkowski was "old Klein in a new bottle"!
robphy said:At that time, Klein had a glimpse of the mathematics of spacetime geometry,
but not the physical application and interpretation provided by Einstein
and the subsequent geometrical interpretation by Minkowski.
p. 13: If the radius of the sphere be 1, as we shall assume throughout the discussion of this general transformation, or its equation when written homogeneously, be :
$$x^{2}+y^{2}+z^{2}-t^{2}=0$$
the equations connecting x, y, z, t and X, Y, Z, T are those indicated in the following scheme
$$(6)\quad \begin{array}{c|c|c|c|c} & X+iY & X-iY & T+Z & T-Z\\ \hline x+iy & \alpha\bar{\delta} & \beta\bar{\gamma} & \alpha\bar{\gamma} & \beta\bar{\delta}\\ \hline x-iy & \gamma\bar{\beta} & \delta\bar{\alpha} & \gamma\bar{\alpha} & \delta\bar{\beta}\\ \hline t+z & \alpha\bar{\beta} & \beta\bar{\alpha} & \alpha\bar{\alpha} & \beta\bar{\beta}\\ \hline t-z & \gamma\bar{\delta} & \delta\bar{\gamma} & \gamma\bar{\gamma} & \delta\bar{\delta} \end{array}$$
p. 15: The general transformation (6) represents the totality of those projective transformations or collineations of space for which each system of generating lines of the sphere, ##x^{2}+y^{2}+z^{2}-t^{2}=0##, is transformed into itself, and among which all rotations of the sphere are obviously included as special cases. This is the geometrical meaning of the equation
$$\zeta=\frac{\alpha Z+\beta}{\gamma Z+\delta}$$
for unrestricted values of ##\alpha,\beta,\gamma,\delta##.
But the transformation admits also of a very interesting kinematical interpretation which I shall consider at length in my third lecture. With respect to it our sphere of radius 1 plays the role of the fundamental surface or "absolute " in the Cayleyan or hyperbolic non-Euclidian geometry.
At the top of p. 13Histspec said:At least Klein had the Lorentz transformation expressed in terms of PSL(2,C). For instance, he wrote in 1896, The Mathematical Theory of the Top. New York: Scribner; pp. 13ff)
and then on p.16 he closes the chapter with (italics by Klein)Klein(1896) said:We shall consider ##t## also as capable of complex values, not for the sake of studying the behavior of a fictitious, imaginary time, but because it is only by taking this step that it becomes possible to bring about the intimate association of kinetics and the theory of functions of a complex variable at which we are aiming.
Klein(1896) said:First, there is nothing essentially new in the considerations with which we have been occupied thus far. I have merely attempted to throw a method already well known into the most convenient form for application to mechanics.
Second, the non-Euclidean geometry has no meta-physical significance here or in the subsequent discussion. It is used solely because it is a convenient method of grouping in geometric form relations which must otherwise remain hidden in formulas.
No, Galilean spacetime is not Euclidean either. What suggests Minkowski spacetime in particular is the existence of an invariant speed.accdd said:Isn't the fact that the spacetime metric is not Euclidean due to the fact that there is a maximum velocity in our universe?
maximum velocity ##\implies## everyone agrees on its value ##\implies## derivation of spacetime interval (##ds^2=-dt^2+dx^2+dy^2+dz^2##)
Is this reasoning wrong? Doesn't the fact that spacetime is not Euclidean derive from the fact that there is a maximum velocity?
Orodruin said:No, Galilean spacetime is not Euclidean either. What suggests Minkowski spacetime in particular is the existence of an invariant speed.
IMO, "infinity" is not a speed, invariant or otherwise.robphy said:Both Galilean and Minkowski spacetimes have an invariant maximum signal speed:
infinity for Galilean,
a finite speed for Minkowski.
PeroK said:IMO, "infinity" is not a speed, invariant or otherwise.
That’s just word play. I would not call ”infinity” as such an invariant speed.robphy said:Both Galilean and Minkowski spacetimes have an invariant maximum signal speed:
infinity for Galilean,
a finite speed for Minkowski.
Infinity is not a valid eigenvalue, if that's what you mean.robphy said:Call it what you want.
Compute the eigenvectors of the Galilean boost.
I can use this as a parameter to move through all Euclidean, Galilean, and Minkowski geometries.
it is not. He means that the tangent vector of a world line of an object moving at that speed is an eigenvector of the boost.PeroK said:Infinity is not a valid eigenvalue, if that's what you mean.
To me, saying the maximum signal speed in a Galilean Spacetime is infinityOrodruin said:That’s just word play. I would not call ”infinity” as such an invariant speed.