I Why Is Minkowski Spacetime Non-Euclidean?

[Trysse]

In the meantime: What is your answer to the question:
Why Is Minkowski Spacetime Non-Euclidean?

 

[Nugatory]

In the meantime: What is your answer to the question:
Why Is Minkowski Spacetime Non-Euclidean?

Are you asking why we believe it to be non-Euclidean, or why it is?

The answer to the first question is empirical and pragmatic: experiments have confirmed that the universe behaves in a way that is most easily understood by treating the geometry of spacetime as non-Euclidean.

The second question has no good answer: Physics can discover through experiments the laws that govern how our universe works, but it cannot tell us why we ended up in a universe that obeys those laws and not some others.

 

[Dragon27]

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".

 

[Nugatory]

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".

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.

 

[Trysse]

Are you asking why we believe it to be non-Euclidean, or why it is

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.

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?

That is the reason why I was interested in the paper mentioned in the original post

https://www.physicsforums.com/threa...-euclidean-by-cronkhite.1016381/#post-6645881

 

[martinbn]

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?

Do you know what Euclidean and non-Euclidean mean?

 

[Nugatory]

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.

For that you will want Taylor and Wheeler’s “Spacetime Physics” - the first edition is altogether satisfactory and available free online.

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?

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.

 

[Thadriel]

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.

Why, may I ask, do you prefer this sign convention over the other one?

 

[Nugatory]

Why, may I ask, do you prefer this sign convention over the other one?

I have no preference, just didn’t see the need to type out both forms.

 

[Ibix]

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.

 

[Thadriel]

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.

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.

 

[student34]

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.

 

[Ibix]

This is a tough question to answer because the Minkowski metric does not seem imaginable or representable to human minds.

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.

 

[PeroK]

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!

 

[robphy]

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.

As I have often noted, the position-vs-time graph in PHY 101
has a [degenerate-metric] geometry (called the "Galilean geometry") which is non-Euclidean.

• (2021) https://www.physicsforums.com/threa...-mathematical-structures.1010756/post-6581363
• (2019) https://www.physicsforums.com/threads/why-is-time-orthogonal-to-space.969731/post-6164201
• (2017) https://www.physicsforums.com/threads/going-from-galilei-to-minkowski.930035/post-5872091 [and onward]
• (2016) https://www.physicsforums.com/threa...introduction-of-spacetime.866669/post-5441163 [and onward]
• (2013) https://www.physicsforums.com/threads/geometrized-Newtonian-gravity.703510/post-4459086 [and onward]
• (2013) https://www.physicsforums.com/threads/spacetime-interval.695760/post-4418768
• (2013) https://www.physicsforums.com/threa...or-contravariant-quantity.666861/post-4243599
• (2007) https://www.physicsforums.com/threads/the-nature-of-time.163781/post-1291238

(A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity by I.M. Yaglom
https://www.amazon.com/dp/0387903321/?tag=pfamazon01-20 )

Both Minkowski spacetime and Galilean spacetime

• satisfy Euclid's Fifth Postulate (the Parallel Postulate) [which is why we can do vector algebra in Minkowski and Galilean geometries, like in Euclidean geometry],
• but they violate Euclid's first postulate (reformulated as the projective-dual of the Playfair's formulation of the parallel postulate) [implicit in Yaglom's book above]

For more details, see my post at https://physics.stackexchange.com/a/436852/148184 in response to
https://physics.stackexchange.com/q...ids-5-postulates-false-in-minkowski-spacetime

So, a strategy to understand the Minkowski metric is
to first understand some aspects Galilean geometry together with its physical interpretation with PHY 101's kinematics, with a goal toward repeating the storyline for Minkowskian geometry with special-relativistic-kinematics [and dynamics].(Here is a related reference: Andrzej Trautman, Comparison of Newtonian and relativistic theories of gravitation, pp. 413–425 in: Perspectives in Geometry and Relativity, Essays in honor of V. Hlavaty, ed. by B. Hoffmann, Indiana Univ. Press, Bloomington, 1966 ( trautman.fuw.edu.pl/publications/Papers-in-pdf/22.pdf from a website that collected Trautman's papers )

I first learned about the Galilean geometry from
Mathematical and Conceptual Foundations of 20th-Century Physics
By G.G. Emch (see page 96 and onward)
https://books.google.com/books?id=eYQHIjkaEroC&printsec=frontcover&dq="Mathematical+and+Conceptual+Foundations+of+20th+Century+Physics"&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjmsPjVgNb4AhU9q4kEHYARAksQ6AF6BAgDEAI#v=onepage&q=galilean&f=false
which pointed me to Yaglom's book.

 

[martinbn]

Well, it was imaginable to Hermann Minkowski!

May be he was not a human.

 

[Orodruin]

May be he was not a human.

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 …

 

[robphy]

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.

A further, possibly interesting, comment:

---

First, the setup:

Felix Klein (Minkowski's grand-PhD-advisor​

• Klein https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7401
• Lindemann https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7404
• Minkowski https://www.genealogy.math.ndsu.nodak.edu/id.php?id=29675

)​

had formulated a group-theoretic definition of "geometry" ( https://en.wikipedia.org/wiki/Erlangen_program )​

in which Minkowski and Galilean geometries were included among the nine Cayley-Klein geometries in two dimensions (Euclidean and the "classic non-euclidean geometries" [hyperbolic and spherical/elliptic] are the original three). So far, this is 5 of 9. The following is from Emch's book (p. 92).​

​

Co-hyperbolic and Doubly-hyperbolic are the (1+1) deSitter and anti-deSitter spacetimes,​

and the remaining two are the Galilean limits of these curved spacetimes.​

---

Now, the comment
(based on
https://www.physicsforums.com/threads/physics-mathematics-and-analogies.813201/post-5105772 )

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]

I believe this is referencing the fact that the Galilean and Lorentz Transformations are not periodic.
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).

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.

If Klein had looked at the position-vs-time graph (used in PHY 101)
and interpreted its physical/kinematic content geometrically (in the Erlangen program)
in spite of the angle-measure (which I call the Galilean-rapidity, in analogy to rapidity in special relativity) not being periodic,
Klein might have been able to predict aspects of special relativity.

This is the motivation behind my approach to developing selected aspects of the spacetime viewpoint starting in PHY 101.

 

[PeroK]

So, you could say that Minkowski was "old Klein in a new bottle"!

 

[George Jones]

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?

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.

We can say that Minkowski spacetime is consistent with observations.

 

[Vanadium 50]

old Klein in a new bottle

Is there a groan emoji?

 

[Thadriel]

Is there a groan emoji?

Probably the closest you can get.

 

[strangerep]

So, you could say that Minkowski was "old Klein in a new bottle"!

At least Klein's tombstone doesn't mention being famous for a "bottle".

(TIL... there is actually a website Find-A-Grave.)

 

[Hornbein]

In their 4D native environment, a Klein bottle will not contain liquid. Maybe that's what Merle Haggard meant when he sang "the bottle let me down."

 

[Histspec]

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.

At least Klein had the Lorentz transformation expressed in terms of $\mathrm{PSL}(2,\mathbb{C})$. For instance, he wrote in 1896, The Mathematical Theory of the Top. New York: Scribner; pp. 13ff)

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.

For others who wrote similar equations in the 19th century, see:
https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(Möbius)

 

[robphy]

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)

At the top of p. 13
( https://archive.org/details/mathematicaltheo00kleiuoft/page/13/mode/1up ),
preceding your quote,
Klein made a curious comment about "a fictitious, imaginary time"

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.

and then on p.16 he closes the chapter with (italics by Klein)

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.

So, it could be that aspects of this construction by Klein (1896)
was known by Minkowski, who later developed the 1907 spacetime viewpoint
to reformulate the physics of special relativity developed by Einstein (1905).

 

[accdd]

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]

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?

No, Galilean spacetime is not Euclidean either. What suggests Minkowski spacetime in particular is the existence of an invariant speed.

 

[robphy]

No, Galilean spacetime is not Euclidean either. What suggests Minkowski spacetime in particular is the existence of an invariant speed.

Both Galilean and Minkowski spacetimes have an invariant maximum signal speed:
infinity for Galilean,
a finite speed for Minkowski.

These speeds correspond to the eigenvectors of their respective boosts.

(The Galilean Spacetime has two degenerate line-elements:
In the (3+1)-case,
$ds^2=dt^2$
$dL^2= dx^2+dy^2+dz^2$
)

 

[PeroK]

Both Galilean and Minkowski spacetimes have an invariant maximum signal speed:
infinity for Galilean,
a finite speed for Minkowski.

IMO, "infinity" is not a speed, invariant or otherwise.

 

[robphy]

IMO, "infinity" is not a speed, invariant or otherwise.

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.

 

[Orodruin]

Both Galilean and Minkowski spacetimes have an invariant maximum signal speed:
infinity for Galilean,
a finite speed for Minkowski.

That’s just word play. I would not call ”infinity” as such an invariant speed.

 

[PeroK]

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.

Infinity is not a valid eigenvalue, if that's what you mean.

 

[Orodruin]

Infinity is not a valid eigenvalue, if that's what you mean.

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.

However, this is not necessarily true in more than one spatial dimension. For example, an object not moving parallel to the boost at c in SR will not have a tangent vector that is an eigenvector of the boost. That’s why we have aberration of light.

 

[robphy]

That’s just word play. I would not call ”infinity” as such an invariant speed.

To me, saying the maximum signal speed in a Galilean Spacetime is infinity
(following the notation in Minkowski’s https://en.m.wikisource.org/wiki/Translation:Space_and_Time)
[as one does in describing the “opening up of the light cone”]
is analogous to describing a plane as a sphere of infinite radius [as one does when describing plane waves].

Alternatively, one could use (say) $1/(c_{max})^2$ to be zero for the Galilean case (as in Ehlers’ Frame theory described in https://arxiv.org/abs/1910.12106 ), which akin to describing a plane as having curvature zero.These descriptions arise from projective geometry and can be written algebraically. One is certainly free to choose terms for these quantities (akin to folks being strict about metrics only being positive-definite).

 

[PeroK]

is analogous to describing a plane as a sphere of infinite radius

You can only push this sort of flaky mathematics so far before you end up with a contradiction.

 

[robphy]

You can only push this sort of flaky mathematics so far before you end up with a contradiction.

Descriptions are for intuition.
Equations are needed to support the intuition… and provide the context behind the description.
Descriptions alone (without equations) can lead to all sorts of misconceptions.

 

[PeroK]

Descriptions are for intuition.
Equations are needed to support the intuition… and provide the context behind the description.
Descriptions alone (without equations) can lead to all sorts of misconceptions.

The sphere and the plane are topologically distinct. The sphere is locally homeomorphic to the plane. Which is a better description than saying the plane is an infinite sphere- which I suggest is mathematically meaningless.

 

[robphy]

The sphere and the plane are topologically distinct. The sphere is locally homeomorphic to the plane. Which is a better description than saying the plane is an infinite sphere- which I suggest is mathematically meaningless.

Like I said, descriptions need context.
They can’t stand on their own.

Here are some equations in the Desmos below.
$E=1/(c_{max})^2$ in the following.
E=1 is Minkowski, E=0 is Galilean.
Call the terms whatever you want.
This construction is consistent with the physics. Feel free to suggest what may be better descriptions.
https://www.desmos.com/calculator/kv8szi3ic8

 

[Orodruin]

is analogous to describing a plane as a sphere of infinite radius [as one does when describing plane waves].

I think I formally disagree here too. The plane wave approximation is a good approximation far from the source. It is not a spherical wave. (Or vice versa)

A plane is indeed the limit (pointwise) of a set of spheres with increasing radius. It is not a sphere of infinite radius.

The entire mathematical concept of infinity is a tricky one and us physicists often use the nomenclature a bit haphazardly. If you assign infinity as the speed of something you open up the door to questions such as what $\infty + 1$ or $2\infty$ are.

But to the Galilean spacetime, the point is that there is no bound on the speed of information transfer.

 

[Sagittarius A-Star]

E=1 is Minkowski, E=0 is Galilean.
Call the terms whatever you want.
This construction is consistent with the physics. Feel free to suggest what may be better descriptions.

Also Einstein used a sloppy formulation. I understand, what he meant:

The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light $c$ in the latter transformation.

Source:
https://en.wikisource.org/wiki/Rela...art_I#Section_11_-_The_Lorentz_Transformation

I think, mathematicians would use a formulation with limit at infinity.
$\lim_{c \rightarrow \infty}{(\gamma (x-vt))} = x-vt, \ \ \ \ \$ $\lim_{c \rightarrow \infty}{(\gamma (t-vx/c^2))} = t$

Edit:

Source 1: Calculation of ...
Limit[(x - v t)/Sqrt[1 - v^2/c^2], c -> Infinity]
https://www.wolframalpha.com/input?i=Limit[(x+-+v+t)/Sqrt[1+-+v^2/c^2],+c+->+Infinity]

Source 2: Calculation of ...
Limit[(t - (v x)/c^2)/Sqrt[1 - v^2/c^2], c -> Infinity]
https://www.wolframalpha.com/input?i=Limit[(t+-+(v+x)/c^2)/Sqrt[1+-+v^2/c^2],+c+->+Infinity]

 

[robphy]

I think, mathematicians would use a formulation with limit at infinity.
$\lim_{c \rightarrow \infty}{(\gamma (x-vt))} = x-vt, \ \ \ \ \$ $\lim_{c \rightarrow \infty}{(\gamma (t-vx/c^2))} = t$

For clarity (and to minimize having to worry about limits), I formulate the idea as
$$
\left( \begin{array}{c} t' \\ \frac{x'}{c_{light}} \end{array} \right)
=
\left(
\begin{array}{cc}
\frac{1}{\sqrt{1-E\beta^2}} & \frac{E\beta}{\sqrt{1-E\beta^2}}
\\
\frac{\beta}{\sqrt{1-E\beta^2}} &
\frac{1}{\sqrt{1-E\beta^2}} & \end{array}
\right)\\
\left( \begin{array}{c} t \\ \frac{x}{c_{light}} \end{array} \right)
$$
where $\beta=v/c_{light}$, where $c_{light}=3\times10^8\ \mbox{m/s}$ [playing the role of a conversion constant].
For $E=0$ (galilean) or $E=+1$ (minkowskian) , one could think of $E$ as if it were $\left(\frac{c_{light}}{c_{max}}\right)^2$.
For $E=-1$, we essentially have a Euclidean rotation matrix.

These equations (based on Appendix A of Yaglom's book from https://www.physicsforums.com/threads/why-is-minkowski-spacetime-non-euclidean.1016402/post-6647305 ) are used in the "formulae" section of my
spacetime diagrammer https://www.desmos.com/calculator/kv8szi3ic8 ,
which visualizes the metric in the Euclidean, Galilean, and Minkowski cases.

(See also: Richter-Gebert's Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry https://www.amazon.com/dp/3642172857/?tag=pfamazon01-20 )

 

[PeroK]

Also Einstein used a sloppy formulation. I understand, what he meant:

Source:
https://en.wikisource.org/wiki/Rela...art_I#Section_11_-_The_Lorentz_Transformation

I think, mathematicians would use a formulation with limit at infinity.
$\lim_{c \rightarrow \infty}{(\gamma (x-vt))} = x-vt, \ \ \ \ \$ $\lim_{c \rightarrow \infty}{(\gamma (t-vx/c^2))} = t$

No matter how large you make $c$ I can choose a large $v$ and we see there is actually no physical difference in the geometry depending on the value of $c$.

In that sense the limit as $c \to \infty$ is not the Galilean geometry.

This is why mathematical analysis is needed so that we know which limits are valid and which are not.

 

[PeroK]

PS the simple and mathematically valid statement is that in Galilean relativity there is no invariant speed. Introducing an invariant speed of "infinity", IMO, is an unnecessary artifice.

 

[Sagittarius A-Star]

In that sense the limit as $c \to \infty$ is not the Galilean geometry.

In that sense, of course!

The "limit-equations" in posting #41 are valid, if they are regarded as purely mathematical equations, without any reference to physics.

For example, no dependency between the mathematical quantities "c" and "v" is assumed, besides the dependencies stated in those equations. You can't write down the GT and assume at the same time, that actual physics, like described in postulate 2 of SR, is valid.

 

[vanhees71]

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?

In a way that's right. You can derive the Lorentz transformation from the assumption that the special principle of relativity holds and assuming that for inertial observers space is Euclidean and time is homogeneous. Using these assumptions you get either the Galilei transformation and from this you can reconstruct Newtonian spacetime (having no limiting speed) or Minkowski spacetime (having a limiting speed). The Euclidean metric also occurs formally in the solutions, but then you want the transformations between inertial frames to form a group, and when using the Euclidean-metric solution then you'd not be able to set up a causality structure. That's possible in Minkowski spacetime, because for time- or light-like separated events the part of the Lorentz group that is continuously connected with the identity, the proper orthochronous Lorentz group $\mathrm{SO}(3,1)$ (in your sign convention for the metric) the time ordering is invariant, and that's why only time- or light-like separated events can be causally connected. For more details, see. e.g.,

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math.
Phys. 10, 1518 (1969), https://doi.org/10.1063/1.1665000

 

[Dale]

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.

For me individually, spacetime is clearly non-Euclidean since spacetime "distances" are measured with both clocks and rulers. I don't know how the distinction could possibly be more blatant.

 

[Dale]

Why, may I ask, do you prefer this sign convention over the other one?

I usually use the convention $ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$ and $c^2 d\tau^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2$, so I can easily switch between the timelike and spacelike conventions.

 

[robphy]

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math.
Phys. 10, 1518 (1969), https://doi.org/10.1063/1.1665000

The $K$ in that paper corresponds to the $E$ in my post #42
in the sense $E=sign(K)$:

cases (pg 1522)
["$K$ is a universal constant having the dimensions of an inverse-square velocity"]
(my $E$ is dimensionless because I have $c_{light}^2=(3\times 10^8\mbox( m/s))^2$ (instead of $1$) in the numerator):

• (A) Berzi K>0 (my E=+1) : Minkowski [eq 40]
• (B) Berzi K=0 (my E=0) : Galilean [eq 41]
• (C) Berzi K<0 (my E=-1): Euclidean [eq 42]

 

[PAllen]

No matter how large you make $c$ I can choose a large $v$ and we see there is actually no physical difference in the geometry depending on the value of $c$.

In that sense the limit as $c \to \infty$ is not the Galilean geometry.

This is why mathematical analysis is needed so that we know which limits are valid and which are not.

Well, while this is true as stated, there is a sense in which it is meaningful to talk about such a limit. Given any physical set up (distances in terms of reference objects, speeds define from these plus a notion of ideal clock), then as c is taken to approach infinity, the error of any observation compared to Galilean approaches zero. This is important because having speeds very low compared to c does not make relativistic effects arbitrarily small. No matter how small the speeds compared to c, if the system is large enough, relativistic effects are also large. Both speeds and distances must be 'small' compared to c or ct for relativistic effects to be insignificant.