I Why Is Minkowski Spacetime Non-Euclidean?

[PeroK]

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.

Why invent something (an infinite or near infinite $c$) that is simply unnecessary?

 

[Sagittarius A-Star]

Why invent something (an infinite or near infinite $c$) that is simply unnecessary?

In the "limit-equations" in posting #41, c is not set to infinite, nor to a finite value "near infinite", as can be see from the definition of limit at infinity.

 

[PeroK]

In the "limit-equations" in posting #41, c is not set to infinite, nor to a finite value "near infinite", as can be see from the definition of limit at infinity.

In post #41 you reach the sort of contradiction I am talking about. You show that, if we choose a fixed $v$, then in the limit as $c \to \infty$, we have $\gamma \to 1$.

But, as $c \to \infty$ it is not the case that for all $v$ we have $\gamma(v) \to 1$. Because, any $v < c$ is allowed.

And, in fact, the whole idea of setting $c =1$ shows that as $c \to \infty$ the physics does not gradually change depending on how large we make $c$.

In those terms Galilean relativity is not a continuous limit of Special Relativity as $c \to \infty$.

 

[Sagittarius A-Star]

In those terms Galilean relativity is not a continuous limit of Special Relativity as $c \to \infty$.

That is what you wrote already in #43 and I replied to in #45.

 

[PeroK]

That is what you wrote already in #43 and I replied to in #45.

I don't agree with post #45. The conclusion that if you increase the invariant speed $c$ then you approach Galilean relativity in some limiting sense is false. The limit argument doesn't work, IMO.

You only get Galilean relativity by plugging $c = \infty$ directly into the equations. That works, but seems an unnecessary, unphysical artifice.

 

[Sagittarius A-Star]

The conclusion that if you increase the invariant speed $c$ then you approach Galilean relativity in some limiting sense ...

I didn't write or conclude this in #45.

You only get Galilean relativity by plugging $c = \infty$ directly into the equations.

That is the sloppy formulation from Einstein, I wanted to avoid.

 

[PeroK]

That is the sloppy formulation from Einstein, I wanted to avoid.

You can't avoid it. That's my point. The limit argument fails because the Lorentz Transformation entails transforming between any IRFs. And, however large you make $c$ you have all $v < c$ to consider.

 

[Sagittarius A-Star]

You can't avoid it. That's my point. The limit argument fails because the Lorentz Transformation entails transforming between any IRFs. And, however large you make $c$ you have all $v < c$ to consider.

That is again, what you wrote already in #43 and I replied to in #45:

... 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.

 

[PeroK]

That is again, what you wrote already in #43 and I replied to in #45:

But, there is a mathematical dependency between $v$ and $c$. Relativity is about the group of Lorentz transformations. Not just a transformation for a given $v$.

What is valid is that, for a fixed $c$, as $v \to 0$ then the LT does tend to the Galilean transformation. That's why Galilean relativity is valid for $v << c$.

But SR definitely does not tend to Galilean relativity as $c \to \infty$.

 

[vanhees71]

The Galilean limit of the Poincare group is indeed not trivial. This becomes particularly clear in quantum theory: While the Poincare group has not non-trivial central charges, and you just have to deal with the unitary irreps, the Galilei group has a central charge, which physically is the mass, and you have to deal with ray representations (or the unitary representations of the corresponding central extension of the "classical" Galilei group).

 

[Sagittarius A-Star]

What is valid is that, for a fixed $c$, as $v \to 0$ then the LT does tend to the Galilean transformation.

If you mean $\lim_{v \rightarrow 0}{}$, then you get only a special case of the GT

$\lim_{v \rightarrow 0}{(\gamma (x-vt))} = x-0 \cdot t = x, \ \ \ \ \$ $\lim_{v \rightarrow 0}{(\gamma (t-vx/c^2))} = t$

 

[PeroK]

If you mean $\lim_{v \rightarrow 0}{}$, then you get only a special case of the GT

$\lim_{v \rightarrow 0}{(\gamma (x-vt))} = x-0 \cdot t = x, \ \ \ \ \$ $\lim_{v \rightarrow 0}{(\gamma (t-vx/c^2))} = t$

You're missing the point. If $v$ is small, then $\gamma \approx 1$. And you can study a subset of SR where $v$ is small and you have approximately the Galilean transformation.

If $c$ is large, then not all gamma factors are approximately $1$. You don't have an approximation to Galilean relativity.

 

[Sagittarius A-Star]

If $v$ is small, then $\gamma \approx 1$. And you can study a subset of SR where $v$ is small and you have approximately the Galilean transformation.

That's correct, except if the $x$ in $\gamma (t-vx/c^2)$ is very large, as @PAllen mentioned in #50.

 

[Dale]

as c is taken to approach infinity, the error of any observation compared to Galilean approaches zero

This is really the scientifically important fact. For SR to be a valid theory the differences between it and Galilean relativity must be smaller than the experimental uncertainty in any domain where Galilean relativity has been experimentally validated. You can express that requirement with several different limits, but the important fact is that in any of those limits the difference must approach zero.