TL;DR

Can a theory on 4D Euclidean space produce exact Lorentz transformations and an invariant signal speed while the underlying metric stays Euclidean and no Wick rotation is used?

Suppose a theory is formulated on a four-dimensional Euclidean space with a positive-definite metric.

Assume that:

• the metric always remains Euclidean;
• no coordinate is analytically continued (t \to it);
• no real coordinate transformation is claimed to convert the Euclidean metric into a Minkowski metric.

I do not mean approximate Lorentz invariance in a low-energy or long-wavelength limit. I mean exact special-relativistic kinematics.

The usual signature argument shows that a positive-definite Euclidean metric cannot be transformed into a Lorentzian metric by a real nonsingular coordinate transformation.

But does it also imply the stronger statement that exact Lorentzian kinematics cannot arise in a theory whose underlying geometric structure remains Euclidean?

In other words, are these two statements equivalent?

1. A Euclidean metric cannot be transformed into a Lorentzian metric.
2. A theory formulated on Euclidean space cannot produce exact Lorentz transformations in its physical sector.

Suppose a theory is formulated on a four-dimensional Euclidean space with a positive-definite metric.

Assume that:

• the metric always remains Euclidean;
• no coordinate is analytically continued (t \to it);
• no real coordinate transformation is claimed to convert the Euclidean metric into a Minkowski metric.

Can such a theory nevertheless contain a physical sector in which the Lorentz transformations hold exactly, with an exact finite invariant signal speed?

Transformartion equation (27) in the article can be derived without the 2nd postulate.
K<0 : can be excluded (causality would be not invariant)
K=0 : makes it the Galilean transformation
K>0 : makes it the Lorentz transformation
An experiment is needed to decide between Galilean- or Lorentz-transformation.

From this with twin paradox-experiments we know, that the Lorentz transformation is physically valid.

With unit system ##c=1##, in the ##xt##-plane:
LT
##(1)\ x'=\gamma (x-vt)##
##(2)\ t'=\gamma (t-vx)##
Add (1) and (2):
##(3)\ x'+t'=\gamma (x+t-v(x+t))=\gamma(1-v)(x+t)##
Subtract (2) from (1):
##(4)\ x'-t'=\gamma (x-t+v(x-t))=\gamma(1+v)(x-t)##
Multiply (3) and (4):
##(5)\ x'^2-t'^2=\gamma^2(1-v^2)(x^2-t^2)##
$$x'^2-t'^2=x^2-t^2$$
The Lorentz transformation is equivalent to Minkowski metric and not to 4D-Euclidean metric.

If you mean a 4d Euclidean space where one direction is timelike, I don't see what would pick out the dimension to call timelike. You can certainly draw such things (it's what a Minkowski diagram is), but there we impose a Minkowski metric on the diagram artificially and just arbitrarily pick a couple of directions to call null.

So my immediate thought is that you can impose Lorentzian dynamics on other geometries, but it has to be put in manually.

Some related work: analogue gravity (Barceló, Liberati & Visser, Analogue Gravity, Living Rev. Relativity 14, 3, 2011), effective Lorentzian metrics emerging from non-relativistic substrates. It's close to the spirit of the question, though not a full answer: it gives effective/emergent Lorentz invariance, not the exact kinematics the OP is asking about.

##\begin{pmatrix}
x'\\
t'
\end{pmatrix}
=
\frac{1}{\sqrt{1+v^2}}
\begin{pmatrix}
1 & -v\\
v & 1
\end{pmatrix}
\begin{pmatrix}
x\\
t
\end{pmatrix}##

##\begin{pmatrix}
x'\\
t'
\end{pmatrix}
=
\begin{pmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{pmatrix}
\begin{pmatrix}
x\\
t
\end{pmatrix},
\qquad
\theta=\arctan v##

The Lorentz transform treats time and space differently. Since in your space there is nothing that distinguishes the different directions any method that ends in the Lorentz transform would not be natural.

The only solution I can think of is something like a Minkowski diagram, which is a Euclidean plane where we impose Lorentzian physics by fiat rather than having it follow from the geometry.

But I think this may actually be the key point of my question.

PeterDonis, Dale and Ibix are right that in a genuinely Euclidean four-dimensional space none of the four directions is timelike. I agree. Simply choosing one Euclidean coordinate and calling it time would not solve anything.

So maybe the question is whether we have to identify the underlying four-dimensional geometry with physical spacetime in the first place.

Could the underlying geometry have four equivalent Euclidean directions, none of which is physical time, while physical time and the event structure arise at another level? And could the resulting structure then obey the Lorentz transformations exactly?

This is also why Roberto Pavani's comment about the problem of time seems relevant. If time has to be fundamental, I don't see how the construction could work. If time itself can emerge, the situation seems less obvious.

What I am trying to understand is whether there is a theorem that rules out this second possibility. The signature argument tells us that a Euclidean metric cannot be transformed into a Minkowski metric. But does it also tell us that a different physical spacetime structure cannot arise on top of an underlying Euclidean one?

Note that in this view, on the ## \mathbb{R}^4 ## side there will be no scale, because the two spaces are separate. The scale belongs to ## \mathbb{R}^{3,1} ##.

About the theorem, I doubt one can exist, simply because there are infinitely many possible mappings between the two spaces.

Note that analytic continuation means the Euclidean region is not an "overlay" on the Minkowski region; it's a different region, just as the imaginary axis in the complex plane is a different region from the real axis.

For example, Alexander Gersten published Euclidean Special Relativity in Foundations of Physics, and J. M. C. Montanus published a proper-time formulation there as well.

But after looking at these papers, I am not sure they solve the problem we are discussing. Gersten starts from the usual Minkowski interval and then introduces Euclidean coordinates. In the proper-time approach, proper time is already used as the fourth Euclidean coordinate.

So in both cases a substantial part of the relativistic structure seems to be there from the beginning. This looks close to Ibix's point that one can put Lorentzian physics on top of another geometry, but then it has basically been put in by hand.

What I am wondering is whether the harder version is possible: can exact SR arise naturally from a genuinely Euclidean starting point, rather than being encoded into it?

Maybe the signature itself is not the main problem. Perhaps the real question is whether time and the physical event structure have to be fundamental in the first place.

I have never seen such a thing, and I don't think it would be possible. Particularly not the "naturally" qualification. There is no natural way to single out one dimension.

@Ans if you have a reference to an actual published paper (meaning a link to where it is posted online, which journal, etc.--arxiv.org preprint pages will have this information if a paper has been published) that claims that what you are asking about is possible, you can provide it to me by PM and the moderators will review it to see if it merits further discussion.