I am an independent researcher with a formal background in physics: I hold an MSc in physics, studied in a postgraduate physics programme, and have earlier peer-reviewed publications from that period. I am currently outside an academic institution, so my opportunities for ordinary academic discussion are limited. For this reason, I would like to ask for technical criticism of a recently published paper.

The paper is:

A. N. Smirnov, Special Relativity as an Emergent Structure in a Timeless Euclidean Model, International Journal of Quantum Foundations, Vol. 12, Issue 2, pp. 272–312, 2026.

Article page:
https://ijqf.org/archives/8065

PDF:
https://ijqf.org/wp-content/uploads/2026/03/IJQF2026v12n2p13.pdf

I understand that IJQF is a specialized foundations journal rather than a high-visibility mainstream physics journal. I am not presenting the publication venue as a substitute for technical criticism. I mention it only to indicate that the paper has passed external peer review and that there is a complete text available for checking.

The point I would like to discuss is deliberately limited. The paper does not claim to replace special relativity or to derive all of relativistic physics. It addresses a narrower question: whether the local kinematic-causal core of special relativity can be reconstructed from a timeless four-dimensional Euclidean model under explicit operational assumptions.

The starting point is not Minkowski spacetime. The fundamental structure is a four-dimensional Euclidean space (E^4) with Euclidean metric, a scalar field (\Phi), and the equation

[
\Delta_{E^4}\Phi = 0.
]

There is no fundamental time coordinate, no fundamental Lorentzian metric, and no pre-given global spacetime event set.

The key idea is that an observer is treated as an internal localized reconstruction system, not as an external observer added to the model. A foliation of (E^4) by three-dimensional hypersurfaces introduces an operational ordering parameter. Events are then not assumed as primitive spacetime points; they are reconstructed as stable registrations in the internal degrees of freedom of such an observer.

The paper distinguishes two types of transformations.

First, there are direct transformations of the underlying Euclidean structure. These remain Euclidean.

Second, there are observed transformations between reconstructed event descriptions. These act not on the bare Euclidean space, but on the operationally reconstructed event-causal structure. The claim is that, under the stated assumptions of admissible reconstruction, these observed transformations acquire the Lorentz form.

This is not intended as a Wick rotation, nor as a coordinate relabelling of Euclidean space into Minkowski space. The Euclidean metric remains fundamental. The Lorentzian structure is claimed to emerge only at the level of reconstructed events and observed transformations.

The broader research programme contains potentially discriminating consequences, but those are not the subject of this particular paper. In this thread I would prefer to focus only on the special-relativistic kinematics step.

I would especially appreciate criticism of the following points:

1. Is the distinction between direct Euclidean transformations and observed transformations mathematically and conceptually clear?

2. Does the Lorentz-like form of the observed transformations genuinely follow from the stated reconstruction assumptions, or is some part of special relativity implicitly assumed?

3. Is the limiting speed (v_{\max}) merely postulated, or is it adequately defined operationally as a property of admissible causal reconstruction?

4. Are the assumptions of local stationarity, stable event reconstruction, and observer-independent event matching too strong to count as a reconstruction of SR kinematics?

5. Is there a clearer way to formulate the role of the internal observer without making it look like an external conscious observer or an extra physical postulate?

I am not asking readers to accept the broader programme. I am mainly interested in whether this first step — the reconstruction of local SR kinematics — is logically coherent, or whether there is a hidden assumption that already contains the Lorentzian structure.

I have not found how to write latex equations here, so post above is not well formatted

Have you read

Einstein's Theory of Relativity

by Max Born ?

He discusses your question starting at p238, but understandably this is in the middle of Max' development of the subject so you may need to look back a bit as well.

A piece of maths you may not know is the difference between finite rotations, which are non commutative and infinitesimal rotations which are commutative.

This and the fact that the coordinate infinitesimals go to zero in the limit allow linear mathematics to be employed in differential geometry.

All treatments rely heavily on these two facts of maths.

Thank you for the reference. I have not read that particular passage recently, but I think the issue in my paper is different.

The paper does not try to transform Euclidean space into Minkowski spacetime. A real coordinate transformation cannot change the signature of a metric, so the Euclidean and Minkowski metrics are not globally equivalent.

In the model, the underlying space remains Euclidean. Special relativity is not fundamental there; it is reconstructed as an effective event-causal structure for an internal observer. The Lorentz form appears only for observed transformations between reconstructed event descriptions, not as a direct transformation of the Euclidean metric.

11 minutes ago, andsm said:

The paper does not try to transform Euclidean space into Minkowski spacetime. A real coordinate transformation cannot change the signature of a metric, so the Euclidean and Minkowski metrics are not globally equivalent.

I agree and I wasn't suggesting that.

The mathematics of Euclidian space of any number of dimensions is linear - which brings a host of computational benefits and why we always try it first.

But the mathematics of Special Relativity is non linear. The simplest non linear geometry that fits the bill is hyperbolic geometry.

But even in non linear geometry the differential components still obey linear maths as I said so your equation (delta presumably refers to these differential components ?) can still be worked with, as can similar Maxwell equations.

Thank you, that helps to clarify the point.

Just one technical clarification: in the paper Δ_E⁴ denotes the Euclidean Laplacian on the underlying four-dimensional Euclidean space. The basic equation is therefore

Δ_E⁴ Φ = 0.

It is not a wave equation and it does not contain a fundamental time parameter.

This is important for the interpretation of the model. At the fundamental level there is no time coordinate, no temporal evolution, no Lorentzian metric, and no pre-given spacetime event set. The ordering parameter that later plays the role of time appears only after choosing a foliation and considering an internal observer capable of stable event reconstruction.

So I agree that standard SR can be related to hyperbolic geometry, rapidity space, and local linearization. But the question in the paper is different. It is not whether one can locally approximate a Lorentzian or hyperbolic geometry. The question is whether an effective Lorentzian event-causal structure can be reconstructed from a timeless Euclidean harmonic-field model.

In this sense, special relativity is not fundamental in the model. It is claimed to arise only as an effective structure of observed transformations between reconstructed event descriptions.

Edited June 26Jun 26 by andsm

2 hours ago, andsm said:

Thank you, that helps to clarify the point.

Just one technical clarification: in the paper Δ_E⁴ denotes the Euclidean Laplacian on the underlying four-dimensional Euclidean space. The basic equation is therefore

Δ_E⁴ Φ = 0.

It is not a wave equation and it does not contain a fundamental time parameter.

This is important for the interpretation of the model. At the fundamental level there is no time coordinate, no temporal evolution, no Lorentzian metric, and no pre-given spacetime event set. The ordering parameter that later plays the role of time appears only after choosing a foliation and considering an internal observer capable of stable event reconstruction.

So I agree that standard SR can be related to hyperbolic geometry, rapidity space, and local linearization. But the question in the paper is different. It is not whether one can locally approximate a Lorentzian or hyperbolic geometry. The question is whether an effective Lorentzian event-causal structure can be reconstructed from a timeless Euclidean harmonic-field model.

In this sense, special relativity is not fundamental in the model. It is claimed to arise only as an effective structure of observed transformations between reconstructed event descriptions.

I don't recall mentioning a wave equation.

However I (and I expect others) would prefer you used conventional notation when referring to the Laplacian, though I did understand that is what you mean.

However the Laplacian is an operator.
As such it only has meaning in terms of what it operates on.
And that is not E, if you are using E as the standard symbol for Euclidian.

You have a missing variable - usually the field variable under consideration.

The point I am trying to make to you is that (your) delta is a linear combination of differentials - a linear form in other terminology - whcih is why it is permissible to use it as you are proposing.

There are however such forms as non linear forms (ie non linear combinations of differentials). you can't use tensors for these either.

This is all meant to help, nothing more.

Thank you, this is helpful.

I did not mean to imply that you had mentioned a wave equation. I mentioned it only to clarify that the basic equation in the model is elliptic, not hyperbolic, and that it contains no fundamental time parameter.

I agree that the notation should be made more explicit. In my notation, [math] E^4 [/math] was not meant as the argument of the operator. It labels the underlying four-dimensional Euclidean space, or equivalently the Euclidean metric with respect to which the Laplacian is defined.

More explicitly, if \(x^A\), \(A=1,\ldots,4\), are Euclidean coordinates and \(\Phi\) is the scalar field, the equation is

\( \Delta \Phi = \delta^{AB}\partial_A\partial_B\Phi = 0 .\)

Equivalently,

\( \sum_{A=1}^{4}\frac{\partial^2\Phi}{\partial (x^A)^2}=0 . \)

So the operator acts on the scalar field \(\Phi\). The subscript \(E^4\) was only meant to indicate that this is the Euclidean Laplacian on the underlying four-dimensional space.

I also agree that this is a linear differential operator. However, I would not describe the Laplacian itself as a differential form. In the paper it is used simply as a linear second-order differential operator acting on the fundamental scalar field.

The main physical point remains that this equation is imposed on a timeless Euclidean structure. There is no fundamental time coordinate or temporal evolution in the starting model. The effective time parameter appears only later, through foliation and stable reconstruction by an internal observer.

Edited June 26Jun 26 by andsm

Perhaps I should restate the intended issue more directly.

At first sight, the claim may look impossible: how can special relativity arise from a Euclidean, timeless model? The usual immediate objections are quite natural: Euclidean and Minkowski metrics have different signatures; a Euclidean space cannot be converted into Minkowski spacetime by an ordinary coordinate transformation; a Laplace equation has no time evolution; and Lorentz transformations should not simply appear from Euclidean rotations.

I agree with these objections in their direct form. The paper does not claim that Euclidean space is globally transformed into Minkowski spacetime. It does not use a Wick rotation, and it does not reinterpret Euclidean rotations as Lorentz transformations.

The claim is narrower. The underlying level remains Euclidean and timeless. Special relativity appears only as an effective structure of reconstructed events and observed transformations for an internal observer. In other words, the Lorentzian structure is not fundamental in the model; it is claimed to arise at the operationally reconstructed level.

One purpose of this thread is to collect the standard objections to this kind of construction. Some of them are already addressed in the paper, but I would like to understand which objections readers regard as decisive, unclear, or insufficiently answered.

So the question is not whether Euclidean space can be directly turned into Minkowski spacetime. It cannot.

The question is whether the stated reconstruction assumptions are sufficient to produce local special-relativistic kinematics as an effective event-causal structure, or whether some Lorentzian structure is still hidden in the assumptions.