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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.

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.

\( L \)

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.

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.

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: Is the distinction between direct Euclidean transformations and observed transformations mathematically and conceptually clear? 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? Is the limiting speed (v_{\max}) merely postulated, or is it adequately defined operationally as a property of admissible causal reconstruction? Are the assumptions of local stationarity, stable event reconstruction, and observer-independent event matching too strong to count as a reconstruction of SR kinematics? 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

I have two types of transformations that follow from the hypothesis. The hypothesis shows their presence, but does not describe these transformations and does not describe their parameters. The first type, transformations from the observer's point of view, preserve events when switching between IFRs. We look to see if there are such transformations, and we easily find them - SR and Lorentz transformations. SR is not derived from the hypothesis, we can only show that the hypothesis is compatible with SR and Lorentz transformations. The second type of transformations, direct transformations, should be described by the theory based on the hypothesis. Finding the parameters and equations of direct transformation is not the task of this hypothesis. As I have written many times, I consider only the principle of causality, without relying on any of the physical theories or any other principles. The principle of causality was formulated long before the advent of SR. For example, in Aristotle's "Metaphysics" one can find what can be called one of the early formulations of the principle of causality. I do not remember the exact formulation according to Aristotle now, I read it several years ago. The link provided does not set forth the principle of causality, but the relativistic principle of causality. This is the principle of causality based on SR. Obviously, it differs from what I am considering. As I have already written, the hypothesis suggests that the causes and cause-and-effect relationships, including history, may differ in different IFRs. Do I understand correctly that when analyzing step-by-step the results of the hypothesis, logical errors, incorrect or unproven conclusions are not visible? I have already answered this. The task of this hypothesis is to show the fundamental possibility of creating a new class of theories. Finding this transformation is the task of such a theory, but not the task of this hypothesis. I don't have time to answer other messages today. I'll write a reply tomorrow.

The idea is not the same as described, although it is remotely similar. The hypothesis assume that a single space-time, with a single set of events and cause-and-effect relationships, common to all IFRs, does not exist. Instead, each IFR has its own space-time, with its independent set of events and cause-and-effect relationships, with its own history. That is, instead of one space-time with its own history, our Universe consists of many space-times with their own events and with their own history of events. An observer observes only in one IFR - the one relative to which he is stationary. An observer can change his speed and move to another IFR, with its own space-time and cause-and-effect relationships. If events in different IFRs were completely independent, then a person would cease to exist when changing speed. Which obviously contradicts everyday experience. Therefore, events in different IFRs cannot be completely independent; there must be some dependence. For a person to exist, it is necessary that the smaller the difference in the speeds of two IFRs, the smaller the difference in events between these two IFRs. Since an observer observes only in one IFR, he always observes a self-consistent picture. When the IFR changes, history, according to the described model, changes. An observer, when receiving information from another observer who is moving with some non-zero relative speed, cannot receive information about events that are not in his IFR. That is, the received signal may differ from the sent signal, and the received signal is always consistent with the cause-and-effect relationships of the IFR in which it is received. What arises is what can be called a weak information barrier between IFRs. The barrier is weak, because information about events that occurred in the IFRs of both the first and second observers can be transmitted. It turns out that an observer, with any exchange of information, with any change in his speed, will receive only information consistent with the events and cause-and-effect relationships of his current IFR. As a result, this means that from the observer's point of view, events do not change when the IFR changes. Two types of transformations arise. The first type, transformations from the observer's point of view. In this type of transformation, events are preserved when the IFR changes. The second type of transformations, direct transformations, describe what actually happens. The causality principle says that based on the previous state, one can obtain the state at subsequent moments of time if all boundary conditions are known. The state here can also be a wave function, based on which one can obtain the probability of the system being in a certain state during measurement. A certain evolution operator arises, let's call it A. For the hypothesis, some properties of this operator are irrelevant as long as it is applied in some IFR. If, according to some physical theory, the operator must have some properties, and this theory implies that events in all IFRs are the same, then everything is fine, we say that this theory is not fundamental, based on transformations from the observer's point of view. If the hypothesis is true, then all existing theories, including SR/GR/QFT, are not fundamental. SR imposes some restrictions on the evolution operator, and SR implies the sameness of events in all IFR. Therefore, it is compatible with this hypothesis. The Minkowski space here is a tool of SR, and has nothing to do with direct transformations. The previous part of the answer consisted mainly of copying what was already written in the article or in other posts. Here I will simply give a link to the previous answer on this topic: My hypothesis is based on the fulfillment of the principle of causality. In SR, when the speed of light is exceeded, as far as I remember, causality is violated. Anything that violates the principle of causality is incompatible with my hypothesis. The first point in the compatibility requirements says exactly this. Therefore, superluminal speed is incompatible with the hypothesis when fulfilling SR. But if we consider Galilean transformations, then superluminal speed does not violate causality, and Galilean transformations with superluminal speed are compatible with the hypothesis.

As I have already written, for any theory to be compatible with a hypothesis, it must satisfy the following two conditions: • Rely on the principle of causality • Assume that any event exists in all IFRs. As is easy to understand, SR satisfies these requirements. Galilean transformations also satisfy these requirements. Generally speaking, it can be argued that any modern physical theory satisfies these requirements. The speed of information transfer, as is easy to notice, is not mentioned in the conditions. The derivation of the above conditions for compatibility is given in the article, and has been written here in the topic many times. Therefore, they can be checked. If there are any comments on any steps of the derivation, they can be written. So far, I have not seen a single comment. If there were any, I would like to see them. Returning to the question about mathematics. There are no comments on the derivation of the conditions described above. It turns out that you want a mathematical proof that SR fulfills the described conditions, that is, relies on the principle of causality and assumes the immutability of events when the IFR changes? I have written, step by step, how the conclusions I write are made. These steps are easy to check, to show that this and that step contain errors. So, as far as I see, evidence is provided. Model was also provided. Not a single error was found. If you disagree with this statement, show the post where these errors are shown.

I'm trying to figure out what kind of mathematics you want to see, and proof of what you want to see. First, what I believe has already been done within the framework of the hypothesis: 1. It is shown that the hypothesis implies that events and causal relationships in different IFRs may differ. This is a direct consequence of independently applying the causality principle to different IFRs 2. It is proven that from the observer's point of view, events in all IFRs are the same, even if in fact they are different 3. It is proven that two types of transformations arise in the hypothesis when changing IFRs. The first type of transformation, the transformation from the observer's point of view, preserves events. The second type of transformation describes how events actually change. 4. From the observer's point of view, events are preserved; the hypothesis derives a type of transformation with preservation of events. It follows that the hypothesis is compatible with all physical theories that fulfill the following conditions: a. Rely, directly or indirectly, on the causality principle b. It is assumed that events in all IRS are the same The principle of causality in the hypothesis is considered in the most general form, without relying on any physical theories or any other principles. It does not follow from the hypothesis that it is SR that describes the transformations from the point of view of the observer. SR satisfies the requirements to be compatible with the hypothesis. Galilean transformations also satisfy the requirements to be compatible with the hypothesis. Many other attempts to construct transformations, for example, with attempts to introduce small modifications to the Lorentz transformations, also satisfy this hypothesis. SR is a well-tested and widely accepted theory. And this theory is compatible with the hypothesis. For these two reasons, I write that SR describes the transformations from the point of view of the observer. SR is not derived from this hypothesis. QM/QFT also satisfies the described requirements. Therefore, it is also compatible with the hypothesis. You can write that such and such a point is incorrect, for such and such reasons. But, if we do not argue with the written points 1-4, then this means that it has been proven that the hypothesis is compatible with SR and with QM/QFT I would like to point out that these are no longer questions about the logical integrity of the hypothesis and how compatible it is with existing physical theories. These are questions about whether it is possible to construct a theory that would describe our Universe based on this hypothesis. In principle, one could try to refute this hypothesis if one could prove that there are no ways to construct such a theory. I would like to point out that such a refutation would be a significant scientific achievement, because it would prove the fundamental nature of space-time. After that, one could continue to compose space-time from some parts, as in dynamic causal triangulation, but it would be clear that nothing deeper than space-time exists. The question talks about the mapping between particles in different IFRs. Where does it follow that such a mapping should exist at all? A few messages above, I wrote equations that show how events in different IFRs are related to each other, given the presence of something more fundamental than space-time. And, as noted there, it does not follow from anywhere that the inverse operator exists at all. If the inverse operator does not exist, then it will be impossible to accurately determine the state of another IFR based on the state of one IFR. Generally speaking, without the inverse operator, it is impossible to even say which space-time point in IFR1 corresponds to a space-time point in IFR2. Can we say that it is impossible to create a theory based on this hypothesis? I do not see such evidence. I note that my task, as the author of the hypothesis, does not include proving that such a theory can be created. My task is only to prove that the hypothesis does not contradict widely accepted theories in their well-tested area, and to show that the hypothesis is, in principle, testable. Which, as it seems to me, I have done. If anyone thinks that a theory based on this hypothesis cannot be created. The article with the theory provides an example of a model of a hypothetical universe in which this hypothesis is fully realized. So, the theory can be created, and this can be considered proven. What is not proven is that based on this hypothesis it is possible to create a theory that would describe our Universe. This is a question for further research, but already within the framework of attempts to build theories based on this hypothesis. If the hypothesis is true, then all existing theories, including QFT and QCD as part of it, are not fundamental. They satisfy only the transformations from the observer's point of view. Therefore, they must be replaced by another, more fundamental theory, which will satisfy both transformations of this hypothesis. And at the same time I prove that my hypothesis does not contradict modern widely accepted theories, including those mentioned. And somehow no one has yet shown where the error is in the proof that the hypothesis does not contradict these theories. The conclusions of the hypothesis are not liked, it is obvious, but it is not possible to show where exactly in the conclusions something is wrong.

I consider causality in the most general way, not relying on anything except causality. This means that I do not rely on either STR or QM/QFT. Therefore, I do not need to derive STR, I only need to show that the hypothesis is compatible with STR. To do this, all that is needed is to show the presence of transformations that preserve events. After reading what you wrote above, I have a question. Is it clear how this hypothesis derives that from the observer's point of view, events in different IFRs are the same? If space-time is fundamental, then the transition between IFRs is just a change of coordinate system. During such a transition, events and cause-effect relationships cannot change. The hypothesis assumes that during the transition between IFRs, events and cause-effect relationships can change. Therefore, this hypothesis and the fundamentality of space-time are incompatible.

In this article, I do not aim to derive the SR. The aim is only to show that the hypothesis is compatible with the SR. But it is not difficult to show how to derive the SR. Let us verify whether the special theory of relativity, taken together with the corresponding transformations, is transformations of space-time-fields from the viewpoint of observer. Let us list the conditions under which it will be possible to assert this univocally: 1. Events are same in all frames of reference, from the viewpoint of observer 2. The principle of causality connects events in all frames of reference, from the viewpoint of observer 3. Physical laws are the identical in all frames of reference 4. The speed of light in vacuum is the same in all frames of reference It can be easily seen that the conditions listed above describe the explicit and implicit postulates of the special theory of relativity. 3 and 4 are implemented through restrictions on the operator A in each of the IFRs The invariants of SR, from the observer's point of view, are fulfilled exactly. Events, from the observer's point of view, are preserved exactly, not approximately, when transitioning between IFRs. You trying to evaluate the hypothesis by some part of its consequences. At the same time, ignoring other consequences that resolve the apparent contradiction to observations. The hypothesis implies that space-time is not fundamental, that there is something more fundamental than space-time. Particles exist in space-time. Since space-time, according to the hypothesis, is not fundamental, it is not clear how particles can be fundamental. It follows that perhaps there is a continuum of options between a photon and a muon. Are there arguments showing that there are errors in the conclusion of the statement that the hypothesis is compatible with SR? If there are, I would like to see them. If there are no such arguments, then answering the second part of the question - this is a direct transformation. The task of finding equations describing these transformations is the task of creating an theory based on this hypothesis. This hypothesis only shows the possibility of constructing such a theory. I suggested considering step by step the conclusion that events, from the observer's point of view, are the same in all IFRs, although in fact the events differ. But you, in fact, rejected such a proposal. In this question, the conclusion of the hypothesis and, in fact, the hypothesis itself are confused. The hypothesis itself, I will write once again, is that the principle of causality is applied to different IFRs independently. Further consequence - Independent application of the principle of causality means that from the fact that such and such an event occurred in some IFR, or that there is such and such a cause-and-effect relationship there, it does not follow that in another IFR there is this event or this cause-and-effect relationship.

I thought about expanding this part of the article, with a description of the causality principle. The article was sent for review by many journals. One of the reviewers wrote in his review that for a radioactive atom, equation 1 from my article is not satisfied, the decay has no cause, therefore the hypothesis as a whole is not true. The journal does not belong to the first quantile of Scopus, but it is still surprising to see such a quality of review. After receiving this review, I expanded the description, explained that the causality principle is not violated here. I consider a more detailed and expanded description unnecessary, after all, the article is not written in defense of the causality principle.

It is shown that the hypothesis is compatible with SR. Modern quantum field theory relies on gauge symmetries. SR and U(1) symmetry are closely related. SR is a transformation from the observer's point of view. From this we conclude that U(1) symmetry is also satisfied only from the observer's point of view. And here an open question arises - can all other gauge symmetries of the Standard Model be derived from symmetries from the observer's point of view?

About invariants. Transformations from the observer's point of view coincide with the transformations of SR, here the invariants are clear. It is necessary to understand that these are not real invariants, but invariants from the observer's point of view. When moving to another IFR, they can be violated, although for the observer everything will look like the invariants are preserved. Now let's consider direct transformations. Direct transformations should describe how everything changes in reality, and not from the observer's point of view. To begin with, what are the restrictions on these transformations? The presence in the hypothesis of transformations from the observer's point of view, under which events are preserved, means that the hypothesis is completely compatible with all existing physical theories. Therefore, direct transformations do not require any restrictions from existing theories. Further, there is a restriction described in the article, deduced from the fact of the existence of a human. If events in different IFRs are completely independent of each other, this means that a person, changing his speed, will cease to exist. When the speed changes, he will move from one IFR with some events to another IFR with completely independent events, and there is no reason for his body to continue to exist. This obviously contradicts everyday experience, so a limitation on the degree of difference of events in the IFR arises. The smaller the difference in the speed of the IFR, the smaller the difference in events should be. As the difference in the speed of two IFRs tends to zero, the difference in events between them should also tend to zero. It turns out that only this limitation affects direct transformations. Now, from the hypothesis it follows that there must be something more fundamental than space-time. Space-time and state must be derived from this something more fundamental, separately and independently for each IFR. Let's write an equation for this: \[ \Psi(t) =B(L,t) \Omega \] Here \[ \Psi \] is a state, \[ \Omega \] is that something more fundamental, \[ B(L,t) \] operator, which allows us to obtain the state at time t for the IFR \[ L \]. For another IFR, for \[ L^{'} \] , the equation will be: \[ \Psi^{'}( t^{'})= B(L^{'},t^{'}) \Omega \] In order to determine the state in another IFR from the state in the first IFR, we need to obtain \[ \Omega \]. But for this, the inverse operator \[ B^{-1} \] must exist, which does not follow from anywhere. For an invariant to exist, this inverse operator must exist. But since its existence does not follow from anywhere, this means the absence of invariants and direct transformation. Invariants cannot exist in the general case, but for some \[ \Omega \], where the inverse operator exists, they can exist. We consider causality in the most general form, so the conclusion arises about the absence of invariants in the direct transformation. Are there any restrictions on \[ \Omega \] ? Yes, there are. Although, as a consequence of the hypothesis, modern widely accepted theories are satisfied only within each IFR. Within each IFR, the causality principle is satisfied: \[ \Psi (t+\Delta t) = A(t+ \Delta t) \Psi (t) \]. For the known physical theories to be satisfied, the operator \[ A \] must be subject to appropriate restrictions, although only within each IFR, without restrictions on the transition between IFRs. As a result, it must be true: \[ \Psi (t+\Delta t) = A(t+ \Delta t) \Psi (t) = A(t+ \Delta t) \Psi (t)= A(t+ \Delta t) B(L,t) \Omega \] Since \[ \Psi (t+\Delta t) = B(L,t+\Delta t)) \Omega \] then \[ B(L,t+\Delta t) \Omega = A(t+\Delta t) B(L,t) \Omega \] As a result, a number of restrictions on \[ \Omega \] appear, based on which it is possible to calculate what is more fundamental than space-time.