arxiv: 2604.07014 · v1 · submitted 2026-04-08 · ⚛️ physics.gen-ph

Recognition: 2 theorem links

· Lean Theorem

Relativity: A matter of causality

Antonio Pineda

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:06 UTC · model grok-4.3

classification ⚛️ physics.gen-ph

keywords causalityinertial framesLorentz groupdilatationsreference framesfinite signal speedspecial relativitytransformations

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The pith

Causality plus a purely geometric definition of inertial frames fixes their transformations to the orthochronous inhomogeneous Lorentz group times dilatations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper begins with distinct observers who observe unique events and can communicate only after a finite time, which imposes a maximal information-transfer speed between any reference frames. It defines inertial frames strictly by the geometry of spatial distances, with no appeal to physical laws, electromagnetism, or relativity itself. Once causality is imposed on these frames, the allowed transformations are forced to be exactly those of the orthochronous inhomogeneous Lorentz group extended by dilatations. This shows that the causal structure of special relativity can be recovered from communication constraints and distance geometry alone. A reader cares because the result suggests the Lorentz symmetries are not an extra assumption but follow directly from requiring that events stay unique while signals travel at finite speed.

Core claim

By defining inertial reference frames through the geometrical properties of spatial distance without reference to any physical laws, and requiring that causality holds with finite communication time between observers, the allowed transformations between such frames are fixed to be the orthochronous inhomogeneous Lorentz group times dilatations.

What carries the argument

The causality condition, which enforces that distinct observers communicate with finite maximal velocity while preserving unique events, applied to geometrically defined inertial frames.

Load-bearing premise

Inertial reference frames can be defined by fixing only the geometrical properties of spatial distance, with no reference to relativity, electromagnetism, or any laws of physics.

What would settle it

An observed transformation between two inertial frames that preserves event uniqueness and finite communication speed yet lies outside the orthochronous inhomogeneous Lorentz group extended by dilatations would falsify the claim.

read the original abstract

We take causality and uniqueness of events observation as our driving forces. They are built in in the way we define distinct observers, which then require a finite time to communicate between each other. This unavoidably leads to the existence of maximal transfer-information velocity between arbitrary (not necessarily inertial) reference frames. Inertial reference frames are defined by fixing the geometrical properties of (spatial) distance without any reference to relativity, electromagnetism, or laws of physics in general. For these inertial reference frames, the causality condition fixes the causal group to be the orthochronous inhomogeneous Lorentz group times dilatations. The mathematics we will use are quite basic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper re-derives the orthochronous inhomogeneous Lorentz group plus dilatations from causality and event uniqueness, but the geometric definition of inertial frames likely supplies the key structure rather than deriving it.

read the letter

The paper starts from causality and the uniqueness of events, builds in finite communication time between observers, and then defines inertial frames by fixing the geometrical properties of spatial distance with no reference to physics. From that setup it concludes that causality forces the transformations to be the orthochronous inhomogeneous Lorentz group times dilatations. The math is kept elementary, which is a plus for accessibility, and the starting assumptions are stated plainly in the abstract. That minimalism is the main thing the work does well. It avoids dragging in electromagnetism or dynamics and tries to let communication and event uniqueness do the heavy lifting. Similar derivations exist in the foundations literature, so the result itself is not new, but the particular framing around purely geometric inertial frames is a modest variant worth noting. The soft spot is exactly where the stress-test note points: the geometric fixing of distance properties has to carry enough structure (homogeneity, isotropy, affinity) for causality to select only the Lorentz group. The abstract gives no intermediate equations or explicit steps showing how the selection happens without those properties already being present. If the definition quietly encodes Minkowski-like features, the uniqueness claim weakens. Without the full derivation visible in the abstract, it is hard to tell whether the argument is independent or circular on this point. This is the sort of paper that belongs in a reading group on axiomatic special relativity. Readers who care about minimal postulates and alternative starting points will get something out of it, even if they end up disagreeing with the uniqueness claim. It is not strong enough to cite in most research contexts, but the topic is serious enough that a careful referee could usefully press on the geometric definition and ask for the missing steps. I would send it to peer review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper claims that causality and uniqueness of events, implemented via observers requiring finite communication time, imply a maximal information-transfer velocity between arbitrary frames. Inertial frames are defined solely by fixing the geometrical properties of spatial distance, independent of any physical laws. For these frames, the causality condition is asserted to fix the causal group as the orthochronous inhomogeneous Lorentz group extended by dilatations, using only basic mathematics.

Significance. If the derivation avoids circularity in the geometric definition of inertial frames and supplies a complete, non-circular argument from finite communication time to the stated group, the result would be significant: it would derive the Lorentz structure from causality plus a minimal, physics-independent geometric setup, offering a potential foundation for special relativity that does not presuppose electromagnetism or other dynamical laws.

major comments (2)

[Definition of inertial frames] Definition of inertial reference frames (abstract and opening sections): The central claim requires that a purely geometric fixing of spatial-distance properties (without reference to relativity or physics) supplies sufficient structure—homogeneity, isotropy, and affine character—for causality alone to select exactly the orthochronous inhomogeneous Lorentz group plus dilatations. The manuscript must explicitly derive or justify how these properties emerge from the stated geometric definition rather than being implicitly assumed; otherwise the uniqueness of the group is not established and the argument risks circularity by presupposing Minkowski-like features.
[Main derivation] Derivation of the causal group (sections presenting the main argument): The abstract states that causality 'unavoidably leads' to the group, yet supplies no intermediate equations or verification steps. The full derivation from finite communication time and event uniqueness to the precise group structure must be exhibited with explicit transformations and preservation conditions; without this, the load-bearing step from the geometric definition to the final group cannot be assessed for correctness.

minor comments (2)

Clarify the precise mathematical definition of the 'causal group' and how dilatations are adjoined to the orthochronous inhomogeneous Lorentz group; notation for the group action on events should be introduced explicitly.
The manuscript should briefly situate the result against existing literature on causality-based derivations of the Lorentz group (e.g., Zeeman-type theorems) to highlight the novelty of the geometric-frame approach.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying areas where the presentation of our geometric setup and derivation can be strengthened. We address the two major comments point by point below. In both cases we agree that additional explicit steps are needed to eliminate any appearance of circularity or incompleteness, and we will revise the manuscript accordingly.

read point-by-point responses

Referee: [Definition of inertial frames] Definition of inertial reference frames (abstract and opening sections): The central claim requires that a purely geometric fixing of spatial-distance properties (without reference to relativity or physics) supplies sufficient structure—homogeneity, isotropy, and affine character—for causality alone to select exactly the orthochronous inhomogeneous Lorentz group plus dilatations. The manuscript must explicitly derive or justify how these properties emerge from the stated geometric definition rather than being implicitly assumed; otherwise the uniqueness of the group is not established and the argument risks circularity by presupposing Minkowski-like features.

Authors: We accept that the current text does not spell out how the purely geometric definition of spatial distance (based on unique events and finite communication) yields homogeneity, isotropy, and affine structure. In the revised version we will insert a short, self-contained subsection that starts from the minimal geometric axioms—unique events, finite propagation time between observers, and the requirement that distance is a symmetric, positive-definite function on pairs of events—and derives the listed properties step by step using only elementary set-theoretic and metric arguments. This will make clear that no Minkowski structure is presupposed. revision: yes
Referee: [Main derivation] Derivation of the causal group (sections presenting the main argument): The abstract states that causality 'unavoidably leads' to the group, yet supplies no intermediate equations or verification steps. The full derivation from finite communication time and event uniqueness to the precise group structure must be exhibited with explicit transformations and preservation conditions; without this, the load-bearing step from the geometric definition to the final group cannot be assessed for correctness.

Authors: The referee is correct that the present manuscript states the conclusion without displaying the intermediate transformations and the precise preservation conditions. We will expand the central argument to include: (i) the general form of a coordinate transformation compatible with the geometric distance definition, (ii) the explicit causality constraint (preservation of the temporal order of events connected by finite signals), and (iii) the algebraic verification that the only transformations satisfying both the geometric and causal requirements are the orthochronous inhomogeneous Lorentz transformations extended by dilatations. The added steps will use only the basic mathematics already referenced in the paper. revision: yes

Circularity Check

0 steps flagged

Derivation from causality and geometric definition of inertial frames is self-contained with no reduction to inputs by construction

full rationale

The paper defines inertial reference frames purely via geometrical properties of spatial distance, independent of physics laws, then applies the causality condition (finite communication time and event uniqueness) to derive the orthochronous inhomogeneous Lorentz group times dilatations. No equations or steps are shown that reduce the final group to a fitted parameter, self-referential definition, or load-bearing self-citation. The central claim rests on the stated independence of the geometric definition supplying sufficient structure for causality to select the group, without evidence of smuggling in the target result via ansatz or renaming. This is a normal non-circular derivation from stated primitives.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The derivation rests on three explicit domain assumptions: causality together with uniqueness of observed events, finite communication time between observers, and a purely geometric definition of inertial frames that excludes any physical law.

axioms (3)

domain assumption Causality and uniqueness of events observation as driving forces
Built directly into the definition of distinct observers who require finite time to communicate.
domain assumption Existence of a maximal transfer-information velocity between arbitrary reference frames
Claimed to follow unavoidably from finite communication time.
domain assumption Inertial reference frames defined solely by geometrical properties of spatial distance, independent of any physical laws
Stated explicitly as the starting point before causality is applied.

pith-pipeline@v0.9.0 · 5385 in / 1359 out tokens · 59101 ms · 2026-05-10T18:06:14.042190+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction (spacetime-emergence certificate) echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

For these inertial reference frames, the causality condition fixes the causal group to be the orthochronous inhomogeneous Lorentz group times dilatations.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking / light-cone classification echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

if d²_M(x, y) ≥ 0 then d²_M(x′(x), y′(y)) ≥ 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 2 canonical work pages

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Observer/emitter (OE) 3
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Causality and space 6 B

Messengers 5 A. Causality and space 6 B. Discussion: Instant interaction between OEs located at different points in space 8 III. Universe 9 IV. Rigid reference frame (RRF) 10
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Clock synchronization within the same RRF 11
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Geodesic Coordinates 12
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Inertial RRF (IRRF) 17
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Family of IRRFs that are causally connected 18 A

The RRFs are smooth manifolds 17 V. Family of IRRFs that are causally connected 18 A. IRRF generated by messengers (boosts) 21 B. Weaker versions of Theorem 1 23 C. Extra remarks 24 VI. Conclusions 25 References 27 2 ”Observo, luego existo” I. INTRODUCTION Since the seminal derivation of special relativity by Einstein [1], based on the constancy of the sp...
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Reference frames and inertial reference frames are often defined in a somewhatvague wayusing statements such that: ”the laws of physics” ... ”take a simpler form”/”are invariant” .. ..”when no forces act on the particles”. This is unsatisfactory: What are the laws of physics? What does it mean ”simpler form”? What does it mean ”invariant”? What are forces...
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Compared with the points above our stands is the following

Whereas the concept of relativity between reference frames is very intuitive, it is indeed 1 possible to derive special relativity without using it (at least in the way it is usually implemented). Compared with the points above our stands is the following. We take causality and uniqueness of events observation as our driving forces. They are built in in t...
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Humans have the ability to discern between before , after, and same (or organize the description of nature in this way)

Time Time: order relation . Humans have the ability to discern between before , after, and same (or organize the description of nature in this way). Clock. Mechanism/object that is always changing (there is dynamics here). We can choose the change to be in a closed cycle. The clock will have a mechanism to count cycles. This is what we call time . Time is...
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a point in space

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We now give examples, and show how they can potentially define distances between OEs:

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When it reaches the observer OE j, it cuts a string that launches a return arrow that reaches back the observer OE i when its clock markst (f inal) i

The OEi launches an arrow with a well-calibrated mechanism (it does not wear out) when OEi’s clock markst (in) i . When it reaches the observer OE j, it cuts a string that launches a return arrow that reaches back the observer OE i when its clock markst (f inal) i . Then, we define thedistancefrom OE j to OEi associated with this method (a) as d(a) ij (ti...
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Each catapult has a person with an infinitely sharp knife

Each OE i has a catapult system. Each catapult has a person with an infinitely sharp knife. OE i activates the catapult and launches the person with an infinitely sharp knife when its clock readst (in) i . When it reaches the OE j, it cuts a rope that activates another catapult that launches another person back who reaches OE i when its clock readst (f in...
[15]

The OE i turns on the flashlight when his clock readst (in) i

Each OE i has a system of flashlights and mirrors. The OE i turns on the flashlight when his clock readst (in) i . When it reaches OE j it is reflected in the mirror. It returns to OE i when its clock readst (f inal) i . We define thedistancebetween the two OEs using this method (l) as d(l) ij (ti)≡ t(f inal) i (t(in) i ,(l), j)−t (in) i 2 .(4) 5 Overall,...
[16]

We do that in the following

Clock synchronization within the same RRF So far, we have not synchronized the clocks of the different observers of the RRF. We do that in the following. The construction of the RRF has been made by taking a particular OE: OEi and its clock. Let’s proceed to synchronize the clocks of the different OEs. We follow the method by Einstein [1] with the only qu...
[17]

We arbitrarily take one OE and define it to be the OE 0, i.e

Geodesic Coordinates Coordinate representation of the different OEs of the RRF We are now in the position to define the coordinate axes. We arbitrarily take one OE and define it to be the OE 0, i.e. the OE located atx= 0 (the origin) by definition. The coordinates of the other observers can be defined by the speed-limit geodesics, i.e. by the minimal time...
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Definition

Metric space Let us first remind the definition of distance and metric space. Definition. Given any set,A, we will say that inAthere is defined adistanceif ∀x, y,∈Awe can define a real number,d(x,y), with the properties: 1.d(x,y) = 0 iffx=y 2.d(x,y)≤d(x,z) +d(y,z)∀x,y,z∈A. We name such sets endowed with a distancemetric spaces. Using Eq. (6) and synchroni...
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The OEs of the RRF are then characterized by world-lines withxfixed and changingx 0

Pseudo-metric 8 It is convenient to define the (D=d+ 1) vector:x≡(x 0,x). The OEs of the RRF are then characterized by world-lines withxfixed and changingx 0. An eventEhas associated the point coordinatesx E = (x0 E,x E(x0 E)). We consider the following (Lorentzian) pseudo-metric 9 d2 L(x, y)≡(x 0 −y 0)2 −(d(x,y)) 2 .(18) 7 Yes, Pythagoras theorem is not ...
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We defineIRRFs as those RRFs for which the Pythagoras theorem holds

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