Is the Theory of Relativity Self-consistent?
| Scientific Paper | |
|---|---|
| Title | Is the Theory of Relativity Self-consistent? |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | theory of relativity, paradox, contradiction |
| Published | 2001 |
| Journal | Apeiron |
| Volume | 8 |
| Number | 2 |
| No. of pages | 10 |
Read the full paper here
Abstract
It is shown that the comment by Vladimir Onoochin to the mentioned paper contains some irrelevant assertions. At the same time, the comment recalls a recognized statement that the theory of relativity is internally consistent theory, and the revealed in [1] paradox comes into a certain contradiction with this statement. A goal of the present paper is to answer on the entitled question.
Overview
This ten-page note in Apeiron (Vol. 8, No. 2, April 2001) is Alexander L Kholmetskii's reply to a published comment by Vladimir Onoochin on Kholmetskii's earlier paper Remarks about correspondence of relativity and causality principles (Apeiron 8, 1, 2001). The earlier paper had claimed a new relativistic paradox: in a uniformly accelerated rigid frame, a chain of light re-emitters produces a meeting of two light pulses whose location depends on the observer, which Kholmetskii read as a clash between the relativistic postulates and the causality principle. Onoochin objected on three counts and added a general remark that the paper stands against "the majority view that the relativistic theory is an internally consistent theory, which can only be falsified by experiment."
The reply has two halves. The first answers Onoochin's three specific objections and argues that they miss the setup of the original problem. The second, and the substantive part, attempts to locate a logical gap in the foundations of special relativity itself: Kholmetskii argues that the standard derivation of the invariance of the space-time interval quietly uses the causality principle as an extra, unproved premise alongside Einstein's two postulates. The departure from the mainstream account is therefore not experimental but architectural — the claim is that the axiom list conventionally given for special relativity is incomplete, and that once the missing premise is made explicit there is no guarantee it is compatible with the rest.
The argument
Replies to Onoochin
Kholmetskii answers the three specific remarks in order. (1) The metric written as formula (13) of the earlier paper was derived not for a constant homogeneous gravitational field but for a uniformly accelerated rigid frame, from the definition (10) of such a frame; he cites Møller's Theory of Relativity (1972) as using the same definition and the same expression. For the gravitational case he says he introduced no metric at all, using only the standard weak-field clock-rate relation. (2) Onoochin had complained that it is unclear which frames the observer and emitters occupy. Kholmetskii replies that the chain of emitters and the non-inertial observer are in the same rigid accelerated frame, and that the inertial observer is external and precisely defined as the one who sees the simultaneous arrival of the two light pulses at the outer re-emitters. Onoochin instead treated a chain at rest in an inertial frame with an accelerating observer — "a different problem," so "his calculations are irrelevant." (3) The objection that no such experiment is feasible is conceded: the paradox "was not invented for purposes of practical realization." Kholmetskii points instead to his own proposed tests, including a test of the relativity principle using Mössbauer synchrotron radiation.
The two ways for the interval to vanish
The core of the paper begins from the standard route to the Lorentz Transformation. From Einstein's two postulates — equivalence of all inertial frames, and independence of the velocity of light from the velocity of the emitter — one infers the invariance of the interval
- s2 = c2t2 − r2,
and from the invariance of s the Lorentz transformations and all of relativistic kinematics follow. The usual argument runs: light emitted at A arrives at B after time r/c, so s = 0; by the constancy of c every other inertial observer also finds s′ = 0; the null cone is therefore preserved, and the interval is invariant.
Kholmetskii states that this evidence "is not complete." Formally, he says, s can vanish in two distinct ways: (1) t = r/c, and (2) t = 0, r = 0 — and "the second way has been lost in the evidence." Case (2) corresponds not to light propagation but to the intersection of two world lines — the coincidence of two particles or two short light pulses at a point (his Fig. 1). Whether that intersection is seen by every inertial observer, he argues, does not follow from the light postulate; it follows only from the requirement that a collision is an absolute fact — that is, from the causality principle.
He defines the causality principle by two requirements: "1. A cause-consequence order of events is absolute. 2. The events, which could cause essential inferences (for example, collision of particles), are absolute." The finiteness of the light velocity, he says, secures conformity with requirement 1 only; nothing in the postulates secures requirement 2.
Kholmetskii then divides pairs of events into two classes. Two events are "related" if they lie on the same world line, or on world lines that intersected in the absolute past; "unrelated" if their world lines never intersected in the absolute past. He claims these definitions are themselves invariant by the causality principle. Within the class of related events, the invariance theorem can be proved from the postulates alone. Within the class of unrelated events, s can vanish only through t = 0, r = 0, and its simultaneous vanishing in all frames rests on the causality principle. Hence "a possible contradiction between the relativistic postulates and the causality principle may exist only in phenomena dealing with 'unrelated' events."
This, he says, is how the earlier paradox was found: before the hypothetical meeting of the two pulses, the absorption and re-emission events at different re-emitters are unrelated. His Fig. 2 gives a second illustration — a rod of proper length L rotating in an inertial frame, whose ends simultaneously touch two emitters EM1 and EM2, which then emit pulses toward one another. The contact events are unrelated, their interval is space-like, and the meeting of the pulses is a vanishing of s of the second kind.
Conclusion of the paper
The stated conclusion is that "an implicit application of the causality principle in deriving STR has been revealed," so the logical chain of the theory is closed only if the compatibility of Einstein's postulates with the causality principle is proved — and, Kholmetskii insists, proved without calculations, since the mathematical apparatus already presupposes both. He argues such a proof is impossible, because whether two pulses intersect cannot be known before a calculation is done.
Notably, he then concedes a limited answer to his own title question: "in a restricted meaning the STR continues to be a self-consistent theory," since a genuine internal contradiction would mean deriving something like an absolute frame from relativistic calculations, which "seems impossible." His charge is instead that relativity is "physically incorrect." The alternative he prefers is his own "covariant ether theories," which he says satisfy all experimental facts gathered to date — an Aether-based space-time kinematics whose starting statements are, by construction, already in accord with causality.
Assessment
What is genuinely attractive here is the level at which the objection is pitched. Kholmetskii is not disputing an experiment; he is auditing an axiom list, asking exactly which premises are doing which work in the derivation of interval invariance. That is a legitimate and unusually disciplined kind of question, and the paper's honesty is notable: it explicitly distinguishes internal self-consistency from physical correctness, and concedes the former. The reply to Onoochin's second point is also fair — the two authors really were analysing different configurations, and Kholmetskii is right to say so.
The central logical claim, however, does not survive inspection. The case "t = 0, r = 0" is not a second, independent branch of s = 0 that was "lost"; it is the r = 0 member of the family ct = ±r. Solving c2t2 − r2 = 0 gives ct = ±r with no residue, and setting r = 0 in the first case yields t = 0 automatically. Nothing was omitted from the standard derivation; a special case was re-labelled as a new one.
Worse for the argument, the invariance of a coincidence of world lines needs no physical postulate at all. If a coordinate change is a one-to-one map of the manifold — which every Lorentz transformation is, and which is a mathematical property of the transformation rather than a causal assumption — then two curves meeting at a point in one chart meet at the corresponding point in every chart. Kholmetskii's "requirement 2" is therefore not an extra premise smuggled into relativity; it is a theorem about invertible mappings. The alleged incompleteness of the logical chain dissolves once that is noticed, and with it the claim that the compatibility of the postulates with causality is an open and unprovable question. This is an instance of a recurring pattern: a claimed hidden assumption that turns out to be an identity.
Two further difficulties are worth naming. First, the illustration in Fig. 2 leans on a rigid rod in rotation. Rigid bodies are precisely the objects relativity forbids: a rotating rigid disc or rod is the subject of the Ehrenfest paradox, and no relativistic rigid-body dynamics exists that allows the construction Kholmetskii draws. An argument against relativity that uses a rotating rigid rod as its exemplar has assumed the contested conclusion in the setup. Second, the paper's "uniformly accelerated rigid frame" carries the same burden, and it is the frame in which the original paradox lives; a frame-dependent answer to "where do the pulses meet" is expected for a non-inertial chart with a horizon, and needs to be shown to be more than a coordinate artefact before it can be read as a causal contradiction.
Finally, the paper's own escape route is unexercised here. The "covariant ether theories" are asserted to reproduce all known experiment but are not developed in these pages, and no observation is named that would separate them from standard relativity — not the (1+z) time dilation of Type Ia supernova light curves, not muon lifetime dilation, not the Ives–Stilwell or Kennedy–Thorndike results. As a reply to a comment the note is disciplined and clear; as a demonstration that special relativity rests on an unproved extra premise, it turns on a case distinction that is not a distinction.