Remarks on the Causality Principle (comment on a previous paper)
| Scientific Paper | |
|---|---|
| Title | Remarks on the Causality Principle (comment on a previous paper) |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | causality principle, emission/absorption of light, transformations between inertial and non-inertial reference frames |
| Published | 2003 |
| Journal | Apeiron |
| Volume | 10 |
| Number | 2 |
| No. of pages | 19 |
Read the full paper here
Abstract
This paper returns to the problem [1] about two light pulses propagating across an accelerated infinite chain of re-emitters (point absorber-re-emitters) of light, where both non-inertial and inertial observers look for possible intersection of these light pulses. It is shown that the author's previous conclusion about violation of the causality principle in this problem contained a mistake. In this connection the present paper analyses a simplified version of the problem, where two reflectors of light substitute for an infinite chain of reflectors. The compatibility of causality and relativity principles is derived.
Overview
In 2001 Kholmetskii published in Apeiron a paper claiming to have found a violation of the causality principle inside relativity theory. Two years later he published this one, which retracts that claim. The retraction is unusually complete: he identifies the error, says who found it, thanks them by name, and then works the problem again in a simplified form where the calculation can be carried through in full. The answer he obtains is that causality and relativity are compatible in the case he had contested.
The paper is therefore of interest less for a new result than for what it does. It is a dissident author publishing, in a dissident journal, a demonstration that his own anti-relativistic result was wrong — while preserving intact the conceptual question that motivated it. Kholmetskii is careful to separate the two: the particular example he chose was wrong, but the general worry, about what happens to the causality criterion at world-line "fracture points" produced by the emission and absorption of light, he still regards as legitimate. He credits Prof. Valeri V Dvoeglazov with organising the discussion that exposed the mistake, thanks the anonymous reviewer who found it, and records that the objection published by Vladimir Onoochin had itself been erroneous — Onoochin had attacked the accelerated-frame solution, which was in fact correct.
The argument
The original problem and the mistake
The setting is a rigid non-inertial frame moving with constant (relativistic) acceleration a along x, defined by x = x′, t′ = 0, a = a(t), with t the proper time at the origin. Point re-emitters RLm are placed along the axis; each absorbs an arriving pulse and re-emits it after a fixed interval Δt0 of its own proper time. A first pulse is emitted from x = 0; a second is emitted from the same point later, when the first has reached coordinate Δx. One asks for t1, the moment RLn emits the right pulse, and t2, the moment the left pulse reaches RLn.
In the accelerated frame the two are equal at a particular xn, so the non-inertial observer sees an absolute event: the two pulses meet. Kholmetskii notes the physical reason — "the different rate of clocks at different points x."
His 2001 argument then claimed that no external inertial observer could ever see them meet. To make this tractable he took the chain to be infinitely long and introduced a second inertial frame Ks, at rest relative to K but displaced by the RL0–RL1 spacing, arguing from the homogeneity of space that this shift merely renumbers the re-emitters. The mistake is located precisely: "the inertial frames K and Ks are not equivalent to each other, if one considers a finite-length chain of re-emitters (an infinitely long chain cannot exist in nature)." A finite chain has an origin and an end whose coordinates the displacement changes, so two spatially shifted inertial observers are not equivalent for this problem despite the homogeneity of space. Treating the problem for a single inertial observer, the momentary velocities of the re-emitters differ because the contraction of the moving chain is time-dependent; hence the time-dilation factor depends on x, hence clock rates differ from point to point in the inertial frame as well — and the pulses may meet there too.
Why the worry was raised at all
Kholmetskii formulates the causality principle (CP) as two requirements: the cause–consequence order of events is absolute, and events capable of causing essential inferences (a collision of particles) are absolute. The finiteness of the light velocity secures the first; the homogeneity of the admissible space-time transformations secures the second, since Δt, Δr = 0 maps to Δt′, Δr′ = 0.
He then argues these are necessary but not sufficient. For two nearby world lines he writes local intersection/non-intersection conditions as inequalities between the difference of the slopes dx(1)/dt − dx(2)/dt and the ratio Δx/Δt, with the corresponding pair for a second observer in X, T. A strong form of CP requires that the two observers' inequalities hold together. For smooth world lines a theorem (which he does not reproduce here) guarantees they do. The proof fails if a world line has a slope discontinuity, since the derivative is then infinite at the fracture point.
Physically such fractures should be impossible — no entity exceeds the speed of light — "with one exception: the cases of absorption (emission) of light." An absorber that holds a pulse for a fixed proper time Δtr and re-emits it can be said to "keep" information about the pulse, so the world lines of absorbed pulse, absorber and re-emitted pulse may be joined into one macroscopic world line with kinks at the absorption and emission events. That is the loophole the original paper tried to exploit.
The two-re-emitter calculation
The retraction is completed by replacing the infinite chain with just two re-emitters, RL0 at the origin and RL1 at x1, so that everything can be computed in both frames.
In the accelerated frame the metric is obtained from the standard rigid-frame relations, giving ds2 = (1 + ax/c2)2(cdt)2 − dx2 − dy2 − dz2, so that g00 = (1 + ax/c2)2 and the coordinate light speed is cx = c(1 + ax/c2). Integrating dx/cx gives the arrival time tR of the right pulse at RL1 as a logarithm, and the coordinate holding time as Δt1 = Δt0/(1 + ax1/c2). Setting t1 = t2 gives the condition tR + Δt1 = Δt0 + tL, where tL is the left pulse's travel time from the origin to x1, and substituting the integrals yields a closed expression for the required holding time Δt0 in terms of c, a, Δx and x1 (his Eq. 23), combining a logarithmic term with a factor (1 + ax1/c2). Kholmetskii notes that this admits an intersection "only for a negative sign of the acceleration a."
He then repeats the calculation for an inertial observer whose momentary relative velocity vanishes at t = 0. The motion of RL0 before emission integrates to TL = (c/a)sinh(aΔt0/c), XL = (c2/a)[cosh(aΔt0/c) − 1]; the pulses propagate as ΔX = cΔT; and the motion of RL1 between absorption and re-emission gives ΔTR and ΔXR as differences of hyperbolic sines and cosines. Assembling the coordinate difference of the two pulses at the moment TR + ΔTR, using cosh x − sinh x = e−x, and substituting the previously derived tR, Δt1 and Δt0, the whole expression collapses to zero. The pulses meet for the inertial observer as well.
The conclusion is stated carefully. There is no contradiction between causality and relativity in emission/absorption processes with fracture points, because although the time derivatives on either side of a fracture are uncorrelated, "stepwise changes of time derivatives of world lines are correlated for different observers. That, perhaps, prevents a violation of causality."
Assessment
What is admirable here is the method. Kholmetskii poses a concrete, computable question — do two pulses meet, yes or no — rather than a verbal paradox, and he answers it in both frames explicitly. The diagnosis of his own error is genuinely illuminating and generalisable: the homogeneity of space in an inertial frame does not license translating an observer when the physical system has a boundary, and idealising a chain to infinite length quietly removes the boundary that carried the physics. That is a trap wider than this problem, and worth remembering. The observation that emission and absorption are the only physical processes producing genuine world-line kinks is also a sharp one, and it is the reason the question was worth asking at all.
The paper's own limits should be stated. Its positive result is narrow: one specific two-re-emitter configuration, one-dimensional, with a simplifying assumption of momentarily zero relative velocity at t = 0, and with the sign of a constrained. The general theorem — that for smooth world lines the intersection inequalities hold simultaneously for all observers — is not proved here but referred to an unpublished conference proceeding, so the pivot on which the fracture-point argument turns is taken on trust. The final sentence is correspondingly hedged: "that, perhaps, prevents a violation of causality." No demonstration is given that correlated stepwise derivative changes always secure CP; what is shown is that they did so in this case.
None of this is a defect of honesty, and the paper should not be read as a defence of relativity so much as a refusal to keep an argument that did not work. That is worth more to the dissident literature than another unretracted paradox would have been. A reader wanting the substantive critique of relativistic kinematics should look to Kholmetskii's other work rather than to this note, which is, by design, a correction.