Two Clocks and One Scale: Coordinate Time dt, Proper Time dtau and Entropy dS Are Three Different Logical Types.
An hour with someone you love passes in an instant; an hour in a dull meeting feels eternal – and the clock on the wall reads the same number. ALMOST EVERYONE HAS THIS EXPERIENCE, AND YET THE INTERPRETATION OF TIME HAS ADVANCED REMARKABLY LITTLE, AND EVEN THE INTERPRETATION OF RELATIVITY REMAINS DIVIDED. This paper’s diagnosis is simple: ONE WORD COVERS SEVERAL QUANTITIES, AND WE HAVE KEPT TRYING TO ELECT ONE OF THEM AS “REAL TIME”. The Bergson-Einstein exchange in Paris in 1922 was not a dispute over who was right; it was two people DEFENDING QUANTITIES OF DIFFERENT TYPE.
A BRAKE IS PLACED IMMEDIATELY: THIS PAPER DOES NOT EXPLAIN FELT TIME. Physics holds three quantities – dt, dtau, dS – and the “instant” and the “eternity” above are none of them. They are A FOURTH QUANTITY belonging to psychology and neuroscience, and are left untouched. What everyday experience supplies here is NOT AN ANSWER BUT THE SHAPE OF A QUESTION, and only that is taken up, to be answered inside physics. To identify felt time with dS would be precisely the error of type this paper forbids.
Paper 82 DECLARED that coordinate time, proper time and thermodynamic time are three distinct times that do not fuse. THIS PAPER TURNS THAT DECLARATION INTO A VERDICT.
FIRST: THE THREE DIFFER IN LOGICAL TYPE. dt IS A COORDINATE AND HAS NO DOMAIN – it is a chart label, changed by any diffeomorphism, and it becomes a clock only once pasted onto a congruence of worldlines (in the vocabulary of Paper 95, a CONVENTION – the same line Paper 95 drew for c: that it is finite is a fact, that its value is 299792458 is a convention). dtau IS A FUNCTIONAL OF A CURVE: gamma -> (1/c) integral sqrt(g_uv dx^u dx^v), defined on worldlines in a Lorentzian manifold; without a worldline it has no value. dS IS A FUNCTIONAL OF A MEASURE: rho -> -k_B integral rho ln rho, defined on probability measures on phase space; a single trajectory does not give it a value. HENCE dtau AND dS ARE NOT TWO NUMBERS THAT DIFFER IN VALUE – THEY ARE FUNCTIONALS LIVING ON DIFFERENT SPACES.
SECOND: THE SEPARATION IS EXHIBITED IN BOTH DIRECTIONS. (i) dtau = 0 WHILE dS > 0: along a null worldline ds^2 = c^2 dt^2 – dx^2 = 0, so no proper time accumulates; yet a radiation field carries S = (4/3) a T^3 V, and isothermal expansion gives dS/dV = (4/3) a T^3 > 0 (both thermodynamic identities verified here). (ii) dtau > 0 WHILE dS = 0: a bound reversible Hamiltonian system accumulates dtau = sqrt(1 – v^2/c^2) dt > 0, while Liouville’s theorem keeps the Gibbs entropy EXACTLY CONSTANT WITHOUT ANY COARSE-GRAINING (checked via the conserved phase-space area 2 pi E / omega of a harmonic oscillator). Elect either one as “the real time” and the other rises as a counterexample.
THIRD, A DECISIVE BRAKE: in (i), dtau = 0 is a statement about ONE PHOTON’S WORLDLINE while dS > 0 is a statement about A PHOTON GAS, a statistical state. THE TWO CLOCKS ARE NOT ATTACHED TO THE SAME OBJECT. But this shift does not spoil the counterexample – IT IS THE COUNTEREXAMPLE: a functional of a curve and a functional of a measure cannot in principle be attached to the same object, and that is precisely why they are independent. Hide the shift and say in one breath “light has dtau = 0 yet dS > 0”, and it becomes an error of category; write the address, and it becomes a counterexample.
FOURTH: THE COUPLED REGIME. In dissipative systems (Paper 78) motion along dtau leaks into unresolved degrees of freedom and produces dS > 0; relativistic dissipative hydrodynamics (Eckart; Landau-Lifshitz) writes this covariantly. At a black-hole horizon (Paper 73) the area theorem dA >= 0 together with S_BH = k_B c^3 A / (4 G hbar) LOCKS A GEOMETRIC QUANTITY TO A STATISTICAL ONE ([c^3 A / (G hbar)] is dimensionless, checked here). BUT A LOCK IS NOT AN IDENTITY: what is tied is AREA to ENTROPY, not PROPER TIME to entropy – area is not dtau. And by Paper 88, S_BH and coarse-grained entropy are themselves distinct roots. THIS PAPER THEREFORE RESTRAINS ITSELF TWICE.
FIFTH: THE BOOKKEEPING IS GIVEN A NAME – THE EINSTEIN-YAMAGISHI DUAL-CLOCK FRAMEWORK: one process, recorded simultaneously in two ledgers, the dtau ledger and the dS ledger. A BRAKE ON THE NAME: this is A DISCIPLINE OF ACCOUNTING, NOT A THEORY AND NOT AN EQUATION. General relativity is not altered by one character and no new prediction is generated. IF THE WORD “COMPLEMENT” IS USED, IT MEANS JUXTAPOSITION, NOT CORRECTION – RELATIVITY IS COMPLETE ABOUT CLOCK A AND NEVER CLAIMED ANYTHING AT ALL ABOUT CLOCK B, and what is never claimed cannot be wrong. There is nothing to repair inside relativity; there is only a second ledger to place beside it. The name serves one practical purpose: to force a declaration of WHICH LEDGER was used when one says how much time has passed.
SIXTH: THE REACH OF “ORTHOGONAL” IS LIMITED. It means LOGICAL INDEPENDENCE – one may vanish while the other does not – and NOT orthogonality in an inner-product space; dtau and dS are not elements of a common vector space, so there is no arena in which an inner product could be taken. Reading (dtau, dS) as “coordinates on a plane” is a metaphor: no metric, no inner product and no rotation is defined there.
SELF-AUDIT (the difference from Paper 82): Paper 82 opened a framework; THIS PAPER CLOSES ONE DOOR INSIDE IT. What is added is (a) separating the three BY LOGICAL TYPE, (b) exhibiting the separation constructively IN BOTH DIRECTIONS, and (c) exposing the shift of address inside the counterexample itself. A further consequence for Paper 82: a “brick” of event-based time can be counted in either dtau or dS, BUT THE COUNTS WILL NOT AGREE – light lays down no brick in dtau, a reversible oscillator none in dS – so BEFORE COUNTING A STEP ONE MUST DECLARE WHICH CLOCK. This is not a correction to Paper 82 but a type declaration for the framework it opened.
IMPORTANT (scope): a structural interpretation and roll-call of established facts at Layer 2 (proper time as metric arc length, the Clausius inequality, Boltzmann’s formula, Liouville’s theorem, the thermodynamics of radiation, Bekenstein-Hawking, the area theorem). NO NEW THEOREM IS PROVED AND NOT ONE CHARACTER OF AN EXISTING EQUATION IS CHANGED. Neither Einstein, nor Clausius, nor Boltzmann is contradicted; what is done is to write the address of what each was counting. THE THREE ARE NOT REDUCED TO ONE “REAL TIME” – that is the point. dtau is not derived from dS nor the converse (the restraint of Paper 74). The origin of the arrow of time is NOT solved (the past hypothesis; Paper 98) – that the arrow lives in coarse-graining is only half. No new evolution equation is proposed. The horizon lock is not read as an identity. No geometry is imported onto the (dtau, dS) plane. Metaphors – ageing, consciousness, a “Universe OS”, the inference of an AI – appear NOWHERE in the body and are quarantined in the Layer-3 personal note under a non-scientific label.
[WHAT THE SECOND EDITION ADDS] – AN EXPLICIT ATTRIBUTION AND A RESTATEMENT OF SCOPE; NO CLAIM, THEOREM OR COUNTEREXAMPLE IS ALTERED, AND NOTHING IS RETRACTED.
FIRST ADDITION – ATTRIBUTION. THE SEPARATION OF dt FROM dtau IS NOT A DISCOVERY OF THIS PAPER. It is Minkowski’s, from 1908, given as the INVARIANT INTERVAL ds^2 = c^2 dt^2 – dx^2 – dy^2 – dz^2, and within this series it is already treated by PAPER 108, which states that although time and length change with the observer this interval is common to everyone, and which sorts the geometry of the invariant interval, the hyperbolic geometry of velocity space, and proper time as DISTINCT ROOTS. The first edition described this paper’s contribution as “separating the three times by logical type”, which left room to read the dt/dtau separation itself as novel here. WHAT THIS PAPER NEWLY PLACES IS THE THIRD QUANTITY (dS) UNDER THE SAME MEASURING STICK, AND THE COUNTEREXAMPLES IN BOTH DIRECTIONS.
SECOND ADDITION – A RENUMBERING. The first edition called felt time “A FOURTH QUANTITY” and wrote that “PHYSICS HOLDS THREE”. BOTH ARE NOW INACCURATE. Paper 119 identified A FOURTH TYPE ON THE PHYSICAL SIDE – the parameter of Ricci and renormalisation-group flow, of dimension L^2, whose domain is a path in the space of metrics Met(M) – and Paper 121 treated the geometry of that arena, showing the positive-definite L^2 metric (Clarke; CAT(0) completion) and the DeWitt supermetric (signature -,+,+,+,+,+) to be DISTINCT ROOTS. This edition therefore STOPS GIVING FELT TIME A NUMBER and restates the scope as “THIS PAPER LINES UP THREE”. Felt time is none of the four either, and remains untouched (psychology and neuroscience). NO CLAIM, THEOREM OR COUNTEREXAMPLE CHANGES AS A RESULT – what changed is only a manner of writing that made the roll-call look closed.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Two Clocks and One Scale — Second Edition: Coordinate Time $dt$, Proper Time $d\tau$ and Entropy $dS$ Are Three Different Logical Types — [Paper 118]. Zenodo.
DOI: 10.5281/zenodo.22036237
ORCID: 0009-0006-8587-3450