Relativity of Time in Earthquake Physics

A.V. Guglielmi; O.D. Zotov

REVIEW 4 major objections 6 minor 21 references

Relativity of Time in Earthquake Physics

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Two foreshock phases of the Tohoku earthquake emerge when events are counted as the source's own clock.

desk verdict A small empirical case study that re-labels event order as 'proper time' and finds a two-phase foreshock pattern in Tohoku, but the phase split is fixed by the M7.3 and the statistics are too weak to prove a regime change. read the letter →

arxiv 2509.04858 v1 pith:5MOHKORT submitted 2025-09-05 physics.geo-ph

A.V. Guglielmi , O.D. Zotov This is my paper · ORCID

classification physics.geo-ph

keywords earthquakesourcepropertimeTohokuforeshocksaftershocksdeactivationcoefficientaftershockdecaylawundergroundclock

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the 166 foreshocks of the 2011 Tohoku earthquake, ordered by the ordinal number of events rather than by universal time, fall into two regimes: the first 103 events occur at a constant rate, and the remaining 63 decelerate exponentially with a deactivation coefficient of 0.065. This uses the idea that each tremor is one tick of an 'underground clock' measuring the source's proper time, borrowed metaphorically from relativity. The two-phase structure resembles the previously found three-phase aftershock relaxation, starting with a degenerate phase of constant activity and switching abruptly to a classical hyperbolic decay. If correct, the result shows that the source parameters were non-stationary in the 200 days before the main shock, and that a powerful magnitude-7.3 foreshock 51 hours before the main shock may have triggered its own aftershock-like sequence.

What carries the argument

The central mechanism is the 'underground clock': each catalogued shock (foreshock or aftershock) is treated as one unit of the source's proper time x, so the clock ticks at every event. The deactivation coefficient sigma is recovered from the data through the relation sigma = d ln T / dx, where T is the mean interval between successive events in universal time. In the phenomenological theory, sigma is the coefficient in the hyperbolic decay equation dn/dt + sigma n^2 = 0, so that the standard aftershock decay law corresponds to constant sigma; the clock's purpose is to make this comparison possible without assuming any universal-time law in advance.

What would settle it

Take the same foreshock catalog and recompute the deactivation coefficient using equal windows of world time instead of event number, or randomize the event order and re-run the fit; if the straight-line/exponential split at event 104 disappears or the coefficients change widely across shuffles, the claimed two phases depend on the chosen clock rather than on source physics.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes a two-phase evolution of the Tohoku foreshocks. Numbering foreshocks by x = 1..166 and plotting their occurrence times t(x) gives a straight line t = 46.5x - 485 for x = 1..103 (R^2 = 0.98) and an exponential trend exp(0.065 x) for x = 104..166 (R^2 = 0.97). Equivalently, the deactivation coefficient sigma = d ln T / dx is zero in the first phase and 0.065 in the second. The authors interpret this as a sudden transition from a degenerate constant-rate regime to a classical hyperbolic decay regime, and note the similarity to the aftershock pattern: the initial aftershock phase also has sigma = 0 before it jumps to a positive constant. The proper time of

Load-bearing premise

The entire analysis assumes that simply numbering successive earthquakes creates a physical clock, so that each event marks one equal unit of the source's proper time; if the event number is not a meaningful time variable, the two phases are an artifact of how the data are reordered.

Editorial extensions

If this is right

The Tohoku foreshock sequence is not stationary: it switches abruptly from a constant-rate regime (sigma = 0) to an exponentially slowing regime (sigma = 0.065) at the 104th foreshock.
The decelerating phase likely reflects a cascade of aftershocks triggered by the magnitude-7.3 foreshock 51 hours before the main shock, meaning the main shock was preceded by a detectable secondary sequence.
The same ordering by event number reveals an analog in aftershock relaxation, where sigma also starts at zero and then jumps to a positive constant, suggesting a common regime-change pattern in source evolution.
The uneven flow of proper time relative to world time gives a direct experimental measure of source non-stationarity: sigma(t) is not constant when expressed in world time.
Applying the procedure to other 'complete classical triad' earthquakes is the next planned test of whether this two-phase foreshock structure is generic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the pattern reproduces on other events, the event-count clock would provide a simple, parameter-free time axis for comparing foreshock and aftershock evolution across different magnitudes and tectonic settings.
The transition at x = 104 could be detected in real time if the event count and magnitude are known, offering a possible short-term precursor indicator: a switch from steady to hyperbolic slowing before a large event.
The exponential segment is fit with a single coefficient; a cleaner falsification is to test on shuffled event order or to compare the exponential fit against a two-parameter power law on the same interval.
The vortex, or Umov-energy, reading of the epicenter path is speculative, but it implies that foreshock epicenters should trace a systematically curved path in space-time, which is measurable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. This is a case study, not a new theory. The paper re-labels the event ordinal number as 'proper time' and then shows that for Tohoku foreshocks, the occurrence time grows roughly linearly for the first 103 events and then exponentially for the next 63 after the M7.3 foreshock. That empirical pattern is real as a description of this catalog. The authors deserve credit for reporting the fits and for being honest that this may be unique to Tohoku. The acknowledgement that the second phase may just be aftershocks of the M7.3 is also honest.

The central claim, though, is statistically weak. The breakpoint is not estimated; it is fixed at the M7.3 event. The high R² values are unsurprising because t(x) is a cumulative sum of waiting times, so neighboring points are highly autocorrelated. The paper never compares a one-phase model with the two-phase model, gives no confidence intervals, and does not test whether the split improves prediction. The sigma values in Table 1 are just the slopes of the two fitted segments, so they cannot be used as independent evidence for a regime change. In short, the two-phase structure may be real, but the paper does not demonstrate it beyond visual inspection.

The 'proper time' framing adds little physical content; the uneven flow of proper time relative to world time is a restatement of the fact that the inter-event times change. The spatial 'vortex' and Umov energy flow speculation should be ignored.

Is the paper serious? Yes. It is clearly written, uses public data, and the calculations are simple enough to reproduce. But as it stands, it is an interesting observation that needs proper statistical treatment. I would not cite it as evidence for a two-phase foreshock process, but I'd bring it up in a reading group as an example of how easy it is to over-interpret re-parameterized data. If it landed on my desk as an editor, I'd send it to a referee only if the authors were willing to add a real model comparison and breakpoint estimation; otherwise it's a conference poster.

Referee Report

4 major / 6 minor

Summary. The paper proposes a phenomenological concept of 'proper time' for an earthquake source, defined operationally as the ordinal number x of events in a catalog ('underground clock'). For the 2011 Tohoku earthquake, the authors plot world-time occurrence times t(x) for 166 foreshocks and identify two phases: a linear relation t(x)=46.5x−485 for x=1–103 (Eq. 5) and an exponential relation t(x)∝exp(0.065x) for x=104–166 (Eq. 6). They report deactivation coefficients σ=0 and σ=0.065 for the two phases (Table 1) and compare these with earlier aftershock phases. The paper also presents preliminary spatial analyses (foreshock convergence, aftershock divergence, and a 'vortex movement' interpretation). The central claim is that ordering events by this proper time reveals a previously unnoticed regime change in foreshock evolution.

Significance. If the two-phase foreshock structure is real, it would complement the authors' earlier three-phase aftershock relaxation picture and could indicate a sudden change in source deactivation about 51 hours before the main shock. The paper is clearly written, uses a public catalog, provides explicit functional fits, and honestly acknowledges that the result may be specific to this earthquake. The analogy to relativity is explicitly stated to be symbolic, avoiding overclaiming on that front. However, the statistical support for the central regime-change claim is currently weak: the phase split is imposed at the time of the known M7.3 foreshock rather than estimated, no competing model is tested, and the reported σ values are derived from the same fits. The paper is better viewed as a suggestive phenomenological description than as a statistically validated discovery.

major comments (4)

[Section 2, Eqs. (5)–(6) and Table 1] The two-phase claim is not statistically validated. The breakpoint is fixed at the time of the M7.3 foreshock (x≈103/104) rather than estimated from the data, and the paper does not compare the two-phase model against a single-phase alternative (e.g., one exponential or a power-law acceleration). Because t(x) is a cumulative sequence of waiting times, consecutive points are strongly autocorrelated, so high R² values (0.98 and 0.97) are expected for many smooth curves and do not by themselves establish a regime change. The authors should estimate the breakpoint (or at least test sensitivity to its location), provide confidence intervals for the slopes/rates, and use a model-comparison criterion such as AIC or a likelihood-ratio test.
[Section 2, Eq. (7) and Table 1] The deactivation coefficients in Table 1 are not independent measurements; they are direct consequences of the fits in Eqs. (5) and (6). For the first phase, t(x) is linear, so T=dt/dx is constant and Eq. (7) gives σ=0 by construction. For the second phase, t(x)∝exp(0.065x), so T∝exp(0.065x) and σ=d ln T/dx=0.065 exactly. Thus Table 1 restates the phase fits rather than confirming them. The paper should be explicit that σ is a fitted parameter of the piecewise model, not an independently estimated quantity.
[Section 2, definition of proper time] The 'underground clock' is defined so that each event increments proper time by one unit. Under this definition, the statement that 'proper time flows unevenly relative to world time' is a restatement of the fact that inter-event times vary. The physical content of the paper therefore reduces to the empirical claim that the inter-event time pattern changes at x≈103/104. This may be a legitimate empirical finding, but the manuscript should not present it as a measurement of an independent source clock without justifying that ordinal numbering carries physical meaning. A concrete test would be to apply the same procedure to other earthquake sequences and show that the piecewise structure is reproducible and preferable to a single-phase null model.
[Section 3, Figs. 6 and 7] The spatial results (foreshock convergence, aftershock divergence, 'vortex movement') are presented without quantitative uncertainty or significance testing. The moving average over 20 points is not accompanied by confidence bounds, and the curvature/torsion of the epicenter path in Fig. 7 is not measured. Since these claims are secondary to the main temporal phase analysis, they could be relegated to clearly labeled preliminary observations, or supported with appropriate statistical measures.

minor comments (6)

[References, Ref. [6]] The journal title contains a typo: 'Solid Evarth' should be 'Solid Earth'.
[Section 2, Eq. (1) and Fig. 3] The symbol τ is used in Eq. (1), but the main text and figures use x for proper time. Please define the relation between τ and x explicitly.
[Figures 4 and 5 captions] Fig. 4 refers to a green dot marking 'a distinct change' but the caption does not specify its coordinates or how it was determined. Please clarify whether this dot is the M7.3 foreshock.
[Section 2, Eq. (5)] The intercept -485 in Eq. (5) is dimensionally ambiguous. Please specify the reference time (e.g., days before the main shock) and the time units.
[Section 2, data description] The catalog source is given as USGS/NEIC, but the retrieval date and version are not stated. This would aid reproducibility.
[Section 3, Fig. 6] The moving-average window is described as '20 points' in the text, but it would be helpful to state whether this is a centered or trailing window and whether the smoothing changes results for different window lengths.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the two-phase result is an empirical fit, not a prediction derived from the theory.

full rationale

The paper's derivation chain is observational: it defines proper time x as the event ordinal, plots the catalog occurrence times t_k against x, fits a straight line and an exponential to different segments, and defines the deactivation coefficient sigma from T = dt/dx via Eq. (7). Nothing in this chain is equivalent to its inputs by construction: the clock definition (event ordinal as proper time) does not force t(x) to be piecewise linear/exponential, and the fitted slopes are not predicted from Eqs. (1)–(4); they are estimated from the USGS/NEIC catalog. The Table 1 sigma values are transformations of the fitted curves, but the paper does not present them as an independent prediction from theory. The split at x=103/104 is chosen from the known M7.3 foreshock, which is a statistical/validation weakness (no breakpoint estimation, no model comparison), not a circularity. Self-citations to [1,2,6,7] supply the phenomenological framework and prior aftershock results, but the key equations are stated and applied directly rather than imported as an unexamined black box. Therefore no circular step is present.

Assumptions & free parameters 6 free parameters · 4 assumptions · 4 invented entities

The paper adds no free physical constants but introduces several fitted parameters (slopes, exponents, phase boundary) and ad hoc conceptual entities (proper time, underground clock). The framework rests on unvalidated domain assumptions about source dynamics and the event-count clock.

free parameters (6)

Slope of first foreshock phase (a1) = 46.5 (per event, units unspecified)
Fitted by least squares to the first 103 foreshocks in eq. (5), with R^2=0.98.
Exponential growth rate of second foreshock phase (beta) = 0.065 per event
Fitted to the next 63 foreshocks in eq. (6), with R^2=0.97; yields sigma=0.065.
Deactivation coefficient for first phase (sigma1) = 0
Derived from sigma = d ln T / dx for the linear phase.
Deactivation coefficient for second phase (sigma2) = 0.065
Derived from sigma = d ln T / dx for the exponential phase.
Aftershock deactivation coefficients = 0 and 0.0014
From the authors' prior work [2], used for comparison in Table 2.
Moving-average window length = 20 points
Used in Fig. 6 to smooth spatial distances.

assumptions (4)

domain assumption The aftershock (and foreshock) frequency n is related to the deactivation coefficient sigma by the differential equation dn/dt + sigma*n^2 = 0 (eq. 2).
Core postulate of the phenomenological theory, motivated by Omori's law and the change of variable g=1/n. Stated as 'it is natural to base the elementary theory of aftershocks on the axiom sigma = dg/dt' (Section 1).
domain assumption The source can be described as a dynamic system with a well-defined deactivation coefficient sigma(t).
The entire framework rests on this, but it is not directly verifiable from data.
ad hoc to paper Each event in the catalog serves as a tick of the source's proper time, i.e., the ordinal number x is a uniform measure of proper time.
This is the 'underground clock' idea stated in Section 2: 'we agreed to consider the excitation of the next underground shock as evidence that a unit of the proper time of the source has passed.' It is an ad hoc normalization.
domain assumption The deactivation coefficient can be estimated from the logarithm of the inter-event time gradient: sigma = d ln T / dx (eq. 7).
Derived from the smoothing assumption and the definition of T as the average interval, but relies on the validity of the continuous approximation (eq. 4).

invented entities (4)

Proper time of the earthquake source (tau or x)
purpose: To provide a time coordinate tied to the source's internal evolution rather than world time.
A theoretical construct defined by eq. (1) and operationalized as the event ordinal number. There is no independent measurement of it outside the earthquake catalog itself.
Hypothetical underground clock
purpose: A metaphor to justify counting events as ticks of proper time.
Purely a conceptual device; no physical clock exists.
Deactivation coefficient sigma
purpose: To quantify the rate of change of inter-event times and characterize source relaxation.
A free parameter fitted to each phase; it is not measured independently.
Umov energy flow and vortex movement
purpose: To interpret the spatial pattern of foreshock epicenters.
The 'string of pearls' pattern is an informal observation, and the vortex interpretation is admitted by the authors as speculative ('Our own interpretation, right or wrong').

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Cite this review

Pith. "Pith review of Relativity of Time in Earthquake Physics." pith.science (2026). https://pith.science/paper/5MOHKORT

@misc{pith2026250904858,
  author       = {Pith},
  title        = {Pith review of: Relativity of Time in Earthquake Physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MOHKORT}},
  note         = {Machine review of arXiv:2509.04858}
}

read the original abstract

In this paper we develop an understanding of the proper time of the Tohoku earthquake source. The paper is dedicated to the 120th anniversary of Einstein's theory of relativity, but the dedication is symbolic, since we are investigating a purely non-relativistic geophysical object. We still found it possible to borrow the terms time relativity and proper time from the theory of relativity. The concept of the proper time of the source is developed by us within the framework of the phenomenological theory of earthquakes. The paper describes a procedure for measuring proper time using observation data of foreshocks and aftershocks. We synchronized the imaginary underground clock that counted the proper time and the clock that showed world time. The idea of a hypothetical underground clock turned out to be effective. We have shown that the proper time of the source flows unevenly relative to the flow of world time. Two phases of the evolution of the source before the main shock of the earthquake were discovered. This complements the picture of three-phase relaxation of the source after the main shock, which we established earlier. The uneven flow of proper time relative to world time indicates the non-stationarity of the source parameters. The overall conclusion is that the concept of the proper time of the source has enriched the possibilities of experimental study of earthquakes.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

On the elementary theory of earthquake aftershocks

Guglielmi A.V., Zotov O.D. On the elementary theory of earthquake aftershocks // arXiv:2505.10414 [physics.geo-ph]

work page Pith review arXiv

[2] [2]

Evolution of the Tohoku earthquake aftershocks in the framework of the phenomenological theory of aftershocks

Zotov O.D. Guglielmi A.V . Evolution of the Tohoku earthquake aftershocks in the framework of the phenomenological theory of aftershocks // arXiv:2508.04323 [physics.geo-ph]

work page Pith review arXiv

[3] [3]

On he Omori Law in the physics of earthquakes // Appl

Zavyalov A., Zotov O., Guglielmi A., Klain B. On he Omori Law in the physics of earthquakes // Appl. Sci. 2022, vol. 12, issue 19, 9965. https://doi.org/10.3390/app12199965. 11

work page doi:10.3390/app12199965 2022

[4] [4]

The fundamentals of a phenomenological theory of earthquakes // J

Guglielmi, A.V ., Klain, B.I., Zavyalov, A.D., Zotov, O.D. The fundamentals of a phenomenological theory of earthquakes // J. V olcanol. Seismol., 2023, vol. 17, no. 5, pp. 428–438

work page 2023

[5] [5]

A method for processing and analysis of aftershocks due to a tectonic earthquake: A new look at an old problem // J

Guglielmia A.V ., Zavyalov A.D., Zotov O.D., Klain B.I. A method for processing and analysis of aftershocks due to a tectonic earthquake: A new look at an old problem // J. V olcanology and Seismology. 2025. V . 19. No. 2. P . 196–202

work page 2025

[6] [6]

Interpretation of the Omori Law

Guglielmi A.V. Interpretation of the Omori Law // Izv. Phys. Solid Evarth. 2016. V . 52. P . 785–786 // arXiv:1604.07017 [physics.geo-ph]

work page Pith review arXiv 2016

[7] [7]

Omori's law: a note on the history of geophysics // Phys

Guglielmi A.V. Omori's law: a note on the history of geophysics // Phys. Usp. 2017. V. 60. P . 319–324. DOI: 10.3367/UFNe.2017.01.038039

work page doi:10.3367/ufne.2017.01.038039 2017

[8] [8]

On the aftershocks of earthquake // J

Omori F. On the aftershocks of earthquake // J. Coll. Sci. Imp. Univ. Tokyo. 1894. V . 7. P. 111–200

work page