Counterintuitive properties of relativistic relative motion for accelerated observers

Markus P\"ossel

Counterintuitive properties of relativistic relative motion for accelerated observers

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For accelerated observers, a constant radar distance does not imply zero relative speed, and two observers exchanging light can each infer the other is approaching while the other infers recession.

desk verdict A clear, correct pedagogy paper that honestly flags its own definition-dependence; the stress-test's main objection is overstated. read the letter →

arxiv 2411.19802 v1 pith:LJ4FV66L submitted 2024-11-21 gr-qc

Markus Pössel This is my paper · ORCID

classification gr-qc

keywords specialrelativityacceleratedobserversrelativevelocityradardistancehyperbolicmotiongravitationalredshiftDopplereffecteventhorizons

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

This paper tries to show that two classical intuitions about relative motion fail once observers accelerate: constant radar distance does not force zero relative speed, and two observers can each be justified in saying the other is approaching (or receding). The argument rests on an event-based definition of relative speed, comparing the inertial frames in which each observer is instantaneously at rest at one chosen event on each worldline. Under that definition, two uniformly accelerated observers with constant radar separation have relative speed $v_{\rm rel}=c\,\tanh(a|T_2-T_1|/c)$, and the radial velocity inferred from exchanged light has opposite signs for the two directions. If the reasoning is correct, gravitational redshift in static spacetimes can be read as a Doppler shift and horizons correspond to $v_R=c$, which would give a unified, intuitive picture of redshift and event horizons.

What carries the argument

The central object is the event-pair definition of relative speed, $\gamma(v_{\rm rel})=-\eta(u_1,u_2)/c^2$, applied to the family of uniformly accelerated observers defined by Eqs. (6)-(7). In their radar coordinates $X,T$, light propagates as $X(T)=X_0\pm c(T-T_0)$ and the metric is $ds^2=e^{2aX/c^2}(dX^2-c^2dT^2)$, so constant $X$ means constant radar distance. Comparing the observers' four-velocities at different $T$ values produces the nonzero relative speed, while comparing them at the emission and reception events of a light signal produces the sign-dependent radial velocity. In general relativity, the same scalar product is evaluated after parallel transporting one four-velocity to the other event, which yields the Schwarzschild radial velocity formula and the horizon condition.

What would settle it

Place two clocks on the hyperbolic trajectories (6)-(7) with fixed $X_1\ne X_2$ and exchange a laser pulse train: the paper predicts a constant frequency shift $\exp(a[X_2-X_1]/c^2)$ in one direction and its inverse in the other, so observing no shift, or a shift of the opposite sign, would contradict Eq. (20).

Watch

Extended reading notes

Core claim

The paper's central claim is that when relative speed is defined via the four-velocities of two observers evaluated at two distinct events, accelerated observers who maintain a constant radar distance are not at relative rest. For the family of uniformly accelerated observers with worldlines $x=(c^2/a)e^{aX/c^2}\cosh(aT/c)$ and $t=(c/a)e^{aX/c^2}\sinh(aT/c)$, the relative speed between observers at $X_1$ and $X_2$ evaluated at times $T_1$ and $T_2$ is $v_{\rm rel}=c\,\tanh(a|T_2-T_1|/c)$, which vanishes only when the two events are simultaneous in the radar coordinates. For light signals exchanged between them, the radial relative velocity is $v_R=c\,\tanh(a(X_2-X_1)/c^2)$, so one direction of signal propagation yields a redshift and the other a blueshift. The apparent contradiction is resolved because relative radial velocity is a property of a chosen pair of events, not of the two worldlines alone. Transferred to general relativity by parallel transport, the same event-pair dependence lets gravitational redshift be interpreted as a Doppler shift and identifies horizons with the limit $v_R=c$.

Load-bearing premise

The load-bearing premise is the choice to define relative speed by comparing the two observers' momentary inertial frames at two selected events; if one instead defines relative rest through constant radar distance or through a continuously transported standard of rest, the same two observers have zero relative speed.

Editorial extensions

If this is right

Two observers who keep a constant radar distance will still measure a nonzero Doppler shift between them, so radar-stationarity does not imply zero relative speed under the four-velocity definition.
The sign of radial relative velocity depends on which event pair is chosen, so 'approaching' and 'receding' can both be correct descriptions for the same pair of accelerated observers.
In static gravitational fields, gravitational redshift can be interpreted as a Doppler shift arising from the nonzero relative radial velocity of static observers.
The equivalence principle need not be framed as motion versus gravity, because relative motion and its associated Doppler shift appear in both descriptions.
Schwarzschild and cosmological horizons are characterized by $v_R=c$, giving a concrete physical meaning to the boundary beyond which light can never reach an observer.

Reading between the lines

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

Editor's inference: the counterintuitive results are definition-dependent; if one adopts radar-coordinate or transported-frame notions of rest, the same physical setup yields zero relative velocity, so the paper is best read as showing that 'relative rest' is not an invariant concept rather than as a new physical effect.
Editor's inference: the $v_R=c$ horizon criterion could be applied as a pedagogical device to Rindler, Schwarzschild, and de Sitter horizons alike, though the paper explicitly discusses only Schwarzschild and cosmological cases.
Editor's inference: a practical test could come from spacecraft maintaining constant separation with laser ranging; the predicted frequency shift $\exp(a[X_2-X_1]/c^2)$ between identical clocks is in principle measurable with current optical-clock technology, although tiny for realistic accelerations.

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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

Thanks for the report. I read the paper in full and mostly agree with your ACCEPT. The stress-test note is half right but I don't think it lands as a load-bearing objection. The two "counterintuitive properties" are real consequences of the event-pair definition in Eq. (3), and the paper is transparent that they are definition-dependent; it says explicitly that at equal T the relative speed is zero. What's new is the pedagogical framing: static coordinates or constant radar distance don't imply zero relative speed under the definition tied to light emission/reception, and the sign of v_R can flip depending on which event pair you use. That precise statement is not in the references I know.

The algebra in Sections III and IV checks out. I verified the metric (9), proper time (11), four-velocities, Eq. (15), and Eq. (20). The Doppler link (5) is standard. The GR part leans on Narlikar and Synge without rederiving (22), which is fine for an AJP-style paper; the interpretation as Doppler shift is clearly attributed. The horizon discussion is a neat extension: v_R = c as the light-speed limit on a null path.

The soft spot the stress-test points to is real, but minor: Section II calls Eq. (3) "natural" and doesn't spend much time comparing it to radar-based or Fermi-Walker definitions of rest. Someone could reasonably prefer a definition under which the two observers are always at rest, and the paper doesn't fully adjudicate that dispute. But the paper's claims are explicitly conditional on the event-pair definition, and it never claims a physical paradox. For a pedagogical article, this is acceptable; more argument would help, but the absence isn't a fatal gap.

No circularity problem: no fitting, no free parameters; the only input is the acceleration scale a. The citation pattern is honest and relevant. The title is slightly stronger than the content—these are properties of a particular definition, not of all relativistic relative motion—but the body makes the scope clear.

Who gets value: instructors teaching relativity, people who use Doppler/redshift interpretations, and anyone who has been tripped up by Bell's spaceship paradox or cosmological horizon talk. It deserves a serious referee. I'd bring it to a reading group on relativity pedagogy, though I wouldn't cite it in my own research.

Referee Report

0 major / 5 minor

Summary. This paper argues that two intuitive statements about relative motion fail for accelerated observers in special relativity: (a) constant radar distance and "relative rest" do not imply zero relative speed under the event-based definition introduced in Eq. (3), and (b) two observers can each infer, from Doppler-shifted light signals, that the other is respectively approaching and receding. The author uses a family of uniformly accelerated observers in 1+1 Minkowski space, derives the radar coordinate metric (9), proper time (11), four-velocities (12)-(13), relative speed (15), and radial relative velocity (20), then generalizes to Schwarzschild via parallel transport and cites Narlikar's formula (22) for a Doppler interpretation of gravitational redshift and horizons.

Significance. All central derivations in Sections III and IV check out algebraically, and the paper is transparent about the event-pair dependence of v_R. The definitional concern about Eq. (3) is real but does not undermine the results: the author explicitly defines relative speed as the speed between momentarily comoving inertial frames at two selected events, and this is precisely the quantity that enters the longitudinal Doppler formula (5). Under an alternative radar-distance-rate definition the "counterintuitive" properties vanish, and the paper would benefit from saying so explicitly; however, the manuscript makes its convention clear and applies it consistently. The GR discussion is a well-referenced reinterpretation rather than a new derivation, and it is appropriately cautious. For an American Journal of Physics-style pedagogical contribution, this is a useful and largely correct set of examples with no fitted parameters and no hidden assumptions beyond the stated convention.

minor comments (5)

[Section V A, Eq. (22)] The Schwarzschild relative-velocity formula (22) is quoted from Ref. 14 without derivation; since the GR section's conclusions rest on it, a short derivation or a specific equation reference in Narlikar's paper would improve self-containedness.
[Sections II and IV A] Add one sentence in Section II or IV A noting that Eq. (3) is a convention and that alternatives such as the rate of change of radar distance give zero relative velocity for the same observers; this would prevent the abstract from being read as making a definition-independent statement.
[Reference 14] The citation for Narlikar appears to have the wrong volume: Am. J. Phys. 92, 903-907 (1994) should presumably be volume 62, so please correct the bibliographic details.
[Section V D] The statement that cosmological horizons are marked by v_R=c should specify that this uses the light-like parallel-transport definition of v_R and should refer explicitly to Ref. 21; as written it could be mistaken for a new result.
[Section I] In the abstract and introduction, "student's everyday experience" should be "students' everyday experience".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's results are explicit consequences of its stated definition of relative speed and independent Doppler/GR formulas, with self-citations confined to non-load-bearing discussion.

full rationale

The derivation chain is self-contained. Equation (3) defines relative speed via four-velocities at two specified events, and the paper explicitly states that this requires two events; Eqs. (12)-(15) then follow by direct substitution of the Lass-Minguzzi observer four-velocities. The radar-distance 'at rest' conclusion in Section IV A uses the metric (9) and light propagation (10), which are independent inputs, and the paper explicitly notes that the relative speed is zero for T1=T2 and nonzero only for different-time event pairs, so the apparent paradox is a disclosed property of the chosen event-pair definition rather than a hidden equivalence. The sign asymmetry in Section IV B is derived from the independent longitudinal Doppler formula (5), the proper-time relation (11), and radar light propagation (10), and the paper transparently explains that the two observers use disjoint sets of event pairs. In the GR section, the Schwarzschild relative velocity (22) is quoted from Narlikar/Synge, and the author's self-citations (refs. 18 and 21) appear only in the closing discussion of cosmological redshifts and horizons; they are not needed for the paper's central SR results and do not constitute load-bearing circular support. There are no fitted parameters presented as predictions, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation. The strongest caveat is that the 'counterintuitive' properties depend on a particular, event-based definition of relative speed, but the paper states that definition openly and does not claim it is the only possible one. Therefore no significant circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivations use only standard special and general relativistic formalism: four-velocity kinematics, radar coordinates for uniformly accelerated observers, the longitudinal Doppler formula, and parallel transport in GR. The acceleration parameter a is an input, not fitted. No new entities are introduced, and no free parameters are tuned to data.

assumptions (5)

domain assumption Relative speed between two objects is defined by gamma(v_rel) = -eta(w,u)/c^2, using momentarily comoving inertial frames at two selected events.
Adopted in Sec. II as 'a natural way of defining relative speed'; the counterintuitive properties are consequences of this event-based definition.
standard math The family of observers defined by x=(c^2/a)e^{aX/c^2}cosh(aT/c), t=(c/a)e^{aX/c^2}sinh(aT/c) is a uniformly accelerated family whose radar coordinates yield the metric (9).
Standard Rindler-type coordinate system, with references to Lass and Minguzzi; all SR derivations rely on it.
standard math The longitudinal Doppler formula 1+z = sqrt((1+v_R/c)/(1-v_R/c)) connects radial relative velocity to wavelength shift.
Used in Sec. IV B to assign sign and magnitude to v_R and to interpret gravitational redshift as Doppler.
domain assumption In general relativity, relative velocity is obtained by parallel-transporting four-velocities between events along a path.
Generalization in Sec. V A, following Synge and Narlikar; the path dependence is explicitly acknowledged.
domain assumption Narlikar's formula (22) for the Schwarzschild relative radial velocity is correct.
Quoted from prior literature and used in Sec. V B to discuss horizon at v_R=c; not re-derived in the paper.

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Cite this review

Pith. "Pith review of Counterintuitive properties of relativistic relative motion for accelerated observers." pith.science (2026). https://pith.science/paper/LJ4FV66L

@misc{pith2026241119802,
  author       = {Pith},
  title        = {Pith review of: Counterintuitive properties of relativistic relative motion for accelerated observers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJ4FV66L}},
  note         = {Machine review of arXiv:2411.19802}
}

read the original abstract

A challenge in teaching about special relativity is that a number of the theory's effects are at odds with the intuition of classical physics, as well as student's everyday experience. The relativity of simultaneity, time dilation and length contraction are prominent examples. This article describes two additional, less well-known counter-intuitive properties, both of which follow from the relativistic definition of relative motion, in situations with accelerated observers: (a) when two objects have a constant radar distance and are by that standard ``at relative rest,'' their relative speed is not necessarily zero, and (b) for two observers A and B, a situation is possible where A considers the two to be approaching each other, while B considers them to be moving away from each other. In general relativity, the generalisations of these two properties prove helpful for understanding static gravitational fields, and they also provide some insight into the nature of horizons.

Figures

Figures reproduced from arXiv: 2411.19802 by the authors.

**Figure 2.** FIG. 2. Two of our accelerated observers ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Reza Rashidi and Fatemeh Ahmadi, ``Reciprocity principle and relative accelerations in the theory of relativity,'' Am. J. Phys. 87 , 758--762. doi:10.1119/1.5115338

work page doi:10.1119/1.5115338

[2] [2]

L. McL. Marsh, ``Relativistic Accelerated Systems,'' Am. J. Phys. 33 , 934--938 (1965). doi:10.1119/1.1971081

work page doi:10.1119/1.1971081 1965

[3] [3]

Dolby and Stephen F

Carl E. Dolby and Stephen F. Gull, ``On radar time and the twin `paradox','' Am. J. Phys. 69 , 1257--1261 (2001). doi:10.1119/1.1407254

work page doi:10.1119/1.1407254 2001

[4] [4]

Ettore Minguzzi, ``Differential aging from acceleration: An explicit formula,'' Am. J. Phys. 73 , 876--880 (2005). doi:10.1119/1.1924490

work page doi:10.1119/1.1924490 2005

[5] [5]

Desloge, ``Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field,'' Am

Edward A. Desloge, ``Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field,'' Am. J. Phys. 57 , 1121--1125 (1989). doi:10.1119/1.15802

work page doi:10.1119/1.15802 1989

[6] [6]

Preston Jones, Gerardo Mu\ n oz, Michael Ragsdale, and Douglas Singleton, ``The general relativistic infinite plane,'' Am. J. Phys. 76 , 73--78 (2008). doi:10.1119/1.2800354

work page doi:10.1119/1.2800354 2008

[7] [7]

Gerardo Mu\ n oz and Preston Jones, ``The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field,'' Am. J. Phys. 78 , 377--383 (2010). doi:10.1119/1.3272719

work page doi:10.1119/1.3272719 2010

[8] [8]

Hermann Bondi, Relativity and Common Sense (Doubleday, 1963)

work page 1963