Operational Schwarzschild Radius at the Light Cylinder

Operational Schwarzschild Radius at the Light Cylinder Max Karson December 23, 2025 Abstract Rigid rotation requires an identity postulate: a co-rotating observer treats chosen rod markers as non-drifting “places” and implies the rod is a straight, persistent spatial line. We demonstrate that a two-way radar definition of this straight radius necessarily selects two distinct radial tracks when mapped into the laboratory frame: a forward-winding spiral for the outward leg and a backward-winding spiral for the return leg. When null signals are constrained to satisfy this identity condition (intercepting the specific rod markers), the outward signal on the forward-winding spiral and the inward signal on the backward-winding spiral suffer a linear collapse of effective radial velocity as r→rL=c/ω. This produces logarithmically divergent one-way travel times. Consequently, the induced optical line element develops a second-order pole at r=rL, indistinguishable in the radial null sector from the Schwarzschild radius. We interpret rLas an operational Schwarzschild radius for the radar-defined geometry of a rigid rotating rod. 1 Introduction Einstein famously utilized rotating rods and clocks to argue that a rotating reference frame implies non-Euclidean geometry, motivating the equivalence principle [1]. In the present work, we construct the inertial diagram of this thought experiment. However, we invert the standard derivation: rather than assuming the lab is Euclidean and finding curvature in the rotating frame, we grant the rotating observer a Euclidean, static worldview and calculate the resulting geometry in the inertial background. We adopt the perspective of a rotating observer who stipulates that their frame is globally Euclidean and static. Just as an observer in a gravitational field enforces the rule that light travels in straight lines at cthroughout the domain, our rotating observer enforces that the rod is a persistent, straight line. We ask: what becomes of the optical data if the observer strictly maintains this stipulation? We find that this operational insistence forces the “straight” radius to map to a spiral trajectory in the inertial background, creating a horizon structure at the light cylinder rL=c/ω with infinite optical depth, operationally indistinguishable from a gravitational black hole. The argument has two logically distinct stages: (i) construction, in which the observer’s twoway radar straightness/identity rule defines the straight rod as a pair of one-way spiral tracks in the laboratory frame; and (ii) constrained propagation, in which null signals are required to remain on these rotating spiral tracks. 2 Operational postulates We consider a single physical observer at the pivot point of a rotating rod. An inertial cylindrical chart K= (t, r, θ) on Minkowski space serves as the geometric background. 1 •Non-drifting place identity. The observer treats a marked segment of the rod as “the same place” over time. •Straight radius by two-way radar. The observer defines the radial line using the set of physical rod markers. A “radial” light signal is operationally defined not by a vacuum geodesic, but by the requirement that it intercepts the specific sequence of non-drifting rod markers that the observer calls “the radius.” 3 The inertial background Kand the light cylinder In K= (t, r, θ), the Minkowski metric is: ds2=−c2dt2+dr2+r2dθ2.(1) Rod markers rotate as θrod(t)=ωt +θ0. The light-cylinder radius is rL≡c/ω. 4 Defining the spiral rod in the laboratory frame We determine the K-description of what the central observer calls “the straight rod.” Fix a central proper-time label τ. During a small radial advance dr along the rod, the non-drifting marker advances by dθ =ω dt in the laboratory coordinates. The radar protocol uses cto convert between flight time and radial increment: dt =±dr c.(2) Substituting (2) into the angular advance gives dθ/dr =±ω/c. Integrating this yields the track geometry: θ±(t, r)=ωt ±ω cr+θ0.(3) The term ±(ω/c)ris the spatial twist required to maintain contact with the rotating rod during finite signal transit, and ωt is the rigid rotation of that geometry. 5 Constrained propagation and linear stall We now constrain a null signal to the tracks defined in (3). 5.1 Constraint and null condition For the signal to satisfy the identity condition θ±(t, r), it must satisfy: dθ dt =ω±ω c dr dt .(4) Substituting into the null condition ds2= 0 (Eq. 1) yields a quadratic equation for the normalized radial velocity x= (dr/dt)/c. 2 5.2 Branch selection Let β=r/rL. The physical roots corresponding to the radar intent (outgoing on the forwardwinding track, incoming on the backward-winding track) are: dr dt out =c1−β2 1+β2,(5) dr dt ret =−c1−β2 1+β2.(6) 5.3 Linear stall and logarithmic divergence The magnitude of radial progress is:  dr dt  =c1−β2 1+β2.(7) As r→rL(i.e., β→1), the term (1 + β2)→2 and (1 −β2)≈2(1 −β). Thus:  dr dt  ∼c1−r rL.(8) This is a linear collapse of velocity. The one-way travel time is the integral of the inverse velocity: t1w(r) = Zr 0 dr′ |dr′/dt|∼Zdr′ 1−r′/rL ∼ − ln 1−r rL.(9) The exact integration yields: t1w(β) = rL c−β+ ln 1+β 1−β.(10) This diverges as r→rL. Unlike the vacuum chord (Appendix A), which reaches the horizon in finite time, the radar-defined radius places the light cylinder at infinite temporal depth. 6 Operational Schwarzschild Radius From (8), the effective line element near the horizon scales as: c2dt2≈dr2 1−r rL2.(11) This is a second-order pole. Compare this to the radial sector of the Schwarzschild metric (rs= 2M): dt2=dr2 1−rs r2.(12) Both systems exhibit the same optical pole structure. The rotating observer, by enforcing rigidity via radar, induces an optical metric where rLfunctions exactly as rs: an event horizon at finite coordinate distance but infinite optical depth. 3 7 Conclusion The horizon derived above arises directly from the observer’s axioms. In General Relativity, a Schwarzschild horizon emerges because an observer stipulates that light moves in straight lines at speed c—a stipulation that, when applied in a curved spacetime, generates a horizon. Here, we show that the same stipulation made by a rotating observer produces the operational structure of a Schwarzschild radius at the light cylinder. In both cases, identical time dilation arises from operational blindness to non-local changes to the velocity of light. The rotating observer’s rod is perceived as straight and static, yet it is actually spiral-shaped, and light traveling along it never follows a lab-straight path. Operationally, rigid rotation generates the same horizon as gravitational curvature. AI Disclosure The author used AI language models to assist with drafting, derivations, and algebraic checks. A Control calculation: naive rotating spoke gives finite time Define the naive rotating spoke by θ′=θ−ωt = const, i.e. dθ/dt =ω. The null condition (1) becomes dr dt 2 + (ωr)2=c2⇒dr dt =±cp1−β2, β =ωr c.(13) Integrating to rLgives t(r) = Zr 0 dr′ cq1−r′2 r2 L =rL carcsinr rL,(14) which is finite at r=rL. This is the square-root (finite) case. It corresponds to a different operational meaning of “radius” than the two-way radar-defined non-drifting rod used in the main text. B Algebra: solving the constrained null condition on θ±(t, r) Let x≡(dr/dt)/c and β≡ωr/c. From (4) and (1): Forward-winding track (+). x2+β2(1+x)2= 1 ⇒(1 + β2)x2+ 2β2x+ (β2−1) = 0, whose roots are x= (1 −β2)/(1+β2) and x=−1. Backward-winding track (−). x2+β2(1 −x)2= 1 ⇒(1 + β2)x2−2β2x+ (β2−1) = 0, whose roots are x= 1 and x=−(1 −β2)/(1+β2). 4 References [1] A. Einstein, “Die Grundlage der allgemeinen Relativit¨atstheorie,” Annalen der Physik 49, 769 (1916). [2] Ø. Grøn, “Relativistic description of a rotating disk,” American Journal of Physics 43, 869 (1975). [3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th ed. (Pergamon, Oxford, 1975). 5