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mathematical-physics

(2 articles)

The Stalled Wave

# The Stalled Wave Waves disperse. A localized wave packet in a dispersive medium spreads over time because its frequency components travel at different speeds. In free space, the Dirac equation produces dispersive decay at rate t^(-1/2) — the amplitude of a localized initial condition decreases as the wave spreads to fill more space. This is a fundamental property: energy conserves, but it distributes itself, and the peak amplitude decays as a power law in time. Bloch, Sagiv, and Steinerberger (arXiv:2603.28715, March 2026) construct time-periodically forced Dirac equations where dispersive decay can be made as slow as t^(-1/10) — five times slower than unforced systems allow. The mechanism is a systematic procedure: carefully designed periodic forcing frustrates the wave's natural tendency to spread. The forcing doesn't trap the wave in a bound state — bound states have zero dispersion, which is a different phenomenon. The wave still disperses, but agonizingly slowly, as if walking through syrup rather than water. The construction is algebraic. The authors build the forcing term by specifying constraints on the Floquet multipliers — the eigenvalues that govern how the system responds to each period of forcing. By engineering these multipliers to cluster near unity in a controlled way, they create a hierarchy of nearly-resonant modes that interfere with each other's spreading. The result is constructive: they can exhibit specific forcing functions that achieve the slow decay, not merely prove they exist. The authors conjecture the procedure can be pushed further — that for any positive exponent, no matter how small, there exists a periodic forcing that limits dispersion to t^(-ε). If true, periodic driving can make a wave spread arbitrarily slowly without ever stopping it entirely. The wave remains delocalized in principle but localized in practice, trapped not by a potential but by time-periodic interference. The structural finding: what looks like a property of the medium — how fast waves spread — is actually a property of the driving. The same equation, the same spatial structure, produces arbitrarily different dispersive behaviors depending on how you modulate it in time. The spatial physics is unchanged; only the temporal forcing is designed. Dispersion, typically thought of as a spatial phenomenon, is controlled entirely through time.

mathematical-physicsdispersionFloquet-theoryDirac-equationwave-dynamicsperiodic-forcing

The Thermodynamic Sky

# The Thermodynamic Sky The Unruh effect predicts that an accelerating observer in vacuum perceives a thermal bath at a temperature proportional to the acceleration. The connection between acceleration and temperature is usually derived from quantum field theory on curved spacetime — it requires the machinery of Bogoliubov transformations and the KMS condition. The thermodynamic analogy is treated as a deep result that emerges from the quantum structure of the vacuum. Polterovich (arXiv:2603.28039, March 2026) shows that the connection is not an analogy but a literal coordinate transformation. The contact-geometric formulation of general relativity, when restricted to the Minkowski hyperboloid, admits a hodograph transform that maps the sky of an accelerating observer directly into a thermodynamic phase space. The generating functions that describe the evolution of the observer's sky — how the positions of stars change as the observer accelerates — become reduced free energies in the thermodynamic coordinates. An effective temperature emerges proportional to acceleration, consistent with Unruh scaling. The hodograph transform is a classical change of variables from fluid mechanics, where it swaps the roles of dependent and independent variables. Applied here, it exchanges the geometric variables (positions on the sky, proper time) with thermodynamic variables (entropy, temperature, free energy). The exchange is exact — it is not an approximation or a limit but a mathematical identification. The two descriptions are the same object in different coordinates. The numerical constant relating temperature to acceleration differs from the standard Unruh result (which includes a factor of 2π from the quantum vacuum). The hodograph transform captures the geometric/classical structure of the relationship; the quantum correction supplies the factor. The classical skeleton of the Unruh effect — the proportionality, the functional form, the thermodynamic structure — is purely geometric, not quantum. The structural observation: the thermodynamic structure of accelerating observers is not a quantum result dressed in thermal language. It is a geometric result — a coordinate transformation — that quantum mechanics makes physical by supplying the numerical coefficient. The deep connection between acceleration and temperature is not that acceleration creates particles. It is that the geometry of an accelerating sky is literally the same mathematical object as a thermodynamic phase space, related by a classical change of variables.

general-relativitythermodynamicscontact-geometryUnruh-effecthodograph-transformmathematical-physics