A stochastic field — turbulence, electromagnetic noise, density fluctuations in a relativistic plasma — has both a spatial power spectrum and a temporal power spectrum. The spatial spectrum describes how the field varies across space at a single instant; the temporal spectrum describes how it varies at a single point over time. In non-relativistic physics, these are related through the dispersion relation: if you know how fast waves travel, you can translate between frequency and wavenumber, and the spatial and temporal spectral indices are simply connected.
In special relativity, the connection becomes constrained by Lorentz invariance. Tevzadze (arXiv 2602.23195, February 2026) demonstrates that for Lorentz-homogeneous stochastic fields, the temporal spectral index differs from the spatial index by a universal geometric factor that depends only on the effective dimensionality of momentum space. The relationship is not a derivable consequence of a specific dispersion relation — it's a consequence of the symmetry itself.
The temporal spectral index is symmetry-protected. An observer moving at any velocity through the stochastic field measures the same temporal spectral index. The power spectrum's slope in time is Lorentz-invariant: it doesn't change with the observer's frame. This is not obvious. The temporal power spectrum involves sampling the field along a worldline, and different observers have different worldlines. But the symmetry of the underlying field distribution ensures that the statistical properties measured along any timelike worldline are the same.
The universality has a limit. It breaks when the spectrum is anisotropic — when different spatial directions contribute differently — or when dispersion is strong enough that the effective speed of propagation depends on frequency. In those cases, the geometric factor becomes scale-dependent, and the relationship between temporal and spatial indices ceases to be universal.
The result spans astrophysics, plasma physics, and field theory. Any measurement of temporal fluctuations in a relativistic medium constrains the spatial spectrum through the geometric factor, without needing to resolve the spatial structure directly. The clock measures the map.