Formulate a relativistically consistent fractal matter model

Construct a mathematically consistent formulation that incorporates a genuine fractal matter distribution into the Einstein field equations despite the incompatibility between the differentiable framework of general relativity and fractal geometry.

Background

The thesis models fractal-like matter distributions using simplified Lemaître–Tolman–Bondi profiles, but concludes that these profiles do not provide a complete description of genuinely fractal matter. A rigorous treatment would require addressing the nondifferentiable structure of fractal geometry within a theory whose field equations assume differentiable manifolds.

References

The procedure to perform this inclusion of fractal dimension in cosmological analysis is still an open question, since the nature of fractal geometry is contrary to the differential nature of general relativity.

— Cosmology without the cosmological principle: A study of the large-scale structure effects in the background universe  (2608.15983 - Pastén, 17 Aug 2026) in Chapter 3, Section “Discussion”