Galaxies

For the past fifty years, the physical foundations of black hole (BH) thermodynamics have remained widely contested because their original derivations rest at the theoretical intersection of general relativity and quantum field theory (QFT). This work provides an alternative macroscopic formulation by treating the physical vacuum in and around a BH as an elastic continuum strained by the presence of mass. Utilizing Hooke’s law of elasticity, we find that the radial strain at the event horizon of every Schwarzschild BH saturates at the same constant value. Under this principle of universal maximum strain, the laws of BH thermodynamics can be formulated within macroscopic continuum mechanics. This dispenses with the conventional QFT derivation in curved spacetime, paralleling the recent macroscopic descriptions of the Casimir effect. Our calculations are carried out using the Reformulated Planck System (RPS), which relies exclusively on Planck’s constant h. This h-native approach prevents the dimensional contamination of 3D continuum mechanics caused by the 2D geometric factor inherently embedded in Dirac’s reduced constant ℏ, a pivotal misstep first recognized in the iconic definitions of the fine-structure constant $\mathrm{\alpha }$ and the gravitational coupling constant ${\mathrm{\alpha }}_{\mathrm{g}}$, which long precluded the identification of the $\mathrm{\alpha }$-dependence of the weak coupling constant ${\mathrm{\alpha }}_{\mathrm{w}}=\sqrt{\mathrm{\alpha }}$. In BH thermodynamics, this RPS-based approach yields higher Hawking temperatures and lower Bekenstein entropies, both by a factor of $2\pi$.

This paper examines the thermal properties of a Bardeen-type black hole (BH) embedded in a cloud-of-strings and a quintessence background within the Tsallis non-extensive entropy framework. Within the extended thermodynamic phase space, generalized expressions are derived for the Tsallis thermodynamic temperature, heat capacity, equation of state, Gibbs free energy, and the associated critical parameters. Particular attention is paid to how the cloud-of-strings parameter, the magnetic charge and the quintessence field influence the resulting thermodynamic behaviour. A divergence in the heat capacity marking the boundary of local thermodynamic stability is identified from the heat-capacity analysis. The equation of state is further examined using the standard inflection-point conditions. For the representative parameter set considered in this work, the resulting mathematical critical solution has negative critical temperature and pressure and therefore lies outside the physical thermodynamic domain. Hence, no physically admissible Van der Waals-type critical point is obtained for the chosen parameters. Furthermore, the thermodynamic geometry is investigated using the HPEM metric, whose Ricci scalar provides an independent geometric characterization of the phase-transition structure. The results demonstrate that the cloud-of-strings parameter, magnetic charge, and quintessence field significantly influence the thermodynamic stability and phase structure of the BH, whereas the Tsallis entropy modifies the quantitative thermodynamic behaviour while preserving the overall qualitative phase characteristics.

Chromosome maps (ChMs) reveal multiple stellar populations in globular clusters (GCs), but radial trends can depend on how overlapping structures are partitioned. We analyse 56 Galactic GCs with Hubble Space Telescope UV Globular Cluster Survey (HUGS) photometry, separating the stability of star-by-star Gaussian-mixture assignments from that of radial statistics. Strict two-family assignments are stable in only 11/55 eligible clusters, whereas the posterior-weighted radial contrast is reproducible for 34/55. In this radial-stability sample, none of four pre-specified broad-family tests shows a robust association with compactness or half-mass relaxation time. By contrast, the label-free absolute median-radius separation between the lowest and highest 30 per cent of normalized ChM pseudocolour increases with compactness ($N=34$, $\rho =0.500$, permutation $p=0.00256$) and remains positive after F814W matching and common-footprint restrictions. Secondary tests show no comparable monotonic ordering with remaining-mass fraction, a present-day dynamical-age proxy, or the atlas-based global second-generation fraction. Larger amplitudes instead occur at greater compactness, higher central density, and shorter core-relaxation time. Because these central variables are strongly collinear, we interpret them as a common structural-state axis rather than unique causal drivers. Thus, the magnitude—not a universal direction—of extreme-tail radial differentiation is linked to present-day central structural state.