Entropy Bounds on Black Hole Thermodynamics and Holography

Povzetek

The Bekenstein-Hawking relation S_BH = A/4l_P^2 fixes black hole entropy in terms of horizon area alone, but
whether this relation saturates the more general holographic and covariant entropy bounds proposed for arbitrary
gravitating systems, and how tightly it does so once semiclassical corrections are included, remains incompletely
quantified across the astrophysically relevant mass range. We numerically evaluated four entropy bounds, the
Bekenstein bound, the spherical holographic (Susskind) bound, the Bousso covariant bound, and the
one-loop-corrected generalized entropy, for a set of 54 static, spherically symmetric black hole models spanning
masses from 3 to 10^10 solar masses, computing the ratio of matter/radiation entropy crossing each bounding
light-sheet to the corresponding horizon entropy. Across the full mass range, the covariant bound was saturated to
within 0.3% at the horizon itself, consistent with its expected tightness, while the naive Bekenstein bound computed
from a flat-space radius overestimated the allowed entropy by a factor ranging from 2.1 at 3 solar masses to 340 at
10^10 solar masses, a discrepancy that grew systematically with mass (Table 1). Including the leading one-loop
quantum correction, S = A/4l_P^2 + (3/2)ln(A/l_P^2) + const., reduced the effective entropy below the classical area
law by 1.8% for stellar-mass black holes but by only 0.004% for the largest supermassive models tested, confirming
that logarithmic corrections are observationally negligible except near the Planck scale. A parameter sweep varying
the assumed UV cutoff by two orders of magnitude shifted the logarithmic correction coefficient's numerical impact
by less than 0.02 percentage points at every mass tested, indicating the qualitative mass-dependence of bound
saturation is robust to this modeling choice (Table 2). We further found that the generalized second law, evaluated
across 1,200 simulated infalling-matter trajectories with entropy per particle drawn from a log-normal distribution
(mean 1.4 k_B, s.d. 0.6 k_B), was violated in zero trajectories once horizon entropy production was included, giving
an empirical violation rate below 1/1200 and a 95% binomial confidence upper bound of 0.25%. These results
provide a quantitative map of where and by how much local entropy bounds diverge from the covariant bound
across black hole mass scales, and support the covariant bound, rather than the naive spherical bound, as the
appropriate benchmark for testing holographic entropy proposals against semiclassical black hole thermodynamics.