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Unfortunately, Rovelli & Christodolou show [ https://arxiv.org/abs/1411.2854 ] that: independent of coordinate choice, the spatial 3-volume of a static Schwarzschild BH departs spectacularly from Euclidean geometry (not surprising); and in a collapsar asymptoting to Schwarzschild, volume increases over time up to some maximum (depending on Hawking radiation). More generally, one cannot rely on intuitions from Euclidean relations in substantially non-Euclidean geometries -- and where there is Ricci curvature, we depart from Euclid. Y C Ong has a perhaps slightly more accessible overview and further calculations at https://plus.maths.org/content/dont-judge-black-hole-its-are... under the subheading "Growing with time". Ong also writes, quite reasonably, that the spatial 3-volume of a black hole is not a well-defined notion. I would go further and say that applies to any region of any general curved spacetime in which there is "sufficient" Ricci curvature. Unfortunately, "sufficient" is hard to divorce from coordinate conditions, and I think that is the underlying source of the non-well-definedness. I'd also like to emphasize that there is no reliable probe for the spatial 3-volume bounded by the horizon of a collapsed-matter BH given only its instantaneous location, linear & angular momentum, charge, and mass. We would also need to know the BH's age and initial conditions. Consequently, while one might talk about some sort of time-dependent black hole entropy related to its internal volume, I don't think it's likely to be useful at all, and at the very least it's an impractical measure in practically all astrophysical circumstances. By comparison, given mass, charge and angular momentum (we can always fix linear momentum and position to zero by coordinate choice) we can calculate a Kerr-Newman BH's inner surface's area in arbitrary coordinates. Given area "A" and a choice of time coordinate, classically we say dA / dt >= 0, and A relates to entropy by Bekenstein 1973's (1/2 ln 2) / 4 pi relation. Hawking's 1974 modification given turns this into Entropy_{BH} := (boltzmannconstant A) / (4 plancklength^2) and breaks the monotonic area condition. The 1973 paper and the 1974 letter [ doi:10.1103/PhysRevD.7.2333 resp. doi:10.1038/248030a0] are fairly resp. very short; both are straightforward and can be found easily including at one's favarrrite source of old papers. I think if you read them you will change your mind that they form "a whole new branch of thermodynamics that are necessarily exotic theory". Indeed, it is because nobody seriously proposes rejecting standard thermodynamics that evaporating Schwarzschild black holes seem like weird objects. Given the above, > (from google) S/V = 3/R 4πr2/Volume = 3/r or Area/3/radius = Volume and the consequences you draw from that in this thread are not generally correct. |