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On the Size of a Black Hole: The Schwarzschild is the Biggest

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arxiv 1911.02019 v1 pith:AXZFLRZC submitted 2019-11-05 gr-qc hep-th

classification gr-qchep-th
keywords blackholeinequalitiesbiggestenergygivenholeshorizon
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abstract

We consider static black holes in Einstein gravity and study parameters characterizing the black hole size, namely the radii of the horizon $R_+$, photon sphere $R_{\rm ph}$ and black hole shadow $R_{\rm sh}$. We find a sequence of inequalities $\frac32 R_+\, \le\, R_{\rm ph}\, \le \,\frac{1}{\sqrt3} R_{\rm sh}\,\le\, 3 M$, where $M$ is the black hole mass. These are consistent with and beyond the previously known upper bounds in literature. The Schwarzschild black hole saturates all the inequalities, making it the biggest of all for given mass. The inequalities include an upper bound of entropy for any quantum system with given energy. We also point out that some black holes satisfying the dominant energy condition can trap photons to form a stable photon shield outside the event horizon, but the shadow hides it from an observer at infinity.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bounds on the radius of outermost photon sphere and black hole shadow in $n$-dimensional Einstein gravity

    gr-qc 2026-06 unverdicted novelty 7.0 of 10

    Proves model-independent lower bound r_sh ≥ ((n-1)/2)^{1/(n-3)} sqrt((n-1)/(n-3)) r_H under WEC and upper bound r_sh ≤ sqrt((n-1)/(n-3)) [(n-1)M]^{1/(n-3)} under WEC+SEC+decay for nD black holes.

  2. Static spheres in black hole spacetimes: pairing, energy conditions, and an upper bound on the innermost radius

    gr-qc 2026-07 conditional novelty 6.0 of 10

    Static spheres around black holes come in unstable/stable pairs, need negative radial pressure, and their innermost radius obeys a new upper bound tied to strong-energy-condition violation.

  3. The existence and upper bound for stable photon spheres in static spherically symmetric black holes

    gr-qc 2025-08 conditional novelty 6.0 of 10

    For static spherical black holes with monotone m(r)/r^3, every stable photon sphere obeys r < 6M.

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