Universal bounds on the size of a black hole
- PDF / 773,771 Bytes
- 16 Pages / 595.276 x 790.866 pts Page_size
- 59 Downloads / 222 Views
Regular Article - Theoretical Physics
Universal bounds on the size of a black hole Run-Qiu Yanga , H. Lüb Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Yaguan Road 135, Jinnan District, Tianjin 300350, People’s Republic of China
Received: 14 July 2020 / Accepted: 3 October 2020 © The Author(s) 2020
Abstract For static black holes in Einstein gravity, if matter fields satisfy a few general conditions, we conjecture that three characteristic parameters about the spatial size of black holes, namely the outermost photon sphere area Aph,out , the corresponding shadow area Ash,out and the horizon area AH satisfy a series of universal inequalities 9 AH /4 ≤ Aph,out ≤ Ash,out /3 ≤ 36π M 2 , where M is the ADM mass. We present a complete proof in the spherically symmetric case and some pieces of evidence to support it in general static cases. We also discuss the properties of the photon spheres in general static spacetimes and show that, similar to horizon, photon spheres are also conformal invariant structures of the spacetimes.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Spherically symmetric case . . . . . . . . . . . . . 2.1 Proof of upper bounds . . . . . . . . . . . . . 2.2 Proof of lower bound . . . . . . . . . . . . . . 3 General static cases . . . . . . . . . . . . . . . . . . 3.1 Generalization of photon spheres . . . . . . . . 3.2 Outermost photon sphere and conjectures about its size . . . . . . . . . . . . . . . . . . . . . . 4 Proofs without spherically symmetry . . . . . . . . 4.1 Bondi–Scahs formalism . . . . . . . . . . . . . 4.2 Proof of 9AH /4 ≤ Ash,out /3 . . . . . . . . . . 4.3 Proof of Aph,out ≤ Ash,out /3 . . . . . . . . . . 4.4 Proof of Ash,out /3 ≤ 36π M 2 . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . Appendix A: Static gauge in Bondi–Scahs formalism . Appendix B: Boundedness of functional B . . . . . . . Appendix C: Critical surfaces on equal-t submanifold and u = 0 null sheet . . . . . . . . . . . . . . . . . Appendix D: An intuition about finding upper bound . a e-mail:
[email protected] (corresponding author)
b e-mail:
[email protected]
. . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction Black holes are fundamental objects in Einstein’s general relativity. The spatial size of a black hole is usually characterized by its horizon; however, the horizon cannot be directly observed in classical theories either locally or from asymptotic infinity. A few of recent arguments (e.g. see Refs. [1,2]) suggest that quantum effects may render the horizon locally observable, but this topic remains controversial. There is another special surface named “photon sphere” where gravity is also so strong that photons are forced to travel in orbits [3– 5]. Differing from the horizon, some photons can escape from the photon sphere, making it observable. The photon sphere plays a key role for gravitational lensing [6,7] or ringdown of waves around
Data Loading...
