Boundary terms and Noether current of spherical black holes

Abstract In this paper, we use the “complexity equals action” (CA) conjecture to evaluate the holographic complexity in some multiple-horzion black holes for the gravitational theory coupled to a first-order source-free electrodynamics. Motivated by the vanishing result of the purely magnetic black hole founded by Goto et al., we investigate the complexity in a static charged black hole with source-free electrodynamics and find that this vanishing feature of the late-time rate is universal for a purely static magnetic black hole. But this result shows some unexpected features of the late-time growth rate. We show how the inclusion of a boundary term for the first-order electromagnetic field to the total action can make the holographic complexity be well-defined and obtain a general expression of the late-time complexity growth rate with these boundary terms. However, the choice of these additional boundary terms is dependent on the specific gravitational theory as well as the black hole geometries. To show this, we apply our late-time result to some explicit cases and show how to choose the proportional constant of the additional boundary term to make the complexity be well-defined in the zero-charge limit. Typically, we investigate the static magnetic black holes in Einstein gravity coupled to a first-order electrodynamics and find that there is a general relationship between the proper proportional constant and the Lagrangian function $$h(\mathcal {F})$$h(F) of the electromagnetic field: if $$h(\mathcal {F})$$h(F) is a convergent function, the choice of the proportional constant is independent on explicit expressions of $$h(\mathcal {F})$$h(F) and it should be chosen as 4/3; if $$h(\mathcal {F})$$h(F) is a divergent function, the proportional constant is dependent on the asymptotic index of the Lagrangian function.

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INFORMATION RECOVERY FROM BLACK HOLES

International Journal of Modern Physics D ◽

10.1142/s0218271806009765 ◽

2006 ◽

Vol 15 (12) ◽

pp. 2285-2292 ◽

Cited By ~ 10

Author(s):

VIJAY BALASUBRAMANIAN ◽

DONALD MAROLF ◽

MOSHE ROZALI

Keyword(s):

Black Holes ◽

Black Hole ◽

Finite Number ◽

Planck Scale ◽

Black Hole Entropy ◽

Information Loss ◽

Discrete Energy ◽

Information Recovery ◽

Conserved Charges ◽

Boundary Terms

We argue that if black hole entropy arises from a finite number of underlying quantum states, then any particular such state can be identified from infinity. The finite density of states implies a discrete energy spectrum, and, in general, such spectra are non-degenerate except as determined by symmetries. Therefore, knowledge of the precise energy, and of other commuting conserved charges, determines the quantum state. In a gravitating theory, all conserved charges including the energy are given by boundary terms that can be measured at infinity. Thus, within any theory of quantum gravity, no information can be lost in black holes with a finite number of states. However, identifying the state of a black hole from infinity requires measurements with Planck scale precision. Hence, observers with insufficient resolution will experience information loss.

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