Finite Upper Bound for the Hawking Decay Time of an Arbitrarily Large Black Hole in Anti-de Sitter Spacetime

In an asymptotically flat spacetime of dimension d>3 and with the Newtonian gravitational constant G, a spherical black hole of initial horizon radius r_h and mass M ~ r_h^{d-3}/G has a total decay time to Hawking emission of t_d ~ r_h^{d-1}/G ~ G^{2/(d-3)}M^{(d-1)/(d-3)} which grows without bound as the radius r_h and mass M are taken to infinity. However, in asymptotically anti-de Sitter spacetime with a length scale l and with absorbing boundary conditions at infinity, the total Hawking decay time does not diverge as the mass and radius go to infinity but instead remains bounded by a time of the order of l^{d-1}/G.

Hawking radiation [1,2] causes a black hole to decay away if there is not sufficient incoming radiation to prevent this. In four-dimensional asymptotically flat spacetime with no incoming radiation, a nonrotating black hole of at least a solar mass M ⊙ (which emits essentially only photons and gravitons, assuming the lightest neutrino is not greatly lighter than the next lightest) has a lifetime 1.1589 × 10 67 (M/M ⊙ ) 3 years [3,4,5], which grows as the cube of the mass and hence diverges when the mass it taken to infinity. A common toy model for preventing black hole decay is to put it into asymptotically anti-de Sitter spacetime (AdS, which for brevity I shall use not just as an abbreviation for the anti-de Sitter spacetime but also for an asymptotically anti-de Sitter spacetime) with a length scale ℓ not too large and imposing reflecting or thermal boundary conditions at spatial infinity [6,7,8]. Although AdS has an infinite volume, the gravitational potential that rises indefinitely as one goes to large distances acts effectively like a finite confining box for the black hole and its Hawking radiation. Massless radiation can escape to infinity, but one can impose reflecting boundary conditions there, and then the radiation reflects back inward in a finite time as seen by observers at finite distances from the center or from a black hole.
Alternatively, one can postulate a sufficiently large thermal bath at or beyond the boundary at radial infinity (though there is not space for a large enough bath within the AdS spacetime itself) that emits thermal radiation inward to keep a sufficiently large black hole from evaporating.
Much less attention has been paid to the possibility of imposing absorbing boundary conditions at infinity in AdS. Here I consider the case in which one starts in AdS with a large black hole (horizon ra-dius r h much larger than the AdS length scale ℓ) and has its Hawking radiation being absorbed by the boundary at infinity, so that no radiation comes back to keep the black hole from evaporating. I shall then calculate the time for the black hole radius to drop from one initial large value to another. Surprisingly, it turns out that this time does not diverge even when the initial black hole size (and hence mass) is taken to infinity. This calculation is simplified by the fact that the geometric optics approximation is good for the bulk of the thermal Hawking radiation emitted by a large black hole, r h ≫ ℓ. The geometric optics approximation breaks down for small black holes, r h < ℓ, so without doing numerical calculations that are left for the future, I am not able to give a precise estimate for the time for the black hole to evaporate all the way down to zero size. However, the time to evaporate from r h = ℓ to r h = 0 can be estimated to be of the same order of magnitude as the time to evaporate from r h = ∞ to r h = ℓ, namely t ∼ ℓ d−1 /G, so that the total time to evaporate from infinite size to zero size will be finite, of the order of t ∼ ℓ d−1 /G. where with r h being the value of the radial coordinate r (the circumference divided by 2π for the (d−2)-spheres of symmetry) at the event horizon, where V vanishes, giving and where dΩ 2 (d−2) is the metric on a unit round (d − 2)-sphere, which has (d − 2)-dimensional area that I shall denote by the same symbol without the d and without the exponent 2, though this is a numerical value rather than a metric, The surface gravity κ of the static black hole is the value on the horizon of the derivative of √ −g 00 = V 1/2 with respect to the proper radial distance √ g rr dr = V −1/2 dr and hence is (1/2)dV /dr evaluated on the horizon, r = r h . Then the Hawking temperature T of the black hole is the surface gravity divided by 2π, which for this Schwarzschild-AdS black hole is The first term is larger for a 'small' black hole, one with r h < (d − 3)/(d − 1)ℓ, and gives the Hawking temperature for a Schwarzschild black hole in asymptotically flat spacetime, which is the limit of AdS with ℓ → ∞. Such a 'small' black hole has negative specific heat, Hawking temperature T decreasing with increasing r h (which is monotonically increasing with the mass M that is proportional to µ), and hence is unstable when one imposes thermal boundary conditions at the corresponding value of T at infinity [6].
The second term is larger for a 'large' black hole, one with r h > see [6].
The Bekenstein-Hawking entropy S for the black hole is its horizon By integrating dM = T dS, one can then get that the black hole mass Now we need to calculate the Hawking emission rate. Massive particles cannot reach infinity in AdS but fall back into the black hole, so only massless particles contribute to the decay. We shall make a geometrical optics approximation and then show that it is valid for the Hawking radiation from a very large black hole, r h ≫ ℓ.
In the geometric optics approximation, massless quanta move along null geodesics. One can orient the angular coordinates so that a null geodesic has only t, r, and one angular coordinate, say θ with g θθ = r 2 , changing. If one normalizes the affine parameter λ so that d/dλ is the momentum p of the quantum, then the conserved energy is ω = −p 0 and the conserved angular momentum is L = p θ . Then one gets If the null geodesic is coming from infinity to approach the black hole, so that r is decreasing, the geodesic will enter the hole if there is no turning point where (dr/dλ) 2 vanishes. This will be the case if the square of the impact parameter b, that is b 2 = L 2 /ω 2 , is smaller than the maximum value of r 2 /V , which is the square of the critical impact parameter, b 2 c . The maximum value of r 2 /V occurs at the minimum value of which is at and gives the critical impact parameter as A quantum with typical energy of the order of the Hawking temperature, ω = T , which has the maximum angular momentum L = b c ω that can fall into the black hole with that energy, will have angular momentum L = b c T . For r h < ℓ, this is of the order of unity, so that there are not many different angular momentum modes that can fall into the black hole freely with energy ω ∼ T , and even the ones that can have wavelengths comparable to the size of the sphere at r = r c .
That implies that the geometric optics approximation is not valid for This implies that the geometric optics approximation is valid for quanta with energy comparable to the Hawking temperature, ω ∼ T , as there are many different values of the angular momentum that can freely fall into the black hole.
To put it another way, for r h ≫ ℓ, b c T = r c T c ≫ 1, where T c = T V (r c ) −1/2 is the local temperature measured by a static observer at r c , which is also the radius of the circular photon orbits. The Hawking quanta at r = r c will have a typical wavelength ∼ 2π/T c , which is much less than the circumference 2πr c of the sphere at the location of the circular photon orbits. Therefore, one can view this   Then the Hawking emission power is where and f is the thermal-averaged cross section divided by the geometric . For x ≡ r h /ℓ ≫ 1, the geometric optics approximation is good, so y ≈ z ≈ f ≈ 1 and the Hawking emission power is proportional to x d ≡ (r h /ℓ) d . Now when one evaluates dM = T dS either from Eq. (6) or from Eq. (7) with T = [(d − 1)/(4πℓ)]xy, one gets When one sets this equal to the formula given for the power in Eq.
(15), one gets For x ≡ r h /ℓ ≫ 1 so that y ≈ z ≈ f ≈ 1, a black hole that started at infinite initial size (r h ≡ ℓ x = ∞) and infinite initial mass and to a finite mass, However, f will have a positive minimum value, which I shall denote as f m , since black holes do evaporate even for small x (e.g., x = 0 for asymptotically flat spacetime with no negative cosmological constant).
Presumably this lower limit occurs at x = 0, since it is plausible that the geometric optics approximation gets better and better as x increases, so that f increases monotonically to approach unity for large x. My old numerical calculations [3,4,5] for photons and gravitons in four-dimensional Schwarzschild spacetime (x = 0) give the value f m ≈ 0.13395 for d − 4, assuming that indeed the minimum is at One can easily show that z < y for all x, so that if one defines then replacing z by the larger value y = 1 + u 2 in Eq. (20) and f by the smaller value f m gives, for positive du (e.g., for a black hole that is evaporating), One can then integrate this from u = 0 (infinite black hole size) to u = ∞ (zero black hole size) to get a finite upper bound on the lifetime ∆t of an initially infinitely large and massive black hole evaporating in asymptotically anti-de Sitter spacetime with absorbing boundary conditions at infinity: If one now goes to four-dimensional spacetime and uses my numer- One can fairly easily do somewhat better in d = 4 by using the exact expression for z instead of replacing it by its upper bound y, though still replacing f by f m , giving which is (8 √ 3 − 9)/6 ≈ 0.8094 times as large as the previous more crude limit. The actual limit might be of the order of two or more times smaller, since f is expected to be greater than f m for positive x and approach 1 ≈ 7.4655f m as x gets large. However, it would require extensive numerical calculations to evaluate f (x) and do the integral to get a precise value for the upper bound on the total decay time even in just four-dimensional spacetime, which is beyond the scope of this present paper that just establishes the existence of a finite decay time for an initially infinitely large and infinitely massive black hole in asymptotically anti-de Sitter spacetime without giving a precise numerical value for this decay time.
In conclusion, arbitrarily large black holes in asymptotically antide Sitter spacetime do not take an arbitrarily long time to evaporate away; instead the total decay from infinite initial mass to zero final mass takes a finite time that has the form where for the spherically symmetric black holes considered in this paper, C is a finite constant that depends on the spacetime dimension