Thermodynamics of black holes in anti-de Sitter space

TL;DR: In this paper, it was shown that the canonical ensemble exists for asymptotically anti-de-Sitter space, unlike the case for the case of asymPTotically flat space.

Abstract: The Einstein equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. Like black holes in asymptotically flat space, these solutions have thermodynamic properties including a characteristic temperature and an intrinsic entropy equal to one quarter of the area of the event horizon in Planck units. There are however some important differences from the asymptotically flat case. A black hole in anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the anti-de Sitter space. For larger black holes the red-shifted temperature measured at infinity is greater. This means that such black holes have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. It also implies that the canonical ensemble exists for asymptotically anti-de Sitter space, unlike the case for asymptotically flat space. One can also consider the microcanonical ensemble. One can avoid the problem that arises in asymptotically flat space of having to put the system in a box with unphysical perfectly reflecting walls because the gravitational potential of anti-de Sitter space acts as a box of finite volume.

Summary

Summary

• Changes in pharmacokinetic parameters of critical ill patients make the treatment of infections challenging, particularly when multidrug-resistant bacteria are involved.
• This target is also related to a better clinical response [9].
• Six adult female sheep weighing from 63 to 81 kg were included in this study.
• Animals were fed with hay ad libitum and with alfalfa pellets and given free access to water.
• The French National Agency of Drug Safety recommends not to administer another dose of amikacin if the Cmin is not below the threshold of 2.5 µg/mL [16].
• The pharmacokinetic analysis was based on a compartmental approach using a two- compartment model, as described for amikacin in patients with renal replacement therapy [17].
• The analysis was performed using the non-parametric modeling software Pmetrics ® (LAPKB, Hollywood, CA) [18].
• In the current study, the authors obtained high peaks, with median Cmax close to 210 µg/mL. Rybak MJ, Abate BJ, Kang SL, Ruffing MJ, Lerner SA, Drusano GL.
• Mise au point sur le bon usage des aminosides administrés par voie injectable : gentamicine, tobramycine, nétilmicine, amikacine.
• Hemodialysis of amikacin in critically ill patients.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Thermodynamics of black holes in anti-de sitter space" ?

In this paper, it was shown that the canonical ensemble exists for asymptotically anti-de-Sitter space, unlike the case for the case of asymPTotically flat space. 

Q2. What is the effect of the gravitational potential in anti-de Sitter space?

However in anti-de Sitter space the gravitational potential has the effect of reflecting back all particles of non-zero rest mass. 

Q3. What is the reason that the Green functions are periodic?

The reason that the Green functions are periodic is that particles pass right around the Einstein cylinder and return to their original positions in space after a time y. 

Q4. What is the implication of these results for the canonical ensemble?

The implication of these results for the canonical ensemble is that the lower mass black hole at a given temperature, which has a mass M<M0, is unstable but contributes to the tunneling amplitude for the formation or disappearance of black holes. 

Q5. What is the partition function of Z()?

The partition function Z(β) is the Laplace transform of N(E\\Z(β)=\\N{E)e~βEdE. (4.1) oThus N(E) is the inverse Laplace transform1 + ίOON(E)=— j Z{βyEdβ. (4.2) ^ π ^ -ΐooThe contour of integration is taken parallel to the imaginary β axis and to the right of any singularities in Z(β). 

Q6. What is the nonconformal mode for the black hole?

The Schwarzschild-anti-de Sitter solution has a negative nonconformal mode for small values of M as in the asymptotically flat case. 

Q7. What is the metric of the covering space of anti-de Sitter space?

The anti-de Sitter vacuum state for conformally invariant fields is then the state induced from the natural vacuum state in theEinstein universe. 

Q8. What is the simplest solution of the classical equations?

The Schwarzschild-anti-de Sitter solution is probably the only other nonsingular positive-definite solution of the classical equations that satisfies the periodic boundary conditions. 

Q9. What is the way to get a real density of states?

if one uses the micro-canonical ensemble, one has to rotate the contour in the relation between the density of states and the partition function in order to obtain convergence. 

Q10. What is the free energy of the black hole?

IfT 0<T<T 1=(πfc)~ 1, (3.10)the free energy of the black hole is positive so this configuration would reduce its free energy if the black hole evaporated completely. 

Q11. How can one construct thermal states in anti-de Sitter space?

One can construct thermal states in anti-de Sitter space by periodically identifying the imaginary time coordinate τ with period β = T-1. 

Q12. How can one verify the temperature of a black hole?

In the case of conformally invariant particles, one can verify this by taking a thermal state on the Einstein universe and conformally transforming. 

Q13. What is the probability of a black hole in the universe?

IfEί<E<E2^m 2 pb, (4.14)the pure radiation and the black hole states will be locally stable but the black hole state will be more probable.