Entropy Monitor Anesthesia

Remo Garattini

Universit`a degli Studi di Bergamo, Facolt`a di Ingegneria,

Abstract

The entropy for a black hole in a de Sitter space is approached within the framework of spacetime foam. A simple model, made by N wormholes in a semiclassical approximation, is taken under exam- ination to compute the entropy for such a case. An extension to the extreme case when the black hole and cosmological horizons are equal is discussed.

Introduction

Black holes have many properties analogous to those of thermodynamics. In particular, four laws of black holes combined with the generalized second law make up a main framework of the black hole thermodynamics. In these

(1)

where A is the area of black hole horizon. This formula is known as Bekenstein- Hawking formula, since the concept of black hole entropy was first introduced by Bekenstein as a quantity proportional to the horizon area and the pro- portionality coefficient was fixed by Hawking’s discovery of thermal radiation

Kbtbh = ℏκ

2πc.

1

κ is the surface gravity of a background black hole. This thermal radiation and its temperature are called Hawking radiation and Hawking temperature, respectively. Let us recall basic properties of the black hole thermodynamics by taking the example of a one-parameter family of Schwarzschild black holes, parameterized by the mass M. The first law of thermodynamics, in this

(3)

where EBH, SBH and TBH are quantities that are identified with the energy, the entropy and the temperature of a black hole, respectively. The energy of the black hole is simply given by EBH = MBH. A simple relation exists also for the temperature TBH. Hawking showed that a black hole with surface gravity κ emits thermal radiation of a quantum matter field (which plays the rˆole of a thermometer) at temperature given by Eq. (2) . Since κ = c4/4GMBH, it is natural to define the temperature of a Schwarzschild black

8Πgmbh

.

Where A ≡16Πg2M2

BH/c4 is the area of the event horizon and C is some constant. Since a value of C is not essential in our discussions, we shall set hereafter C = 0. This is a special case of the Bekenstein-Hawking formula.

It can be shown that classically the area of the event horizon cannot decrease in time (the area law or the second law of black hole) just as the ordinary thermodynamical entropy. This observation was the real motivation of intro- ducing a black-hole entropy. Moreover, when quantum effects are taken into account, it is believed that a sum of the black hole entropy and matter entropy does not decrease (the generalized second law). The zeroth law of black hole thermodynamics states that surface gravity of a Killing horizon is constant throughout the horizon. This is expected for the Schwarzschild black hole, because it is a static black hole. What is unexpected is that the same result is valid also for a Kerr black hole, which is dependent by its tem- perature, while the surface gravity is not. Of course we could just check this result but the point is that it is. This supports the choice of the black-hole temperature. The third law does hold in the sense of Nernst: it is impossible by any process, no matter how idealized, to reduce the surface gravity to zero

2

in a finite sequence by operations . Thermodynamics has a well-established microscopic description: the quantum statistical mechanics. In the thermo- dynamical description, information on each microscopic degree of freedom is lost, and only macroscopic variables are concerned. However, the number of all microscopic degrees of freedom is implemented in a macroscopic variable: entropy S is related to the number of all consistent microscopic states N as S = kB ln N.

(6)

In analogy, it is expected that there might be a microscopic description of the black hole thermodynamics, too.

In Particular, It Is Widely Believed

that the black hole entropy might be related to a number of microscopic states. This microscopic description seems to require a yet to be developed quantum theory of gravity. Actually a microscopic derivation of the black hole entropy was given in superstring theory[5, 6, 7] by using the so-called D-brane technology. In this approach, the black hole entropy is identified with the logarithm of the number of states of massless strings attached to D-branes, with D-brane configuration and total momentum of the strings along a compactified direction fixed to be consistent with the corresponding black hole[8, 9]. The analysis along this line was extended to the so-called M-theory. Recently a different approach based on a foamy structure of space-time has been proposed [11, 12]. In this approach space-time foam is described by a collection of N coherent wormholes, whose energy density

Λ4

e . Λ is an U.V. cut-off, V is the volume of the space and Nw is the wormholes number. When we apply the wormhole model to the area, we obtain the mass quantization of the Schwarzschild black hole, namely1

2Lp

.

(7)

A second consequence is that in de Sitter space, the cosmological constant is quantized in terms of lp, i.e.

3

l2pN = Λc.

(8)

1Units in which ℏ= c = k = 1 are used throughout the paper.

3

In this paper we would like to apply the same wormhole model of spacetime foam, to compute the entropy of a black hole embedded in a de Sitter space whose line element is described by the Schwarzschild-de Sitter metric (SdS).

We will also look at its extreme version, the so-called Nariai metric. The plan of the paper is the following: in section 2, we will briefly report our model of space-time foam, in section 3, we give a simple example of application of the resulting discretized (foamy) spacetime to the computation of the entropy in the Schwarzschild and in de Sitter case; in section 4, we discuss the entropy quantization in the case of the SdS case. We summarize and conclude in section 5.

Constructing The Foam

When we try to merge General Relativity with Quantum Field Theory at the Planck scale, spacetime could be subjected to topology and metric fluc- tuations 2. Such a fluctuating spacetime is known under the name of “spacetime foam” which can be taken as a model for the quantum gravi- tational vacuum. At this scale of lengths (or energies) quantum processes like black hole pair creation could become relevant. To establish if a foamy spacetime could be considered as a candidate for a Quantum Gravitational vacuum, we can examine the structure of the effective potential for such a spacetime. There are some examples showing that flat space cannot be con- sidered as the true ground state for General Relativity [16, 17, 18, 19]. In the case of Ref., the whole spacetime has been considered as a black hole- anti-black hole pair formed up by a black hole with positive mass M in the coordinate system of the observer and an anti black-hole with negative mass −M in the system where the observer is not present. In this way we have an energy preserving mechanism, because flat space has zero energy and the pair has zero energy too. However, in this case we have not a cosmological force producing the pair: we have only pure gravitational fluctuations. The black hole-anti-black hole pair has also a relevant pictorial interpretation: the black hole with positive mass M and the anti black-hole with negative mass −M can be considered the components of a virtual dipole with zero total energy created by a large quantum gravitational fluctuation. Note that this is the only physical process compatible with the energy conserva- tion. The importance of having the same energy behaviour (asymptotic) is related to the possibility of having a spontaneous transition from one space- 2It is interesting to note that there are also indications on how a foamy spacetime can be tested experimentally.

4

time to another one with the same boundary condition . This transition is a decay from the false vacuum to the true one[22, 23]. However, if we take account of a pair of neutral black holes living in different universes, there is no decay and more important no temperature is involved to change from flat to curved space. To see if this process is realizable we need to compute quantum corrections to the energy stored in the boundaries. These quantum corrections are pure gravitational vacuum excitations which can be measured

(9)

where E0 is the zero-point energy and ∂M is a boundary.

We Begin To

consider the following line element (Einstein-Rosen bridge) related to a single

⟨Ψ|Ψ⟩

.

(11)

by perturbing the three-dimensional spatial metric gij = ˜gij +hij. ∆E (M) is computed in a WKB approximation, by looking at the graviton sector (spin 2 or TT tensor) in a Schr¨odinger representation with trial wave functionals of the Gaussian form by means of a variational approach.

(12)

where △is the Laplacian on a Schwarzschild background and Ra



.

(13)

The total energy at one loop, i.e., the classical term plus the stable and

5

where ∆Eq.l. is the quasilocal energy. For symmetric boundary conditions with respect to the bifurcation surface S0 (such as this case Eq.l. = E+−E−= 0. When the boundaries go to spatial infinity E± = MADM. The Stable



.

(14)

Λ is a cut-offto keep under control the UV divergence, we can think that Λ ≤mp. For the unstable sector, there is only one eigenvalue in S-wave. This is in agreement with Coleman arguments on quantum tunneling: the presence of a unique negative eigenvalue in the second order perturbation is a signal of a passage from a false vacuum to a true vacuum. The Rayleigh- Ritz method joined to a numerical integration technique gives E2 = −.

175 41/ (MG)2, to be compared with the value E2 = −. 19/ (MG)2 of Ref.. How to eliminate the instability? We consider Nw coherent worm- holes (i.e., non-interacting) in a semiclassical approximation and assume that

I=1Σi, With Σi ∩Σj = ∅When

i̸ = j. Each Σi has the topology S2 × R1 with boundaries ∂Σ±

I With Respect

to each bifurcation surface. On each surface Σi, quasilocal energy is zero because we assume that on each copy of the single wormhole there is sym- metry with respect to each bifurcation surface. Thus the total energy for the

Collection Is Etot

|2 = NwH|2 and the total trial wave functional is the product of Nw t.w.f.

(15)

By repeating the same calculations done for the single wormhole for the

Nw Wormhole System, We Obtain

a) The total Casimir energy (stable modes), at its minimum, is

Λ4

e . The minimum does not correspond to flat space →∆Es (M)̸ = 0. b) The initial boundary located at R± will be reduced to R±/Nw.

c) Since the boundary is reduced there exists a critical radius ρc = 1. 113 4 such that : ∀N ≥Nwc ∃rc s.t. ∀r0 ≤r ≤rc, σ (∆2) = ∅. This means that the system begins to be stable[11, 12]. To be compared with the value ρc = 1.445 obtained by B. Allen in Ref..

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.