Prado Martın-Moruno and Pedro F. Gonzalez-Dıaz- Lorentzian wormholes generalizes thermodynamics...

8/3/2019 Prado Martn-Moruno and Pedro F. Gonzalez-Daz- Lorentzian wormholes generalizes thermodynamics still further

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arXiv:090

4.0099v2

[gr-qc]

4Sep2009

IFF-RCA-09-03

Lorentzian wormholes generalizes thermodynamics still further.

Prado Martn-Moruno1, and Pedro F. Gonzalez-Daz1,

1Colina de los Chopos, Instituto de Fsica Fundamental,

Consejo Superior de Investigaciones Cientficas, Serrano 121, 28006 Madrid, Spain

This paper deals with some thermodynamical aspects of Lorentzian wormholes, including theformulation of the three main laws and the consideration of a possible thermal emission made upof some sort of phantom radiation coming out from the wormhole at a negative temperature. Inorder for these topics to be consistently developed we have used a 2+2 formalism first advanced byHayward for spherically symmetric space-times, where a generalized surface gravity is defined on thetrapping horizon. Our results generalize still further those of the already generalized gravitationalthermodynamics.

PACS numbers: 04.62.+v, 04.70.Dy

Twenty-one years ago, Morris, Thorne and Yurtsever[1] presented the first classically consistent solution for awormhole that was stabilized by exotic matter and be-came furthermore convertible into a time machine. Thatpaper meant a real breakthrough in that it inauguratedthe history of these space-time tunnels as real scientific

objects, rather than as science-fiction toys. In spite ofthat, wormholes remain still belonging to the frameworkof technology fiction. However, recent cosmological ob-servations are compatible with the dominance in the uni-versal vacuum of a so called cosmic phantom energy,which has been shown to play the same stabilizing role asthe conventional exotic matter with respect to wormholes[2].

It is well known that the thermodynamical descrip-tion of gravitational vacuum and black holes has providedthese objects with quite a more robust consistency, allow-ing moreover for a deeper understanding of their space-time structure and properties. We think that, since ex-

otic matter can be viewed as the time-reversed version ofordinary matter and hence one may well think of worm-holes as time-reversed black holes, if a similar thermody-namical representation of wormholes would be feasible,then the physical status of such tunneling would likewisegreatly improve, entitling us by the way to get a morecomprehensible account for the exotic matter which, onthe other hand, would thus become the largest energysource in the universe.

The aim of this paper is to construct a complete andphysically consistent wormhole thermodynamics, follow-ing the 2+2 formalism developed by Hayward for dy-namical black holes [3, 4, 5], as that formalism depends

on local variables and can be in this way applicable tospace-times having no event horizon, such as that forwormholes is. In fact, Hayward himself already realizedthat this formalism may also be applied to wormholes [6],as wormholes have an actual generalized surface gravity[7], too. However, the important contribution by Hay-

Electronic address: [email protected] address: [email protected]

ward to that subject lacks of a precise definition of dy-namic wormholes and of the formulation of the laws oftheir thermodynamics, restricting himself to study thethermal radiation just for the case of black holes. Alsocontemplated in the present paper is the study of all theselatter issues.

It is well known that in spherically symmetric space-times the metric can generally be written as

ds2 = 2g+d+d + r2d2, (1)

where are the double-null coordinates and r is theareal radius. In terms of such an areal radius, the expan-sion becomes

=2

rr, (2)

with / the two preferred normal directions,which we shall consider to be future-pointing.

In a spherically symmetric space-time one can intro-duce the Kodama vector [8], which is defined by

k = curl2r (3)

where the subscript 2 means referring to the two-dimensional space normal to the spheres of symmetry.An interesting property of the Kodama vector, whicheventually turns out to be similar to that of the Killingvector, is [4]

k kb

= kb (4)

on a trapping horizon, that is an hypersurface which canbe foliated by marginal spheres (+ = 0) [3], at which

k vanishes. In Eq. (4) is the generalized surface gravity

=1

2div2grad2r, (5)

implying (choosing e.g. + = 0) that the outer (+ 0) trap-ping horizons, respectively have > 0, = 0 and < 0.

In the case of a Morris-Thorne wormhole we wouldhave a bifurcating, i. e. = 0, outer trapping hori-zon at the wormhole throat r0. Although this is a static
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case, no Killing horizon is present. However one can thenobtain a generalized surface gravity given by

|H =1K(r0)

4r0= 2r0(p + )|H, (6)

where K(r) is the shape-function, p the radial pressure, the energy density and |H means evaluation at the hori-

zon. The outward flaring condition implies K

(r0) < 1,therefore the surface gravity is positive (|H > 0 becausep + < 0, equivalently), as it should be since we have anouter horizon.

On the other hand, it is known that in a sphericallysymmetric space-time a generalized first law of thermo-dynamics can be written as [4],

LzE =LzA

8+ LzV, (7)

where A is the surface area, Lz = z and z = z++ +z is tangent to the trapping horizon, E is the Misner-Sharp energy given by

E =1

2r (1 arar) , (8)

and

= 1

2trace2T. (9)

Eq. (6) allows then to introduce an expression for thegeometric entropy, with the familiar dependence on thesurface area

S A. (10)

In order to see how the area of a trapping horizonevolves, we can express the surface area in terms of the2-form area as A =

S

, with = r2 sin dd in thespherically symmetric case. It follows that the evolutionof a trapping horizon area is governed by the integralexpression

LzAH =

H

z, (11)

in which we have chosen + = 0 for the trapping horizonand from the very definition of such a horizon Lz+ = 0should keep taking on vanishing values along the entire

horizon, so implying

z+

z=

+|H++|H

. (12)

The ++ component of the Einstein equations, corre-sponding to metric (1), the definition (2) and an energymomentum tensor component T++, can be written as [9]

++ = 1

22+ ++log(g+) 8T++, (13)

which, when evaluated at the trapping horizon, yields

++|H = 8T++|H. (14)

It can be seen that by considering an energy-momentumtensor of type I according to the Hawking-Ellis classi-fication [10], i. e. an energy momentum tensor whichtakes a diagonal form in an orthonormal basis, we haveT++

p + as expressed in our basis, which actually isa very natural and consistent assumption1.

Now, for a dynamical black hole which is defined bya future ( < 0) outer trapping horizon [3] and sur-rounded by ordinary matter with p+ > 0, Eqs. (12) and(14) will impose that the signs of non-vanishing z+ andz ought to be different, so that the horizon is space-like.Taking z+ > 0, i. e. choosing z with a positive compo-nent along the future-pointing null direction of vanishingexpansion, it then follows from Eq.(11) that LzAH > 0[5]. A similar line of reasoning starting with p + < 0would finally lead to LzAH < 0.

Babichev et al. [11] used a test-fluid approach to studythe evolution of the horizon area of a Schwarzschild black

hole induced by the accretion of dark energy, and showedsimilar results to those that we have just derived. If suchresults are consistently assumed to be originated from aflow of the surrounding matter into the hole, then one canregard both methods to actually describe just the samesingle, physical process. In what follows we shall considerthat the above coincidence is more than a mere analogy,so that it must also hold in the case of wormholes. Infact, when applied to wormholes [12], the Babichev et al.procedure leads to an increase (decrease) of the size of thewormhole throat in case that p + < 0 (p + > 0). Suchresults can only be recovered by using the 2+2 formalismin terms of trapping horizons whereas the outer trapping

horizon of the wormhole is past ( > 0 for + = 0, inwhich case + would be ingoing and outgoing). Or, inother words, in order to recover the same result followingthe 2+2 formalism obtained by the mentioned procedure,i. e.

LzAH 0, (15)

for p + < 0 dominating the environment that surroundsa wormhole, the wormhole must be characterized by apast outer trapping horizon2. We would use in whatfollows this characterization.

1 In general one would have T++ T00 + T11 2T01, where thecomponents of energy-momentum tensor on the r.h.s. are ex-pressed with respect to an orthonormal basis. In our case, weconsider an energy-momentum tensor of type I [10], not just be-cause it represents all observer fields with non-zero rest mass andzero rest mass fields, except in special cases when it is type II[10], but also because if this would be not the case either T++ = 0(for types II and III) which at the end of the day would implyno horizon expansion, or we would be considering the case wherethe energy density vanishes (type IV)

2 The same results can be obtained by using the trapping horizon


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Let us now consider the possible emission process as-sociated with the semi-classical effects, with the parti-cle production rate being given by the WKB approxi-mation for the tunneling probability along a classi-cally forbidden trajectory, exp[2Im(I)]. If thatprobability would take on a thermal form at the hori-zon, exp(/TH), then one could easily obtain anexpression for the temperature of that radiation.

Following then a parallel reasoning to that of Ref.[13] for black holes, we can express the metric given byEq.(1) in terms of the most convenient generalized re-tarded Eddington-Finkelstein coordinates, owing to thefeature that a wormhole possesses a past outer trappinghorizon. That is

ds2 = e2Cdu2 2edudr + r2d2, (16)

where we have again considered u = + related tothe ingoing direction, implying +|H = 0, |H > 0and +|H < 0, d = udu + rdr, e =g+r

> 0 and e2C = 2g+u, with C =

1 2E/r, E defined by Eq. (3). expresses all the

gauge freedom contained in the choice of the null coor-dinate u. It can be noted that the use of retarded co-ordinates ensures that the marginal surfaces, for whichC = 0, are past marginal surfaces. We want to emphasizethat such a change of coordinates is general enough forour present purposes, assuming that the spacetime mustpossess a past outer trapping horizon; therefore it mustbe applicable to wormholes without any restriction aboutits traversability. On the other hand, a possible bad be-haviour of the r coordinate could be expected taking intoaccount that the more natural and well behaved coordi-nate to describe the radial coordinate of a wormhole is l(with l such that gll = 1 in orthogonal coordinates).

Let us also consider a massless scalar field in theeikonal approximation, = 0 exp(iI), with a slowlyvarying amplitude and being governed by a rapidly vary-ing action given by [13]

I =

e

du

kdr, (17)

in which and k should be interpreted as the angularfrequency and wave number for the scalar field , respec-tively; that is uI = e

and rI = k. The fieldwould then describe radially outgoing radiation. Sincethe wave equation 2 = 0 implies gabaIbI = 0, onecan finally obtain

k2C+ 2k = 0. (18)

This equation possesses two solutions: k(1) = 0, which

corresponds to the outgoing modes, and k(2) = 2/C

defined by = 0, evolving according to Lz|H = 0, andemploying Einstein equations which, evaluated at the horizonproduces, |H = 8T|H , since T + pr, too.

for the ingoing modes. k(2) produces a pole in the ac-

tion (17), because C|H = 0 on the horizon. Noting that|H = rC/2, where we have taken = 0, withoutany loss of generality as |H should be gauge-invariant,with expanding C, one obtains k / [|H(r r0)].Therefore the action has an imaginary contribution whichis obtained by deforming the contour of integration in theupper r half-plane, i.e. Im (I) |H =

|H

and has a

thermal form for a temperature

T = |H2

. (19)

Of course, the radiation associated with temperature(19) has a semiclassical origin, being independent ofany possible classically allowed path passing through thetraversable wormhole throat. A rather key property ofsuch a temperature is its characteristic of being alwaysnegative, a property stemming from the positiveness ofthe surface gravity on the outer horizon, i. e. |H > 0.Some authors seem to be rather uncomfortable with theconcept of negative temperatures in gravitational sys-

tems. While this attitude can well be understandablefor classical systems, it is not definitively so in case ofa quantum-mechanical system, where for sure one mustnot be afraid of negative temperatures. In fact, differ-ent experimentally checked devices which are inexorablyinterpreted in terms of quantum-mechanically governedphenomena, have shown the existence and properties ofnegative temperatures. On the other hand, it is knownthat phantom energy has associated a negative temper-ature [14], so manifesting its quite likely deep quantumnature. As we mentioned above, a phantom fluid is oneof the kinds of exotic materials which can be used as thestuff to build up a traversable wormhole [2]. Therefore,Eq.(19) would imply that a wormhole radiates parti-cles with the same properties as its surrounding matter,such as it already occurred with dynamical black holesin relation to ordinary matter.

Eq. (19) allows us to rewrite Eq.(7) in a more familiarform which is given by

LzE = T LzS+ LzV (20)

on a trapping horizon, where

S =A

4. (21)

The first term in the right hand side (r.h.s) of Eq. (20)can be interpreted as an energy-exchange term (in anal-ogy with the heat term of usual thermodynamics) andthe second one as a work-term. The negative sign inthe energy-exchange term would imply that the exoticmatter which supports this space-time gets energy fromthe space-time itself. Parallely, the term in the r.h.s. ofEq.(20) indicated that exotic matter carried up a workin order to support the space-time. So we are alreadyprepared to formulate a first law of wormhole thermody-namics in the following terms: the change in the gravita-tional energy of the wormhole is equal to the sum of the


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energy removed from the wormhole and the work done

in the wormhole. Even more, Eq. (21) confirms rela-tion (10), specifying the involved proportionality con-stant which turns out to get the familiar numerical value1/4, and, since LzA > 0 in an exotic background, onehas LzS > 0. That is to say, we have now a propersecond law of wormhole thermodynamics expressible as:

the entropy of a wormhole, which is given in terms of

the throat surface area, can never decrease, when placedin its most natural dominant-energy-condition violating

environment.

On the other hand, Eq.(5) leads furthermore to a for-mulation of the third law of wormhole thermodynamics,as it implies that an outer horizon has always > 0. Ifwe consider that no dynamical evolution is able to mod-ify the outer property of the horizon, then the generalizedsurface gravity would always remain being positive. Theabove arguments can in fact be expressed by stating: ifno dynamical process can change the outer character of

the trapping horizon, then it is impossible to reach the ab-

solute zero for surface gravity by means of any dynamical

process.It must be noted that if some dynamical process couldchange the outer character of a trapping horizon in such away that it becomes an inner horizon, then the wormholewould converts itself into a different physical object and,therefore, the laws of wormhole thermodynamics wouldno longer be valid.

It has been argued [6] that by replacing the back-ground energy from exotic to ordinary, one also changesthe causal nature of an outer trapping horizon. It canthen be also considered that caused by such a process, orby a subsequent one, a past outer trapping horizon (i. e.a dynamical wormhole) should change into a future outer

trapping horizon (i.e. a dynamical black hole), and viceversa (check, for example, in the case of the Schwarzschildsolution). The thermal radiation can be computed for adynamical black hole surrounded by ordinary matter byusing the advanced Eddington-Finkelstein coordinates,obtaining again Eq. (19) but with a plus sign, [13].Therefore, if such a conversion would be at all possible,we expected the temperature to also change from nega-tive (wormhole) to positive (black hole) in a way which isnecessarily discontinuous due to the holding of the thirdlaw, without passing through the zero temperature.

Actually, black holes and wormholes can be viewedto be the time-reversed version of each other. It couldseem that a white hole is the time-reversed version of ablack hole, but if one considers also the time inversionof the surrounding material, then the surrounding ordi-nary matter would become exotic matter and vice versa,changing the causal nature of the horizon from space-like to time-like and vice versa. Such an idea about therelation between wormholes and black holes under time-inversion is supported not only by the fact that bothare defined by an outer trapping horizon, which is past(wormhole) and future (black holes), being both bifur-cating in the static case where no local dynamical prop-

erties can be rigorously derived. Really, the three laws ofwormhole thermodynamics respectively become those ofblack hole thermodynamics under time inversion. Thiscan be easily checked taking into account that the exoticmatter is nothing but ordinary matter moving backwardin time, so implying a change in the sign of tempera-ture. Moreover, when one considers a time reversed firstlaw of wormholes thermodynamics, the energy-exchange

term in Eq. (20) changes sign and the temperature be-comes positive, so that ordinary matter supplies energyto space-time. On the other hand, we should also takeinto account that a dynamical black hole emits radiationat a positive temperature. It is worth noticing neverthe-less that the other two wormhole thermodynamical lawsremain nevertheless unchanged under time inversion.

Finally, the growth of the wormhole area can be relatedto the accretion of the surrounded exotic matter, but theradiation process at the trapping horizon would producea decrease of the wormhole size, decreasing thereby thewormhole entropy, too. Like in black holes thermody-namics such a violation of the second law of thermody-

namics is only apparent, because it is the total entropy ofthe universe, composed by the wormhole, the surroundedmatter and the thermal radiation, what should increase.In fact, one could perform an analysis similar to that fol-lowed in [15] where a box filled with a black hole and ra-diation was considered, in order to show that the entropyof the whole system should always increase until the ther-mal equilibrium is reached. The case of a box containinga wormhole just differs from what is discussed in Ref.[15], in that all the contents in the box are at negativetemperature, so that any subsystem with lower temper-ature is hotter [16]. Carefully extending that analysis tothe most general case of a box simultaneously containinga black hole, a wormhole and given amounts of ordinaryand exotic radiations at different proportions would leadto the conclusion that the interwoven final effect from allpossible involved thermal processes that can take placewithin the box inexorably implies the holding of a mostgeneralized second law of thermodynamics for which thesum of all four involved entropies always increases.

After completion of this paper we became aware ofa paper by Hayward [17] in which some part of thepresent work was also discussed following partly simi-lar though somewhat divergent arguments. In work [17]he studies the thermodynamic of two-types of dynamicwormholes, characterized by past or future outer trap-ping horizon. Although these two types are completelyconsistent mathematical solutions, we have concentratedon the present work in the first one since we considerthat they are the only physical consistent wormholes so-lution. One of the reasons which support the previousclaim has already been mentioned in this work and isbased on the possible equivalence between the 2+2 for-malism and the Babichev et al. method. On the otherhand, a traversable wormhole must be supported by ex-otic matter and it is known that it can collapse by ac-cretion of normal matter. That is precisely the problem


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of how to traverse a traversable wormhole finding themouth open for the back-travel, or at least avoiding apossible death by a strangulated wormhole throat duringthe trip. If the physical wormhole could be characterizedby a future outer trapping horizon, by Eqs. 11, 12 and14, then it would increase (decrease) its size by accretionof ordinary (exotic) matter and, therefore, it would benot problem to traverse it, even more, it would increase

its size when a traveler would pass through the worm-hole, contrary to what it is expected for the basis of thewormhole physics.

Acknowledgments

The authors are indebted to S. Robles-Perez and A.Rozas-Fernandez for useful discussions. P. M. M. grate-

fully acknowledges the financial support provided by theI3P framework of CSIC and the European Social Fund.This work was supported by a Spanish MEC ResearchProject No. FIS2008-06332/FIS.

[1] M. S. Morris, K. S. Thorne and U. Yurtsever, Phys. Rev.Lett. 61 (1988) 1446.

[2] S. V. Sushkov, Phys. Rev. D 71 (2005) 043520;F. S. N. Lobo, Phys. Rev. D 71 (2005) 084011.

[3] S. A. Hayward, Phys. Rev. D 49 (1994) 6467.[4] S. A. Hayward, Class. Quant. Grav. 15, 3147 (1998).

[5] S. A. Hayward, Phys. Rev. D 70, 104027 (2004).[6] S. A. Hayward, Int. J. Mod. Phys. D 8, 373 (1999).[7] D. Ida and S. A. Hayward, Phys. Lett. A 260, 175 (1999).[8] H. Kodama, Prog. Theor. Phys. 63, 1217 (1980).[9] S. A. Hayward, Phys. Rev. D 53 (1996) 1938

[10] S. W. Hawking and G. F. R. Ellis, The large scale struc-ture of space-time., Cambridge University Press (1973).

[11] E. Babichev, V. Dokuchaev and Yu. Eroshenko, Phys.Rev. Lett. 93 (2004) 021102 E. Babichev, V. Dokuchaevand Y. Eroshenko, J. Exp. Theor. Phys. 100 (2005) 528

[Zh. Eksp. Teor. Fiz. 127 (2005) 597][12] P. F. Gonzalez-Daz, Phys. Rev. Lett. 93 (2004) 071301

P. F. Gonzalez-Daz and P. Martn-Moruno, Proceed-ings of the eleventhMarcel Grossmann Meeting on Gen-eral Relativity, Editors: H. Kleinert, R. T. Jantzen andR. Ruffini, World Scientific, New Jersey (2008)

[13] S. A. Hayward, R. Di Criscienzo, L. Vanzo, M. Nadaliniand S. Zerbini, arXiv:0806.0014 [gr-qc].

[14] P. F. Gonzalez-Diaz and C. L. Siguenza, Nucl. Phys. B697, 363 (2004) E. N. Saridakis, P. F. Gonzalez-Diaz andC. L. Siguenza, arXiv:0901.1213 [astro-ph].

[15] S. W. Hawking, Phys. Rev. D 13 (1976) 191.[16] A. Alonso and P. Martin-Moruno, Work in Preparation.[17] S. A. Hayward, arXiv:0903.5438 [gr-qc].
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