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2014
Kretschmann Invariant and Relations Between Spacetime Singularities Entropy and Information
Ioannis Gkigkitzis East Carolina University
Ioannis Haranas Wilfrid Laurier University, [email protected]
Omiros Ragos University of Patras
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Recommended Citation Gkigkitzis, I., I. Haranas and O. Ragos, 2014. Kretschmann Invariant and Relations Between Spacetime Singularities Entropy and Information. Physics International 5(1): 103-111. DOI: 10.3844/ pisp.2014.103.111
This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. Physics International 5 (1): 103-111, 2014 ISSN: 1948-9803 ©2014 Science Publication doi:10.3844/pisp.2014.103.111 Published Online 5 (1) 2014 (http://www.thescipub.com/pi.toc)
KRETSCHMANN INVARIANT AND RELATIONS BETWEEN SPACETIME SINGULARITIES ENTROPY AND INFORMATION
1Ioannis Gkigkitzis, 2Ioannis Haranas and 3Omiros Ragos
1Departments of Mathematics, East Carolina University, 124 Austin Building East Fifth Street, Greenville NC 27858-4353, USA 2Department of Physics and Astronomy, York University, 4700 Keele Street Toronto, Ontario, M3J 1P3, Canada 3Department of Mathematics, Faculty of Sciences, University of Patras, 26500 Patras, Greece
Received 2014-02-05; Revised 2014-03-28; Accepted 2014-03-28 ABSTRACT Curvature invariants are scalar quantities constructed from tensors that represent curvature. One of the most basic polynomial curvature invariants in general relativity is the Kretschmann scalar. This study is an investigation of this curvature invariant and the connection of geometry to entropy and information of different metrics and black holes. The scalar gives the curvature of the spacetime as a function of the radial distance r in the vicinity as well as inside of the black hole. We derive the Kretschmann Scalar (KS) first for a fifth force metric that incorporates a Yukawa correction, then for a Yukawa type of Schwarzschild black hole, for a Reissner-Nordstrom black hole and finally an internal star metric. Then we investigate the relation and derive the curvature’s dependence on the entropy S and number of information N. Finally we discuss the settings in which the entropy’s full range of positive and negative values would have a meaningful interpretation. The Kretschmann scalar helps us understand the black hole’s appearance as a “whole entity”. It can be applied in solar mass size black holes, neutron stars or supermassive black holes at the center of various galaxies.
Keywords: Kretschmann Scalar (KS), Schwarzschild Black Hole, Reissner-Nordstrom (RN)
1. INTRODUCTION where, Rαβγδ is the Riemann tensor. In principle the derivation of the KS is simple, but in practice to When we study any space time, it is important actually derive it requires a very long algebraic above other things to know whether the spacetime is computation, which can very much be simplified with regular or not. By regular spacetime we simply mean today’s software that perform algebraic and tensorial that the space time must have regular curvature calculations. In order to calculate the above scalar we invariants are finite at all space time points, or contain first need to calculate the Christoffel symbols of the curvature singularities at which at least one such second kind according to the Equation 2: singularity is infinite. In many cases one of the most useful ways to check that is by checking for the ∂g ∂ g ∂ g finiteness of the Kretschmann Scalar (from then on α=1 δα γδ + βδ + γβ Iβγ g β γ δ (2) KS) which sometimes is also called Riemann tensor 2 ∂x ∂ x ∂ x squared, in other words Equation 1: Once the Christoffel symbols are calculated we then = αβγδ K Rαβγδ R (1) calculate the Riemann tensor to be Equation 3:
© 2014 Ioannis Gkigkitzis, Ioannis Haranas and Omiros Ragos. This open access article is distributed under a Creative Commons Attribution (CC-BY) 3.0 license which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. DOI: 10.3844/pisp.2014.103.111 Ioannis Gkigkitzis et al . / Physics International 5 (1): 103-111, 2014
∂Γα ∂Γ α φ αβδ βγ µαµα fields . Actually, more complicated invariants like Rβγδ= − +ΓΓ−ΓΓ βδ µγ βγ µδ (3) µν µναβ µναβ ∂xγ ∂ x δ R Rµν and R Rµναβ and C Cµναβ. In the weak field limit theories described by Equation 6 result to potentials For example in a sphere there are only two nonzero of the form (Capozziello et al ., 2010) Equation 7: 1= 2 θ Riemann tensor components i.e., R212 sin and r 1 2 n = θ G∞ M λ also R221 sin , which exactly characterize the () = − + α k V r1 ∑ k e (7) curvature of the sphere. We usually think of the r k =1 curvature as the Ricci scalar, which can be obtained α by contraction of the Riemann tensor, first Rβδ= R βαδ where, G∞ is the value of the gravitational constant as th α measured at infinity, λk is the interaction length of the k and then R= R α . In the case of a sphere the takes the form component of non-Newtonian corrections and αk (Henry, 2000) Equation 4: amplitude of is term is normalized to the standard. Newtonian term. If somebody considers the first term =αβγδ = 4 in the series of Equation 9, we obtain the following Ks Rαβγδ R 4 (4) a potential Equation 8:
Because it is a sum of squares of tensor components, r G∞ M λ () = − + α 1 this is a quadratic invariant. In the case of black holes the V r1 1 e (8) calculation of the scalar is required if somebody wants to r derive and investigate the curvature of a black hole. The need for calculation of the KS emanates from the fact that in The second term is a Yukawa type of correction to vacuum the field equations of general relativity a zero the Newtonian potential and its effect can be Gaussian curvature at and in the black hole, thus giving no parameterized by a1 and λ1 which for simplicity we will information about curvature of the spacetime and thus the K call α and λ. For large distances, i.e., r>> λ the scalar need to be computed. In the case of Schwarzschild exponential term vanishes and the gravitational constant black holes the KS is (D’inverno, 1992 ) Equation 5: is simply G∞. If r<< λ the exponential becomes unity.
48 G2 M 2 K = (5) 3. THE METRIC AND RICCI TENSOR bh c4 r 6 AND THE KRETSCHMANN SCALAR In this contribution we examine the KS of a Yukawa Writing the metric of a spherically symmetric star in type modified Schwarzschild black holes as it is give in the following way: (Haranas and Gkigkitzis, 2013b) and compare this to the Schwarzschild scalar. Furthermore, in an effort to µ()()() ξ ds2222= c ert, dt − e 2, vrt dr 2 −Ω e 2, rt d 2 (9) investigate the relation between entropy, information and geometry we write the Yukawa black hole scalar as a where, , ν, ξ are in general functions of radial function of entropy and information number N. distance r and time t and dΩ2 = dθ2+sin 2θdφ2. For a static metric in Equation 9 the Ricci tensor is diagonal 2. THE YUKAWA POTENTIAL and therefore we obtain the following nonzero components Equation 10 to 13: Following (Capozziello et al ., 2010; Haranas and Gkigkitzis, 2013b) we say that theories derived from Re02=− µ (2ξ&& +++− v&& 2 ξ & 22 v & µξ &( 2 & + v & )) the action: 0 (10) −e−2v ()µ′′ + µ ′()2 ξ ′ + µ ′ v ′ ε =−() ∇∇∇2 kφ −µ v φ φ + 4 A∫ v gfRRRR,, , , g;µ ; v Ldx m (6) 2 1=− 2 µ && + &ξ& −+ µ & & Revv1 ( (2 v )) (11) − Result to Yukawa corrections to the gravitational −e2v ()2ξ′ 2 +− µ ′ 2 v ′′′() 2 ξµ + potential where f(R) is an analytic function of Ricci −ξ − µ scalar, g is the determinant of the metric g ν and Lm is a 2==+ 3 2 2 ξ&& + ξ & ξ & −+ µ & & RRee2 3 ( (2 v )) fluid-matter Lagrangean and where f is an unspecified (12) −−2v ()ξ′′ + ξ ′() ξ ′ +− µ ′ ′ function of curvature invariants R and ∇R and of scalar e2 v
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=ξ&′ + ξξ ′ & −−& µξ ′ & where, Equation 23: R01 2( ( v ) ) (13)
− r Similarly, we find that Equation 15 to 17: = −2GM + α λ T12 1 e (23) c r −()µ + v d d µ µ − KRe= −01 = e v (14) 1 01 dr dr For a black hole of mass M the KS possesses an
ξ µ essential singularity at the value of r = rH where T = 0 =−02 =− 03 = −()2v d d K2 R 02 Re 03 (15) given by Equation 24: dr dr
α 2GM ξ 2GM 2 GM λ 2 12 13 −()v + ξ dξ −v d = + λ c =− =− = rH 2 W 2 e (24) KR2 12 Re 13 e (16) λ dr dr c c
2 −()2ξ − dξ where, W is the Lambert function of the indicated K=− Re23 = + e (2)v (17) 4 23 dr argument. Next, following (Haranas and Gkigkitzis, 2013a) we write the metric of a Yukawa type of And therefore the KS becomes: Schwarzschild black hole that is curved in along both the time and radial coordinate, according to the 2 2 2 2 KK=4( ) + 8( K) + 8( K) + 4 ( K ) (18) equation Equation 25: 1 2 3 4
2GM − λ 22= −() + α r /2 Next in our effort to investigate KS singularities of ds c12 1 e dt various metric let us now proceed with a fifth force c r −1 (25) 2GM − λ metric that incorporates a Yukawa correction as it is −−() +α r / 222 −Ω 12 1 e dr r d given by (Spallicci, 1991) curved only in the time c r coordinate and used for measurements in the solar system experiments namely: where, dΩ2 = d θ2+sin 2θdφ2 and upon comparing
Equation 9 and 14 we obtain that Equation 26 to 28: 2GM − λ 22=−() +α r /2222 −−Ω dsc12 1 e dtdrrd (18) c r −1 r 1 2 GM − v() r=ln 1 − 1 + α e λ (26) 2 c2 r And therefore Equation 19 to 21:
( ) = v r 0 (19) ξ (r) = r (27)
ξ (r) = r (20) 1 2 GM − r µ() = − + α λ rln 12 1 e (28) 2 c r 1 2 GM − r µ()r r=ln 1 − 1 + α e λ (21) 2 2 c r Therefore using Equation 18 the final expression for the KS becomes: − − − λ − λ Omitting orders O(c 8), O(c 6) and e 3r/ and e 4r/ the final expression for the KS becomes Equation 22: 48 G2 M 2 K= K + K = Yuk sch cor c4 r 6 r − 2 − r 2 1 λ r r 16 r r +αe 1 + + 1+αe λ 1 + + 2 2 λ λ 2 2 (29) 48 G M 2 3 12 λ λ K ≅ (22) 4 6 2 2r 2 3 4 c r T − 1r 5 r r r − 2r 16 rr2 r 3 r 4 +α 2e λ ++++ +α 2e λ 1 ++++ λλ2 λ 3 λ 4 2 3 4 2 6 3 12 12λλ 3 λ 12 λ
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The KS equation above Equation 28 does not Furthermore, Equation 33 approaches the KS of the possess an essential singularity. This could be due to Schwarzschild KS if the two RHS terms become zero for the fact that the metric elements g ν cancels with the the values of r given by: −1 corresponding gµν . As a check we see that when α = 0 R3 we immediately obtain the KS of a Schwarzschild r= ± c 2 (35) metric as it is given in (Henry, 2000). K int GM Next, we use the non-rotating but electrically charged source vacuum solution as it is given by the Reissner- 4. KRETSCHMANN SCALAR AND IT Nordstrom solution (Misner and Wheeler, 1973): RELATION TO ENTROPY AND 2GM GQ 2 NUMBER INFORMATION ds2= c 2 1 − + dt 2 cr24πε cr 4 2 0 In general relativity there exist a set of curvature −1 (30) 2GM GQ 2 invariants that they are scalars. They can be formed from −−1 + dr2 −Ω r 2 d 2 2πε 4 2 cr4 0 cr the Riemann, Ricci and Weyl tensors respectively and they describe various possible operations such that, covariant The final expression for the KS becomes: differentiation, contraction. We can obtain various invariants that they are formed from these curvature tensors 22 2 4 =48GM + 2 Q + 7 Q that play an important role in the classification of space- KRN 1 (31) cr46 crM 248 crM 422 times. Invariants useful in distinguishing Riemannian manifolds or manifolds that they have a positive and well As a check we see that when Q = 0 we immediately defined metric tensor. In order to investigate true obtain the KS of a Schwarzschild metric as it is given in singularities one must look at quantities that are (Henry, 2000). Similarly, the KS equation of the Reissner- independent of the choice of coordinates. One of these is the Nordstrom above Equation 29 does not possess an essential singularity but only the r = 0 one. Moreover, KS which, at the origin only or r = 0 possesses a let us consider the internal star metric solution that singularity that is a physical singularity which we can also reads (Stephani, 1990) Equation 32: call gravitational singularity. At r = 0 the curvature becomes infinite indicating the presence of a singularity. 2 3 2GM 1 2 GMr 2 At this point the metric and space-time itself, is no longer 2= 2 − −− ds c 12 1 2 well-defined. For a long time it was thought that such a 2Rc 2 Rc (32) solution was non-physical. However, a greater −1 2GMr 2 understanding of general relativity led to the realization dt2−−1 dr 222 −Ω r d c2 R 3 that such singularities were a generic feature of the theory and not just an exotic special case. Such solutions are now The final expression for the KS becomes: believed to exist as called black holes. This contribution is an effort to connect geometry to 22 22 =48GM + 48 GM information and an attempt to investigate the Kint 4 6 2 c R 2GMr2 2 GMr 2 consequences of essential KS entropy and information c4 R 6 1− − 3 1 − 2 2 singularities that may result from the elimination of mass Rc Rc −8 (33) M that enters the metric. Since α = 3.04×10 and r< λ = 3 3 2 15 − 96 G M r 4.94×10 m (Haranas and Gkigkitzis, 2011) and 2 2 2 considering a 1 solar mass black hole Equation 29 the 6 9 2GMr 2 GMr − λ −2 λ c R 1− − 3 1 − exponential quantities e r/ ≅1 and e r/ ≅1 are both Rc2 Rc 2 approximately equal Equation 22 simplifies to:
The KS equation i.e., Equation 33 possesses an 1 r essential singularity at the following values of r: +α 1 + 2 2 λ ≅ 48 G M 2 K 2 (36) 2 2GM 1 r 9GM− 4 c R c4 r 6 1−() 1 + α +α 2 + r= ± R (34) 2 λ K int GM rc 2
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3 Therefore we find that the KS has an essential c4 k ()1+ α 6 ()S = ± B (42) singularity that is given by the equation: KS αG2 m 2 k p KS 2GM r≅()()1 +=α R 1 + α (37) KS 2 Sch And therefore the number of information N can be H c written as a function of the KS in the following way: where, RSch is the gravitational radius of the 3 c4 ()1+ α 6 Schwarzschild black hole. At this point we see that this N() K ≅ (43) KS α 2 2 particular metric does posses an event horizon and G mpln 2 k KS therefore the entropy cannot be calculated on the horizon. However, the existence of a singular boundary Similarly, taking into account the same assumptions is apparent: the “ Kretcmann horizon ” given by Equation as in Equation 35 we obtain KS of a fully curved 37. To proceed let us assign a definition of entropy, the Yukawa type of metric is given by Equation 28 can be KS entropy, similar to that calculated to (Haranas and approximated in the following way Equation 44: Gkigkitzis, 2011), which reads as follows: 2 2 = + ≅ 48 G M 2 KYuk K sch K cor 4 6 c3 k 4π() 1 + α k G c r =B = B (44) SKS A KS 4Gh c h 16r 2 16 r 1+α 1 + + α 1 + 2 (38) 12λ 12 λ 2 M M2 =4π() 1 + α k B mp Next, u sing expressions for the mass and the gravitational radius for a Yukawa black hole as a functions of entropy for a solar mass black hole, as it where, AKS is the surface area defined by the Equation 37. This definition suggests that the KS horizon is a given in (Haranas and Gkigkitzis, 2013b) we are trying depository of information (or infinite information). Next to express the singularity in the gravitational radius as a we obtain that Equation 40: singularity in essence in entropy S Equation 45 and 46:
mp S mp S M() S = (45) M = KS (39) 2πk () 1 − α ()+ α π B 2 1 kB
=2GM () + α And: rH 1 c2
2G m ()1+ α S Gm S =()1 + α p (46) = p KS 2 π r 2 (40) c2 k KS H ()+ α π B c1 k B S =lp()1 + α π()+ α Substituting Equation 38 in Equation 39 in kB 1 Equation 36 we express KS as a function of the entropy SKS and we find that: Equation 41 has only one singularity at r = 0 and there is no other essential singularity. Using Equation 12c8 k 1 Gm S 41 and taking into account the same assumptions as in K() S =B + p KS KS KS α2422 2 λ()+ α Equation 35 we obtain that the KS as a function of GmSpKS2 c 1 k B (41) entropy S on the horizon of such a black hole up to the 3 2 Gm S O( α ) the most significant contribution comes from +α 1 + P KS 2 2λ()+ α the coefficient of the 1/S term and therefore the KS c1 k B can be written as Equation 47:
For 1 solar mass black hole the most significant 12 π 2G 2 m 22 k 2 ≅p B −α − α 3 contribution comes from the coefficient of the 1/S term and KYuk ()1 (47) c4l 6()1+ α 6 S 2 therefore the KS entropy can be written as Equation 42: p
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Which tends to zero as the entropy approaches infinity. 116Gh S Gk Q4 8 GQ 2 = +B − Solving for the entropy S we obtain an expression of the r 2 2 NR H π πεh πε 8c kB 0 cS 0 entropy as a function of the scalar Kyuk to be Equation 48: (53) 1 εch GS ε k GQ4 GQ 2 ±0GQ 2 ++− 0 B 1/ 2 επ2 εε 2 2c k 16 cSh 2 2πGm k ()1−α 3() 1 +α + α B 0 0 S = ± p B (48) 2l 3 3 cp()1+ α K yuk And therefore substituting for rNRH and M we write the KS of the Reissner-Nordstrom (RN) metric as a function of entropy in the following way Equation 54 and 55: Next using (Haranas and Gkigkitzis, 2013a) we write the entropy to be: S = Nk ln2 , we write the KS as a 2 4 2 B 7G Q 3 Q function of entropy and number of information N K = + KS 2π2828c εΦ c 6 ε Φ 7 according to the equations: 0 0 16cGSh 3 kGQ 234 8 GQ 32 +B − (54) 1/ 2 π3k ckSh επ 23 επ 3 π −α ()+α + α 2 B0 B 0 2 Gm p ()1 3 1 = 2 4 2 N 2 3 3 (49) l + α 3kB GQ 8 k B GQ ln 2 cp()1 K Yuk +16 Gch S + − π4Φ 6h ε 2 ε 4 kcB cS0 0
N = 0 only if alpha imaginary. Both Equation 43 and Where: 44 can exhibit a singularity if and only if the coupling constant becomes negative i.e., α = -1. A negative α kQ2 48 kQS 2 16 ch GS 2 +B − B would introduce a repulsive component in the Yukawa hε2 ε Φ =1c 0 0 + 1 correction something that it is known as a fifth force. 8c2 π kS 2 ε c 2 Next we consider the Reissner-Nordstrom (RN) B 0 (55) metric the KS does not possess a singularity besides the r εG ε kQkQS2 28 2 0Q2+ 0 16 ch S 2 +B − B = 0 case. The event horizon RN metric has the following πh ε2 ε 16 kB S c 0 0 solutions Equation 50: The KS of the Reissner-Nordstrom (RN) metric Gε() Q2+ 4 πε GM 2 possesses a singularity when Ф = 0, for the following values =GM ± 1 0 0 r 2 2 (50) of entropy out of which the first two are real numbers RN H c2ε c π 0 Equation 56 to 59:
k Q2 (2 + c ε ) And therefore the corresponding entropy becomes B 0 S = (56) Equation 51: 1 hε+ ε 4c0() 2 c 0
c3 k 2π k GM 2 S=B A = B k Q2 (2 − c ε ) RNh RN h B 0 4G c S= SOS − ve (57) (51) 2 hε+ ε 4c0() 2 c 0 2 π ε()2+ ε 2 k Q kB M G0 Q4 0 GM +B ± 4εch ε c h 0 0 kQ2 (2 i+ c ε ) B 0 S = (58) 3 4chε() 2 i− c ε Solving for the mass M in terms of the entropy we 0 0 obtain Equation 52: kQ2 (2 i− c ε ) B 0 S = (59) 116hS kQ4 8 Q 2 4 hε()+ ε M +B − (52) 4c0 2 i c 0 π πε2 h πε 8 GkB 0 c GS 0 G And therefore the values of the number of Similarly the event horizon as a function of entropy information N at which Equation 49 is singular are becomes Equation 53: Equation 60 to 63:
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Q2 (2 + c ε ) The KS of the internal metric solution has an essential 0 N = (60) entropy singularity that is given from Equation 68: 1 hε− ε 4c0() 2 c 0 ln 2 π k c 3 S=B ()9 R2 − r 2 (68) KS int 8Gh Q2 (2 − c ε ) 0 N = (61) 2 hε+ ε 4c0() 2 c 0 ln 2 Or an equivalent number of information singularity that takes the value Equation 69:
2 + ε 3 Q(2 i c 0 ) πc = =()2 − 2 N3 (62) N9 R r (69) hε− ε KS int h 4c0() 2 i c 0 ln 2 8G ln 2
To compare we write the KS of the Schwarzschild 2 − ε Q(2 i c 0 ) metric on the horizon an a function of entropy and N = (63) 4 4chε() 2 i+ c ε ln 2 information to be Equation 70: 0 0 2 12Gh 12 l Finally for the internal metric solution of a star there K= S = p S (70) Sch π3 6 π 3 is a metric singularity at r = rH when Equation 64: ckrB k B r
R3 5. DISCUSSION r= ± c (64) H 2GM The Schwarzschild solution was the first physically And therefore the entropy becomes Equation 65: significant solution of the field equations of general relativity. It showed how space time is curved around
spherically symmetric distribution of matter. The KS for ck3 π kc 5 π kR 3 a Schwarzschild black hole is inversely proportional to S=B A = B R 3 = B (65) KSh KS 2 h l 2 the distance. In Equation 18 a formula for the KS is 4G 2 GMp R gr Sch given for a general spherically symmetric metric which we apply to a fifth force metric that incorporates a where, Rgrsch is the Schwarzschild gravitational radius. Yukawa correction and notice that KS possesses an Therefore solving for the mass M from Equation 66 essential singularity at the horizon. Then we write the we obtain that: metric of a Yukawa type of Schwarzschild black hole that is curved in along both the time and radial π k c2 R 3 coordinate and we notice in Equation 29 that the M == B 2 (66) singularity disappears. This is also the case for the non- l GS p rotating but electrically charged source vacuum solution as it is given by the Reissner-Nordstrom solution in And thus Equation 33 of the corresponding KS can be Equation 29-31. In considering the internal star metric written in terms of the entropy S as follows Equation 67: solution we see again the occurrence of an essential singularity in Equation 34. The introduction of the 12 π 2c 6 k 2 concept of entropy leads to the definition of “KS = B Kint horizon”, beyond the distinction between gravitational G2h 2 S 2 singularities and essential geometric singularities. This 12 π 2c 6 k 2 + B allows us to define and investigate singularities in 2 (67) πkcR32 π kcr 32 essence in entropy S. The equations for the entropy, such G2h 2 S 2 1−B − 3 1 − B GSh GS h as 42, 48, 50, 58 and 59 as well as the equations for the corresponding information may result in negative and 12 π 3c 9 k 3 r 2 − B complex values for the entropy, which makes it necessary πkcr32 π kcR 32 to seek an appropriate framework for the interpretation of G3h 3 S 3 1−B − 3 1 − B these results in an intersection between or combination of GSh GS h general relativity and quantum mechanics.
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The scalar gives the curvature of the spacetime as a information possession of a system in the total state function of the radial distance r in the vicinity as well (Horodecki et al ., 2005). It is possible for a black hole to as inside of the black hole. Furthermore, the correlate with its environment in ways such as quantum Kretschmann scalar helps us understand the black entanglement using procedures such as quantum hole’s appearance as a whole entity from a geometric teleportation where qubits (the basic units of quantum as physical point of view. It can be applied in solar information) can be transmitted exactly (in principle) mass size black holes, neutron stars or supermassive from one location to another, without the qubits being black holes at the center of various galaxies. transmitted through the intervening space. Therefor it Finding a setting in which the entropy’s full range of can be correlated to a reference system that regulates values would have a meaningful interpretation is an maximally entangled states of a system which can be essential step in interpreting our results on entropy. used as a teleportation protocol to transmit quantum Negative entropy appears as an interesting feature of states from a source to a receiver. higher derivative gravity (Cvetic et al ., 2002) Landauer’s principle states that the erasure of data (Schwarzschild-Anti-deSitter and Schwarzschild-deSitter stored in a system has an inherent work cost and therefore gravity black holes) and has been suggested that this dissipates heat. However, this consideration assumes that appearance of negative entropy may indicate a new type the information about the system to be erased is classical instability where a transition between SdS (SAdS) black and does not extend to the general case where an observer holes with negative entropy to SAdS (SdS) black holes may have quantum information about the system to be with positive entropy would occur and where the classical erased, for instance by means of a quantum memory thermodynamics would not apply any more. It was also entangled with the system. The study cost of erasure is suggested that such a new type phase transition is determined by the entropy of the system, conditioned on the presumably related with the known fact that strong energy quantum information an observer has about it. The more an condition in higher derivative gravity maybe violated observer knows about the system, the less it costs to erase it. (Cvetic et al ., 2002). On the other hand, entropy can be Entropies can become negative in the quantum case, when negative and erasing bit from a system can result in a net an observer who is strongly correlated with a system may gain of work, leading to the result of the effect that a gain work while erasing it, thereby cooling the environment quantum computer may cool itself by erasing bits of (Del Rio et al ., 2011). information (Del Rio et al ., 2011). It will be interesting if The notion of complex-valued as well as negative such negative entropy idea can apply to black holes. This information entropy remains to be given some physical can change Hawking radiation-maybe under some sense interpretation through possibly an interaction range condition, the black hole can also be cooling system (not consideration. We can only conjecture that the complex only radiating) or other unknown as of now effects may entropy may be the result of a certain background free be happen. Modeling our universe as just a big computer, energy and/or scale invariance. From a quantum deleting information in a black hole may occur through mechanics point of view, this might indicate the presence accretion matter to black hole and the black hole does of a non Hermitian Hamiltonian (Rotundo and Ausloos, not radiate but cools the universe. This seems to be in 2013). Certain observations indicate this direction. An accordance with another approach of suggesting that attempt to a dynamical interpretation of a classical Hawking radiation, via quantum tunneling, carries entropy complex free energy is found in (Zwerger, 1985) where out of the black hole, a process that represents a net gain of the author pointed out that the problem is to determine a information drawn out from the black hole (Zhang et al ., characteristic ”relaxation” time for some process in 2011). So, Hawking radiation carries not only entropy, but which a dynamical equation (Langevin or Fokker- also information, out of the black hole. Planck) is connected to some Hamiltonian or some Negative values for the entropy point towards a corresponding transfer matrix. A probability current can quantum entropy which can be understood operationally be written, in fact, in terms of some unstable mode times as quantum channel capacity (Horodecki et al ., 2005). In an equilibrium factor which is the imaginary part of the the language of communication theory, the amount of free energy. The imaginary part of the free energy (or information originating from a source is the memory largest eigenvalues) give some information about the required to faithfully represent its output and this amount ”nucleation stage” of the dynamics, i.e., an initial decay is given by its entropy. Negative entropy has a physical rate out of a metastable state of a system after a sudden interpretation in terms of how much quantum change in an effective field (Zwerger, 1985). This communication is needed to gain complete quantum metastable state in our may correspond to a physical
Science Publications 110 PI Ioannis Gkigkitzis et al . / Physics International 5 (1): 103-111, 2014 gravitational singularity, independent of coordinates, Del Rio, L., J. Aberg, R. Renner, O. Dahlsten and V. where instead of “relaxation time”, one may consider Vedral et al ., 2011. The thermodynamic meaning of the “spatial aspect of the phenomenon”, e.g., through negative entropy. Nature, 476: 476-476. DOI: some correlation range length ξ that corresponds to a 10.1038/nature10123 certain differential distance of the horizon. The real Haranas, I. and I. Gkigkitzis, 2011. Temperature, entropy part, of course of such a free energy, determines the and heat capacity of a non-newtonian black hole equilibrium energy state. resulting from a yukawa correction to the metric. Astrophys Space Sci., 337: 693-702. 6. CONCLUSION Haranas, I. and I. Gkigkitzis, 2013a. Bekestein bound of Invariance plays a very essential role in mathematics information number n and its relation to cosmological and physics. It is one of most important tools of tensor parameters in a universe with and without analysis as well as of the theory of relativity and cosmological constant. Mod. Phys. Lett. A, 28: 19-19. cosmology. Black hole is perhaps the least exemplary of Haranas, I. and I. Gkigkitzis, 2013b. Entropic gravity physical entities. Indeed, despite their appearance at the resulting from a Yukawa type of correction to the center of most if not all spiral galaxies, black holes are metric for a solar mass black hole. Astrophys entities for which it is most often said “physics breaks Space Sci., 347: 77-82. DOI: 10.1007/s10509- down”. In addition to this, the physics of black holes has 013-1492-4 been revolutionized by developments that grew out of Henry, R.C., 2000. Kretschmann scalar for a kerr- Jacob Bekenstein's realization that black holes have newman black hole. Astrophysical. J. E, 353: 535- entropy. Steven Hawking raised profound issues 350. DOI: 10.1086/308819 concerning the loss of information in black hole Horodecki, M., J. Oppenheim and A. Winter, 2005. evaporation and the consistency of quantum mechanics Partial quantum information. Nature, 436: 673-676. in a world with gravity. In this direction, we have studied DOI: 10.1038/nature03909 Kretschmann scalar curvature invariant which may be of Misner, C.W. and J.A. Wheeler, 1973. Gravitation. 1st great utility in the study and classification of black hole Edn., Freeman, New York, ISBN-10: 0716703440, singularities. We have discussed the notion of negative pp: 1279. and complex-valued information entropy for which there Rotundo, G. and M. Ausloos, 2013. Complex-valued is a need to give some physical sense interpretation information entropy measure for networks with through possibly quantum considerations. These new directed links (digraphs). Application to citations by findings may open the door to solving many previously community agents with opposite opinions. Eur. intractable problems in quantum information theory of Phys. J. B, 86: 1-10. DOI: 10.1140/epjb/e2013- black holes. It is concluded that such generalizations are 30985-6 not only interesting and necessary for discussing information theoretic properties of black hole gravity, Spallicci, A.D.A.M., 1991. The fifth force in the but also may give new insight into conceptual ideas schwarzschild metric and in the field equations: The about entropy and information. concept of parageodesic motions. Annalen der Physik, 503: 365-368. DOI: 7. REFERENCES 10.1002/andp.19915030509 Stephani, H., 1990. General Relativity: An Introduction Capozziello, S., G. Cristofano and M. De Laurentis, to the Theory of the Gravitational Field. 2nd Edn, 2010. Astrophysical structures from primordial Cambridge, University Press, pp: 124. quantum black holes. Eur. Phys. J. C, 69: 293-303. Zhang, B., Q.Y. Cai, M.S. Zhan and L. You, 2011. An DOI: 10.1140/epjc/s10052-010-1387-2 interpretation for the entropy of a black hole. Cvetic, M., S. Nojiri and S.D. Odintsov, 2002. Black General Relat. Gravit., 43: 797-804. DOI: hole thermodynamics and negative entropy in 10.1007/s10714-010-1097-y desitter and anti-desitter einstein-gauss-bonnet Zwerger, W., 1985. Dynamical interpretation of a gravity. Nucl. Phys. B. 628: 295-330. DOI: classical complex free energy. J. Phys. A, 18: 2079- 10.1016/S0550-3213(02)00075-5 2085. DOI: 10.1088/0305-4470/18/11/028 D’inverno, R., 1992 . Introducting einstein’s relativity. Am. J. Phys., 65: 897-897.
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