Advances in High Energy Physics

Advances in High Energy Physics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 918490 | https://doi.org/10.1155/2013/918490

Arindam Lala, "Critical Phenomena in Higher Curvature Charged AdS Black Holes", Advances in High Energy Physics, vol. 2013, Article ID 918490, 18 pages, 2013. https://doi.org/10.1155/2013/918490

Critical Phenomena in Higher Curvature Charged AdS Black Holes

Academic Editor: Dandala R. K. Reddy
Received22 Jul 2013
Accepted20 Aug 2013
Published04 Nov 2013

Abstract

In this paper, we have studied the critical phenomena in higher curvature charged AdS black holes. We have considered Lovelock-Born-Infeld-AdS black hole as an example. The thermodynamics of the black hole have been studied which reveals the onset of a higher-order phase transition in the black hole in the canonical ensemble (fixed charge ensemble) framework. We have analytically derived the critical exponents associated with these thermodynamic quantities. We find that our results fit well with the thermodynamic scaling laws and consistent with the mean field theory approximation. The suggestive values of the other two critical exponents associated with the correlation function and correlation length on the critical surface have been derived.

1. Introduction

Constructing gravity theories in higher dimensions (i.e., greater than four) has been an interesting topic of research for the past several decades. One of the reasons for this is that these theories provide a framework for unifying gravity with other interactions. Since string theory is an important candidate for such unified theory, it is necessary to consider higher-dimensional space-time for its consistency. In fact, the study of string theory in higher dimensions is one of the most important and challenging sectors in high energy physics. String theory requires the inclusion of gravity in order to describe some of its fundamental properties. Other theories like brane theory are theory of supergravity are also studied in higher dimensions. All these above mentioned facts underscored the necessity for considering gravity theories in higher dimensions [1]. The effect of string theory on gravity may be understood by considering a low energy effective action that describes the classical gravity [2]. This effective action must include combinations of the higher curvature terms and found to be ghost-free [3]. In an attempt to obtain the most general tensor that satisfies the properties of Einstein’s tensor in higher dimensions, Lovelock proposed an effective action that contains higher curvature terms [4]. The field equations derived from this action consist of only second derivatives of the metric and hence are free of ghosts [5]. In fact, these theories are the most general second-order gravity theories in higher dimensions.

Among various higher dimensional theories of gravity, it is the seven-dimensional gravity that has earned repeated interests over the past two decades because of the interesting physics associated with them. For example, addition of certain topological terms in the usual Einstein-Hilbert action produces new type of gravity in seven-dimensions. For a particular choice of the coupling constants it is observed that these terms exist in seven dimensional gauged supergravity [6]. Apart from these aspects of the supergravity theories in 7 dimensions, there are several other important features associated with gravity theories in these particular dimensions. For example, there exists an octonionic instanton solution to the seven-dimensional Yang-Mills theory having an extension to a solitonic solution in low energy heterotic string theory [7]. Emergence of -duality in seven dimensions involves some nontrivial phenomena which are interpreted in -theory [8]. Also, study of black holes in 7 dimensions is important in the context of the AdS/CFT correspondence [911].

There has been much progress in the study of various properties of the black holes both in four as well as in higher dimensions (greater than four) over the past few decades. Black holes in higher dimensions possess interesting properties which may be absent in four dimensions [1216] (for an excellent review on black holes in higher dimensions see [17]). Among several intriguing properties of black holes we will be mainly concerned with the thermodynamics of these objects. This motivation primarily arises from the fact that the study of black hole thermodynamics in higher dimensions may provide us with information about the nature of quantum gravity. The limits of validity of the laws of black hole mechanics [18] may address some interpretations in this regard. Various quantum effects can be imposed in order to check the validity of these laws. For example, if we want to study the effect of backreaction of quantum field energy on the laws on black hole mechanics, we are required to consider higher curvature interactions [19]. From this point of view it is very much natural to study the effect of higher curvature terms on the thermodynamic properties of black holes. In fact, the higher-dimensional Lovelock theory sets a nice platform to study this effect. Moreover, the higher curvature gravity theories in higher dimensions are free from the complications that arise from the higher derivative theories. All these facts motivate us to study the higher curvature gravity theories in higher dimensions.

The study of thermodynamic properties of black holes in anti-de Sitter (AdS) space-time has got renewed attention due to the discovery of phase transition in Schwarzschild-AdS black holes [20]. Since then a wide variety of researches have been commenced in order to study the phase transition in black holes [15, 16, 2133]. Nowadays, the study of thermodynamics of black holes in AdS space-time is very much important in the context of AdS/CFT duality. The thermodynamics of AdS black holes may provide important information about the underlying phase structure and thermodynamic properties of CFTs [34]. The study of thermodynamic phenomena in black holes requires an analogy between the variables in ordinary thermodynamics and those in black hole mechanics. Recently, Banerjee et al. have developed a method [28, 29], based on Ehrenfest’s scheme [35] of standard thermodynamics, to study the phase transition phenomena in black holes. In this approach one can actually determine the order of phase transition once the relevant thermodynamic variables are identified for the black holes. This method has been successfully applied in the four dimensional black holes in AdS space-time [2932] as well as in higher dimensional AdS black holes [15].

The behavior of the thermodynamic variables near the critical points can be studied by means of a set of indices known as static critical exponents [36, 37]. These exponents are to a large extent universal and independent of the spatial dimensionality of the system and obey thermodynamic scaling laws [36, 37]. The critical phenomena have been studied extensively in familiar physical systems like the Ising model (two and three dimensions), magnetic systems, elementary particles, hydrodynamic systems, and so forth. An attempt to study the critical phenomena in black holes was commenced in the last twenty years [3852]. Despite all these attempts, a systematic study of critical phenomena in black holes was still lacking. This problem has been circumvented very recently [16, 53]. In these works the critical phenomena have been studied in (3 + 1) as well as higher dimensional AdS black holes. Also, the critical exponents of the black holes have been determined by explicit analytic calculations.

All the researches mentioned above were confined to Einstein gravity. It would then be interesting to study gravity theories in which action involves higher curvature terms. Among the higher curvature black holes, Gauss-Bonnet and Lovelock black holes will be suitable candidates to study. The thermodynamic properties and phase transitions have been studied in the Gauss-Bonnet AdS (GB-AdS) black holes [22, 5458]. Also, the critical phenomena in the GB-AdS black holes were studied [59]. On the other hand, Lovelock gravity coupled to the Maxwell field was investigated in [60, 61]. The thermodynamic properties of the third order Lovelock-Born-Infeld-AdS (LBI-AdS) black holes in seven space-time dimensions were studied in [60, 62, 63]. But the study of phase transition and critical phenomena have not yet been done in these black holes.

In this paper we have studied the critical phenomena in the third-order LBI-AdS black holes in seven dimensions. We have also given a qualitative discussion about the possibility of higher order phase transition in this type of black holes. We have determined the static critical exponents for these black holes and showed that these exponents obey the static scaling laws. We have also checked the static scaling hypothesis and calculated the scaling parameters. We observe that these critical exponents take the mean field values. From our study of the critical phenomena we may infer that the third order LBI-AdS black holes yield results consistent with the mean field theory approximation. Despite having some distinct features in the higher curvature gravity theories, for example, the usual area law valid in Einstein gravity does not hold in these gravity theories, the critical exponents are found to be identical with those in the usual Einstein gravity. This result shows that the AdS black holes, studied so far, belong to the same universality class. We have also determined the critical exponents associated with the correlation function and correlation length. However, the values of these exponents are more suggestive than definitive. As a final remark, we have given a qualitative argument for the determination of these last two exponents.

The organization of the paper is as follows. In Section 2 we discuss the thermodynamical variables of the seven-dimensional third order LBI-AdS black holes. In Section 3 we analyze the phase transition and stability of these black holes. The critical exponents, scaling laws, and scaling hypothesis are discussed in Section 4. Finally, we have drawn our conclusions in Section 5.

2. Thermodynamic Variables of Higher Curvature Charged AdS Black Holes

The effective action in the Lovelock gravity in dimensions can be written as (here we have taken the gravitational constant ) [4]

where is an arbitrary constant and is the Euler density of a -dimensional manifold. In dimensions all terms for which are equal to zero, the term is a topological term, and terms for which contribute to the field equations. Since we are studying third-order Lovelock gravity in the presence of Born-Infeld nonlinear electrodynamics [64], the effective action of (1) may be written as

where is the cosmological constant given by , being the AdS length, and are the second- and third-order Lovelock coefficients, is the usual Einstein-Hilbert Lagrangian, is the Gauss-Bonnet Lagrangian,

is the third-order Lovelock Lagrangian, and is the Born-Infeld Lagrangian given by

In (4), , , and is the Born-Infeld parameter. In the limit we recover the standard Maxwell form .

The solution of the third order Lovelock-Born-Infeld anti de-Sitter black hole (LBI-AdS) in -dimensions can be written as [63]

where

and determines the structure of the black hole horizon (). At this point of discussion it must be mentioned that the Lagrangian of (2) is the most general Lagrangian in seven space-time dimensions that produce the second-order field equations [60]. Thus, we will restrict ourselves in the seven space-time dimensions.

The equation of motion for the electromagnetic field for this ()-dimensional space-time can be obtained by varying the action (2) with respect to the gauge field . This results in the following [63]:

which has a solution [63]

Here is the abbreviation of the hypergeometric function [65] given by

In (8), and is a constant of integration which is related to the charge () of the black hole. The charge () of the black hole may be obtained by calculating the flux of the electric field at infinity [60, 63, 66]. Therefore,

where and are the time-like and space-like unit normal vectors to the boundary , respectively, and is the determinant of the induced metric on having coordinates . It is to be noted that in deriving (10) we have only considered the component of .

The quantity is the volume of the () sphere and may be written as

The metric function of (5) may be written as [63]

where

Here we have considered the special case [60, 63].

Since, in our study of the critical phenomena in the third-order LBI-AdS black holes, the thermodynamic quantities like “quasilocal energy” (), Hawking temperature (), entropy (), and so forth will play important roles, we will now focus mainly on the derivations of these quantities.

The “quasilocal energy” of asymptotically AdS black holes may be obtained by using the counterterm method which is indeed inspired by the AdS/CFT correspondence [6769]. This is a well-known technique which removes the divergences in the action and conserved quantities of the associated space-time. These divergences appear when one tries to add surface terms to the action in order to make it well defined. The counterterm method was applied earlier for the computation of the conserved quantities associated with space-time, having finite boundaries, de-Sitter (dS) space-time and asymptotically flat space-time in the framework of Einstein gravity [7076]. On the other hand, this method was applied in Lovelock gravity to compute the associated conserved quantities [60, 61, 63, 77, 78]. However, for the asymptotically AdS solutions of the third order Lovelock black holes the action may be written as [61, 63, 78]

where is the determinant of the induced metric on the time-like boundary of the space-time manifold . The quantity is a scale length factor given by

where

The boundary terms in (14) are chosen such that the action possesses well-defined variational principle, whereas, the counterterm makes the action and the associated conserved quantities finite.

The boundary terms appearing in (14) may be identified as [61, 63, 78]

where is the trace of the extrinsic curvature . In (18) is the Einstein tensor for in -dimensions, and is the trace of the following tensor:

In (19) is the second order Lovelock tensor for in -dimensions which is given by

whereas is the trace of

Using the method prescribed in [79], we obtain the divergence-free energy-momentum tensor as [61, 63, 78],

The first three terms of (23) result from the variation of the boundary terms of (14) with respect to the induced metric , whereas the last term is obtained by considering the variation of the counterterm of (14) with respect to .

For any space-like surface in which has the metric we can write the boundary metric in the following form [61, 63, 78, 79]:

where are coordinates on and and are the lapse function and the shift vector, respectively. For any Killing vector field on the the space-like boundary in , we may write the conserved quantities associated with the energy momentum tensor () as [61, 63, 78, 79]

where is the determinant of the metric on , is the time-like unit normal to , and is the time-like Killing vector field. For the metric (5) we can write , and , , where is the space-like unit normal to the boundary.

With these values of and the only nonvanishing component of becomes . Hence, corresponds to the “quasilocal energy” of the black hole. Thus, from (25) the expression for the “quasilocal energy” of the black hole may be written as

Using (12) and (23), (26) can be computed as

where the constant is expressed as the real root of the equation

Using (11) and substituting from (28) we finally obtain from (27)

The electrostatic potential difference between the black hole horizon and the infinity may be defined as [63]

where

It is to be noted that in obtaining (31) we have used (10).

The Hawking temperature for the third order LBI-AdS black hole is obtained by analytic continuation of the metric. If we set , we obtain the Euclidean section of (12) which requires to be regular at the horizon (). Thus we must identify , where (, being the surface gravity of the black hole) is the inverse of the Hawking temperature. Therefore the Hawking temperature may be written as

It is interesting to note that in the limit , the corresponding expression for the Hawking temperature of the Born-Infeld AdS (BI-AdS) black hole can be recovered as [16]

The entropy of the black hole may be calculated from the first law of black hole mechanics [80]. In fact, it has been found that the thermodynamic quantities (e.g., entropy, temperature, “quasilocal energy,” etc.) of the LBI-AdS black holes satisfy the first law of black hole mechanics [60, 61, 63, 66, 77, 81]:

Using (34) the entropy of the black hole may be obtained as

where we have used (29) and (32). At this point it is interesting to note that identical expression for the entropy was obtained earlier using somewhat different approach [8286]. In this approach, in an arbitrary spatial dimension , the expression for the Wald entropy for higher curvature black holes is written as

where is the th order Lovelock Lagrangian of and the tilde denotes the corresponding quantities for the induced metric . If we put in (36) we obtain the expression of (35) (the entropy of black holes both in the usual Einstein gravity and in higher curvature gravity can also be obtained by using the approach of [8789], respectively. The expression for the entropy of the third order Lovelock black hole given by (35) is the same as that of [89]). Thus, we may infer that the entropy of the black hole obtained from the first law of black hole mechanics is indeed the Wald entropy.

From (35) and (36), we find that the entropy is not proportional to the one-fourth of the horizon area as in the case of the black holes in the Einstein gravity. However, if we take the limit , we can recover the usual area law of black hole entropy in the BI-AdS black hole [16] as

In our study of critical phenomena, we will be mainly concerned with the spherically symmetric space-time. In this regard we will always take the value of to be . Substituting in (32) and (35) we finally obtain the expressions for the “quasilocal energy,” the Hawking temperature, and the entropy of the third order LBI-AdS black hole as

3. Phase Transition and Stability of the Third-Order LBI-AdS Black Hole

In this section we aim to discuss the nature of phase transition and the stability of the third order LBI-AdS black hole. A powerful method, based on Ehrenfest’s scheme of ordinary thermodynamics, was introduced by the authors of [28] in order to determine the nature of phase transition in black holes. Using this analytic method, phase transition phenomena in various AdS black holes were explored [2932]. Also, phase transition in higher dimensional AdS black holes has been discussed in [15].

In this paper we have qualitatively discussed the phase transition phenomena in the third order LBI-AdS black hole following the arguments presented in the above mentioned works. At this point it must be stressed that the method presented in [28] has not yet been implemented for the present black hole. However, we will not present any quantitative discussion in this regard.

From the plot (Figures 1 and 2) it is evident that there is no discontinuity in the temperature of the black hole. This rules out the possibility of first order phase transition [15, 2932].

In order to see whether there is any higher order phase transition, we calculate the specific heat of the black hole. In the canonical ensemble framework the specific heat at constant charge (this is analogous to the specific heat at constant volume () in the ordinary thermodynamics) () can be calculated as [16, 53]

where

In the derivation of (41) we have used (39) and (40).

In Figures 3, 4, 5, 6, 7, 8, 9, and 10, we have plotted against the horizon radius (here we have zoomed in the plots near the two critical points () separately). The numerical values of the roots of (41) are given in Tables 1 and 1. For convenience we have written the real roots of (41) only. From our analysis it is observed that the specific heat always possesses simple poles. Moreover, there are two real positive roots () of the denominator of for different values of the parameters , , and . Also, from the plots it is observed that the specific heat suffers discontinuity at the critical points . This property of allows us to conclude that at the critical points there is indeed a continuous higher order phase transition [16, 53].

(a)


15 0.6 1.59399 7.96087 −7.96087 −1.59399
8 0.2 1.18048 7.96088 −7.96088 −1.18048
5 0.5 1.25190 7.96088 −7.96088 −1.25190
0.8 20 1.01695 7.96088 −7.96088 −1.01695
0.3 15 0.975037 7.96088 −7.96088 −0.975037
0.5 10 0.989576 7.96088 −7.96088 −0.989576
0.5 1 0.989049 7.96088 −7.96088 −0.989049
0.5 0.5 0.987609 7.96088 −7.96088 −0.987609
0.05 0.05 0.965701 7.96088 −7.96088 −0.965701

(b)


15 0.6 1.76824 7.74534 −7.74534 −1.76824
8 0.2 1.53178 7.74535 −7.74535 −1.53178
5 0.5 1.51668 7.74535 −7.74535 −1.51668
0.8 20 1.40553 7.74535 −7.74535 −1.40553
0.3 15 1.40071 7.74535 −7.74535 −1.40071
0.5 10 1.40214 7.74535 −7.74535 −1.40214
0.5 1 1.40214 7.74535 −7.74535 −1.40214
0.5 0.5 1.40213 7.74535 −7.74535 −1.40213
0.05 0.05 1.39992 7.74535 −7.74535 −1.39992

Let us now see whether there is any bound in the values of the parameters , , and . At this point it must be stressed that a bound in the parameter values () for the Born-Infeld-AdS black holes in ()-dimensions was found earlier [53, 57]. Moreover, this bound is removed if we consider space-time dimensions greater than four [16]. Thus, it will be very much interesting to check whether the third-order LBI-AdS black holes possess similar features. In order to do so, we will consider the extremal third order LBI-AdS black hole. In this case both and vanish at the degenerate horizon [16, 53, 57]. The above two conditions for extremality result in the following equation:

In Tables 2 and 2 we give the numerical solutions of (44) for different choices of the values of the parameters and for fixed values of . From this analysis we observe that for arbitrary choices of the parameters and we always obtain atleast one real positive root of (44). This implies that there exists a smooth extremal limit for arbitrary and and there is no bound on the parameter space for a particular value of . Thus, the result obtained here (regarding the bound in the parameter values) is in good agreement with that obtained in [16].

(a)


15 0.6 −66.4157 +0.632734
8 0.2 −66.4157 +0.121590
5 0.5 −66.4157 +0.200244
0.8 20 −66.4157 −0.408088 +0.268514
0.3 15 −66.4157 −0.304225 +0.145685
0.5 10 −66.4157 −0.349764 +0.192519
0.5 1 −66.4157 +0.0221585
0.5 0.5 −66.4157 +0.00625335
0.05 0.05 −66.4157 +6.57019 10

(b)


15 0.6 −66.1628 +0.50473
8 0.2 −66.1628 +0.0546589
5 0.5 −66.1628 +0.110054
0.8 20 −66.1629 −0.563564 +0.236186
0.3 15 −66.1629 −0.514815 +0.116834
0.5 10 −66.1629 −0.531635 +0.155024
0.5 1 −66.1629 −0.542417 +0.00640486
0.5 0.5 −66.1629 +0.00163189
0.05 0.05 −66.1629 +1.64256 × 10

We will now analyse the thermodynamic stability of the third order LBI-AdS black hole. This is generally done by studying the behaviour of at the critical points [16, 30, 32, 53, 57]. The plots show that there are indeed three phases of the black hole. These phases can be classified as, Phase I (), Phase II (), and Phase III, (). Since the higher mass black hole possesses larger entropy/horizon radius, there is a phase transition at from smaller mass black hole (Phase I) to intermediate (higher mass) black hole (Phase II). The critical point corresponds to a phase transition from an intermediate (higher mass) black hole (Phase II) to a larger mass black hole (Phase III). Moreover, from the plots we note that the specific heat is positive for Phase I and Phase III whereas it is negative for Phase II. Therefore Phase I and Phase III correspond to thermodynamically stable phases (), whereas Phase II corresponds to thermodynamically unstable phase ().

We can further extend our stability analysis by considering the free energy of the third order LBI-AdS black hole. The free energy plays an important role in the theory of phase transition and critical phenomena. We may define the free energy of the third order LBI-AdS black hole as

Using (38), (39), and (40) we can write (45) as

In Figures 11, 12, 13, and 14 we have given the plots of the free energy () of the black hole with the radius of the outer horizon . The free energy () has a minima at . This point of minimum-free energy is exactly the same as the first critical point , where the black hole shifts from a stable to an unstable phase. On the other hand has a maxima at . The point at which reaches its maximum value is identical with the second critical point , where the black hole changes from unstable to stable phase. We can further divide the plot into three distinct regions. In the first region the negative-free energy decreases until it reaches the minimum value () at . This region corresponds to the stable phase (Phase I: ) of the black hole. The free energy changes its slope at and continues to increase in the second region approaching towards the maximum value () at . This region corresponds to Phase II of the plot, where the black hole becomes unstable (). The free energy changes its slope once again at and decreases to zero at and finally becomes negative for . This region of the plot corresponds to the Phase III of the plot where the black hole finally becomes stable ().

4. Critical Exponents and Scaling Hypothesis

In thermodynamics, the theory of phase transition plays a crucial role to understand the behavior of a thermodynamic system. The behavior of thermodynamic quantities near the critical point(s) of phase transition gives a considerable amount of information about the system. The behavior of a thermodynamic system near the critical point(s) is usually studied by means of a set of indices known as the critical exponents [36, 37]. These are generally denoted by a set of Greek letters: , , , ,,, , and . The critical exponents describe the nature of singularities in various measurable thermodynamic quantities near the critical point(s).

In this section we aim to determine the first six static critical exponents (, , , , , and ). For this purpose we shall follow the method discussed in [16, 33, 53]. We shall then discuss the static scaling laws and static scaling hypothesis. We shall determine the other two critical exponents ( and ) from two additional scaling laws.

Critical Exponent . In order to determine the critical exponent which is associated with the singularity of near the critical points (), we choose a point in the infinitesimal neighborhood of as

where . Let us denote the temperature at the critical point by and define the quantity

such that .

We now Taylor expand in the neighborhood of keeping the charge constant (), which yields

Since the divergence of results from the vanishing of at the critical point (41), we may write (49) as

where we have neglected the higher order terms in (49).

Using (47) and (48) we can finally write (50) as

where

The detailed expression of is very much cumbersome and we will not write it for the present work.

If we examine the plots (Figures 1 and 2), we observe that near the critical point (which corresponds to the “hump”) so that , and on the contrary, near the critical point (which corresponds to the “dip”) implying .

Substituting (47) into (41) we can write the singular part of as

where is the value of the numerator of (42) at the critical point and critical charge . The expression for is given by

where

It is to be noted that while expanding the denominator of , we have retained the terms which are linear in , and all other higher order terms of have been neglected.

Using (53) we may summarize the critical behavior of near the critical points ( and ) as follows:

where

We can combine the right-hand side of (56) into a single expression, which describes the singular nature of near the critical point , yielding

where we have used (48). Here and are the abbreviations of and , respectively.

We can now compare (58) with the standard form

which gives .

Critical Exponent . The critical exponent is related to the electric potential at infinity () by the relation

where the charge () is kept constant.

Near the critical point the Taylor expansion of yields

Neglecting the higher order terms and using (30) and (51) we may rewrite (61) as

Comparing (62) with (60) we finally obtain .

Critical Exponent . We will now determine the critical exponent which is associated with the singularity of the inverse of the isothermal compressibility () at constant charge near the critical point as

In order to calculate we use the standard thermodynamic definition

where in the last line of (64) we have used the identity

Using (30) and (39) we can write (64) as

where is the denominator identically equal to (43) (the denominator of ), and the expression for may be written as

where

From (66) we observe that possesses simple poles. Moreover and exhibit common singularities.

We are now interested in the behavior of near the critical point . In order to do so we substitute (47) into (66). The resulting equation for the singular part of may be written as

In (69), is the value of the numerator of (67) at the critical point and critical charge , whereas was identified earlier (54).

Substituting (51) in (69) we may express the singular nature of near the critical points ( and ) as

where

Combining the right-hand side of (70) into a single expression as before, we can express the singular behavior of near the critical point as

Comparing (72) with (63) we find .

Critical Exponent  . Let us now calculate the critical exponent which is associated with the electrostatic potential () for the fixed value of temperature. The relation can be written as

In this relation is the value of charge () at the critical point . In order to obtain we first Taylor expand around the critical point . This yields

Neglecting the higher order terms we can write (74) as

Here we have used the standard thermodynamic identity

and considered the fact that at the critical point , vanishes.

Let us now define a quantity

where . Here we denote and by and , respectively.

Using (47) and (77) we obtain from (75)

where

The expression for is very much cumbersome, and we shall not write it here.

We shall now consider the functional relation

from which we may write

Using (76) we can rewrite (81) as