Asymptotically Lifshitz Brane-World Black Holes
We study the gravity dual of a Lifshitz field theory in the context of a RSII brane-world scenario, taking into account the effects of the extra dimension through the contribution of the electric part of the Weyl tensor. We study the thermodynamical behavior of such asymptotically Lifshitz black holes. It is shown that the entropy imposes the critical exponent to be bounded from above. This maximum value of corresponds to a positive infinite entropy as long as the temperature is kept positive. The stability and phase transition for different spatial topologies are also discussed.
Quantum field theories and their holographic duals in the context of AdS/CFT have been the subject of numerous studies in recent years. The AdS/CFT correspondence provides a well established method for obtaining dual descriptions of strongly coupled conformal field theories in terms of weakly coupled gravitational theories where the gravitational background symmetries are realized as the symmetries of the dual CFT. For example, the conformal group of a dimensional CFT arises as the isometry group of .
The AdS/CFT seems to extend to some systems which do not exhibit Lorentz invariance. Recently, attempts have been made to apply the holographic principle to study condensed matter systems near a critical point [1, 2] where there are many strongly coupled systems which have to be studied non-perturbatively. In this situation, one encounters many scale-invariant field theories which are not Lorentz invariant. In such theories, space and time are not considered in the same footing, so they can scale in different ways under dilatation
where is the so-called dynamical critical exponent showing relative scale dimension of space and time. For , this scaling symmetry reduces to the familiar relativistic scale invariance
In this paper, we focus our attention on the Lifshitz theory as an anisotropic scale-invariant theory. A toy model which exhibits such anisotropic scale invariance is the Lifshitz field theory represented by
The theory describes a line of fixed points parametrized by and respects the scaling invariance (1) with . Although, this anisotropic scaling theory is mostly applied to condensed matter theories, it has been shown recently that some models for particle physics could benefit of Lifshitz symmetry in their Lagrangians .
where is a characteristic length scale.
The Einstein-Hilbert (EH) action cannot admit a Lifshitz geometry [5, 7]. However, there are generically three ways to construct actions which admit such geometry (4). One way is by adding some nontrivial matter terms to the EH action  which can be shown to be equivalent to adding a massive gauge field . It is generally difficult to obtain analytic solutions to the field equations generated by these actions. Attempts have been made to find analytic and numeric black hole/brane solutions which asymptotically admit Lifshitz geometry [10, 11, 12, 13, 14, 15, 16, 17, 18]. The embedding of such theories in the string theory and supergravity has been also discussed in .
As a second approach but not completely independent of the former, one may add higher order curvature terms to generate dynamics for parameter to ensure that the Lifshitz geometry (4) satisfies the resulting field equations of the new action [20, 21, 22, 23]. We refer to these procedures as a renormalization of . Also recently it has been shown that stationary black hole solutions can be obtained in the context of this approach .
There is however another approach which introduces a dilaton field together with some gauge fields instead of a massive gauge field . This approach has an advantage respect to others, since finding analytic solutions are more probable. But from the holographic point of view, this approach has some difficulties in the sense that the dilaton field diverges on the boundary . Analytic solutions for asymptotically Lifshitz black holes/branes of this model have been obtained in .
Also recently, a non-abelian Lifshitz Chern-Simons gauge theory in dimensions with dynamical critical exponent has been proposed  to explain the possibility of a phase transition from an isotropic to anisotropic fractional quantum hall systems. The gravitational dual of such theory was obtained in .
We propose an alternative to renormalize the dynamical exponent , considering the global effects of the extra dimensions. Consequently, asymptotically Lifshitz black hole solutions can be obtained in the context of brane-world scenarios which is the main purpose of the present paper. For a review on the brane world scenarios see .
In order to obtain the equations of motion governing gravity on the brane one can use the method initiated by Shiromizu, Maeda and Sasaki (SMS) . They assumed general bulk and brane geometries and obtained equations of motion of the matter field on the brane by use of the Gauss-Codazzi equations and Israel junction conditions . The resulting field equations are and include the global bulk structure through the electric part of the Weyl tensor. The main feature of these equations is the introduction of a new tensor which is quadratic in the energy-momentum tensor.
The brane generalization of the vacuum Einstein field equations (4) does not admit the Lifshitz geometry. Roughly speaking, this arises from the dependence of matter components to the couplings of the theory. This implies that a Lifshitz space-time cannot be admitted as a solution by the field equations on the brane in the absence of an energy-momentum tensor. The calculations lead to this result are presented in the Appendix. There, we show that the Lifshitz space-time can be embedded on the brane in the presence of matter field. As we will show below, the brane field equations can admit asymptotically Lifshitz geometry. This is done with the aid of the electric part of the Weyl tensor which ensures the renormalizability of the dynamical exponent . The generalization of this work to the case that we have matter fields are straightforward. It is worth mentioning that attempts have been made to show that the Lifshitz space-time can be a solution of a brane-world model in the presence of an anisotropic perfect fluid energy-momentum tensor on a thick brane [32, 33].
The paper is organized as follows: In the next two sections we obtain the asymptotically Lifshitz solution to the brane geometry and show that the value of the dynamical exponent must be bounded from below to represent a black hole solution. We interpret the constants of the solution in terms of the mass and charge of a black hole using the fact that in the case the solutions reduce to the AdS-Reissner-Nordström black hole. In section 4 we study the thermodynamical properties of such black holes. As we will see, the entropy restricts the value of the dynamical exponent from above. This result shows that in vacuum, the asymptotically Lifshitz black holes in Randall-Sundrum brane scenario only admit a narrow range for . The stability of the black hole and the possibility of phase transition in the most interesting case of will also be discussed. The reasons for the lack of a vacuum Lifshitz solution in the present framework are discussed in the Appendix. Conclusions are drawn in the last section.
2 The setup
We begin with the field equations of the brane 
and is the energy-momentum tensor of the matter fields residing on the brane and is the electric part of the Weyl tensor. The Greek indices in the above equations are defined on the brane and run from 0 to 3. It is worth mentioning at this early stage that as has been mentioned in , equation (5) is not sufficient for obtaining the whole dynamics of the bulk. There are some other equations from which one may obtain the bulk dynamics . However, equation (5) is sufficient to obtain the dynamics of the brane, and the 5-dimensional equations are then used to obtain the -dependence of the bulk metric and the electric part of the Weyl tensor where is the coordinate of the fifth dimension. In the present paper, we shall consider the dynamics of the brane without worrying about the -dependence of the metric since this can be trivially found by a Taylor series as in .
Let us use the following irreducible decomposition for the electric part of the Weyl tensor 
where , and the spatial metric is orthogonal to the chosen normalized time-like vector field with the property . In this decomposition, we introduce three terms, known as dark radiation
dark energy flux
and dark pressure
In the case of a static and isotropic space-time we have , and
where is the unit radial vector .
3 Asymptotically Lifshitz Black hole Solution
The asymptotically Lifshitz space-time can be described in general by
where is the portion of the metric which depends on the curvature index of the space-time
and, as was mentioned in the Introduction, shows the relative scaling of the space and time. This form of the metric guarantees that the brane geometry approaches the Lifshitz geometry, equation (4), up to the re-scaling and if as .
We are interested in the vacuum solutions of the model, so we assume that the energy-momentum tensor is zero. As a result, the brane equations of motion reduce to the form
Assuming and , the electric part of the Weyl tensor can be computed from metric (12) as
These equations can be solved for the metric coefficient with the result
where we have defined
with both and being constants. Solution (3) will also satisfy the Bianchi identities as it should.
To have an asymptotic Lifshitz black hole, the leading term of the function must be equal to unity and the constants and must be well defined and positive. This means that in a brane-world scenario the dynamical exponent is bounded from below,
Also, the cosmological constant could be read off as a function of the dynamical exponent
The calculations would become much simpler if one writes in the following form
It is now possible to calculate dark radiation and pressure from equation (16) with the result
Since the integration constants and are independent of they can be obtained by considering the behavior of the solution at . This is the case where the relativistic scaling invariance is realized and the resulting solution can be compared with the known black hole solutions. In this case, the asymptotic Lifshitz solution of the brane reduces to the form
where is given by equation (13) and . Comparing with the standard topological Reissner-Nordström-AdS black hole solution
one can read off the constants as
From now on, and are referred to as the mass and charge parameters of the general asymptotic Lifshitz black hole (27). It is worth to mention that the charge here does not mean a real charge because the theory does not contain the electromagnetic source. However, one can expect that the electric part of the Weyl tensor is responsible for the appearance of such term.
We also mention that the position of the horizon of the black hole can be obtained through imposing the condition , which for the solution (23) leads to
where we have defined
One can easily verify numerically, that the above equation has solutions for all allowed values of . Hereafter, this premise is eligible for future use in thermodynamics.
4 Thermodynamical properties
In this section we study the thermodynamical behavior of the asymptotically Lifshitz black holes (12). After calculating the corresponding thermodynamical quantities for such black holes we discuss their stability.
The gravitational mass of a black hole, , can be obtained by the equation , governing the position of the horizon. In the case of black hole (12) we find
where is associated with the outer most horizon.
The Hawking temperature associated with a black hole is obtained by requiring the absence of a conical singularity at the black hole horizon in the Euclidean sector of the asymptotically Lifshitz black hole . One then has
where the surface gravity on the horizon admits the following useful representation
where is a killing vector normal to the surface of the horizon. As a consequence, the surface gravity of a black hole could be written in terms of the temporal component of the metric as
The Hawking temperature of the asymptotically Lifshitz black hole (12) is now given by
One should note that for asymptotically flat space-times, the normalization of the killing vector field, , in (36) is imposed at the spatial infinity. However, it has been shown by Brown et al. that the choice of the normalization elsewhere results in the appearance of an additional redshift factor, the so-called Tolman redshift factor, for a given temperature in a stationary gravitational field . For non-asymptotically flat space-times, one should impose the normalization at a finite space-like boundary surface. Nevertheless, one can assume that observations take place at the boundary surface where the redshift factor equals unity. So, in what follows we set the Tolman redshift factor equal to “one” without loss of generality.
Another important thermodynamical quantity is the entropy of black holes. In Einstein gravity the entropy of a black hole is obtained by the so-called Bekenestein area formula. This means that the entropy is equal to one-quarter of the horizon area. It has also been shown that when higher curvature terms are present, this statement is no longer true [37, 38]. However, Wald  has shown that in any case the entropy of a black hole is a function of the horizon geometry.
Alternatively, given the temperature of a black hole as a function of energy (or as a function of mass ), we can integrate the first law of thermodynamics, , to which a black hole as a thermodynamical system should conform, to define an “entropy” for the black hole
It is suitable to write it as an expression which is useful for future application, so we change the integration variable to radial coordinate
where the upper limit of integration is the horizon radius.
It is worth to mention that the term containing the charge does not appear in the first law of thermodynamics, because we assume that the spurious charge introduced in the theory is kept constant among the thermodynamical analysis.
The positivity condition on the entropy implies an upper bound on the dynamical critical exponent , that is
From equation (41) we see that corresponds to an infinitely large entropy. It then implies that the dynamical exponent must lie near this value by the maximum entropy principle. This behavior however resembles some condensed matter systems such as Rokhsar-Kivelson dimer model  and the strongly correlated electron systems , which is well described by the action (3) at . From the view point of duality, our result is expectable since our gravitational model becomes strongly stable near the point for which its dual theory becomes stable and physical. It is worth mentioning that the limit (42) is obtained in the context of asymptotically Lifshitz space-times which may induce a perturbation term on the dual field theory (3). This perturbation term can cause the critical exponent admitting values up to , supporting our gravitational model. As a result, our model restricts the physically reasonable solution to those which admit a dynamical exponent in the range
However, if we do not take the condition (42) ab initio, the entropy will contain negative values. This happens, for example, in the case of the Gauss-Bonnet black holes for negative spatial curvature. It has been emphasized that negative values for the entropy is related to the lower limit of the integral (40). This problem could be resolved by considering a minimum radius corresponding to the zero gravitational mass as the lower limit of the integral [37, 39]. In the case of the black hole (12), the entropy is independent of the spatial curvature so that it can be negative for all topologies. In our case instead of introducing the lower limit of integral we remove negative values for the entropy by restricting the dynamical exponent .
In order to study the thermodynamical stability of black holes, one should calculate the heat capacity. The local thermal stability of a black hole is determined by the sign of the heat capacity. If the heat capacity is positive, the black hole is locally stable to thermal fluctuations. Negative values for the heat capacity result in unstable black holes. We define the heat capacity for a constant charge black hole as
For the black hole (12), the heat capacity in constant charge can be written as
This expression may be converted to a more familiar thermodynamical relation
The value of the free energy determines the strength of the stability of a black hole; a more stable black hole corresponds to less free energy.
Moreover, the singularity in the heat capacity corresponds to a phase transition provided that the temperature and free energy possess a local minimum and a local maximum at the specified points respectively.
Since the Lifshitz action (3) is scaling invariant for , we are interested in focusing on this case. Many condensed matter systems which contain quantum critical phenomena such as strongly correlated electron systems are described by the Lifshitz action (3) in . In addition, from the thermodynamical point of view, the case mimics the behavior of black holes with maximum entropy.
In this case the normalized temperature, the free energy and the heat capacity reduce to
where we define
The thermodynamical behavior of the black hole differs considerably depending on the charge of the black hole. Black holes with a small amount of charge ( ) show a phase transition, unlike black holes with a large amount of charge which do not.
We first consider the case . The general behavior of black holes with a large amount of charge are very similar to this case. Figure (1) shows the behavior of the heat capacity for different spatial curvatures in this case. One can see that for all horizon topologies with , black holes are stable beyond . As mentioned above, this value for the charge of a black hole does not imply a phase transition which can be inferred from the smoothness of the heat capacity function. Figure (2) shows the normalized free energy for all spatial topologies. One can see from the figure that a larger black hole is more stable.
The most interesting case is when the normalized charge is less than . Among such black holes the most interesting ones are topological black holes () where a phase transition could take place in principle.
It is appropriate to remark on the conditions for the occurrence of any physical phase transitions in Lifshitz theories. It has been shown in  that in order to have a physical phase transition in Lifshitz theories which is described by , one must have another length scale in addition to the Lifshitz length scale. This additional length scale was achieved by introducing new Maxwell fields into the theory. Nonetheless, the brane-world scenario discussed here has an intrinsic length scale introduced by the ratio of the brane to the bulk Planck scales. In this context therefore, it is possible to have a physical phase transition.
For such black holes, there are two singularities in heat capacity. However the regions between these two singular points have negative temperature. The issue of a negative temperature for black holes has been investigated, for example, in [43, 44] where the physical meaning of the phase transition for such black holes is also discussed. The behavior of this type of a black hole is shown in figure (3), for the normalized heat capacity, and figure (4), for the normalized free energy . In cases and , black holes with a horizon larger than the approximate value are stable so that the larger they get the more stable they become. For the case , black holes with a horizon radius in the outer region are more stable.
The maximum possible value for , corresponds to the infinitely positive entropy and this indicates that the critical exponent must eventually go close to the maximum value because of the maximum entropy principle. In this case the free energy is minus infinity everywhere as long as the temperature is positive. This suggests that in this case black holes are stable for positive temperatures, independent of the horizon radii.
The Lifshitz space-time is inconsistent with standard general relativity. Attempts have been made to generalize GR in such a way as to admit Lifshitz space-times. In this paper we have introduced another generalization, i.e. higher dimensions, to deal with this problem. Alas, the Lifshitz space-time is not the solution of the vacuum brane field equations. However Lifshitz space-time can be admitted by the brane-world scenario with some ordinary matter fields as we will show in the appendix. We have shown that one can obtain an asymptotically Lifshitz solution in a brane-world scenario with the aid of the electric part of the Weyl tensor. The electric part of the Weyl tensor reflects the global geometry of the bulk in brane-world scenarios whose effect is to enable the brane admitting an asymptotically Lifshitz geometry. The electric part of the Weyl tensor on the other hand, can be realized as a probable source of some spurious charge. This tensor is however geometrical, so one can expect that the charge is not physical. Therefore, one can assume that this charge is fixed in the thermodynamical considerations. From the thermodynamical considerations we found that the dynamical exponent must be bounded from above. Together with the lower limit for the dynamical exponent which is implied by the existence of the metric, we found that the dynamical exponent of the asymptotically Lifshitz space-time cannot possess any arbitrary values. As a result, the maximum value for , that is, corresponds to the case where the entropy becomes infinite. From the maximum entropy principle, we then expected that an asymptotic Lifshitz black hole should approach the maximum value of where the entropy is maximum. The case was used to find the behavior of black holes in this case. In this case, for the black hole encounters a phase transition. It is worth to mention that the occurrence of phase transition in asymptotically Lifshitz space-time lies on the existence of another length scale beyond the characteristic Lifshitz length scale . In our model, it can be understood by the existence of a new length scale provided by the ratio of the bulk to the brane Planck scales. The case also has another interesting property for in that the region between the two singularities in heat capacity has a negative temperature. The physical meaning of these cases are not clear yet.
The standard Einstein field equations do not admit the Lifshitz space-time even in the presence of a perfect fluid. We first note that, as we mentioned in the Introduction, the Lifshitz solution is obtained in thick brane models with an anisotropic perfect fluid. In this Appendix we show that the RS brane-world scenario can admit the Lifshitz space-time as a solution, in the presence of a perfect fluid.
The energy-momentum tensor of a perfect fluid is defined as
where . Using equation (6), the tensor reduces to
The solution for the above system is
This result implies that a RS brane, in the presence of a perfect fluid, admits the Lifshitz space-time. However, the Lifshitz space-time is not a solution to the vacuum field equations. This becomes more clear if one notices that the equation related to the energy density and pressure of the perfect fluid depends on the coupling constants of the theory, which is exactly the reason why the Lifshitz space-time cannot be a vacuum solution in brane-world scenarios.
-  S. Sachdev and M. Mueller, J. Phys. Condens. Matter 21 (2009) 164216 [arXiv:0810.3005 [cond-mat]].
-  S. A. Hartnoll, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246 [hep-th]].
-  D. T. Son, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972 [hep-th]].
-  K. Balasubramanian and J. McGreevy, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053 [hep-th]].
-  S. Kachru, X. Liu and M. Mulligan, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725 [hep-th]].
-  J. Alexandre, [arXiv:1109.5691 [hep-th]]; D. L. Lopez Nacir, F. D. Mazzitelli and L. G. Trombetta, [arXiv:1111.1662 [hep-th]].
-  A. Adams, A. Maloney, A. Sinha and S. E. Vazquez, J. High. Energ. Phys. 03 (2009) 097 [arXiv:0812.0166 [hep-th]].
-  S. Schäfer-Nameki, M. Yamazaki and K. Yoshida, J. High. Energ. Phys 05 (2009) 038 [arXiv:0903.4245 [hep-th]].
-  K. Balasubramanian and J. McGreevy, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263 [hep-th]].
-  U. H. Danielsson and L. Thorlacius, J. High. Energ. Phys. 03 (2009) 070 [arXiv:0812.5088 [hep-th]].
-  R. B. Mann, J. High. Energ. Phys. 06 (2009) 075 [arXiv:0905.1136 [hep-th]].
-  G. Bertoldi, B. A. Burrington and A. Peet, Phys. Rev. D 80 (2009) 126003 [arXiv:0905.3183 [hep-th]].
-  D. W. Pang, JHEP 1001 (2010) 116. [arXiv:0911.2777 [hep-th]].
-  E. Ayon-Beato, A. Garbarz, G. Giribet, M. Hassaine, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347 [hep-th]].
-  H. Maeda and G. Giribet, JHEP 1111 (2011) 015 [arXiv:1105.1331 [gr-qc]].
-  I. Amado and A. F. Faedo, JHEP 1107 (2011) 004 [arXiv:1105.4862 [hep-th]].
-  E. J. Brynjolfsson, U. H. Danielsson, L. Thorlacius and T. Zingg, J. Phys. A 43 (2010) 065401 [arXiv:0908.2611 [hep-th]].
-  W. G. Brenna, M. H. Dehghani, R. B. Mann, Phys. Rev. D 84 (2011) 024012 [arXiv:1101.3476 [hep-th]].
-  J. Maldacena, D. Martelli and Y. Tachikawa, JHEP 0810 (2008) 072. [arXiv:0807.1100 [hep-th]]; W. Chemissany and J. Hartong, [arXiv:1105.0612 [hep-th]]; K. Balasubramanian, K. Narayan, JHEP 1008, 014 (2010). [arXiv:1005.3291 [hep-th]]; A. Donos and J. P. Gauntlett, JHEP 1012 (2010) 002 [arXiv:1008.2062 [hep-th]]; D. Cassani, A. F. Faedo, JHEP 1105, 013 (2011) [arXiv:1102.5344 [hep-th]]; N. Halmagyi, M. Petrini and A. Zaffaroni, [arXiv:1102.5740 [hep-th]]; I. Amado and A. F. Faedo, [arXiv:1105.4862 [hep-th]].
-  R. G. Cai, Y. Liu and Y. W. Sun, J. High. Energ. Phys. 10 (2009) 080 [arXiv:0909.2807 [hep-th]].
-  D. W. Pang, J. High. Energ. Phys. 10 (2009) 031 [arXiv:0908.1272 [hep-th]].
-  E. Ayn-Beato, A. Garbarz, G. Giribet and M. Hassaine, J. High. Energ. Phys. 04 (2010) 030 [arXiv:1001.2361 [hep-th]].
-  M. H. Dehghani, R. B. Mann, J. High. Energ. Phys. 07 (2010) 019 [arXiv:1004.4397 [hep-th]].
-  O. Sarioglu, [arXiv:1109.4721 [hep-th]].
-  M. Taylor, [arXiv:0812.0530 [hep-th]].
-  J. Tarrio Ì and S. Vandoren, JHEP 1109 (2011) 017 [arXiv:1105.6335 [hep-th]].
-  M. Mulligan, C. Nayak and S. Kachru, [arXiv:1004.3570 [cond-mat.str-el]].
-  K. Balasubramanian and J. McGreevy, [arXiv:1111.0634 [het-th]].
-  R. Maartens and K. Koyama, Living Rev. Relativity, 13 (2010) 5.
-  T. Shiromizu, K. Maeda, M. Sasaki, Phys. Rev. D 62 (2000) 024012.
-  W. Israel, Nouvo Cimento B 44 (1966) 1.
-  P. Koroteev and M. Libanov, J. High. Energ. Phys. 02 (2008) 104 [arXiv:0712.1136 [hep-th]].
-  I. Gordeli and P. Koroteev, Phys. Rev. D 80 (2009) 126001 [arXiv:0904.0509 [hep-th]].
-  N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania, Phys. Lett. B 487 (2000) 1 [arXiv:hep-th/0003061v3].
-  V. Frolov and I. Novikov, Black Hole Physics: Basic Concepts and New Developments, Kluwer Academic Publishers, 1989.
-  J. D. Brown, J. Creighton and R. B. Mann, Phys, Rev. D 50 (1994) 6394 [arXiv:gr-qc/9405007].
-  T. Clunan, S.F. Ross and D. J. Smith, Class. Quant. Gravity 21 (2004) 3447;
-  S. Nojiri,S. D. Odintsov and S .Nojiri, Phys. Rev. D 65 (2002) 023521 [arXiv:hep-th/0108172]; S. Nojiri and S. D. Odintsov, Phys. Rev. D 66 (2002) 044012 [arXiv:hep-th/0204112].
-  M. Cvetic, S. Nojiri and S. D. Odintsov, Nucl. Phys. B 628 (2002) 295 [arXiv:hep-th/0112045];
-  R. M. Wald, Phys. Rev. D 48 (1993) R3427 [arXiv:gr-qc/9307038].
-  D. Rokhsar and S. Kivelson, Phys. Rev. Lett. 61 (1988) 2376.
-  E. Fradkin, D. Huse, R. Moessner, V. Oganesyan and S. Sondhi, Phys. Rev. B 69 (2004) 224415; A. Vishwanath, L. Balents and T. Senthil, Phys. Rev. B 69 (2004) 224416; E. Ardonne, P. Fendley and E. Fradkin, Annals. Phys. 310 (2004) 493 [arXiv:cond-mat/0311466].
-  Q. J. Cao, Y.X. Chen and K. N. Shao [arXiv:1010.5044 [hep-th]].
-  P. C. W. Davies, Proc. Royal Soc. London. A, 353 (1977) 499; P. C. W. Davies, Rep. Prog. Phys., 41 (1978) 1313.