the theory of massive gravity recently introduced by de rham , gabadadze , and tolley  ( see also de rham and gabadadze  ) , has led to an explosive increase of interest in the possibility of having a graviton with a non - vanishing mass , revitalizing the old topic of gravity .the principal reason for this renewed interest has been that this theory is free of the otherwise problematic sixth degree of freedom , the boulware deser ghost  ; even at the nonlinear level and with a generic reference metric  ( see also  ) .cosmologies with a minkowski reference metric , which were used to construct the first version of this theory  , have been considered in reference  , where the authors showed that one can not have a flrw physical metric compatible with such a reference spacetime .this result is a direct consequence of the bianchi - inspired constraint .( there is one minor exception using the milne spacetime as reference metric  ; this is a particular open flrw universe which can be viewed as a particular chart on a portion of minkowski spacetime .the results of references  about massive gravity with a flat reference metric have been recovered in references  . )the incompatibility of a physical cosmological space with a minkowski reference metric had already been suggested in earlier work on massive gravity  , where the author also pointed out that in massive gravity any arbitrary cosmology can be accommodated in the physical sector by suitably fitting the required reference metric . this point has been recently stressed in references  and  .on the other hand , some authors have suggested that one should focus attention only on the particular branch of solutions where the two spatial metrics , foreground and background , are strictly proportional to each other , with a fixed constant proportionality factor  .this restriction would imply elimination of all nontrivial interaction terms , those which are not equivalent to a cosmological constant , and such a constraint could only be understood if one is trying to preserve continuity in the parameter space of the theory  .moreover , it has been argued that the solutions of this restricted branch might be unstable  .it must be pointed out that the original reason of paying attention only to this particular restricted set of solutions , which are equivalent to simply introducing a cosmological constant , could be misleading .the other branch has been suggested to be trivial  , by arguing that the physical metric would be completely determined from the reference metric , through the bianchi - inspired constraint , and not by the matter content .such a conclusion can be extracted only by assuming some simple relation between the cosmic times of the two metrics , since a nontrivial function could radically change the behavior of the physical cosmology .this has been pointed out in reference  in the case of bimetric gravity ( reference  contains a similar study in the framework of extensions of bimetric gravity  ) , and for massive cosmologies we will explicitly demonstrate the corresponding result in section [ s2 ] .in fact , interesting cosmological solutions with a de sitter reference metric have recently been presented in reference  .overall , we shall see that while the choice of reference metric certainly influences the evolution of the physically observable foreground metric , the effect of matter _ can not _ be neglected .following the lead of reference  , where fasiello and tolley have studied the classical stability of homogeneous and isotropic solutions of this nontrivial branch , one should of course take into account the possible tension between the vainshtein mechanism  , ( originally introduced to avoid the van dam veltman zakharov ( vdvz ) discontinuity  ) , and the higuchi bound  ( ) .nonetheless , these solutions should certainly not be excluded from the very beginning . in order to avoid that tension we will also consider the possibility of having an anti de sitter ( ads ) reference metric  .such a massive gravity model would share some interesting features with that based on assuming a de sitter ( ds ) reference metric ; that is it implies trivial curvature when there is no material content . as we show in section [ s3 ] , this is not a common characteristic of all massive gravity models .nevertheless , an ads reference metric can not successfully describe our own physical universe , at least assuming that it is homogeneous and isotropic at large scales , see section [ s4 ] .the already mentioned conclusions are summarized in section [ s5 ] , where additional comments are also added .in this section we will show that given a particular model of massive gravity , that is , once the reference metric is fixed , adding matter necessarily changes the dynamics of the universe in the nontrivial cosmological branch . to see this ,let us consider a homogeneous and isotropic reference metric ,\ ] ] to agree with the symmetry of our physical cosmological spacetime ,\ ] ] where we have assumed that both spacetimes have the same sign of spatial curvature .we allow an arbitrary relation between the two time coordinates , thereby excluding particular solutions which would not be compatible with both metrics being diagonal in the same coordinate patch .thus , we can express the relation between the two cosmic times as  here ; we have explicitly written the two possible signs instead of absorbing them into .it is already well known  , that the modified friedmann equation describing the dynamics of the universe in this model can be written as where the effects due to a non - vanishing graviton mass have been described as an effective fluid with given by  here , , and are parameters of the theory ; while is the standard matter content filling the universe , which obeys the usual conservation equation , \,\rho_\mathrm{m}=0 ; \qquad w_\mathrm{m}(a ) = p_\mathrm{m}(a)/\rho_\mathrm{m}(a);\ ] ] and includes the common cosmological constant term plus some terms coming from the theory of massive gravity  .the effective fluid which appears in the modified friedmann equation is also conserved  , and the corresponding conservation equation leads to the nontrivial part of the bianchi - inspired constraint . expressing and , it can be written as  taking this expression into account in the modified friedmann equation ( [ fried ] ) , and then simplifying , we have here we have explicitly shown the dependence on due to equation ( [ materia ] ) .equation ( [ pol ] ) can be seen as an implicit algebraic ( non - derivative ) equation for , with the quantity furthermore implicitly depending on .( in the specific case that is a constant , which is often a good approximation for significant epochs of cosmological time , equation ( [ pol ] ) can more specifically be seen as a polynomial equation for , with , and so with the degree of the polynomial depending on . ) in either case , the particular function can only be obtained by specifying , so we shall write it as to make this point explicit .explicitly , there will be some function such that differentiating with respect to , we can now obtain the function through where we have used equation ( [ consev ] ) to get expression ( [ t ] ) .note the implicit dependence on and hence on the matter content .thus , despite the fact that the function is determined ( up to an integration constant ) by the reference metric through the bianchi - inspired constraint ( [ consev ] ) , the influence of the matter content can not be neglected matter matters .furthermore , the behavior of the observable foreground universe is described by , which can describe an evolution completely different to that given by , and which can be obtained only once the matter content is known .  on the other hand, one could proceed the other way around and reconstruct any particular by carefully fixing and , thus , any cosmology could be described by changing the reference metric .therefore , it would be desirable to have some additional criteria to choose a particular reference metric , so that the theory maintains its predictive character .if one now considers a massive gravity theory suitable for describing our own physical universe , it is natural to assume that the reference metric used to define the theory would be homogeneous and isotropic .this particular theory , with that fixed reference metric , should also be consistent in other situations , for example in vacuum .so , in order to better understand the implications of the various models of massive gravity , let us go back to the modified friedmann equation ( [ fried ] ) , which in the absence of matter can be simplified as this equation is a cubic polynomial in , and a first - order differential equation in , which has different implications depending on the symmetry of the reference metric .   as is well known ( see for example , reference  )both the ds and ads spacetimes can be written using flrw charts ; open , flat , or closed for the former space , but necessarily open for the latter . in either of these cases one would have with for ds and for ads .( minkowski space corresponds to and , and has already been discussed in the introduction . )it must be noted that we are not imposing any dynamic friedmann equation for the reference metric ; equation ( [ eq ] ) is a purely algebraic equality fulfilled for the particular of these background spaces .thus , in both these cases equation ( [ friedvacio ] ) is simply a constant coefficient polynomial in , with .so , the solution is with the constant corresponding to one of the positive roots of the the cubic equation which we shall not write out explicitly to keep the analysis as clean as possible . in the casethat there is no real positive solution of the cubic , then there is no well - defined physical metric for those particular values of the parameters , , , , and .since any such solution implies , then equation ( [ t ] ) leads to , where without loss of generality we can set .therefore , thus , the solution of the physical metric in vacuum is a spacetime of the same type as the reference metric , that is ds or ads , with an effective cosmological constant , with the cosmological constant corresponding to the reference metric thus it is also a vacuum solution .the high symmetry of these particular reference spaces is so constraining that the only solution ( in vacuum ) for the physical space must be compatible with that same symmetry , with the only effect of the theory being the change of the value of the cosmological constant with respect to that of the reference spacetime .the introduction of matter , of course , would change this situation , since would no longer be a general solution for constant .   in the case of a generic flrw reference metric , equation ( [ friedvacio ] ) would not simplify so beautifully , since we will now have in this case the cubic equation implied by equation ( [ friedvacio ] ) has a coefficient which is now a function of time . to be more explicit about this ,note that the relevant cubic is now let the positive real roots ( if any ) be denoted then the effective fluid which we have defined through the energy density ( [ rho ] ) would not be equivalent to a cosmological constant , since now is time dependent .that will , lead at the end of the day , to a modified friedmann equation : therefore , in the nontrivial branch of solutions a non - vanishing graviton mass would generally curve the foreground spacetime in some nontrivial manner even in vacuum , leading to a situation where there are gravitational effects evolving in time in the absence of matter .in some sense , there would be gravitation even without matter .the old formal question , dating back to the ancient greeks , concerning the possible existence of space itself in the absence of matter would here be changed to the possible existence of nontrivial gravitational effects without material content .the ds and ads spaces would be the only consistent reference metrics from that point of view , since the massive graviton in those models does not curve the spacetime in the absence of matter in any nontrivial way , ( merely adjusting the value of the constant curvature ) .that is , in both the ds and ads massive gravity models the effects of the non - vanishing graviton mass would become evident only in the presence of matter , hiding as a simple cosmological constant contribution in any vacuum situation .following the spirit of the previous section , we shall now consider if an ads spacetime can be the reference metric leading to a model of massive gravity capable of describing our universe .( models with a ds reference metric have been considered in references  . )we have just seen that the vacuum solution for the physical metric in a model with an ads reference metric is an ads spacetime , equation ( [ sol - ads ] ) .thus , this solution would have both a phase of decelerated expansion and a contracting phase .the addition of a matter component to equation ( [ friedvacio ] ) would then decelerate the expansion phase , ( and accelerate the contracting one ) , at least if this matter fulfills the usual energy conditions  .( in particular , if the matter component satisfies the strong energy condition , sec  . )thus , the model would not be suitable for describing our universe , at least if the matter content fulfills the energy conditions .one might think that the situation could be improved by increasing the value of in equation ( [ friedvacio ] ) , but this is not the case since one would still have a linear relation between the scale factors , as it is shown by equation ( [ cubic ] ) . in order to show this in a simple way, we fix some of the parameters of the model . for example , set , , and take into account that in order to be compatible with the ads reference metric. then we have and , with by increasing the value of we eventually obtain a complex value of , so the solution is not well defined .the ads reference space protects the physical space against acceleration in two ways ; not only by inducing a deceleration , but also an ads reference metric is not compatible with well defined solutions for large values of a positive cosmological constant .this result is a consequence of the linearity between the scale factors in vacuum and , as we have argued above , it can not be changed by adding matter which fulfils the sec .  therefore , we can conclude that there is no accelerating homogeneous and isotropic solutions for models with an ads reference metric in the absence of matter violating the energy conditions , at least if we assume that the metrics take the form given by equations ( [ metricf ] ) and ( [ metricg ] ) , which means that both metrics have the same spatial curvature and can be written in a diagonal way in the same coordinate patch .as we have emphasized in this paper : in massive cosmologies there is no reason to restrict our attention only to that branch of solutions where both scale factors are strictly proportional with a fixed constant of proportionality leading to a situation which is equivalent to a cosmological constant .although there is a great deal of freedom in the set of solutions to this theory , once we have restricted to a particular model with a given reference metric , the dynamics of the physical universe would of course be determined by the matter content of that universe .the reference metric will certainly affect the particular solutions , but so will do the matter content .  on the other hand , we have shown that in those models of massive gravity with ds and ads reference metrics all the vacuum solutions behave as the solutions of the trivial branch , implying strictly proportional scale factors and cosmic times .this result can be understood by observing that the graviton should hide the effect of a non - vanishing mass in vacuum , in order to avoid the presence of nontrivial gravitational effects in the absence of matter .if one were tempted to follow that line of thinking , he / she would conclude that these are the only consistent massive gravity models . as the models with ds reference metric ( as well as those with flrw reference metric )have been suggested to be affected by some tension between the higuchi bound and the vainshtein mechanism for this branch of solutions even when the background metric and the reference metric are distinct ( which leads to metric fluctuations which are not of the fierz - pauli form ) , a promising possibility is to consider an ads reference metric , where those particular problems are not present  .nevertheless , as we have demonstrated in this paper , the attractive effect induced by such a reference metric would be so strong that it can not describe the current accelerated expansion of our universe , even if we were to introduce a positive cosmological constant with large value .  the non - existence of homogenous and isotropic accelerating solutions for models with an ads reference metric is a general result which is valid for any model where both metrics can be written in a diagonal way in the same coordinate patch , having the same spatial curvature ( ) , and assuming the absence of matter violating the energy conditions .the dynamics of the observable physical metric is determined by both the reference metric and the material content , and , therefore , the visible causal structure is affected by the reference metric . in the case of an ads reference metricthis effect is strong enough to forbid the existence of accelerating solutions ( for any matter content ) . for other cosmological models fulfilling the diagonal assumption ,the causal structure of the physical metric can be studied by noting that this assumption is a particular case of the generalized gordon ansatz ( as introduced and discussed in ref .thus , in view of the flrw symmetries common to both metrics the light cones of both metrics have the same axes , those of the physical metric being inside or outside the light cones of the reference metric depending on the particular solution obtained when the material content is considered .it should be stressed that we are not suggesting that there are not massive cosmologies with flrw reference metrics able to describe our universe .in fact , we have proven that any universe can be described by massive gravity if one were to carefully choose an appropriate reference metric .nevertheless , on one hand , the resulting universe would be affected by the already mentioned higuchi - vainstein tension , giving not a lot of chance for a stable universe in which general relativity can be recovered at some range , and , on the other hand , one should accept the presence of nontrivial gravitational effects in vacuum .finally , it should be pointed out that a similar result concerning nontrivial curvature in the absence of material content for some reference metrics _ can not _ be extracted in the framework of ghost - free bimetric gravity  . in bimetric gravity , if we obtain an effective fluid which is not equivalent to a cosmological constant in some gravitation sector where there is no material content , then all we can say for sure that there would be some material content in the other gravitational sector .that is , some material content is automatically present in the theory curving the spacetime , even if that material content can not be directly observed in the foreground sector .this reinforces the idea that , in spite of the mathematical similarities , massive gravity and bimetric gravity should be considered as conceptually very different theories  .it must be emphasised that we are indicating that an effective fluid inequivalent to a cosmological constant is possible in bimetric gravity only if there is some material content , but not the negation of this statement .we can of course quite easily have an effective fluid equivalent to a cosmological constant in the presence of matter , which would ( as has been long known  and has been recently pointed out in the ghost - free theory  ) only require that the effective fluid of the other gravitational sector must also be equivalent to a cosmological constant .pmm wishes to dedicate this article to the memory of pedro f. gonzlez daz , whose views about physics were and will ever be an inspiration .+ pmm acknowledges financial support from a fecyt postdoctoral mobility contract of the spanish ministry of education through national programme no .2008 - 2011 .+ mv acknowledges support via the marsden fund and via a james cook fellowship , both administered by the royal society of new zealand .99 c. de rham , g. gabadadze , and a. j. tolley , `` resummation of massive gravity '' , phys . rev .* 106 * ( 2011 ) 231101 [ arxiv:1011.1232 [ hep - th ] ] . c. de rham and g. gabadadze , `` generalization of the fierz - pauli action '' , phys .d * 82 * ( 2010 ) 044020 [ arxiv:1007.0443 [ hep - th ] ] .d. g. boulware and s. deser , `` can gravitation have a finite range ? 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Abstract: massive gravity is a theory which has a tremendous amount of freedom to describe different cosmologies ; but at the same time the various solutions one encounters must fulfill some rather nontrivial constraints . most of the freedom comes not from the lagrangian , which contains only a small number of free parameters ( typically 3 depending on counting conventions ) , but from the fact that one is in principle free to choose the reference metric almost arbitrarily which effectively introduces a non - denumerable infinity of free parameters . in the current paper we stress that although changing the reference metric would lead to a different cosmological model , this does not mean that the dynamics of the universe can be entirely divorced from its matter content . that is , while the choice of reference metric certainly influences the evolution of the physically observable foreground metric , the effect of matter _ can not _ be neglected . indeed the interplay between matter and geometry can be significantly changed in some specific models ; effectively since the graviton would be able to curve the spacetime by itself , without the need of matter . thus , even the set of vacuum solutions for massive gravity can have significant structure . in some cases the effect of the reference metric could be so strong that no conceivable material content would be able to drastically affect the cosmological evolution .  28 june 2013 ; latex - ed .   keywords : massive gravity , graviton mass , cosmology , background geometry , foreground geometry . _ dedicated to the memory of professor pedro f. gonzlez daz _


the relationship of quantum mechanics and gravity has been a problematic one since the early attempts to quantize the theory of general relativity .there are a plethora of _ prima facie _ difficulties ( see for a summary ) that one faces even before laying down any specific approach to the subject .this has discouraged many people away from the field . on the other hand , it is not healthy to just not attack a problem because some obvious difficulty is forecast to appear based on general considerations .it is much more reasonable to devote some effort to a detailed analysis , since in many cases the particular details of how the expected difficulty appears can lead to insights towards its cure .  partly with this philosophy as a motivation ,a group of people have been pursuing several aspects of the canonical quantization of gravity .the canonical approach faces some a priori problems of its own , like how to recover a spacetime picture from three dimensional notions quantum mechanically , or the related issue of the `` problem of time '' ( see for a review ) .yet , it also seems as an appropriate setting for elucidating issues like what is exactly the space of states of the theory or how degrees of freedom and symmetries interlace in the theory .some of these latter issues were also considered a stumbling block for the theory early on : how to characterize the space of states given that spatial diffeomorphism invariance was a symmetry and how to solve the nonpolynomial constraints of the theory quantum mechanically were one of the deterrents of early progress of the subject in the 60 s ( see for instance ) .unlike the difficulties mentioned at the beginning ( problem of time , observables ) some progress on the issue of diffeomorphism invariance , space of states and solutions to the constraints was achieved through the introduction of a new set of variables that allows to describe general relativity in terms of notions closer to yang mills theory .these new variables were introduced by ashtekar about ten years ago and led to a large amount of new insights and perspectives on canonical quantum gravity . in the canonical approachone describes spacetime through the initial data on a spatial slice consisting of a three - dimensional metric and the extrinsic curvature .the new variables replace these by a set of ( densitized ) triads and a canonically conjugate momentum that transforms under space and triad transformations as an connection .the spacetime diffeomorphism invariance of general relativity translates itself into four constraints that the variables have to satisfy , three of them representing spatial diffeomorphism .the formalism is also invariant under triad rotations and this translates itself into three additional constraints that have exactly the form of an gauss law .therefore the reduced phase space of the theory is exactly that of an yang mills theory with four additional constraints .an additional element that enters into play is the fact that the connection is a complex quantity .if one constructs the connection and triad from a slice of a real four dimensional metric , the formalism assures that the evolution keeps everything real forever .one is simply using a complex coordinatization on a real phase space .however , in the quantum theory one has to ensure that the resulting metric and its time derivatives be real .this can be implemented through a set of `` reality conditions '' , a set of constraints that turns out to be second class .another approach is to forget these constraints and then choose in the quantum theory an inner product that makes the observable quantities self - adjoint .this , in fact , may help to select the correct inner product for the theory , something the dirac quantization procedure says nothing about .an approach that has proved fruitful for the gauge invariant description of yang mills theories is the use of holonomic variables , in which one codes the gauge invariant information of the theory in the holonomies along families of closed curves .one can actually build an entire quantum representation in which the wavefunctions are purely functionals of loops , called the `` loop representation '' .this approach was pioneered in yang mills theories by gambini and trias and was applied in gravity by rovelli and smolin . when applied to gravity, it immediately led to remarkable insights . on the one hand the diffeomorphism invariance of the theorygets naturally embodied into diffeomorphism invariance of closed curves and therefore the quantum states of the theory are _ knot invariants_. moreover the hamiltonian constraint of general relativity , which embodies all the dynamical information of the theory , gets coded into a rather simple operator that acts at intersections of the knots .all these ideas were developed early on . over the last two yearsthere has been considerable progress in the consolidation of many of these ideas , taking them from being `` handwaving '' arguments to either rigorous theorems or rather extensive formulations for doing practical computations given certain assumptions .in particular new connections with knot and graph theory have been drawn . in this talki will briefly summarize three directions in which the formalism is advancing : a ) the implementation of a generalized wick transform " to deal with the issue of the reality conditions ; b ) the development of spin networks as a basis of states in the loop representation ; c ) the use of a lattice regularization for the hamiltonian constraint that suggests unexpected dynamical connection between gravity and knot theory .this paper is only intended as a pointer to current literature , to which we refer the reader for additional details .as we mentioned in the introduction , the new variables are complex variables . specifically , the connection where is the spin connection built from the triad and is the extrinsic curvature with one index raised with the triad . as suggested before ,one possible approach is to treat quantum mechanically and as complex independent variables and then choose an inner product that ensures that one has a real spacetime metric .the details of how this is achieved depend on at which level one wants to introduce an inner product .if one introduces an inner product before imposing the constraint , then one can simply require the metric and its poisson bracket with the hamiltonian to be real .however , a rather general consensus is that one only introduces a physical inner product in canonical quantization after imposing the constraints . since neither the metric nor its poisson bracket with the hamiltonian commute with the constraints, one can not make statements about them after imposing the constraints .what one has to do is to find conditions on the observables of the theory ( quantities that commute with the constraints ) that are equivalent to the reality of the four dimensional metric .unfortunately , a practical impediment is that we do not know explicitly any quantities that commute with the constraints of general relativity that may be used for this purpose , and they are expected to be nonlocal expressions .this has led some people to be suspicious of the whole approach since one is `` sweeping under the rug '' of the impossibility of finding observables the problem of implementing the reality conditions .this feeling is reinforced by the fact that the complexity of the variables is crucial : one can build real connection - based formulations but they have high order nonpolynomial constraints .it therefore seems one is concealing in these unsolved difficulties an important issue .a lot of work on the subject has simply ignored the issue of the reality conditions . at a certain level, it makes sense .suppose one is looking for solutions to the constraints of the theory , or of ways of characterizing the space of solutions .it is natural to start the search ignoring completely the inner product and then , from the solutions found , pick the ones of physical interest by requiring that they be normalizable with respect to the inner product . at that levelone would then worry about having an inner product that respects the reality conditions .the big worry is : will the reality conditions force upon us an inner product such that most of the work done solving the constraints is rendered useless ?an important step away from the above worry was a recent construction introduced ( in slightly different flavors ) independently by thiemann and ashtekar .what follows is an account of their work .the idea is actually simple and attractive .suppose one starts with general relativity formulated canonically in the traditional way , but using triads instead of the metric as fundamental variables .one then introduces a canonical transformation to a new set of variables given by the triad and a real connection ( we use the same notation that for the complex connection used throughout the rest of the paper ) .the theory has a gauss law and diffeomorphism constraints that look in terms of these variables exactly the same as those of the complex ashtekar formulation .however , if one writes the hamiltonian constraint of ordinary general relativity with a lorentzian signature in spacetime , it turns out to be a nonpolynomial expression .this is why the ashtekar variables had to be complex , the complexity achieving the simplification of the hamiltonian constraint to a polynomial expression .if one were interested in general relativity with a euclidean signature , the hamiltonian constraint in terms of the real variables we just introduced is exactly the same as that of the usual ashtekar formulation , .this actually has been known for quite some time .the new insight consists in noticing that one can construct a canonical transformation that maps the constraints of the euclidean theory to those of the lorentzian theory .one can then simply work in the euclidean theory ( which is tantamount to working in the usual ashtekar formulation but assuming that everything is real ) and one knows that the resulting theory can be canonically mapped to real lorentzian general relativity . in a sense , this procedure `` legitimizes '' many calculations that have been done ignoring the reality conditions and assuming the ashtekar variables were real .the generator of the canonical transformation is actually very simple ,   the main drawback of the construction is that if one wishes to implement the canonical transformation quantum mechanically , the operator that materializes it is given by the exponential of , which yields a highly complicated factor ordering for the resulting constraints .it is yet to be investigated what are the implications of this fact for the issue of the constraint algebra and other details of the canonical quantization , like the hilbert space of solutions of constraints . on the positive side, the construction seems to work well not only for vacuum relativity but also for the theory coupled to matter . an interesting point to noticeis that the proposed method maps solutions of the constraints of one theory to the other .it does not , however , map four dimensional riemannian solutions to lorentzian ones .however , it maps the integral curves of the hamiltonian on the constraint surface from one theory to the other .this might lead to new dynamical insights in classical gravity .the main achievement however , is that previous calculations that were only heuristic , because they either ignored the reality conditions or outright treated the variables as real , can now find applicability in the lorentzian domain , possibly in a rigorous way .the idea of a loop representation is to encode all the gauge invariant information of the theory into holonomies along loops .this is easily understood through the loop transform , which is one of the possible ways of defining the loop representation , \psi[a]\ ] ] where  ] is the trace of the holonomy ( wilson loop ) of the connection along the loop , and is a measure in the infinite dimensional set of all connections .this transform is analogous to a fourier transform .one is expanding the functional  ] . apart from the many mathematical subtleties involved in the fact that this is really a functional transform, one important difference stands out : the basis " formed by  ] and = w_{\beta\circ\alpha}[a]$ ] , where denotes composition of curves at an origin , which for simplicity can be taken at the intersection of and .these identities are nonlinear and through combinations of them one can be led to many nontrivial - looking identities among wilson loops .the identities are inherited by the wavefunctions in the loop representation .the presence of these identities makes the space of functions of loops quite nontrivial . as an example , for various reasons one might be interested in wavefunctions that are one on smooth loops and zero on loops with intersections or kinks .it turns out that such functions would easily satisfy the hamiltonian constraint .unfortunately , they fail to satisfy the mandelstam constraints . also consider the following : the diffeomorphism constraint implies that wavefunction must be knot invariants . on the other hand ,very few of the available knot invariants in the mathematical literature satisfy identities like mandelstam s .actually this is not entirely true : the mandelstam identities require that the wavefunctions have defined values on loops with intersections whereas most of the invariants of the mathematical literature are only defined for smooth loops .so one could conceive defining the values of these wavefunctions in such a way as to make them compatible with the mandelstam identities .this procedure is , however , severely constrained .it therefore appears that it would be useful to encounter a subset of wilson loops that would not be related through mandelstam identities and would yet be sufficient to expand all gauge invariant functions .rovelli and smolin noticed that spin networks provide a natural way to tabulate such a basis . in the original version , due to penrose , spin networks are colored trivalent graphs in two dimensions . colored means that to each strand connecting two vertices a number is assigned . what does this have to do with quantum gravity ?a simple example can clarify this .consider the loops depicted in figure 1 , which intervene in the mandelstam identity considered before .now consider a diagrammatic notation as the one introduced in figure 2 .it is clear that and are independent combinations of wilson loops .one can therefore view and as spin networks and as generators of independent wilson loops through the combinations defined . to make a long story short, this happens for all possible loops and networks .that is , given a graph in two dimensions with trivalent intersections , one can construct in a univocous way a basis of combinations of wilson loops , that are not related through mandelstam identities .it turns out that the reason for this is that group - theoretic considerations of recoupling theory are at the foundation of the spin network approach .but for this review we will leave matters here . it suffices to say that one can now construct a representation in which wavefunctions are labelled by spin networks and one can do there whatever one wished to do in terms of loops , like define a hamiltonian constraint , dynamics , define a time and true hamiltonian .moreover , it has been shown rigorously using measures in infinite dimensional spaces that the spin network states constructed as discussed above are orthogonal .in addition , certain operators measuring the area of a surface and volume of a portion of space are naturally diagonalized by the spin network basis .this has profound physical implications , since it means that areas are quantized in quantum gravity .this may , for instance , lead to a new understanding of the thermodynamics of black holes , since there now is a natural discrete structure associated with the horizon and therefore one can count states and define notions of entropy for it .one of the central issues in the construction of a quantum theory of gravity is the definition of a regularized quantum hamiltonian constraint .there are several ab - initio difficulties one can imagine in implementing such an operator . on one hand, one has the problem that the hamiltonian is not diffeomorphism invariant and yet the theory has a diffeomorphism constraint .moreover , the hamiltonian involves products of basic operators , and most regularization procedures are not diffeomorphism invariant .one can try however , to gain some feeling for the structure of the space of solutions of the theory without facing the entire list of difficulties associated with regularizing the hamiltonian .there are several heuristic proposals for the regularization of the action of the hamiltonian constraint in loop space .most of these proposals end up with an operator that acts nontrivially at intersections of loops .there , the action of the operator can be split into two pieces : a diffeomorphism invariant action , consisting of a rerouting of the loops at the intersection and a diffeomorphism- and regularization - dependent prefactor .the topological piece of the action is not unique , it originates in the need to represent the curvature that arises in the hamiltonian as a deformation of the loops and the action of the the triads as a rerouting and there are differing proposals on how to accomplish this .the prefactor absorbs the distributional nature of the functional derivatives that arise in the definition of the quantum triad operators .the implementation of such operators on the space of wavefunctions that are solutions to the diffeomorphism constraint is pretty hopeless , since the hamiltonian by definition will have an action that maps out of that space of functions , since it is not diffeomorphism invariant . in practice , this is assured by the presence of the regularization dependent prefactors .one particular proposal , which appeared in the context of a lattice regularization of the theory , is to ignore the prefactors and concentrate on the topological part of the operator . by doing so, we ensure that the kernel of the operator we are considering is contained in the kernel of the hamiltonian . on the other handthe regularization dependent prefactors can , on rather general grounds , be chosen to be nonzero ( this of course is a delicate issue ) in a first analysis .it turns out that a simple regularization based on lattices yields to a topological action as that shown in figure 3 . in this action the hamiltonian only acts at triple intersections and it displaces one of the strands going through the intersection along a direction determined by one of the other strands back ( ) and forth ( . the action of the operator consists in summing similar contributions per each pair of tangents incoming to the intersection . the contribution is completed by rerouting the subloop determined by the two strands considered in the operation , as indicated by the numbers in the figure that keep track of the orientation .the remarkable property that this regularization has is that the second and third coefficient of the infinite expansion of the jones polynomial considered in are annihilated by this operator . to prove this, one considers the individual coefficient and uses its skein relations ( the relations that define it as a knot invariant ) .one way of viewing this result is that the action of the hamiltonian is a skein relation in itself .it is the skein relation defining the invariant that is the general solution of the quantum einstein equations .notice that this skein relation does not completely characterize an invariant .this is sensible , since we do not expect the quantum einstein equations to have a single solution .it also sheds new light on how to characterize the infinite number of degrees of freedom of general relativity in terms of the topological , discrete notions of knot theory .another upshot of this construction is that one need not worry about the algebra of constraints , since by imposing a skein relation one not only requires that the identity hold , but all possible rearrangements and multiple applications of it hold too .this makes sense in the context of diffeomorphism invariant states , where the hamiltonian should have vanishing commutator with itself . viewing the constraint as a skein relation therefore allows us a new perspective on the issue of the algebra of constraints .it is also worwhile remarking that in the regge ponzano approach to gravity there is an analogous situation : the constraint appears as a relation among graphs that uniquely determines the invariant that is a ( in that case actually the only ) solution .it might come as a surprise to the reader that both the second and third coefficients of the infinite expansion of the jones polynomial ( which by the way are vassiliev invariants ) appear as solutions . in calculations in terms of extended loops ,the third coefficient was not a solution .the answer to this is that this regularization is different .the calculations have some common elements with the extended loop ones but in the end lead to different results . recall that the operator we are dealing with here is finite whereas the extended loop one required a regularization , that in turn required a counterterm to have the second coefficient be a solution .we have checked that other possible candidates ( for instance the square of the second coefficient ) are not annihilated by the operator , which therefore has a nontrivial action .it would be interesting to see if the operator manages to annihilate all the vassiliev invariants stemming from the jones polynomial .the interest of these results for quantum gravity is still questionable , since only the second coefficient satisfies the mandelstam identities and therefore can be a candidate to state of the quantum gravitational field .these results however , highlight how different regularization choices can lead to different properties of the quantum theory .it should be emphasized that both in the extended loop case and in the skein relation calculations , the fact that the knot invariants we are discussing are annihilated by the hamiltonian constraint arises as a consequence of a very complex cancellation , involving in a very strong way properties of the knot invariants .it is remarkable that one can find two regularizations that differ in simple details and yet lead to theories with so distinct properties and of so rich a structure .in summary , progess in this area is steady .it is very encouraging to see physics emerging from these considerations , as the first steps towards calculations of the bekenstein bound show .it is also worthwhile noticing that the various sets of ideas discussed are entangling themselves in a nontrivial way , yielding a rather unexpected picture of quantum gravity .one of the most exciting aspects is that we will soon be able to tie physics deeply into the mathematics developed : for instance it is unlikely that all the regularizations of the hamiltonian constraint proposed will lead to the correct bekenstein bound for black holes .it is also quite likely that the mathematical developments that allow to put the theory on a more rigorous setting will act as a guideline as to what particular choices one has to make to construct the theory .the goal of a final theory of quantum gravity probably is still afar , but the fact that one is able to establish a framework that is finding its way towards concrete physical applications and yet retaining some of the original ideas that motivated it is a quite encouraging development .i wish to thank the organizers of icgc 95 for inviting me to speak , for travel support and hospitality at poona .this work was supported in part by grants nsf - int-9406269 , nsf - int-9513843 , nsf - phy-9423950 , by funds of the pennsylvania state university and its office for minority faculty development , and the eberly family research fund at penn state .the author also acknowledges support from the alfred p. sloan foundation through an alfred p. sloan fellowship .  c. j. isham , gr - qc/9310031 , in `` canonical gravity : from classical to quantum '' , ed j. ehlers , h. friedrich , lecture notes in physics 434 , springer - verlag , berlin ( 1994 ) ; `` structural issues in quantum gravity '' gr - qc/9510063 .k. kucha , in `` proceedings of the 4th canadian meeting on relativity and relativistic astrophysics '' , editors g. kunstatter , d. vincent , j. williams , world scientific , singapore ( 1992 ) .b. dewitt , phys . rev . * 162 * , 195 ( 1967 ) .a. ashtekar , phys .lett . * 57 * , 2244 ( 1986 ) ; phys . rev . *d36 * , 1587 ( 1987 ) 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variational derivations of exact skein relations for chern simons theories '' hep - th/9602165 . c. rovelli , l. smolin , nucl* b442 * , 593 , ( 1995 ) .r. penrose , in `` quantum theory and beyond '' , ed e. bastin , cambridge university press , cambridge ( 1971 ) .l. kauffman , s. lins `` temperley lieb recoupling theory and invariants of 3-manifolds '' , annals of mathematics studies 134 , princeton university press , princeton ( 1994 ) ; r. di pietri , c. rovelli , gr - qc/9602023 , to appear in phys . rev .* d*. r. borissov , `` graphical evolution of spin network states '' , preprint gr - qc/9606013 ; r. borissov , s. major , l. smolin , `` the geometry of quantum spin networks '' , preprint gr - qc/9512043 . c. rovelli , j. math .phys.*36 * , 6529 , ( 1995 ) .j. baez , adv .math . * 117 * , 253 ( 1996 ) . c. rovelli , l. smolin phys .rev.*d52 * , 5743 , ( 1995 ) . c. rovelli ,`` black hole entropy from loop quantum gravity '' , preprint gr - qc/9603063 ; m. barreira , m. carfora 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''preprint gr - qc/9603019 .h. fort , r. gambini , j. pullin , in preparation .r. gambini , j. griego , j. pullin , in preparation .b. brgmann , r. gambini , j. pullin , nucl . phys . *b385 * , 587 ( 1992);gen. rel . grav . * 25 * , 1 ( 1993 ) .m. reisenberger , and private communication .d. bar - natan , topology 34 ( 1995 ) 423 - 472 ; l. kauffman in `` knots and quantum gravity '' , j. baez editor , oxford , clarendon press ( 1994 ) ; j. baez , lett .* 26 * , 43 ( 1992 ) .j. griego , phys .* d53 * 6966 , ( 1996 ) . c. di bartolo , r. gambini , j. griego , phys* d51 * , 502 ( 1995 ) .

Abstract: this is a brief and updated summary of a talk given at the international conference on gravitation and cosmology that took place in poona in december 1995 . it is very brief and is mostly intended as a guide to current literature , or to keep people updated only in very broad terms on the latest developments in the subject .   psfig  cgpg-96/6 - 5 + gr - qc/9606061 +


sonic crystals ( scs ) are materials consisting of periodic distributions of acoustic scatterers in air .the periodicity in these materials give rise to sonic band gaps , a range of frequencies for which sound propagation is forbidden inside the crystal for some ( pseudo gap ) or all ( full gap ) wave vectors .these stop bands have induced various application proposals such as sound shields and acoustic filters nature , sanchezperezprl , zhenyeprl , caballeropre99,rubiojolt , caballeroprb01,wmrobertson , lsanchisjasa , kushwahaapl97,zhenyepre01,goffauxapl03,sanchezperezapl02 .recently , the properties of scs in the low frequency region , where the dispersion relation is linear , have also been of interest for device applications . for these frequenciesthe sc behaves as a homogeneous medium with an acoustic impedance larger than air .this property has allowed the construction of two refractive devices , a convergent lens and an acoustic interferometerfcerveraprl02 . on one hand , a comprehensive study of acoustic interferometers has been reported by some of us . on the other hand ,various works have been published about focusing sound and image formation in the light of the seminal work by cervera _ _ et al.__ .different sc lenses have been proposed and a discussion has aroused about whether refraction or diffraction is the dominant mechanism for focalization . focusing sound can also be obtained by using time - reversal techniquesmfink00,mfink99 , but the discussion of these techniques is out of the scope of the present work .  the goal of this paper is to answer the following question : which is the optimal sc structure that acts as an acoustic lens focusing the sound at a predetermined focal distance ?from this point of view we will discuss what is the physics that control the focusing process in several acoustic lenses , which were optimized for different frequencies and focal distances .our approach to the problem consists in using a population based stochastic search algorithm , specifically a genetic algorithm ( ga ) in conjunction with the multiple scattering theory ( mst ) vtwerskyjasa51 .this approach has been previously used by us to design a spot - size converter in the field of two - dimensional photonic crystalslsanchis04 .gas have been proved to be very efficient , especially in optimization problems having a large set of discrete parameters .to speed - up the stochastic search , here we have implemented a symmetric multiple scattering theory ( smst ) that decreases the calculation time needed for predicting the scattered acoustic wave from a symmetric system .evolutionary computation is a family of algorithms that are related with the optimization process used by nature itself , the evolution .the evolution is very powerful in adapting individuals to a given environment , and is able to tackle enormous complex problems with fairly simple means . throughout generationsthe individuals chromosomes ( the genotypes ) are mixed and new individuals with different characteristics are born .those individuals that adapt better to the environment have the best chance of survival and hence give birth to more offspring creating a new generation more fit for survival than the previous one .we here use one of the most popular algorithms from this family , a simple - ga , introduced by holland .the gas are population based stochastic search algorithms normally used to solve discrete problems .their functionality is based on their ability to find small parts of the individual ( the solution ) that reflects a good quality in the result , these fractions are called building blocks ( bbs ) . in other words ,the ga creates , mixes and finally uses these bbs to build a global optimal solution .unfortunately this goal is not always met .if the global optimum consists of large sized bbs , the ga will have problems of constructing them because high - order bbs are very difficult to grow .these problems are referred to as ga - hard problems .nevertheless , this is not of great concern here since this work concentrates on the method itself and not on the algorithm . at this point, we are pleased by finding good solutions within the boundaries of the optimal .  the simple - ga works with three operators ; selection , crossover and mutation that are iteratively applied to a population of individuals .the selection operator culls the population selecting better solutions over worse for mating .the mating act , which is directed by the crossover operator , mixes and constructs the bbs from two or more individuals creating new offspring .the mutation operator makes it possible for the offspring to possess new bbs that none of the parents have and is applied before passing into the next generation ( i.e. , the next iteration ) .the individual is represented by a chromosome ( a digital string ) that is put together by a number of binary genes coding the genotype .each gene corresponds to one specific part of the phenotype and the corresponding value of the gene is called allele .the genotype represents one and only one phenotype that here is a cluster of rigid cylinders .our goal is to find which cluster focuses the sound most efficiently in a fixed predetermined focal point , under a chosen constrain or boundary condition .this cluster of cylinders can be seen as a transparent acoustic lens device . to quantitativelyjudge the quality of such a device , a real number is assigned to each cluster .this number is referred to as the fitness of the solution and is here calculated as the pressure in the focus if not mentioned others .one lens is better than another if its fitness is superior  to predict the fitness we need to calculate the scattered pressure field from a cluster of cylindrical scatterers .we here apply the multiple scattering theory ( mst ) that earlier has been used by some of us with good agreement between theory and experiments .because the search space of the problem is very large , the ga needs to calculate the fitness for numerous individuals before converging .for example , a genotype with 50 one - bit genes could represent up to different phenotypes .thankfully the ga does not get near this number of fitness calculations , but it is crucial to do the fitness calculation as fast as possible . in order to speed up the fitness calculation , we apply a simple symmetric condition on the space of solutions . the scattered pressure field from a symmetric system has a mirrored symmetry with respect to the central axis . in other words ,the field scattered from a cylinder off the symmetry axis is identical , only mirror reversed , as the field scattered from its symmetric associate .this means if using mst a lot of computational resources will be used to calculate the same scattered field twice for almost every cylinders in the symmetric cluster . by including this mirror symmetry in the mst the fitness calculation and the ga searchare considerably accelerated .   the complete self - consistent procedure including all order of multiple scattering that follows the seminal work of twerskyvtwerskyjasa51 has been reported by chen and ye to study the transmission spectra of large structures .the smst trimmed to handle symmetric systems will be introduced below .consider a cluster of n scatterers located at the positions , all placed above or on the symmetry axis ( the x - axis in fig .the complete symmetric crystal is represented if each cylinder placed off the _x_-axis is copied and placed at the same _ x_-position and with the same , but negative , coordinate .all cylinders placed below the symmetry axis are indicated with a + in the index ( see in fig .1 ) .  if a symmetric external wave with temporal dependence impinges the cluster , the total field around the cylinder is a superposition of the external field and the radiation scattered by the cluster :    where if , and if . and is the field scattered by cylinder and its mirror image respectively .these four fields can be expanded into series bessel functions .the total incident wave to the cylinder , which consists of a superposition of the external incident wave and the scattered waves from all the other cylinders in the cluster , is expressed in terms of the bessel function of the first kind as    where is the wavenumber calculated as , where is the sound velocity in air , i.e. 340 m / s .  the external incident plane wave , impinging to cylinder follows the same expansion principle but with the multipole coefficients .the scattered waves from cylinder and are expressed in a series of outgoing hankel functions rather than regular wave functions :    using the multipole coefficients , , and for , , and respectively , the expression above can be cast into the following relation between coefficients :     and being the propagator from cylinder and to respectively , whose components are    where is the kronecker delta .  notice that the coefficients are known , but , and are not .the boundary conditions at a rigid cylinders surface states that the normal pressure gradient equals zero. this equation relates with and the symmetric condition relates with . the rigid cylinder approximation has been demonstrated to reproduce fairly well experiments made of hollow aluminum cylinders .now the transition matrix can be calculated and gives the proper transformation of the total incident field to cylinder expressed by the coefficients , to the scattered field expressed using the coefficients :    , and the mirror symmetry relates and     introducing the coefficients from eq .[ eq4 ] and [ eq6 ] in eq .eq2 will result in    by truncating the angular momentum within , eq .[ eq7 ] reduces to a linear equation where the dimension of the relevant matrix is .note that is not the total number of cylinders , only the cylinders placed on or above the _ x_-axis . for a system of 100 cylinders truncated to , the cpu - time needed to calculate the coefficientsis reduced by a factor of 4 by using the smst instead of the standard mst .all results presented are calculated with approach is supported by the fact that no error higher than 1% was observed in any of the following calculations of the pressure field when including higher angular momentums in the series expansion .the coefficients in the transfer matrix give a good estimation of the convergence of the method .here , the most significant matrix coefficient with an index higher than three is of the order  . in all optimizationprocesses the angular momentum was truncated at due to computational resources .since this will not guarantee that the simulated fitness is within an proper error , a local search was later done with .the local search was done using a ga with a much smaller population where one individual in the initial population correspond to the one with maximal fitness in the search with to confirm the quality of this two - step method a ga - optimization was done using directly , with the result almost identical to the result received with the two - step optimization method , but significantly more time consuming .figure 2 illustrates a typical ga - run using the smst for designing an acoustic lens .the problem is coded on 100 one - bit genes . as can be seen in the figurethe algorithm has converged after approximately 50,000 fitness calculations , this equals one part of the total number of phenotypes .guided by recent works fcerveraprl02,zhenyepre03,ngarciapre03,chkuo04,ahakansson04 , where a discussion has aroused concerning the shape of scs devices for focusing sound , we will here as the first step in our study treat the problem of finding the optimal shape of a sc cluster for different focal distances and for different symmetry constrains .firstly the ga was implemented to find the optimal sc cluster , symmetric with respect to both axes , that focuses the sound on the -axis for a fixed focal distance . to do this ,the device s genotype was implemented using 10 two - bit genes , where each gene corresponds to a different layer in the cluster . the first gene on the chromosome corresponds to the central layer at = 0 and the four possible alleles of the gene ( 00 , 01 , 10 , 11 ) code the width of this layer , 20 , 18 , 16 or 14 cylinders respectively .this can be interpreted as ( 00 ) keeps the width ( 20 cylinders for ) , ( 01 ) , ( 10 ) and ( 11 ) remove one , two and three cylinder respectively , at each extreme ( leaving the 18 , 16 or 14 cylinders for ) . the next gene onthe chromosome follows this rule and is related to the two layers next to the central one . since the crystal is symmetric the same gene code both neighboring layers .each following gene on the chromosome relates to the next neighboring layer and the allele tells us if this layer will decrease in width ( 01 , 10 , 11 ) or be kept constant ( 00 ) with respect to the previous layer .we have considered a hexagonal array of rigid cylinder with radius r = 2 cm and a lattice constant = 6.7 cm .systems having these parameters have been studied earlier both experimentally and theoretically in refs .we also consider an incident sound plane wave traveling in the positive -direction at 1700 hz .this wave is perfectly symmetric with respect to the _x_-axis , which is the only criteria for using the smst and it will be assumed as incident wave in all the following calculations if not mentioned others .four different lenses for four different focal distances were designed using the ga .the same boundary conditions were applied , i.e. the central layers maximal width is 20 cylinders and the maximum number of layers equals 19 ( 18 layer off - axis plus the central layer on - axis ) .the results are shown in fig .3 , where the amplification [ in decibels ( db ) ] was calculated as    where defines the focal point , and equals the pressure measured in free space that here equals 1 for the plane wave .the amplification in the optimized focal point for the four structures vary from 8.1 db to 8.6 db .the highest gain was achieved for the smallest focal length .a noteworthy observation is that the structure increases in width at the same time as the curvature of the surface decrease with longer focal length , a similar characteristic that for optical lenses and should be the case for homogenous refractive devices .this can be considered as a support of the work presented in refs . and , i.e. the sc is seen as a homogeneous material and the dominant effect in focalization is due to refraction . if comparing the designed structure in fig . 3c with the sc used in the experimental set - up in ref .fcerveraprl02 one can clearly observe a close similarity in the curvature of the different surfaces .the boundary conditions constraining the pool of solutions are very strong in this first approach simplifying the optimization process . drawn by the determination to increase the amplification of the device we lifted some of these constrains .figure 4 shows three different devices design by the ga to maximize the focalization at _ x _ = 1.05 m ( the same focal distance as in fig .3c ) . the upper crystal has the most restrictions and the lower the least .this means that the number of possible solutions increases with the freedom of the ga and results in a harder problem to solve .the device in fig .4a is the result from a genotype including 10 three - bit genes , where each gene is related to the phenotype in a similar way as earlier . herethe three bit gene corresponds to eight different alleles that directly code the number of cylinders in each layer ; for the nine central layers in the crystal : from 20 or 19 cylinders for the allele ( 000 ) down to 6 or 5 for ( 111 ) ; for the ten outer layers ( five to the left and five to the right ) in the crystal : from 14 or 13 cylinders for the allele ( 000 ) down to no cylinders for ( 111 ) .the boundary condition is set in such a way that the crystal from fig .3c is included in the search space and hence mark a minima for the expected optimization result .the increase in the genotype size gives rise to an increase in the total number of possible solutions from  in fig .3c to  in 4a .the resulting device is of the same size as in fig . 3c butnow the crystal is not lens - shaped but formed out of three thinner sections . as expected the amplification increases , from 8.1 db to 8.3 db .  in the next modification of the boundary conditionwe leave the homogeneous sc cluster and give freedom to the algorithm to introduce point - defects in the cluster .we here leave the homogenization discussion and let the ga play with the complex procedure of multiple scattering inside the cluster of scatterers , though the double symmetric condition still holds .the genotype now equals 100 one - bit genes where each one - bit gene represents the presence ( 1 ) or the absence ( 0 ) of one ( and ) , two ( or ) or four ( and ) cylinder , due to the symmetry condition . the search space is increased from to ( possible solutions and the optimized amplification is increased from 8.3 db to 8.5 db .finally , in the last run , the symmetry with respect to the y - axis is removed and the number of one - bit genes in the chromosome is increased to 200 which result in the gigantic number of possible solution equal ( ) .4c shows that an enhancement of sound focalization is obtained .  in principle, one would expect that giving more degrees of freedom in the inverse design process and so increasing the search space will increase the quality in the result after the optimization , but this can be a misleading assumption . even though the results presented here follow this supposition , it is interesting to compare the results in fig . 4a and fig .4b.the qualities differ only 0.2 db while the total number of possible solutions increases from to .since the number of solutions increase exponentially ( base 2 ) with the size of the chromosome it is very important to define the genotype in a good way keeping down the size of the problem and coding the bb in a way that minimizes their size .here we have kept the ga - parameters constant during the different runs . the result in generalcan be enhanced trimming the ga - parameters to handle larger and harder problems .when problems of this size need to be tackled , a more advanced algorithm should be at hand , e.g. cga ( compaq ga ) , fmga ( fast messy ga ) , boa ( bayesian optimization algorithm ) that are capable of dynamically finding larger sized bbs .following the path of the previous analysis we now continue our study using a reduced sc framework consisting of five layers and 98 cylinders , compared to earlier nineteen layers and 371 cylinders .the degree of freedom is set at the previous highest level using 50 two - bit genes , each gene coding one cylinder .the four possible alleles of the gene are representing three different radius of the cylinders ; ( 01 ) : 1 cm , ( 10 ) : r=1.5 cm and ( 11 ) : r=2 cm and ( 00 ) : no cylinder present .figures 5 and 6 illustrate eight different devices for focusing the sound at eight different distances along the symmetry axis .these flat lenses represented by considerably few cylinders is perfectly competitive with the thick lenses earlier presented ( e.g. compare fig .4c . with fig . 6c , two lenses with the same focal distance ) . although the really impacting results are for short focal lengths were the amplification reaches up to 10.3 db . for longer focal lengths presented in fig .6a homogenous cluster of cylinders , where no point defect is present , can be observed as a central piece of the device .this cluster is more or less formed as a trapezoid where the shorter of the two parallel sides always is closer to the focus . that the cluster is homogeneous indicate that the focusing mechanism for this part is mostly due to refraction or diffraction of the clusterwe can clearly see in fig .6e that for this large focal distance ( 2 m ) the flat lens , almost free of defects , works because of the diffraction at the borders of the cluster and the focal - spot is almost completely lost .figure 7 displays the eight results put together in order to show how the amplification decreases with an increase in the focal distance .this is a surprising result since lenses working due to refraction and follow lensmaker s formula , need to be very thick to achieve high f - numbers .the f - number , calculated as the focal length divided by the diameter of the lens , is for the structure in fig .5a which is a very low number in conventional optics .it is also much lower than the sc refractive lens device reported by cervera _et al.fcerveraprl02_ and the optimized homogeneous cluster of cylinders in fig .3c , which have a corresponding f - number close to one one explanation for this is that the basic physics of the functionality is different leaving the homogenization approach were the shape of the lens is the controlling parameter , and instead using every single scatterer for bending the sound towards the focus maximizing the amplification for the set frequency .  to see how the result holds for other frequencies , fig8 shows the sound amplification for eight different structures designed to maximize the pressure at the same focus ( located at m on the symmetry axis ) for eight different frequencies , from 300hz to 1700hz . there is a peak in the frequency spectra confining the amplification for a small range of frequencies near the optimized ( indicated with a diamond symbol in the figure ) . in other words ,a ga lens designed to focus a given frequency is not robust against small variations of the optimized frequency .a substantial decrease in the amplification for lower frequency is also observed .this result is physically understood since for larger wavelength the cluster of cylinders appear more homogeneous to the acoustic incident wave and the ga looses its capability of controlling and directing the incident acoustic wave towards the focus using single scatterers .the previous result motivated the following question : can we create a device that amplify the sound equally over a wide range of frequencies ?this inverse problem can be solved by modifying the fitness function in such a way to include this condition in its definition . in what follows, we will see how to design a device working in the range from 1000 hz to 2000 hz .thus , we have selected 6 frequencies in the range and have modify the fitness function in the optimization process accordingly .a word of caution , a homogenous amplification over the frequency spectra is not guaranteed if the average pressure of the six frequencies is simply used as the fitness for the device . to meet the homogeneous amplification criterion one has to include the standard deviation of the six measurements .the fitness is then calculated using the following expression    where    and is a positive constant .other alternative fitness functions could be equally used .  by multiplying the average pressure with one minus the standard deviation we will end up with device that will not only max out the pressure at the focus but will have similar amplification for all six frequencies .the is a factor that controls the strength of the second condition .the time needed to calculate this design is six times greater since the smst is called six times for each fitness evaluation , one time for each frequency .the inset of fig .8 shows the resulting device for .the corresponding amplification spectrum ( circles ) demonstrated that the maximal amplification is lowered from 8 db to 4 db , but a relative smooth amplification is accomplished in the requested range .   to improve the results achieved for the 5 layers flat lenses we increased the number of cylinders in the sc structure .figure 9 plots the amplification for six different ga - designed 9 layers lenses with different focal distances . in order to compare with the previous results for thinner lenses ,four extra layers were added to the left of the sc structure ; in this way the distance between the last layer and the focus is held constant .the comparison of figs . 7 and 9 shows that an enhancement of approximately 2 - 3 db in the amplification is obtained when introducing the extra rows of cylinders .this result gives an indication that very efficient lenses can be designed by increasing the number of scatterers .the multifrequency focalization device was also designed with 9 layers thick clusters .the resulting device is shown in fig .10 together with its amplification in the range 1000 hz-2000 hz .notice that an enhancement in the amplification of db is obtained in comparison with the 5-layers thick device . even if the six frequencies have a very leveled amplification we can observe oscillations in the spectra between the optimized frequencies used in the fitness function .these oscillations could be eliminated by including more frequencies in the ga optimization .some characteristics are remarkable when the multifrequency lenses having 5 and 9 layers are compared ( see figs .8 and 10 ) .first of all , the central parts of the devices are similar .they are of triangular shape with one edge directed at the focus and does not contain any point defects .this gives an indication that a crystalline cluster , free from point defects , gives rise to amplifications in a broader frequency range .and secondly , the first layers ( i.e. , the ones that impinges the sound first ) of the central cluster mostly contain the cylinders having smaller radius while the triangular cluster is mostly made of the thickest cylinders .this characteristic results in a lower reflectance at the interface and , consequently , the transmission across the cluster is enhanced .an issue of crucial interest is to know if the designed device is robust against changes of its parameters , such as fluctuations in the radius of the cylinders or small displacements of their position in the lattice .it has earlier been shown that the focalization of sound using homogenous 2d sc clusters depends on the displacement of the cylinders .here we analyzed , as a typical example , the lens shown in fig .5c , a flat 5 layer lens having the focus at _ _x__=0.5 m for an impinging wave of 1700hz .table [ table1 ] shows that the amplification decreases when the error in the cylinders radius increases .the amplification is calculated for five different degrees of relative error , from 3% to 60% .twenty structures was generated and the maximal , minimal , mean and standard deviation of these is printed out in the table . for small errors we surprisingly have a small raise in the amplification for some structures and for a relative error of60% we still have solutions that conserve 100% of the pressure in the focus , but a mean value of 69% for the 20 structures is calculated .  the other important issue is the sensibility of the design to changes in the cylinders positions .table [ table2 ] gives the amplification for five different levels of random displacements in the lattice .the cylinders are randomly moved from their original position in the lattice with the relative error as 1% to 15% of the lattice parameter .the quality of the device shows a rather high robustness since more than 90% of the amplitude of the pressure in the focus is conserved for a relative displacement of 12% with respect to the lattice parameter .in this work we have examined how to solve the inverse problem of designing acoustic lenses for airborne sound by using an optimization algorithm together with a multiple scattering theory .we have introduced a symmetric multiple scattering method to speed up the search of solutions .we have presented different devices ; e.g. thick lenses based on refractive effects and flat lenses based on multiple scattering phenomenon .two lenses with flat surfaces were also designed to focus multifrequency sound waves in the range 1 khz to 2 khz .the robustness were also studied against either small variations of the cylinders radius as well as displacements of their positions .we hope that these predictions motivated future experimental work that confirm the simulations .let us stress that the procedures here introduced can be employed to design efficient sound shields , sound filters to a predetermined range of frequency , and many others acoustic devices for airborne sound .also , its application in the range of ultrasounds seems very promising to get improved ultrasonic devices ..robustness of the lens in fig .5c against errors ( ) of the cylinders radius r. twenty random structures were calculated to do a statistical representation .the columns are ( from left to right ) ; the relative error with respect to the cylinders radius ; the maximum amplification among the 20 calculated ; the minimal amplification ; the mean value ; the standard deviation .all expressed as parts of the amplification for the perfect structure ( 8.3 db ) . [ cols="^,<,<,<,<",options="header " , ]

Abstract: a survey of acoustic devices for focusing airborne sound is presented . we introduce a new approach to design high quality acoustic lenses based on arrays of cylindrical rigid scatterers in air . a population based stochastic search algorithm is used in conjunction with the multiple scattering theory to optimize a cluster of cylinders that focuses the sound in a prefixed focal point . various lenses of different sized clusters , for different frequencies and with different focal lengths are presented . in general three focusing phenomena are remarked , focusing due to refraction , diffraction and focusing due to multiple scattering . the dependency on the frequency of the incident sound and the focal distance is analyzed indicating that higher frequencies and smaller focal distances favour larger amplifications in thin lenses based on multiple scattering . furthermore , the robustness of a designed acoustic lens is studied , examining the focusing effect against errors in the cylinders positions and their radius .


in the last decades , modeling of dynamical systems , especially low - dimensional ones , becomes one of the most challenging areas of interest among mathematicians , physicists and many other sciences .depending on both the initial conditions as well as control parameters , such dynamical systems may present a very rich and hence complex dynamics , therefore leading to a variety of nonlinear phenomena .the dynamics can be considered either in the dissipative or non - dissipative regime yielding into new approaches , new formalisms therefore moving forward the progress of nonlinear science .  since the so called boltzmann ergodic theory , the assembly between statistical mechanics and thermodynamics has produced remarkable advances in the area leading also to progress in experimental and observational studies .indeed , statistical tools can be used for a complete analysis of the dynamical behavior of such type of systems .depending on the control parameters , phase transitions and abrupt changes in the phase space can be observed in time as well as in parameter space while many results can be described by using scaling laws approach . in this paperwe revisit the 1-d impact system aiming to obtain and describe the behavior of average properties in the chaotic dynamics focusing in the stationary state , _ i d est _ , for very long time , where transient effects are not influencing the dynamics anymore .analytical expressions will be presented in order to calculate statistical properties for the average velocity , its deviation and the average squared velocity , when these variables reach the stationary state .the developed formalism , allows us to obtain the numerical values for these variables , without doing the numerical simulations .we will show a remarkable agreement between numerical simulations and theoretical analysis considering either statistical and thermal variables , giving so robustness , to the developed theory .the impact system is described by a free particle , or an ensemble of non interacting particles , moving under the presence of a constant gravitational field and experiencing collisions with a heavily vibrating platform . for elastic collisions ,the dynamics leads to a mixed phase space , described in velocity and time , and two main properties are observed according to the control parameter range .if the parameter is smaller than a critical one , invariant spanning curves , also called as invariant tori , are present in the phase space hence limiting the velocity of the particle in a chaotic diffusion for certain portions of the phase space . on the other hand , for a parameter larger than the critical one , invariant spanning curves are not present anymore and unlimited diffusion in velocity , for specific ranges of initial conditions , can be observed .the scenario is totally different when inelastic collisions are considered . in this case , dissipation is in course , hence contracting area in the phase space , therefore leading to the existence of attractors . for strong dissipation and control parameter beyond the critical one ,attractors are most periodic . forweak dissipation and large control parameter , chaotic attractors , characterized by a positive lyapunov exponent , dominate over the phase space .giving the attractors are far away from the infinity ( in velocity axis ) , dissipation has proved to be a powerful way of suppress unlimited diffusion . because of limited diffusion in phase space , the behavior and properties for both average velocity , average squared velocity or the deviation around the average velocity , known also as roughness , are the following .they grow to start with from a low initial velocity value and , eventually , they bend towards a stationary state at very long time .the scenario is scaling invariant with respect to the control parameters and number of collisions with the moving platform . by the use of equipartition theorem , the steady state , obtained in the asymptotic state , can be used to make a connection with the thermal equilibrium of the system .therefore in the present paper , we evaluate numerically , for long time series , the behavior of : ( i ) the average velocity ; ( ii ) the averaged squared velocity ; and ( iii ) the deviation around the average velocity , both for the dissipative impact system .we then compare the numerical results with analytical expressions at the equilibrium , obtained via statistical and thermodynamics analysis by using the dynamical equations .a comparison between the results obtained using numerical simulation and theoretical investigation is remarkable , hence giving robustness to the connection between statistical mechanics , thermodynamics and the modeling of dynamical systems .it also improves the theoretical formalism that can be extended to other different types of systems including the time dependent billiards .the paper is organized as follows : in sec .[ sec2 ] we describe the dynamics of the impact system and some of its properties .section [ sec3 ] is devoted to the discussion of the numerical investigation .the results using the dynamical equations and connection with the thermodynamics in the stationary state and the discussions of the results are presented in sec .finally , sec .[ sec5 ] brings some final remarks and conclusions .the model we consider consists of a particle of mass moving under the action of a gravitational field and experiences collisions with a heavy periodically moving wall .this model is also referred to as a bouncer or bouncing ball model .it backs to pustylnikov and has been studied for many years , with several applications in different areas of research such as vibration waves in a nanometric - sized mechanical contact system , granular materials , dynamic stability in human performance , mechanical vibrations , chaos control , crises between attractors , among many others .  as usual , the dynamics of the system is described by a two - dimensional , non - linear discrete mapping for the variables velocity of the particle and time ( will be measured latter on as function of the phase of moving wall ) immediately after a collision of the particle with the moving wall . for an analysis as function of the time .the investigations are made based on two main versions of the model : _( i ) _ complete , which takes into account the whole movement of the vibrating platform ; and _ ( ii ) _ a static wall approximation . in this version ,the nonlinear mapping assumes the wall is static but that , as soon as the particle hits it , there is an exchange of energy as if the wall were moving .this is then a simplified version and shows to be a very convenient way to find out analytical results in the model where transcendental equations do not need to be solved , as they have to be in the complete version .the two versions can be used either to investigate non - dissipative and dissipative dynamics .dissipation here is introduced by using a restitution coefficient  ] ; ( ii ) transforming the equation of the velocity in the discrete mapping into a differential equation and solve it using an ensemble of different initial phases  ] are considered .the root mean square velocity is obtained as the procedure is the same as running eqs .( [ eq5 ] ) and ( [ eq6 ] ) but using rather than .finally , the deviation around the average velocity , , see for instance , is obtained from   as it is known , for large , unlimited diffusion in velocity can be observed .because of the dissipation , the unlimited diffusion is not allowed anymore .the average dynamics , no mater the initial velocity , will converge to an asymptotic state for large time . if the initial condition is large , the velocity of the particle decreases until reaches the stationary state .it is known in the literature for a similar system , that the decay of velocity is given by an exponential function and the speed of the decay depends on the strength of the dissipation .stronger the dissipation , faster the decay .  in opposite way , starting with a small initial velocity , the dynamics leads the average velocity to experience an initial growth as a function of the number of collisions of the type ^{\beta} ] .a comparison of the saturation of the three observables , and is better seen in figs .[ fig2](c , d ) .important to mention is that a change in the parameter leads to different saturation and it does not affect the crossover time . however , the parameter changes both the saturation ( stationary state ) and the crossover times . with a scaling approach , as done previously in the literature , see for instance refs . , a rescale can be done and overlap both curves , of the same observable , into an universal plot . however , in the scenario where high dissipation is considered , and we have low values for the parameter , the scaling invariance is very difficult to be observed , since we have successive boundary crisis between manifolds and crisis between attractors .  as we will see in the next section, the numerical values of the saturation plateaus play an important role in the thermodynamics analysis .the values of the plateaus for different values of the control parameters are shown in tables [ tab2 ] and [ tab1 ] .we see the saturation plateaus for the complete version are higher as compared to the static wall approximation .this is close connected to the probability distribution function of the velocity in the phase space . for short ,the particle _ prefers _ to stay with high energy in the complete version while compared to the static wall approximation .although the phase space is similar for both versions , their occupation are different ..simplified mapping : numerical values for the stationary state for , and considering some pairs of ( , ) .[ cols="^,^,^,^,^,^",options="header " , ]the dynamics of a dissipative impact system was described by nonlinear mappings for two different versions , complete and simplified , for the velocity of the particle and the phase of the vibrating wall .dissipation was introduced via inelastic collisions leading the existence of attractors in the phase space .a numerical and statistical investigation for the variables , and ( deviation of the average velocity ) was made for both versions of the mappings . for long time series, these observables bend towards a saturation plateau which marks the stationary state .such a regime varies as the control parameters associated with the dissipation and ratio between acceleration are changed .  at the stationary state, the square velocity can be obtained . from equipartition theorem ,such observable can be interpreted as an equilibrium temperature .we obtained analytical equations for the , and variables in the equilibrium state as functions of the parameters of the model , with these equations we were able to calculate the numerical values of those variables without doing the simulations .a remarkable assembly was obtained considering both numerical and theoretical investigation , between statistical and thermal variables .this result gives robustness to the formalism , and opens new doors for similar analysis in other more complex dynamical systems , particularly in time dependent billiards .gdi thanks to the brazilian agency capes .alpl acknowledges fapesp ( 2014/25316 - 3 ) and cnpq for financial support .edl kindly acknowledges support from cnpq ( 303707/2015 - 1 ) , fapesp ( 2012/23688 - 5 ) and fundunesp .

Abstract: the behavior of the average velocity , its deviation and average squared velocity are characterized using three techniques for a 1-d dissipative impact system . the system a particle , or an ensemble of non interacting particles , moving in a constant gravitation field and colliding with a varying platform is described by a nonlinear mapping . the average squared velocity allows to describe the temperature for an ensemble of particles as a function of the parameters using : ( i ) straightforward numerical simulations ; ( ii ) analytically from the dynamical equations ; ( iii ) using the probability distribution function . comparing analytical and numerical results for the three techniques , one can check the robustness of the developed formalism , where we are able to estimate numerical values for the statistical variables , without doing extensive numerical simulations . also , extension to other dynamical systems is immediate , including time dependent billiards .